Atlas › Modular forms › Weight 10 › Level 34 › 34.10.a › 34.10.a.d

Newform orbit 34.10.a.d

Trace form

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

$4q + 64q^{2} + 226q^{3} + 1024q^{4} - 1656q^{5} + 3616q^{6} + 2654q^{7} + 16384q^{8} + 94468q^{9} - 26496q^{10} + 128106q^{11} + 57856q^{12} - 16840q^{13} + 42464q^{14} + 505764q^{15} + 262144q^{16} - 334084q^{17} + 1511488q^{18} + 505964q^{19} - 423936q^{20} + 2043368q^{21} + 2049696q^{22} - 404838q^{23} + 925696q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 1430x^{3} - 6730662x^{2} - 8501880936x + 2537598862440$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
17	$+1$

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
$\operatorname{tr} a_p$	64	226	−1656	2654	128106	−16840	−334084	505964	−404838	4590792	−4952578	11290592	−11496912	−53686900	−57601536	−98104464	1116948	−62897560	69022280	202453914	312667256	886063442	1760033004	100343712	63193280

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.329 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, 17, 34]: [1, 3, 14, 42] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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(34, 10)[3]   # 34.10.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 34:10 1000