Atlas › Modular forms › Weight 2 › Level 955 › 955.2.a › 955.2.a.d

Newform orbit 955.2.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 4q^{2} - q^{3} + 10q^{4} - 14q^{5} - 10q^{6} - 2q^{7} - 12q^{8} + 5q^{9} + 4q^{10} - 17q^{11} + q^{12} + 6q^{13} - 8q^{14} + q^{15} + 6q^{16} - 7q^{17} - q^{18} - 18q^{19} - 10q^{20} - 28q^{21} + 7q^{22} - 6q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 5x^{13} - 20x^{12} - 113x^{11} + 155x^{10} + 983x^{9} - 580x^{8} - 4102x^{7} + 938x^{6} + 8356x^{5} - 107x^{4} - 7295x^{3} - 691x^{2} + 1562x - 191$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
191	$+1$

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
$\operatorname{tr} a_p$	−4	−1	−14	−2	−17	6	−7	−18	−6	−53	−16	−14	−39	5	−7	−45	−9	−32	12	−23	16	−11	−7	−40	49

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 5, 191, 955]: [0, 1, 17, 98] (certified at ell = 2147476141)
T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 7 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(955, 2)[3]   # 955.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 955:2 1000