Atlas › Modular forms › Weight 2 › Level 757 › 757.2.a

Newspace 757.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(757))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
757.2.a.a	29	degree 29	$29q - 12q^{2} - 21q^{3} + 26q^{4} - 11q^{5} - 2q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
757.2.a.b	33	degree 33	$33q + 10q^{2} + 19q^{3} + 34q^{4} + 7q^{5} + 8q^{6} + 6q^{7} + O(q^{8})$	−	$+1$

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 $T$
757.2.a.a	$x^{29} + 33x^{28} + 448x^{27} + 3023x^{26} + 7887x^{25} - 24935x^{24} - 238720x^{23} - 472466x^{22} + 1256333x^{21} + 7049546x^{20} + 4844554x^{19} - 33273301x^{18} - 70584748x^{17} + 47014821x^{16} + 281419743x^{15} + 123543980x^{14} - 538092861x^{13} - 589007699x^{12} + 476830936x^{11} + 975273994x^{10} - 43093729x^{9} - 803818465x^{8} - 256683136x^{7} + 314482475x^{6} + 184529903x^{5} - 36685867x^{4} - 41821800x^{3} - 4317219x^{2} + 1651455x + 259002$
757.2.a.b	$x^{33} - 29x^{32} + 294x^{31} - 595x^{30} - 11008x^{29} + 80396x^{28} - 17490x^{27} - 1814249x^{26} + 5741366x^{25} + 13197875x^{24} - 101672216x^{23} + 56707769x^{22} + 788554961x^{21} - 1555258587x^{20} - 2786735709x^{19} + 10864017551x^{18} + 1382114171x^{17} - 38913950742x^{16} + 22902300393x^{15} + 79533059554x^{14} - 83904939912x^{13} - 93720599504x^{12} + 140097416494x^{11} + 60498341577x^{10} - 127541424298x^{9} - 18809795286x^{8} + 63381223004x^{7} + 2641635648x^{6} - 16018185238x^{5} - 724551105x^{4} + 1813747062x^{3} + 151081785x^{2} - 62152787x - 5685974$

Orbit

Characteristic polynomial of

T

757.2.a.a

x^{29} + 33x^{28} + 448x^{27} + 3023x^{26} + 7887x^{25} - 24935x^{24} - 238720x^{23} - 472466x^{22} + 1256333x^{21} + 7049546x^{20} + 4844554x^{19} - 33273301x^{18} - 70584748x^{17} + 47014821x^{16} + 281419743x^{15} + 123543980x^{14} - 538092861x^{13} - 589007699x^{12} + 476830936x^{11} + 975273994x^{10} - 43093729x^{9} - 803818465x^{8} - 256683136x^{7} + 314482475x^{6} + 184529903x^{5} - 36685867x^{4} - 41821800x^{3} - 4317219x^{2} + 1651455x + 259002

757.2.a.b

x^{33} - 29x^{32} + 294x^{31} - 595x^{30} - 11008x^{29} + 80396x^{28} - 17490x^{27} - 1814249x^{26} + 5741366x^{25} + 13197875x^{24} - 101672216x^{23} + 56707769x^{22} + 788554961x^{21} - 1555258587x^{20} - 2786735709x^{19} + 10864017551x^{18} + 1382114171x^{17} - 38913950742x^{16} + 22902300393x^{15} + 79533059554x^{14} - 83904939912x^{13} - 93720599504x^{12} + 140097416494x^{11} + 60498341577x^{10} - 127541424298x^{9} - 18809795286x^{8} + 63381223004x^{7} + 2641635648x^{6} - 16018185238x^{5} - 724551105x^{4} + 1813747062x^{3} + 151081785x^{2} - 62152787x - 5685974

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(757))$	62
Old forms	0
Cusp forms $S_{2}(\Gamma_0(757))$	62
Eisenstein series $E_{2}(\Gamma_0(757))$	1
Modular forms $M_{2}(\Gamma_0(757))$	63

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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, 757]: [0, 63] (certified at ell = 2147478481)
T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 6 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(757, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 757:2 1000

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