Atlas › Modular forms › Weight 2 › Level 614 › 614.2.a › 614.2.a.f

Newform orbit 614.2.a.f

Trace form

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

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

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

$x^{11} + 6x^{10} - 363x^{9} - 756x^{8} + 46026x^{7} - 63164x^{6} - 1969831x^{5} + 7653068x^{4} + 5623463x^{3} - 43799550x^{2} + 2829008x + 39270748$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
307	$+1$

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	−3	6	11	3	6	17	12	−3	2	−3	−10	19	−2	−21	−17	−7	1	−2	−14	27	6	−13	15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 307, 614]: [0, 1, 26, 79] (certified at ell = 2147478013)
T = [(3, 1), (5, 3)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 4 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(614, 2)[5]   # 614.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 614:2 1000