Atlas › Modular forms › Weight 2 › Level 951 › 951.2.a › 951.2.a.d

Newform orbit 951.2.a.d

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{19} - 19x^{18} + 84x^{17} + 549x^{16} - 5695x^{15} + 6781x^{14} + 83510x^{13} - 282353x^{12} - 298117x^{11} + 2582625x^{10} - 1711469x^{9} - 8955346x^{8} + 13653737x^{7} + 8797900x^{6} - 26112850x^{5} + 5745370x^{4} + 11680980x^{3} - 4559872x^{2} - 1034059x + 427715$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
317	$+1$

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
$\operatorname{tr} a_p$	8	19	11	0	23	2	23	3	7	18	−8	−9	43	−12	17	13	26	−14	−5	14	−12	−7	78	46	−17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 3, 317, 951]: [0, 1, 27, 108] (certified at ell = 2147479753)
T = [(2, 1), (5, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(951, 2)[3]   # 951.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 951:2 1000