Atlas › Modular forms › Weight 2 › Level 241 › 241.2.a › 241.2.a.b

Newform orbit 241.2.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 3q^{2} + q^{3} + 13q^{4} + 6q^{5} - q^{6} + 3q^{7} + 9q^{8} + 15q^{9} - 7q^{10} + 22q^{11} - 7q^{12} - 5q^{13} + 6q^{14} + 13q^{15} + 15q^{16} - 4q^{17} - q^{18} - 6q^{19} + 10q^{20} - 14q^{21} - 12q^{22} + 32q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 4x^{11} - 35x^{10} + 151x^{9} + 362x^{8} - 1959x^{7} - 458x^{6} + 9510x^{5} - 8056x^{4} - 9303x^{3} + 15123x^{2} - 6111x + 691$

Atkin–Lehner signs

$p$	$w_p$
241	$-1$

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
$\operatorname{tr} a_p$	3	1	6	3	22	−5	−4	−6	32	6	8	−8	−1	−2	34	5	26	−26	6	94	−22	9	−8	−3	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.167 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, 241]: [0, 20] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 2 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(241, 2)[1]   # 241.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 241:2 1000