Atlas › Modular forms › Weight 2 › Level 934 › 934.2.a › 934.2.a.i

Newform orbit 934.2.a.i

Trace form

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

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

Coefficient field

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

$x^{12} - 14x^{11} - 327x^{10} + 5363x^{9} + 21275x^{8} - 551448x^{7} + 519887x^{6} + 15373994x^{5} - 26074383x^{4} - 121856300x^{3} + 184009938x^{2} + 133658994x - 185076529$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
467	$+1$

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	2	4	7	18	−10	−6	1	17	20	−3	5	−18	36	21	21	4	−16	22	30	−23	8	−23	7	−30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.304 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, 467, 934]: [0, 1, 40, 119] (certified at ell = 2147483179)
T = [(3, 1), (5, 3)]; h of degree 38 = 1 x 38 (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(934, 2)[8]   # 934.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 934:2 1000