Atlas › Modular forms › Weight 2 › Level 341 › 341.2.a › 341.2.a.c

Newform orbit 341.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + q^{2} + 4q^{3} + 13q^{4} - 5q^{5} + 9q^{6} + 7q^{7} + 6q^{8} + 4q^{9} - 2q^{10} - 8q^{11} + 11q^{12} + 8q^{13} - 17q^{14} - q^{15} + 23q^{16} + 4q^{18} + 16q^{19} - 31q^{20} + q^{21} - q^{22} + 4q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 13x^{7} - 83x^{6} + 1382x^{5} + 1227x^{4} - 42529x^{3} + 29686x^{2} + 289193x - 413445$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
31	$-1$

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
$\operatorname{tr} a_p$	1	4	−5	7	−8	8	0	16	4	8	8	8	3	27	−21	−1	16	23	−10	−3	8	32	−7	−49	−18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 11, 31, 341]: [0, 2, 3, 34] (certified at ell = 2147482951)
T = [(2, 1), (3, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(341, 2)[2]   # 341.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 341:2 1000