Atlas › Modular forms › Weight 2 › Level 997 › 997.2.a

Newspace 997.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(997))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
997.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	−	$+1$	997c1 rank 2
997.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	+	$-1$	997a1 rank 1
997.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - 4q^{5} - 2q^{7} + O(q^{8})$	−	$+1$	997b1 rank 2
997.2.a.d	4	degree 4	$4q + 2q^{2} - q^{3} - 2q^{4} - 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+	$-1$
997.2.a.e	5	degree 5	$5q - 4q^{2} - 6q^{3} + 4q^{4} + 2q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
997.2.a.f	5	degree 5	$5q - q^{3} + 6q^{4} - 10q^{5} + 6q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
997.2.a.g	23	degree 23	$23q - 8q^{2} - 13q^{3} + 24q^{4} + 2q^{5} - 15q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
997.2.a.h	42	degree 42	$42q + 14q^{2} + 24q^{3} + 48q^{4} + 16q^{5} + q^{6} + 15q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11} + 9T_{13}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
997.2.a.a	$x + 48$
997.2.a.b	$x + 30$
997.2.a.c	$x + 72$
997.2.a.d	$x^{4} + 44x^{3} - 1030x^{2} + 4132x - 3131$
997.2.a.e	$x^{5} - 36x^{4} - 2162x^{3} + 68511x^{2} + 175888x + 102481$
997.2.a.f	$x^{5} + 104x^{4} + 96x^{3} - 194064x^{2} - 4105620x - 22572344$
997.2.a.g	$x^{23} + 405x^{22} + 63766x^{21} + 4281187x^{20} - 3531886x^{19} - 18611406350x^{18} - 976241406101x^{17} + 3891316282896x^{16} + 2062509547130750x^{15} + 59902489982389788x^{14} - 866625475594199083x^{13} - 80516588578807660901x^{12} - 1138406358619024116200x^{11} + 26020222770053603148628x^{10} + 1029023729849043317073241x^{9} + 8233182917660758622454143x^{8} - 146368928416988704887059229x^{7} - 3613700039610305402553900072x^{6} - 26366041839015964111793545711x^{5} - 10298667765951873050625227091x^{4} + 718012941283691268386348986770x^{3} + 1946803897938677028272522185828x^{2} - 5099900397439548830401677181088x - 12082660270222699673292140335120$
997.2.a.h	$x^{42} - 585x^{41} + 132066x^{40} - 11373222x^{39} - 623054164x^{38} + 218135782298x^{37} - 14346610183473x^{36} - 616844733683896x^{35} + 137581580428822056x^{34} - 4925693267975406542x^{33} - 368885546398740662680x^{32} + 36500779723890717664722x^{31} - 328433786834578905236443x^{30} - 89885593433005491250968872x^{29} + 3846507899578002378626492055x^{28} + 67099086240867526894563443802x^{27} - 8655065485640679527021601460272x^{26} + 113717175831079936355090904725126x^{25} + 8739060562868840274851102120986612x^{24} - 303075277518135522322354569778467939x^{23} - 2916878179403808766903613232351272233x^{22} + 294278797201594027507558308589457083803x^{21} - 2102809208876772477047256728450248279729x^{20} - 148270611109600128775694888254463802417386x^{19} + 2610064537490180035607986565576031291517422x^{18} + 39661554676293041442913542527747989228457312x^{17} - 1226562918245586616271158599498427373036920887x^{16} - 4529413937282117713692385950686047375510308539x^{15} + 329392472331786667658225457202055956514874301069x^{14} - 244523777108577761353594065421902388772443152586x^{13} - 55764582900106702370156310805264898273155841808784x^{12} + 125793756686188989507775480725117236821960626606372x^{11} + 6069037882488091394449065152118252686824302537907828x^{10} - 11792058312741163751129593671178476090641042060061420x^{9} - 396643435855465459875895292506465330601768091543242340x^{8} + 262312522835133072849269110264657524249456187908519992x^{7} + 12265877041595088829962490282073858974429779959152386536x^{6} - 1713539306938268417776446761710411450294566372163654984x^{5} - 162760552500091277761574703495469247775813307876281877424x^{4} + 16936282187543586083420936497146729698404724865291226688x^{3} + 744739661178482844886653247794549557130581869840079416160x^{2} - 213567872376636993418337056046253662884077078905903130144x - 7078839275233764470708298558485493326232925434988564224$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(997))$	82
Old forms	0
Cusp forms $S_{2}(\Gamma_0(997))$	82
Eisenstein series $E_{2}(\Gamma_0(997))$	1
Modular forms $M_{2}(\Gamma_0(997))$	83

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.719 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 997]: [0, 83] (certified at ell = 2147475601)
T = [(2, 1), (3, 3), (7, 5), (11, 7), (13, 9)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 20 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(997, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 997:2 1000

← 996.2.a · 998.2.a →