Atlas › Modular forms › Weight 2 › Level 923 › 923.2.a › 923.2.a.b

Newform orbit 923.2.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 5x^{9} - 9x^{8} - 62x^{7} + 7x^{6} + 230x^{5} + 27x^{4} - 349x^{3} + 13x^{2} + 197x - 67$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
71	$-1$

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
$\operatorname{tr} a_p$	−1	−4	−6	−9	−1	10	−11	−5	−12	−3	2	−41	0	−20	4	−12	9	−49	−11	10	−9	0	3	9	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.269 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, 13, 71, 923]: [0, 1, 7, 86] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (phi(ord eps) x dim S_k^new); 7 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(923, 2)[1]   # 923.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 923:2 1000