Atlas › Modular forms › Weight 2 › Level 149 › 149.2.a › 149.2.a.b

Newform orbit 149.2.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - q^{2} + 6q^{3} + 13q^{4} - q^{5} + q^{6} + 3q^{7} - 6q^{8} + 9q^{9} - 6q^{10} + 5q^{11} - q^{12} + 7q^{13} - 8q^{14} + 4q^{15} + 13q^{16} - 5q^{17} - 15q^{18} + 30q^{19} - 10q^{20} - 6q^{21} - 7q^{22} - 4q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 5x^{8} - 22x^{7} + 145x^{6} - 17x^{5} - 949x^{4} + 1295x^{3} + 704x^{2} - 1838x + 533$

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
$\operatorname{tr} a_p$	−1	6	−1	3	5	7	−5	30	−4	−16	22	−7	6	4	−6	−2	43	1	33	15	−11	1	−4	−19	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 149]: [0, 13] (certified at ell = 2147481629)
T = [(2, 1), (3, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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, 2)[1]   # 149.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 149:2 1000