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

Newform orbit 293.2.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 13x^{15} + 25x^{14} + 333x^{13} - 1506x^{12} - 1763x^{11} + 19186x^{10} - 11752x^{9} - 95072x^{8} + 121316x^{7} + 204263x^{6} - 325649x^{5} - 209259x^{4} + 335174x^{3} + 109821x^{2} - 111476x - 20716$

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 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	10	1	15	9	10	−4	19	4	−7	7	19	−12	57	3	−4	−5	−8	90	−24	0	11	21	−1	3

$\operatorname{tr} a_n$ 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)[1]   # 293.2.a.b

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

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