Atlas › Modular forms › Weight 2 › Level 913 › 913.2.a › 913.2.a.d

Newform orbit 913.2.a.d

Trace form

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

$12q + 2q^{2} + 4q^{3} + 20q^{4} - q^{5} - 9q^{6} + 5q^{7} + 6q^{8} + 24q^{9} - 12q^{10} - 12q^{11} + 25q^{12} - 12q^{13} + 14q^{14} - 7q^{15} + 36q^{16} + 16q^{17} + 16q^{18} - q^{19} + 13q^{20} + 4q^{21} - 2q^{22} + 53q^{23} - 16q^{24} + O(q^{25})$

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

$x^{12} - 6x^{11} - 25x^{10} + 152x^{9} + 279x^{8} - 1230x^{7} - 2061x^{6} + 3162x^{5} + 7414x^{4} + 1938x^{3} - 3897x^{2} - 2848x - 535$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
83	$-1$

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
$\operatorname{tr} a_p$	2	4	−1	5	−12	−12	16	−1	53	2	24	26	36	−11	49	−4	4	−2	10	12	−14	11	12	3	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 83, 913]: [0, 2, 8, 86] (certified at ell = 2147478731)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (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(913, 2)[3]   # 913.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 913:2 1000