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

Newform orbit 13.10.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 15q^{2} + 161q^{3} + 361q^{4} + 1803q^{5} + 5693q^{6} + 10099q^{7} + 23151q^{8} + 61060q^{9} + 84505q^{10} + 121746q^{11} + 113389q^{12} + 142805q^{13} + 8475q^{14} + 105973q^{15} - 322463q^{16} - 495669q^{17} - 656228q^{18} - 840738q^{19} - 1595607q^{20} - 1599467q^{21} - 2023594q^{22} - 592152q^{23} - 2295657q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 176x^{4} - 71403x^{3} + 10141242x^{2} + 1035735264x - 113786339328$

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	161	1803	10099	121746	142805	−495669	−840738	−592152	10678182	12885296	7171823	9294012	12831975	43354215	93231780	246496182	−132232612	−369388534	212150457	−252729806	−1247271728	1696894296	−753854382	3824606

$\operatorname{tr} a_n$ 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)[1]   # 13.10.a.b

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

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