Atlas › Modular forms › Weight 6 › Level 34 › 34.6.a › 34.6.a.b

Newform orbit 34.6.a.b

Trace form

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

$2q - 8q^{2} - 6q^{3} + 32q^{4} - 36q^{5} + 24q^{6} - 2q^{7} - 128q^{8} + 774q^{9} + 144q^{10} + 1206q^{11} - 96q^{12} - 224q^{13} + 8q^{14} + 1764q^{15} + 512q^{16} + 578q^{17} - 3096q^{18} - 2300q^{19} - 576q^{20} + 420q^{21} - 4824q^{22} + 1614q^{23} + 384q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{69})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 42x - 2940$

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
$\operatorname{tr} a_p$	−8	−6	−36	−2	1206	−224	578	−2300	1614	6732	4546	−1196	−6828	8428	9432	31596	−21012	−30668	19720	14754	−59492	72646	159828	115944	−56348

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 2, 8, 24] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(34, 6)[1]   # 34.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 34:6 1000