Atlas › Modular forms › Weight 2 › Level 451 › 451.2.a › 451.2.a.e

Newform orbit 451.2.a.e

Trace form

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

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

Coefficient field

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

$x^{12} - 4x^{11} - 43x^{10} + 163x^{9} + 570x^{8} - 1833x^{7} - 3657x^{6} + 7561x^{5} + 12477x^{4} - 10091x^{3} - 18148x^{2} - 1644x + 2536$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
41	$-1$

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
$\operatorname{tr} a_p$	3	1	13	−7	−12	0	9	−13	6	42	8	9	12	−8	2	36	2	9	1	0	−21	−47	12	21	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.192 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, 11, 41, 451]: [0, 2, 4, 44] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (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(451, 2)[4]   # 451.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 451:2 1000