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

Newform orbit 22.12.a.d

Trace form

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

$3q + 96q^{2} + 370q^{3} + 3072q^{4} - 1928q^{5} + 11840q^{6} + 58472q^{7} + 98304q^{8} + 362233q^{9} - 61696q^{10} + 483153q^{11} + 378880q^{12} + 1349138q^{13} + 1871104q^{14} + 3956502q^{15} + 3145728q^{16} + 4446434q^{17} + 11591456q^{18} + 13084936q^{19} - 1974272q^{20} + 10649624q^{21} + 15460896q^{22} + 9885678q^{23} + 12124160q^{24} + O(q^{25})$

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

$x^{3} + 1558x^{2} - 95149604x - 215164247832$

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	370	−1928	58472	483153	1349138	4446434	13084936	9885678	283172750	−21681514	−400193332	−883050098	−3189107056	−1877680024	−3405155206	14124901902	7808412854	871235950	−11743560686	12566307382	16783237028	34191801392	13083391472	51624300308

$\operatorname{tr} a_n$ 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)[3]   # 22.12.a.d

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

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