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

Newform orbit 22.12.a.b

Trace form

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

$3q - 96q^{2} - 70q^{3} + 3072q^{4} - 5624q^{5} + 2240q^{6} - 30576q^{7} - 98304q^{8} + 533833q^{9} + 179968q^{10} + 483153q^{11} - 71680q^{12} + 423050q^{13} + 978432q^{14} - 581042q^{15} + 3145728q^{16} - 16823942q^{17} - 17082656q^{18} - 12737072q^{19} - 5758976q^{20} - 64101968q^{21} - 15460896q^{22} - 11397034q^{23} + 2293760q^{24} + O(q^{25})$

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

$x^{3} + 5694x^{2} + 5500220x - 633213048$

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 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
$\operatorname{tr} a_p$	−96	−70	−5624	−30576	483153	423050	−16823942	−12737072	−11397034	−136849122	242041022	450661244	97726086	−712479248	−3330549288	−3777184886	−9293353002	2647736806	1632055702	−2119547430	−5284631794	8982892548	11489211392	181048875488	−17159174540

$\operatorname{tr} a_n$ 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)[1]   # 22.12.a.b

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

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