Atlas › Modular forms › Weight 2 › Level 437 › 437.2.a › 437.2.a.g

Newform orbit 437.2.a.g

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 5q^{3} + 10q^{4} + 2q^{5} - 3q^{6} + 9q^{7} + 9q^{9} - 2q^{10} + 7q^{11} - 11q^{12} + 8q^{13} + 13q^{14} + 10q^{15} + 26q^{16} - 10q^{17} + 3q^{18} - 8q^{19} + 2q^{20} + 17q^{21} + 3q^{22} + 8q^{23} - 39q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 15x^{7} - 40x^{6} + 1251x^{5} - 1222x^{4} - 29411x^{3} + 43307x^{2} + 219484x - 254500$

Atkin–Lehner signs

$p$	$w_p$
19	$+1$
23	$-1$

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
$\operatorname{tr} a_p$	0	5	2	9	7	8	−10	−8	8	−8	13	27	4	19	0	1	−15	−10	48	−17	−21	66	−11	−7	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.386 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, 19, 23, 437]: [0, 2, 3, 42] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 33 = 1 x 33 (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(437, 2)[6]   # 437.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 437:2 1000