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

Newform orbit 437.2.a.f

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} + 13x^{4} - x^{3} - 350x^{2} - 228x + 8$

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 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
$\operatorname{tr} a_p$	−1	−4	−1	−9	−1	−10	5	−5	−5	10	−4	−14	−16	−3	−5	0	10	−15	−16	−20	17	0	2	−4	−14

$\operatorname{tr} a_n$ 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)[5]   # 437.2.a.f

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

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