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

Newform orbit 437.2.a.h

Trace form

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

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

Coefficient field

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

$x^{12} - 23x^{11} + 14x^{10} + 2996x^{9} - 15703x^{8} - 113661x^{7} + 961272x^{6} + 483368x^{5} - 17474995x^{4} + 30757657x^{3} + 34242020x^{2} - 86809372x + 17654544$

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
$\operatorname{tr} a_p$	2	7	−1	14	4	2	9	12	−12	2	17	9	−12	42	13	−15	9	1	64	−11	28	−28	1	−29	0

$\operatorname{tr} a_n$ 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)[7]   # 437.2.a.h

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

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