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

Newform orbit 23.12.a.b

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 32q^{2} - 20q^{3} + 11264q^{4} + 1034q^{5} - 22385q^{6} + 159584q^{7} + 115497q^{8} + 611943q^{9} - 627650q^{10} - 771396q^{11} - 1720771q^{12} + 3433434q^{13} + 4585896q^{14} + 5551840q^{15} + 18802384q^{16} + 29035398q^{17} + 26169127q^{18} + 21398428q^{19} + 72466260q^{20} + 61896432q^{21} + 100463524q^{22} - 70799773q^{23} + 161844076q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} - 12x^{10} - 1274719x^{9} - 131292363x^{8} + 549872365729x^{7} + 121019089265197x^{6} - 81166447960286183x^{5} - 27245722792085231849x^{4} + 133002797872639003650x^{3} + 410605855091268250301971x^{2} - 22611899694894988018433374x + 326358227706495136860784000$

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
$\operatorname{tr} a_p$	32	−20	1034	159584	−771396	3433434	29035398	21398428	−70799773	226699042	251932328	573876170	−1733596378	647370308	−5436527248	−3387203910	15113662084	23895772578	46806014468	45541532768	63786612542	21847812496	40153340788	37300228382	−243602730

$\operatorname{tr} a_n$ 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)[1]   # 23.12.a.b

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

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