Atlas › Modular forms › Weight 8 › Level 14 › 14.8.a › 14.8.a.c

Newform orbit 14.8.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 16q^{2} + 70q^{3} + 128q^{4} + 126q^{5} + 560q^{6} + 686q^{7} + 1024q^{8} + 2014q^{9} + 1008q^{10} - 3420q^{11} + 4480q^{12} - 6398q^{13} + 5488q^{14} - 31032q^{15} + 8192q^{16} - 38472q^{17} + 16112q^{18} - 43358q^{19} + 8064q^{20} + 24010q^{21} - 27360q^{22} + 89928q^{23} + 35840q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1969})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 196x - 116412$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$-1$

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
$\operatorname{tr} a_p$	16	70	126	686	−3420	−6398	−38472	−43358	89928	159576	−143612	−271832	64848	1527964	485436	−145716	−4183662	−280658	5671648	−619272	3939628	4656616	1235850	−17241420	−740936

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.258 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, 2, 7, 14]: [1, 3, 5, 16] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(14, 8)[2]   # 14.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 14:8 1000