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

Newform orbit 34.6.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 12q^{2} + 4q^{3} + 48q^{4} + 144q^{5} + 16q^{6} - 18q^{7} + 192q^{8} + 883q^{9} + 576q^{10} - 180q^{11} + 64q^{12} + 318q^{13} - 72q^{14} - 2268q^{15} + 768q^{16} - 867q^{17} + 3532q^{18} + 1848q^{19} + 2304q^{20} - 6904q^{21} - 720q^{22} - 5154q^{23} + 256q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 148x^{2} + 6450x - 72876$

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
$\operatorname{tr} a_p$	12	4	144	−18	−180	318	−867	1848	−5154	144	−4458	−7140	13050	22188	16008	17862	−34740	−26316	−34776	26262	3654	−96594	57060	276438	−178830

$\operatorname{tr} a_n$ 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)[3]   # 34.6.a.d

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

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