Atlas › Modular forms › Weight 12 › Level 11 › 11.12.a › 11.12.a.a

Newform orbit 11.12.a.a

Trace form

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

$3q - 393q^{3} + 2856q^{4} - 7305q^{5} + 6864q^{6} - 5082q^{7} - 276672q^{8} - 198198q^{9} - 649440q^{10} + 483153q^{11} - 1760376q^{12} - 2434212q^{13} - 4014960q^{14} - 5760165q^{15} - 2213088q^{16} + 12112122q^{17} + 4748832q^{18} - 8590560q^{19} + 39453000q^{20} + 28021422q^{21} - 3136413q^{23} + 39017088q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 393x^{2} - 100761x - 40559681$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	0	−393	−7305	−5082	483153	−2434212	12112122	−8590560	−3136413	−376441824	−313174893	−454281387	−37614456	162163386	−3182498184	−3000753402	−1843219707	−28094112684	10315312497	3703071657	14017034988	−8104583058	26009027946	−17344395051	−7984545237

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.484 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, 11]: [2, 12] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(11, 12)[0]   # 11.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 11:12 1000