Atlas › Modular forms › Weight 4 › Level 149 › 149.4.a › 149.4.a.a

Newform orbit 149.4.a.a

Trace form

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

$15q - 4q^{2} - 22q^{3} + 26q^{4} - 16q^{5} - 41q^{6} - 46q^{7} - 45q^{8} + 73q^{9} - 70q^{10} - 142q^{11} - 137q^{12} - 199q^{13} - 180q^{14} - 270q^{15} - 106q^{16} - 29q^{17} - 150q^{18} - 599q^{19} - 328q^{20} - 300q^{21} - 451q^{22} - 157q^{23} - 425q^{24} + O(q^{25})$

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

$x^{15} + 26x^{14} + 67x^{13} - 2924x^{12} - 17617x^{11} + 101980x^{10} + 789982x^{9} - 1161723x^{8} - 12275469x^{7} + 2561507x^{6} + 66252288x^{5} + 7965165x^{4} - 68729554x^{3} + 1448928x^{2} + 13990940x + 167351$

Atkin–Lehner signs

$p$	$w_p$
149	$-1$

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
$\operatorname{tr} a_p$	−4	−22	−16	−46	−142	−199	−29	−599	−157	−226	−1004	−220	−260	−304	−50	334	−1602	−1106	−1357	−603	−1505	−1025	−2056	−64	−1520

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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, 149]: [1, 39] (certified at ell = 2147481629)
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(149, 4)[0]   # 149.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 149:4 1000