Atlas › Modular forms › Weight 2 › Level 787 › 787.2.a

Newspace 787.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(787))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
787.2.a.a	28	degree 28	$28q - 12q^{2} - 8q^{3} + 24q^{4} - 29q^{5} - 4q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
787.2.a.b	37	degree 37	$37q + 12q^{2} + 6q^{3} + 42q^{4} + 31q^{5} + 4q^{6} + 9q^{7} + O(q^{8})$	−	$+1$

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 $T$
787.2.a.a	$x^{28} + 20x^{27} + 117x^{26} - 192x^{25} - 4385x^{24} - 9436x^{23} + 51988x^{22} + 226749x^{21} - 192839x^{20} - 2160977x^{19} - 943448x^{18} + 11107653x^{17} + 11907183x^{16} - 33807354x^{15} - 50188115x^{14} + 63609358x^{13} + 113342457x^{12} - 76665345x^{11} - 147487918x^{10} + 62290982x^{9} + 108318027x^{8} - 34850024x^{7} - 40340027x^{6} + 11661431x^{5} + 5906869x^{4} - 1684562x^{3} - 105311x^{2} + 16146x + 299$
787.2.a.b	$x^{37} - 18x^{36} + 21x^{35} + 1440x^{34} - 7400x^{33} - 41468x^{32} + 375833x^{31} + 348513x^{30} - 9430734x^{29} + 8483157x^{28} + 140222126x^{27} - 284566626x^{26} - 1308853874x^{25} + 3975455229x^{24} + 7661017857x^{23} - 33438147279x^{22} - 25944763728x^{21} + 185194975755x^{20} + 31762724681x^{19} - 700232362524x^{18} + 111761792272x^{17} + 1840743391053x^{16} - 591980804109x^{15} - 3391167606467x^{14} + 1227784558422x^{13} + 4356954220228x^{12} - 1357087007000x^{11} - 3800384395595x^{10} + 774251462436x^{9} + 2115849927956x^{8} - 178461872279x^{7} - 678787952054x^{6} + 3551214415x^{5} + 115757873788x^{4} + 3163463288x^{3} - 9415076120x^{2} - 210843664x + 280444288$

Orbit

Characteristic polynomial of

T

787.2.a.a

x^{28} + 20x^{27} + 117x^{26} - 192x^{25} - 4385x^{24} - 9436x^{23} + 51988x^{22} + 226749x^{21} - 192839x^{20} - 2160977x^{19} - 943448x^{18} + 11107653x^{17} + 11907183x^{16} - 33807354x^{15} - 50188115x^{14} + 63609358x^{13} + 113342457x^{12} - 76665345x^{11} - 147487918x^{10} + 62290982x^{9} + 108318027x^{8} - 34850024x^{7} - 40340027x^{6} + 11661431x^{5} + 5906869x^{4} - 1684562x^{3} - 105311x^{2} + 16146x + 299

787.2.a.b

x^{37} - 18x^{36} + 21x^{35} + 1440x^{34} - 7400x^{33} - 41468x^{32} + 375833x^{31} + 348513x^{30} - 9430734x^{29} + 8483157x^{28} + 140222126x^{27} - 284566626x^{26} - 1308853874x^{25} + 3975455229x^{24} + 7661017857x^{23} - 33438147279x^{22} - 25944763728x^{21} + 185194975755x^{20} + 31762724681x^{19} - 700232362524x^{18} + 111761792272x^{17} + 1840743391053x^{16} - 591980804109x^{15} - 3391167606467x^{14} + 1227784558422x^{13} + 4356954220228x^{12} - 1357087007000x^{11} - 3800384395595x^{10} + 774251462436x^{9} + 2115849927956x^{8} - 178461872279x^{7} - 678787952054x^{6} + 3551214415x^{5} + 115757873788x^{4} + 3163463288x^{3} - 9415076120x^{2} - 210843664x + 280444288

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(787))$	65
Old forms	0
Cusp forms $S_{2}(\Gamma_0(787))$	65
Eisenstein series $E_{2}(\Gamma_0(787))$	1
Modular forms $M_{2}(\Gamma_0(787))$	66

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 787]: [0, 66] (certified at ell = 2147472259)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (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(787, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 787:2 1000

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