Atlas › Modular forms › Weight 2 › Level 734 › 734.2.a › 734.2.a.e

Newform orbit 734.2.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{11} - 16x^{10} - 349x^{9} + 6576x^{8} + 29921x^{7} - 914246x^{6} + 1056819x^{5} + 48153014x^{4} - 200986140x^{3} - 635068848x^{2} + 5236156992x - 8412464992$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
367	$-1$

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
$\operatorname{tr} a_p$	−11	−2	6	9	3	−5	10	−1	−2	−15	30	23	8	−4	−12	8	−11	31	27	5	22	21	−6	10	33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 367, 734]: [0, 1, 31, 94] (certified at ell = 2147479381)
T = [(3, 1), (5, 3)]; h of degree 31 = 1 x 31 (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(734, 2)[4]   # 734.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 734:2 1000