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

Newform orbit 734.2.a.f

Trace form

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

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

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

$x^{11} - 18x^{10} - 194x^{9} + 5176x^{8} - 5006x^{7} - 418766x^{6} + 2248346x^{5} + 5624844x^{4} - 71787431x^{3} + 165205720x^{2} - 12399028x - 217741864$

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	3	5	10	14	1	−16	3	12	4	5	21	−5	20	17	4	−7	−11	13	25	−39	23	−18	−47	17

$\operatorname{tr} a_n$ 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)[5]   # 734.2.a.f

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

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