Atlas › Modular forms › Weight 2 › Level 611 › 611.2.a › 611.2.a.e

Newform orbit 611.2.a.e

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 2q^{2} + q^{3} + 26q^{4} + 9q^{5} - 7q^{6} - q^{7} - 3q^{8} + 21q^{9} + 16q^{10} - q^{11} - 4q^{12} + 18q^{13} + 2q^{14} + 5q^{15} + 54q^{16} + 7q^{17} - 8q^{18} + 22q^{19} - 2q^{20} + 8q^{21} + 8q^{22} - q^{23} + 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} + x^{17} - 61x^{16} - 46x^{15} + 1428x^{14} + 1100x^{13} - 16872x^{12} - 15329x^{11} + 107312x^{10} + 119767x^{9} - 351072x^{8} - 490649x^{7} + 474825x^{6} + 926635x^{5} - 29144x^{4} - 627202x^{3} - 287327x^{2} - 3340x + 11647$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
47	$+1$

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
$\operatorname{tr} a_p$	−2	1	9	−1	−1	18	7	22	−1	9	29	7	−8	11	−18	9	17	66	−21	−5	−27	−1	39	40	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 13, 47, 611]: [0, 1, 5, 58] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(611, 2)[4]   # 611.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 611:2 1000