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

Newform orbit 34.6.a.c

Trace form

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

$2q - 8q^{2} - 6q^{3} + 32q^{4} + 32q^{5} + 24q^{6} - 138q^{7} - 128q^{8} - 382q^{9} - 128q^{10} - 1038q^{11} - 96q^{12} - 700q^{13} + 552q^{14} - 956q^{15} + 512q^{16} - 578q^{17} + 1528q^{18} - 396q^{19} + 512q^{20} + 2392q^{21} + 4152q^{22} - 222q^{23} + 384q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{43})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 26x - 3314$

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 odd 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	32	−138	−1038	−700	−578	−396	−222	−544	4886	7984	−20428	12916	−1856	30508	−62628	2584	23120	−17886	80452	−48870	−59676	−235412	−98236

$\operatorname{tr} a_n$ 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)[2]   # 34.6.a.c

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

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