Atlas › Modular forms › Weight 2 › Level 419 › 419.2.a › 419.2.a.b

Newform orbit 419.2.a.b

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 2q^{2} + 2q^{3} + 38q^{4} + 5q^{5} + 5q^{6} + 7q^{7} + 3q^{8} + 38q^{9} + 8q^{10} + 4q^{11} + 2q^{12} + 27q^{13} - 3q^{14} - 7q^{15} + 58q^{16} + 2q^{17} - 6q^{18} + 8q^{19} - 13q^{20} + 29q^{21} + 18q^{22} - q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 4x^{25} - 100x^{24} + 409x^{23} + 4086x^{22} - 16885x^{21} - 91646x^{20} + 378396x^{19} + 1267391x^{18} - 5148302x^{17} - 11461636x^{16} + 44663935x^{15} + 70031581x^{14} - 250890860x^{13} - 293751703x^{12} + 902018479x^{11} + 843646917x^{10} - 1986056002x^{9} - 1601257468x^{8} + 2417405024x^{7} + 1815612916x^{6} - 1249649476x^{5} - 949746305x^{4} + 48045096x^{3} + 75306711x^{2} - 1042780x - 1364582$

Atkin–Lehner signs

$p$	$w_p$
419	$-1$

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
$\operatorname{tr} a_p$	2	2	5	7	4	27	2	8	−1	46	17	45	6	15	−14	17	−15	87	4	−7	10	57	−30	−16	−17

$\operatorname{tr} a_n$ 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, 419]: [0, 36] (certified at ell = 2147482943)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (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(419, 2)[1]   # 419.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 419:2 1000