Atlas › Modular forms › Weight 12 › Level 23 › 23.12.a › 23.12.a.a

Newform orbit 23.12.a.a

Trace form

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

$8q - 32q^{2} - 992q^{3} + 5120q^{4} - 11466q^{5} + 34618q^{6} - 54118q^{7} - 155568q^{8} + 648814q^{9} + 517892q^{10} - 291462q^{11} - 884188q^{12} - 2211306q^{13} - 939584q^{14} - 2205330q^{15} - 8561344q^{16} - 5775330q^{17} - 51349034q^{18} - 21015588q^{19} - 65503576q^{20} - 36171230q^{21} - 83047784q^{22} + 51490744q^{23} - 129286728q^{24} + O(q^{25})$

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

$x^{8} + 1024x^{7} - 554077x^{6} - 616567378x^{5} + 113277746511x^{4} + 107031122948252x^{3} - 11956604766607619x^{2} - 5301927678130101962x + 722503614756205523600$

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 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
$\operatorname{tr} a_p$	−32	−992	−11466	−54118	−291462	−2211306	−5775330	−21015588	51490744	−322285430	−415184840	176642018	357962218	2500461376	261795200	3542935060	930905396	−25338655048	−3123467482	−52612263236	−67014176274	−27683357604	−12253964262	10662817760	−124519454530

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.519 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]: [2, 23] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (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(23, 12)[0]   # 23.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 23:12 1000