Atlas › Modular forms › Weight 2 › Level 291 › 291.2.a › 291.2.a.h

Newform orbit 291.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 12x^{6} - 140x^{5} + 1507x^{4} + 8023x^{3} - 51956x^{2} - 205788x + 161693$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
97	$+1$

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
$\operatorname{tr} a_p$	0	7	4	9	−3	11	−5	6	−10	3	9	4	0	5	−2	−12	−12	12	−5	−24	14	16	−16	−1	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 3, 97, 291]: [0, 1, 8, 34] (certified at ell = 2147483137)
T = [(2, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(291, 2)[7]   # 291.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 291:2 1000