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

Newform orbit 373.2.a.c

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 4q^{2} + 14q^{3} + 18q^{4} + 9q^{5} + 4q^{6} + 3q^{7} + 9q^{8} + 23q^{9} - 7q^{10} + 16q^{11} + 19q^{12} - q^{13} - q^{14} + 3q^{15} + 8q^{16} + 16q^{17} - 5q^{18} + 5q^{19} + 10q^{20} - 10q^{21} - 15q^{22} + 32q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 18x^{16} + 95x^{15} + 128x^{14} - 2854x^{13} + 6924x^{12} + 17361x^{11} - 98178x^{10} + 50931x^{9} + 426324x^{8} - 734186x^{7} - 409436x^{6} + 1976076x^{5} - 1183497x^{4} - 1145884x^{3} + 1805872x^{2} - 846056x + 136693$

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 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
$\operatorname{tr} a_p$	4	14	9	3	16	−1	16	5	32	−7	−6	−13	−5	9	42	16	55	−20	−6	−7	−6	−23	98	4	−13

$\operatorname{tr} a_n$ 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)[2]   # 373.2.a.c

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

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