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

Newform orbit 149.4.a.b

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + 2q^{2} + 20q^{3} + 110q^{4} + 24q^{5} + 43q^{6} + 66q^{7} + 27q^{8} + 262q^{9} + 110q^{10} + 100q^{11} + 199q^{12} + 113q^{13} - 12q^{14} + 90q^{15} + 678q^{16} + 107q^{17} + 120q^{18} + 693q^{19} + 152q^{20} + 204q^{21} - 55q^{22} + 119q^{23} + 583q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 22x^{21} - 372x^{20} + 10506x^{19} + 43744x^{18} - 2029364x^{17} - 894806x^{16} + 214287269x^{15} - 235787718x^{14} - 13810039198x^{13} + 24479763342x^{12} + 566443787816x^{11} - 1134633521632x^{10} - 14814151094368x^{9} + 29228120004077x^{8} + 238405578027866x^{7} - 421507451813190x^{6} - 2151388065845448x^{5} + 3067239610268564x^{4} + 8536701721305100x^{3} - 8156429562611558x^{2} - 3148897443538255x + 363166899029362$

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 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
$\operatorname{tr} a_p$	2	20	24	66	100	113	107	693	119	238	1228	816	−326	1038	230	−910	1436	170	2183	−1223	187	1615	842	2586	2518

$\operatorname{tr} a_n$ 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)[1]   # 149.4.a.b

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

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