Atlas › Modular forms › Weight 4 › Level 151 › 151.4.a › 151.4.a.a

Newform orbit 151.4.a.a

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 6q^{2} - 10q^{3} + 44q^{4} - 61q^{5} - 36q^{6} - 26q^{7} - 93q^{8} + 15q^{9} - 91q^{10} - 139q^{11} - 195q^{12} - 40q^{13} - 192q^{14} - 124q^{15} + 56q^{16} - 357q^{17} - 119q^{18} - 239q^{19} - 501q^{20} - 402q^{21} - 92q^{22} - 256q^{23} - 491q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 16x^{14} - 128x^{13} - 2593x^{12} + 3884x^{11} + 130533x^{10} - 33487x^{9} - 2593052x^{8} + 263152x^{7} + 22644508x^{6} - 4315514x^{5} - 78265780x^{4} + 19548092x^{3} + 58849912x^{2} + 8169600x - 3781600$

Atkin–Lehner signs

$p$	$w_p$
151	$-1$

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 odd 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
$\operatorname{tr} a_p$	−6	−10	−61	−26	−139	−40	−357	−239	−256	−443	−483	−491	−1612	169	−599	−36	−2267	488	1596	−1450	−548	70	−110	−7224	−1947

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 151]: [1, 39] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 5 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(151, 4)[0]   # 151.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 151:4 1000