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

Newform orbit 614.2.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 162x^{4} + 104x^{3} + 7413x^{2} - 3480x - 105396$

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
$\operatorname{tr} a_p$	−6	0	0	−3	−6	12	−9	18	6	21	12	6	−15	−15	−24	24	−15	30	−15	−18	−6	45	6	−3	−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)[4]   # 614.2.a.e

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

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