Atlas › Modular forms › Weight 4 › Level 23 › 23.4.a › 23.4.a.b

Newform orbit 23.4.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 2q^{2} + 7q^{3} + 20q^{4} + 14q^{5} - 17q^{6} + 16q^{7} - 63q^{8} - 33q^{9} - 70q^{10} + 8q^{11} - 67q^{12} + 111q^{13} - 144q^{14} + 10q^{15} + 64q^{16} + 98q^{17} + 49q^{18} + 96q^{19} + 140q^{20} + 180q^{21} + 220q^{22} - 92q^{23} - 188q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 9x^{3} - 6x^{2} + 141x + 25$

Atkin–Lehner signs

$p$	$w_p$
23	$+1$

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
$\operatorname{tr} a_p$	2	7	14	16	8	111	98	96	−92	21	−193	170	−125	2	−677	−230	−1140	754	488	−401	1509	−838	142	2342	1062

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.09 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 engine's checks

dim M^+ at levels [1, 23]: [1, 7] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(23, 4)[1]   # 23.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 23:4 1000