Atlas › Modular forms › Weight 10 › Level 13 › 13.10.a › 13.10.a.a

Newform orbit 13.10.a.a

Trace form

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

$4q - 33q^{2} - 163q^{3} + 1429q^{4} + 471q^{5} - 4529q^{6} - 11241q^{7} - 45543q^{8} - 29953q^{9} - 67831q^{10} - 40140q^{11} - 155479q^{12} - 114244q^{13} - 277653q^{14} + 83307q^{15} + 726609q^{16} + 78717q^{17} + 1691026q^{18} + 209664q^{19} + 870843q^{20} + 1138431q^{21} + 1364090q^{22} - 4257444q^{23} + 3561573q^{24} + O(q^{25})$

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

$x^{4} + 196x^{3} - 2391x^{2} - 923102x + 21397904$

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 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
$\operatorname{tr} a_p$	−33	−163	471	−11241	−40140	−114244	78717	209664	−4257444	−1647936	−11366002	4636891	13859538	−33368081	−3943005	−171019326	−63389388	77050190	−41174072	252460989	594415068	115998984	−79577862	−1152240276	1049098084

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 11] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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, 10)[0]   # 13.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 13:10 1000