Atlas › Modular forms › Weight 2 › Level 455 › 455.2.a › 455.2.a.c

Newform orbit 455.2.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} + x^{3} - 104x^{2} - 162x + 1567$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
7	$+1$
13	$-1$

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
$\operatorname{tr} a_p$	−1	0	−4	−4	2	4	0	18	−12	−2	18	4	12	2	−18	2	30	0	12	−8	34	12	2	24	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 0.316 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, 7, 13, 35, 65, 91, 455]: [0, 1, 1, 1, 6, 8, 10, 60] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (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(455, 2)[2]   # 455.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 455:2 1000