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

Newform orbit 451.2.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 4x^{9} - 34x^{8} + 169x^{7} + 146x^{6} - 1460x^{5} + 745x^{4} + 3491x^{3} - 3616x^{2} + 28x + 280$

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
$\operatorname{tr} a_p$	4	0	6	7	10	4	9	5	7	34	−23	−8	−10	2	−16	26	13	−7	−8	13	−1	27	−28	−24	−27

$\operatorname{tr} a_n$ 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)[3]   # 451.2.a.d

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

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