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

Newform orbit 21.12.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 45q^{2} + 972q^{3} + 4801q^{4} + 13356q^{5} + 10935q^{6} + 67228q^{7} + 182529q^{8} + 236196q^{9} - 383106q^{10} - 369324q^{11} + 1166643q^{12} + 2270240q^{13} + 756315q^{14} + 3245508q^{15} + 4006657q^{16} + 3148884q^{17} + 2657205q^{18} + 27344192q^{19} + 38609262q^{20} + 16336404q^{21} + 32285676q^{22} + 34792812q^{23} + 44354547q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 13401x^{3} - 70137066x^{2} + 946011785040x + 294712626140496$

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
$\operatorname{tr} a_p$	45	972	13356	67228	−369324	2270240	3148884	27344192	34792812	22117320	149896976	−731866096	−1827649764	−785485840	−3718935000	−80694432	2229046920	7252101992	16864668392	−10651955436	21070267808	83041303880	−1205635536	9412854012	205207747088

$\operatorname{tr} a_n$ 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)[4]   # 21.12.a.e

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

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