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

Newform orbit 409.2.a.a

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 6q^{2} - 3q^{3} + 6q^{4} - 10q^{5} - 8q^{6} - 9q^{7} - 15q^{8} + q^{10} - 26q^{11} + 3q^{12} - 3q^{13} - 7q^{14} - 22q^{15} + 4q^{16} - 8q^{17} - 7q^{18} - 15q^{19} - 16q^{20} - 15q^{21} + 19q^{22} - 31q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} + 9x^{12} + 13x^{11} - 80x^{10} - 197x^{9} + 225x^{8} + 707x^{7} - 299x^{6} - 944x^{5} + 363x^{4} + 437x^{3} - 257x^{2} + 32x - 1$

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 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	−3	−10	−9	−26	−3	−8	−15	−31	−12	−24	0	−21	−1	−31	−8	−37	7	12	−71	10	−10	16	−36	18

$\operatorname{tr} a_n$ 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)[0]   # 409.2.a.a

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

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