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

Newform orbit 751.2.a.b

Trace form

The 38 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$38q + 4q^{2} + 6q^{3} + 40q^{4} + 18q^{5} + 13q^{6} + q^{7} + 9q^{8} + 62q^{9} + 20q^{11} + 10q^{12} + 10q^{13} + 17q^{14} - 7q^{15} + 32q^{16} + 61q^{17} - q^{18} + 7q^{19} + 27q^{20} + 13q^{21} - 10q^{22} + 4q^{23} + 38q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 38. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

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

Atkin–Lehner signs

$p$	$w_p$
751	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	4	6	18	1	20	10	61	7	4	53	−5	0	117	−5	32	27	26	37	−25	3	24	−29	9	120	12

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

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)[1]   # 751.2.a.b

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

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