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

Newform orbit 21.12.a.d

Trace form

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

$3q - 33q^{2} - 729q^{3} + 765q^{4} + 3102q^{5} + 8019q^{6} + 50421q^{7} + 20661q^{8} + 177147q^{9} + 258234q^{10} - 323232q^{11} - 185895q^{12} - 1701414q^{13} - 554631q^{14} - 753786q^{15} - 8286783q^{16} - 13285206q^{17} - 1948617q^{18} + 3457092q^{19} - 20182458q^{20} - 12252303q^{21} - 57420132q^{22} - 18776676q^{23} - 5020623q^{24} + O(q^{25})$

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

$x^{3} - 3069x^{2} - 37748730x + 60897667320$

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 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
$\operatorname{tr} a_p$	−33	−729	3102	50421	−323232	−1701414	−13285206	3457092	−18776676	−64656294	−30699048	1046484186	−158666502	746774892	−3828844824	−3245231286	−12392434452	−17915258046	−11801453124	−26571896460	15228295590	8453153688	−31295925492	−16030004742	−87243902370

$\operatorname{tr} a_n$ 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)[3]   # 21.12.a.d

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

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