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

Newform orbit 421.2.a.b

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 4q^{2} + 7q^{3} + 18q^{4} + 7q^{5} + 8q^{6} + 3q^{7} + 9q^{8} + 20q^{9} - 4q^{10} + 31q^{11} + 7q^{12} - 2q^{13} + 5q^{14} + q^{15} + 8q^{16} + 16q^{17} + 2q^{18} + 24q^{19} + 5q^{20} - 2q^{21} - 11q^{22} + 13q^{23} + 23q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 11x^{18} - 14x^{17} + 535x^{16} - 1031x^{15} - 8646x^{14} + 28978x^{13} + 56322x^{12} - 297854x^{11} - 80329x^{10} + 1448541x^{9} - 665130x^{8} - 3284627x^{7} + 2675780x^{6} + 2960543x^{5} - 2515146x^{4} - 1425513x^{3} + 705866x^{2} + 371316x + 18823$

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 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
$\operatorname{tr} a_p$	4	7	7	3	31	−2	16	24	13	9	9	−25	34	15	19	−9	90	−6	−18	18	−6	−14	66	19	−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)[1]   # 421.2.a.b

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

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