Atlas › Modular forms › Weight 2 › Level 421 › 421.2.a › 421.2.a.a

Newform orbit 421.2.a.a

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 6q^{2} - 9q^{3} + 10q^{4} - 11q^{5} - 10q^{6} - 5q^{7} - 15q^{8} + 8q^{9} - 33q^{11} - 5q^{12} - 8q^{13} - 7q^{14} - 7q^{15} + 4q^{16} - 14q^{17} + 2q^{18} - 30q^{19} - 25q^{20} - 18q^{21} + 11q^{22} + q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 15x^{14} + 76x^{13} + 93x^{12} - 408x^{11} - 1203x^{10} + 144x^{9} + 3259x^{8} + 1796x^{7} - 2812x^{6} - 2259x^{5} + 371x^{4} + 593x^{3} + 126x^{2} + 2x - 1$

Atkin–Lehner signs

$p$	$w_p$
421	$+1$

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
$\operatorname{tr} a_p$	−6	−9	−11	−5	−33	−8	−14	−30	1	−21	−15	−7	−26	1	−33	−3	−68	−4	2	−18	18	−6	−64	−29	29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 421]: [0, 35] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 4 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(421, 2)[0]   # 421.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 421:2 1000