Atlas › Modular forms › Weight 2 › Level 914 › 914.2.a › 914.2.a.e

Newform orbit 914.2.a.e

Trace form

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

$12q - 12q^{2} + 12q^{4} - 8q^{5} + 7q^{7} - 12q^{8} + 16q^{9} + 8q^{10} - 3q^{11} + 10q^{13} - 7q^{14} + 2q^{15} + 12q^{16} - 11q^{17} - 16q^{18} + 26q^{19} - 8q^{20} + q^{21} + 3q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{12} + 8x^{11} - 37x^{10} - 417x^{9} - 50x^{8} + 5799x^{7} + 7840x^{6} - 21653x^{5} - 36898x^{4} + 13942x^{3} + 21651x^{2} - 3141x - 2978$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
457	$-1$

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
$\operatorname{tr} a_p$	−12	0	−8	7	−3	10	−11	26	6	4	30	−7	9	44	18	−1	12	44	−4	4	5	38	−11	−10	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 2, 457, 914]: [0, 1, 38, 116] (certified at ell = 2147478649)
T = [(3, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(914, 2)[4]   # 914.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 914:2 1000