Atlas › Modular forms › Weight 2 › Level 311 › 311.2.a › 311.2.a.b

Newform orbit 311.2.a.b

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + 2q^{2} + 2q^{3} + 30q^{4} + q^{5} + 2q^{6} + 8q^{7} + 38q^{9} + 10q^{10} + 6q^{11} - q^{12} + 12q^{13} + 2q^{14} - 2q^{15} + 42q^{16} + 9q^{17} - 5q^{18} + 16q^{19} - 19q^{20} + 13q^{21} + 15q^{22} - 2q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 4x^{21} - 83x^{20} + 344x^{19} + 2721x^{18} - 11867x^{17} - 45429x^{16} + 214882x^{15} + 412401x^{14} - 2251279x^{13} - 1928513x^{12} + 14125266x^{11} + 2893612x^{10} - 52649821x^{9} + 11137652x^{8} + 110674157x^{7} - 51561324x^{6} - 117814489x^{5} + 68112425x^{4} + 54704945x^{3} - 31139582x^{2} - 7575637x + 3738089$

Atkin–Lehner signs

$p$	$w_p$
311	$-1$

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	2	1	8	6	12	9	16	−2	24	10	27	17	15	−20	−10	−33	55	−5	−27	22	66	−16	30	38

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 311]: [0, 27] (certified at ell = 2147479121)
T = [(2, 1), (3, 1)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 3 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(311, 2)[1]   # 311.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 311:2 1000