Atlas › Modular forms › Weight 12 › Level 21 › 21.12.a › 21.12.a.c

Newform orbit 21.12.a.c

Trace form

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

$3q - 68q^{2} - 729q^{3} + 1920q^{4} + 3326q^{5} + 16524q^{6} - 50421q^{7} - 365088q^{8} + 177147q^{9} - 744096q^{10} + 1348256q^{11} - 466560q^{12} - 892158q^{13} + 1142876q^{14} - 808218q^{15} + 20181888q^{16} - 2450550q^{17} - 4015332q^{18} + 11057244q^{19} + 76290688q^{20} + 12252303q^{21} - 71211336q^{22} + 9675972q^{23} + 88716384q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} - 3258x^{2} - 154567152x + 817858898640$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$+1$

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
$\operatorname{tr} a_p$	−68	−729	3326	−50421	1348256	−892158	−2450550	11057244	9675972	196390898	−70964448	−874700478	−386189798	544378572	20129256	4746325602	19835761884	−6480362406	25314754956	41912998956	−18034242474	−33952994712	25737739644	101517768986	3701782446

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.534 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, 3, 7, 21]: [2, 5, 9, 32] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 6 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(21, 12)[2]   # 21.12.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 21:12 1000