Atlas › Modular forms › Weight 2 › Level 445 › 445.2.a › 445.2.a.f

Newform orbit 445.2.a.f

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} + 28x^{6} + 243x^{5} + 288x^{4} - 5216x^{3} - 18307x^{2} - 1176x + 29691$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
89	$-1$

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	−8	7	−16	−10	−7	−13	−7	−13	−4	1	−5	5	−31	−14	−13	−14	3	1	−8	9	9	−42	7	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.289 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, 5, 89, 445]: [0, 1, 8, 46] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(445, 2)[5]   # 445.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 445:2 1000