Atlas › Modular forms › Weight 8 › Level 13 › 13.8.a › 13.8.a.c

Newform orbit 13.8.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 15q^{2} + 80q^{3} + 253q^{4} + 258q^{5} + 1579q^{6} + 1692q^{7} + 1893q^{8} + 3494q^{9} - 4495q^{10} + 1836q^{11} - 3655q^{12} - 8788q^{13} - 18285q^{14} - 29736q^{15} - 36159q^{16} + 11814q^{17} - 2738q^{18} + 27660q^{19} - 30369q^{20} + 69930q^{21} + 59930q^{22} + 172920q^{23} + 26553q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 95x^{3} - 3570x^{2} + 411472x - 3668752$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$

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
$\operatorname{tr} a_p$	15	80	258	1692	1836	−8788	11814	27660	172920	133344	−231748	248026	588108	309304	557916	2022348	1162668	−1340572	−598484	697860	−13725816	20079576	−2024724	17646240	−6329096

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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 engine's checks

dim M^+ at levels [1, 13]: [1, 9] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(13, 8)[2]   # 13.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 13:8 1000