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

Newform orbit 611.2.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 3q^{2} - 2q^{3} + 3q^{4} - 7q^{5} - 3q^{6} - 7q^{7} - 6q^{8} + 5q^{9} - 6q^{10} - 2q^{11} + 4q^{12} - 9q^{13} + 6q^{14} - 4q^{15} - 5q^{16} - 6q^{17} - 3q^{18} - 18q^{19} - 8q^{20} - 30q^{21} - 10q^{22} + 3q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 5x^{8} - 11x^{7} - 72x^{6} + 13x^{5} + 294x^{4} + 132x^{3} - 355x^{2} - 281x - 19$

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 odd 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
$\operatorname{tr} a_p$	−3	−2	−7	−7	−2	−9	−6	−18	3	−5	−20	−25	−12	7	−9	10	−17	−44	−4	9	−22	−16	19	−8	−47

$\operatorname{tr} a_n$ 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)[2]   # 611.2.a.c

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

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