Atlas › Modular forms › Weight 12 › Level 22 › 22.12.a › 22.12.a.c

Newform orbit 22.12.a.c

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 96q^{2} + 106q^{3} + 3072q^{4} + 17344q^{5} - 3392q^{6} + 13204q^{7} - 98304q^{8} - 98447q^{9} - 555008q^{10} - 483153q^{11} + 108544q^{12} - 547898q^{13} - 422528q^{14} + 3940046q^{15} + 3145728q^{16} + 2157614q^{17} + 3150304q^{18} + 11483300q^{19} + 17760256q^{20} + 62926516q^{21} + 15460896q^{22} + 82445286q^{23} - 3473408q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} - 17450x^{2} - 14484096x + 957692196576$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
11	$+1$

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
$\operatorname{tr} a_p$	−96	106	17344	13204	−483153	−547898	2157614	11483300	82445286	94600910	−22911314	194979268	568218922	617580736	511210568	−1857677230	14034825102	615487510	14355694190	36259154866	−30219767470	−95444039732	22089629528	−60399228376	−51279943244

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.359 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, 2, 11, 22]: [2, 4, 12, 35] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 5 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(22, 12)[2]   # 22.12.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 22:12 1000