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

Newform orbit 451.2.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 6x^{4} + 7x^{3} - 3x^{2} - 3x + 1$

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 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
$\operatorname{tr} a_p$	−2	−4	−4	−3	−5	1	−9	7	−3	−18	−1	−10	−5	8	−12	−10	−13	−17	0	−25	5	2	6	−24	9

$\operatorname{tr} a_n$ 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)[1]   # 451.2.a.b

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

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