Atlas › Modular forms › Weight 2 › Level 823 › 823.2.a

Newspace 823.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(823))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
823.2.a.a	30	degree 30	$30q - 10q^{2} - 12q^{3} + 28q^{4} - 20q^{5} - 6q^{6} - 14q^{7} + O(q^{8})$	+	$-1$
823.2.a.b	38	degree 38	$38q + 9q^{2} + 10q^{3} + 39q^{4} + 18q^{5} + 6q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
823.2.a.a	$x^{30} + 22x^{29} + 150x^{28} - 81x^{27} - 5318x^{26} - 15963x^{25} + 57120x^{24} + 350064x^{23} - 80780x^{22} - 3378942x^{21} - 3107763x^{20} + 18188679x^{19} + 29356703x^{18} - 58559088x^{17} - 131202266x^{16} + 113488365x^{15} + 348749104x^{14} - 123603356x^{13} - 589424272x^{12} + 50714910x^{11} + 647512172x^{10} + 40629878x^{9} - 459693711x^{8} - 66269803x^{7} + 203074243x^{6} + 37411810x^{5} - 50752631x^{4} - 10008201x^{3} + 5552326x^{2} + 1051834x - 35531$
823.2.a.b	$x^{38} - 19x^{37} + 41x^{36} + 1390x^{35} - 8912x^{34} - 30949x^{33} + 407587x^{32} - 135711x^{31} - 9087069x^{30} + 19016778x^{29} + 113636427x^{28} - 406427070x^{27} - 788005998x^{26} + 4665199394x^{25} + 2136696137x^{24} - 33691210160x^{23} + 10487985648x^{22} + 161984638602x^{21} - 126576356826x^{20} - 532298664369x^{19} + 580823894114x^{18} + 1216831594989x^{17} - 1563657612732x^{16} - 1981772588733x^{15} + 2676213480631x^{14} + 2389862061309x^{13} - 2956336874840x^{12} - 2210612594575x^{11} + 2047224039389x^{10} + 1548965393739x^{9} - 791212616934x^{8} - 748922897305x^{7} + 96414556484x^{6} + 206282621073x^{5} + 34293826137x^{4} - 19501951112x^{3} - 9067517561x^{2} - 1423291205x - 79855915$

Orbit

Characteristic polynomial of

T

823.2.a.a

x^{30} + 22x^{29} + 150x^{28} - 81x^{27} - 5318x^{26} - 15963x^{25} + 57120x^{24} + 350064x^{23} - 80780x^{22} - 3378942x^{21} - 3107763x^{20} + 18188679x^{19} + 29356703x^{18} - 58559088x^{17} - 131202266x^{16} + 113488365x^{15} + 348749104x^{14} - 123603356x^{13} - 589424272x^{12} + 50714910x^{11} + 647512172x^{10} + 40629878x^{9} - 459693711x^{8} - 66269803x^{7} + 203074243x^{6} + 37411810x^{5} - 50752631x^{4} - 10008201x^{3} + 5552326x^{2} + 1051834x - 35531

823.2.a.b

x^{38} - 19x^{37} + 41x^{36} + 1390x^{35} - 8912x^{34} - 30949x^{33} + 407587x^{32} - 135711x^{31} - 9087069x^{30} + 19016778x^{29} + 113636427x^{28} - 406427070x^{27} - 788005998x^{26} + 4665199394x^{25} + 2136696137x^{24} - 33691210160x^{23} + 10487985648x^{22} + 161984638602x^{21} - 126576356826x^{20} - 532298664369x^{19} + 580823894114x^{18} + 1216831594989x^{17} - 1563657612732x^{16} - 1981772588733x^{15} + 2676213480631x^{14} + 2389862061309x^{13} - 2956336874840x^{12} - 2210612594575x^{11} + 2047224039389x^{10} + 1548965393739x^{9} - 791212616934x^{8} - 748922897305x^{7} + 96414556484x^{6} + 206282621073x^{5} + 34293826137x^{4} - 19501951112x^{3} - 9067517561x^{2} - 1423291205x - 79855915

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(823))$	68
Old forms	0
Cusp forms $S_{2}(\Gamma_0(823))$	68
Eisenstein series $E_{2}(\Gamma_0(823))$	1
Modular forms $M_{2}(\Gamma_0(823))$	69

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 823]: [0, 69] (certified at ell = 2147477467)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (phi(ord eps) x dim S_k^new); 7 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(823, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 823:2 1000

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