Atlas › Modular forms › Weight 2 › Level 817 › 817.2.a

Newspace 817.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(817))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
817.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+ −	$+1$	817a1 rank 2
817.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− −	$-1$	817b1 rank 1
817.2.a.c	13	degree 13	$13q - 7q^{2} + 17q^{4} - 13q^{5} - 2q^{6} - 15q^{7} + O(q^{8})$	− −	$-1$
817.2.a.d	15	degree 15	$15q - 7q^{2} - 4q^{3} + 13q^{4} + q^{5} - 2q^{6} - 13q^{7} + O(q^{8})$	+ +	$-1$
817.2.a.e	15	degree 15	$15q + 9q^{2} + 6q^{3} + 19q^{4} + 3q^{5} + 2q^{6} + 15q^{7} + O(q^{8})$	+ −	$+1$
817.2.a.f	18	degree 18	$18q + 6q^{2} + 2q^{3} + 22q^{4} + 7q^{5} - 2q^{6} + 13q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ 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 $T$
817.2.a.a	$x + 89$
817.2.a.b	$x - 39$
817.2.a.c	$x^{13} + 312x^{12} + 39527x^{11} + 2432288x^{10} + 50326897x^{9} - 2799766596x^{8} - 235797955345x^{7} - 7402933230100x^{6} - 103660818423194x^{5} + 53797755012828x^{4} + 21328677758768846x^{3} + 231647282333074392x^{2} + 274095075764698082x - 5470455563220129899$
817.2.a.d	$x^{15} + 258x^{14} + 23072x^{13} + 550480x^{12} - 39510904x^{11} - 2823391372x^{10} - 44571306420x^{9} + 1293111228292x^{8} + 55535722416467x^{7} + 475384741094788x^{6} - 7739384694942778x^{5} - 158221069331280356x^{4} - 336091009959280770x^{3} + 9325400812181395823x^{2} + 53384550033310229362x + 44500675272428429$
817.2.a.e	$x^{15} - 264x^{14} + 26675x^{13} - 1123606x^{12} - 1558291x^{11} + 1889225992x^{10} - 57157875449x^{9} - 184482654002x^{8} + 39346544381796x^{7} - 577474109462884x^{6} - 4628930440207026x^{5} + 185285795158774384x^{4} - 1187540459064453816x^{3} - 6154624214009754011x^{2} + 85315706702430857000x - 204676932582423476834$
817.2.a.f	$x^{18} - 273x^{17} + 20638x^{16} + 738306x^{15} - 170972538x^{14} + 5339306270x^{13} + 261672202838x^{12} - 15966885491042x^{11} - 116890597414969x^{10} + 18541416974143211x^{9} - 22562199087086278x^{8} - 11942239270782113320x^{7} + 19525740860598216612x^{6} + 4557052065679112035435x^{5} + 8328233297492909126143x^{4} - 884408121105297968884897x^{3} - 5695788442236411468251141x^{2} + 37283500223327417027786714x + 313491175850637373695133786$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(817))$	63
Old forms	8
Cusp forms $S_{2}(\Gamma_0(817))$	71
Eisenstein series $E_{2}(\Gamma_0(817))$	3
Modular forms $M_{2}(\Gamma_0(817))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(817)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.412 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, 19, 43, 817]: [0, 2, 4, 74] (certified at ell = 2147483647)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 15 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(817, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 817:2 1000

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