Atlas › Modular forms › Weight 2 › Level 751 › 751.2.a

Newspace 751.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(751))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
751.2.a.a	24	degree 24	$24q - 5q^{2} - 8q^{3} + 17q^{4} - 16q^{5} - 13q^{6} - 7q^{7} + O(q^{8})$	+	$-1$
751.2.a.b	38	degree 38	$38q + 4q^{2} + 6q^{3} + 40q^{4} + 18q^{5} + 13q^{6} + q^{7} + O(q^{8})$	−	$+1$

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 $T$
751.2.a.a	$x^{24} + 13x^{23} + 29x^{22} - 279x^{21} - 1294x^{20} + 1580x^{19} + 15952x^{18} + 4933x^{17} - 92905x^{16} - 85703x^{15} + 290200x^{14} + 353949x^{13} - 521684x^{12} - 695482x^{11} + 577084x^{10} + 719985x^{9} - 420999x^{8} - 394300x^{7} + 199978x^{6} + 104890x^{5} - 54060x^{4} - 10041x^{3} + 6323x^{2} - 107x - 131$
751.2.a.b	$x^{38} - 10x^{37} - 109x^{36} + 1364x^{35} + 4218x^{34} - 79842x^{33} - 43750x^{32} + 2663957x^{31} - 1831556x^{30} - 56850797x^{29} + 79871999x^{28} + 826048404x^{27} - 1539807810x^{26} - 8498856480x^{25} + 18441152374x^{24} + 63598164970x^{23} - 150451094568x^{22} - 352814344035x^{21} + 869442539078x^{20} + 1470435467368x^{19} - 3620133277754x^{18} - 4640104521813x^{17} + 10896440487095x^{16} + 11096980628149x^{15} - 23502194226605x^{14} - 19931920935341x^{13} + 35498615836986x^{12} + 26235747470086x^{11} - 35931594613852x^{10} - 24080054431262x^{9} + 22454201267031x^{8} + 14006583568918x^{7} - 7334084652264x^{6} - 4202947152172x^{5} + 841002005390x^{4} + 327439200118x^{3} - 55586688469x^{2} + 709293592x - 1085345$

Orbit

Characteristic polynomial of

T

751.2.a.a

x^{24} + 13x^{23} + 29x^{22} - 279x^{21} - 1294x^{20} + 1580x^{19} + 15952x^{18} + 4933x^{17} - 92905x^{16} - 85703x^{15} + 290200x^{14} + 353949x^{13} - 521684x^{12} - 695482x^{11} + 577084x^{10} + 719985x^{9} - 420999x^{8} - 394300x^{7} + 199978x^{6} + 104890x^{5} - 54060x^{4} - 10041x^{3} + 6323x^{2} - 107x - 131

751.2.a.b

x^{38} - 10x^{37} - 109x^{36} + 1364x^{35} + 4218x^{34} - 79842x^{33} - 43750x^{32} + 2663957x^{31} - 1831556x^{30} - 56850797x^{29} + 79871999x^{28} + 826048404x^{27} - 1539807810x^{26} - 8498856480x^{25} + 18441152374x^{24} + 63598164970x^{23} - 150451094568x^{22} - 352814344035x^{21} + 869442539078x^{20} + 1470435467368x^{19} - 3620133277754x^{18} - 4640104521813x^{17} + 10896440487095x^{16} + 11096980628149x^{15} - 23502194226605x^{14} - 19931920935341x^{13} + 35498615836986x^{12} + 26235747470086x^{11} - 35931594613852x^{10} - 24080054431262x^{9} + 22454201267031x^{8} + 14006583568918x^{7} - 7334084652264x^{6} - 4202947152172x^{5} + 841002005390x^{4} + 327439200118x^{3} - 55586688469x^{2} + 709293592x - 1085345

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(751))$	62
Old forms	0
Cusp forms $S_{2}(\Gamma_0(751))$	62
Eisenstein series $E_{2}(\Gamma_0(751))$	1
Modular forms $M_{2}(\Gamma_0(751))$	63

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 751]: [0, 63] (certified at ell = 2147482501)
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(751, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 751:2 1000

← 750.2.a · 752.2.a →