Atlas › Modular forms › Weight 2 › Level 373 › 373.2.a › 373.2.a.b

Newform orbit 373.2.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 19x^{11} + 136x^{10} + 420x^{9} + 265x^{8} - 1547x^{7} - 3429x^{6} - 1037x^{5} + 3337x^{4} + 3049x^{3} + 401x^{2} - 175x + 7$

Atkin–Lehner signs

$p$	$w_p$
373	$+1$

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	−15	−11	−1	−8	−10	−21	−11	−24	−7	−3	−8	2	−19	−38	−14	−36	0	−26	−20	9	29	−63	13	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 373]: [0, 31] (certified at ell = 2147478997)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 3 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(373, 2)[1]   # 373.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 373:2 1000