Atlas › Modular forms › Weight 2 › Level 409 › 409.2.a › 409.2.a.b

Newform orbit 409.2.a.b

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 5q^{2} + q^{3} + 23q^{4} + 8q^{5} - 4q^{6} + 5q^{7} + 12q^{8} + 27q^{9} - 5q^{10} + 30q^{11} - 5q^{12} - 7q^{13} + 17q^{14} + 26q^{15} + 29q^{16} + 6q^{17} - 4q^{18} + q^{19} + 14q^{20} - 3q^{21} + q^{22} + 37q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 6x^{19} - 53x^{18} + 371x^{17} + 884x^{16} - 8541x^{15} - 3292x^{14} + 93301x^{13} - 45060x^{12} - 515110x^{11} + 444762x^{10} + 1447059x^{9} - 1304306x^{8} - 2181912x^{7} + 1332857x^{6} + 1858280x^{5} - 294748x^{4} - 603955x^{3} - 24502x^{2} + 61633x + 6929$

Atkin–Lehner signs

$p$	$w_p$
409	$-1$

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
$\operatorname{tr} a_p$	5	1	8	5	30	−7	6	1	37	12	10	−4	−7	−1	51	10	43	−21	4	97	−36	16	28	2	−36

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 409]: [0, 34] (certified at ell = 2147477809)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (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(409, 2)[1]   # 409.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 409:2 1000