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

Newform orbit 241.2.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 7x^{6} + 12x^{5} - 9x^{4} - 40x^{3} - 33x^{2} - 10x - 1$

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 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
$\operatorname{tr} a_p$	−4	−3	−8	−7	−18	−1	−2	−6	−22	−16	−18	8	−15	14	−10	15	−18	4	18	−50	0	−15	−24	−13	1

$\operatorname{tr} a_n$ 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)[0]   # 241.2.a.a

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

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