Atlas › Modular forms › Weight 2 › Level 293 › 293.2.a › 293.2.a.a

Newform orbit 293.2.a.a

Trace form

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

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

Coefficient field

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

$x^{8} + 11x^{7} + 36x^{6} + 18x^{5} - 78x^{4} - 93x^{3} + 3x^{2} + 40x + 13$

Atkin–Lehner signs

$p$	$w_p$
293	$+1$

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
$\operatorname{tr} a_p$	−3	−8	−1	−13	−9	−8	−2	−15	−8	7	−9	−23	12	−43	−9	6	3	10	−50	0	−14	5	−3	−3	−27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.171 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, 293]: [0, 25] (certified at ell = 2147478961)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(293, 2)[0]   # 293.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 293:2 1000