Atlas › Modular forms › Weight 2 › Level 427 › 427.2.a › 427.2.a.g

Newform orbit 427.2.a.g

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 5q^{2} - q^{3} + 13q^{4} + 9q^{5} - 4q^{6} - 9q^{7} + 15q^{8} + 10q^{9} + 3q^{10} + 3q^{11} + 5q^{12} + 13q^{13} - 5q^{14} + 2q^{15} + 21q^{16} + 9q^{17} + 14q^{18} + q^{19} + 2q^{20} + q^{21} - 3q^{22} + q^{23} - 27q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} - 2x^{8} - 168x^{7} + 201x^{6} + 9052x^{5} - 7005x^{4} - 175428x^{3} + 129957x^{2} + 924662x - 1134608$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
61	$-1$

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
$\operatorname{tr} a_p$	5	−1	9	−9	3	13	9	1	1	18	−7	13	3	10	16	38	−34	9	5	−6	4	−14	14	−4	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 7, 61, 427]: [0, 1, 5, 42] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 31 = 1 x 31 (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(427, 2)[6]   # 427.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 427:2 1000