Atlas › Modular forms › Weight 6 › Level 102 › 102.6.a › 102.6.a.f

Newform orbit 102.6.a.f

Trace form

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

$2q - 8q^{2} - 18q^{3} + 32q^{4} - 53q^{5} + 72q^{6} + 130q^{7} - 128q^{8} + 162q^{9} + 212q^{10} + 115q^{11} - 288q^{12} + 925q^{13} - 520q^{14} + 477q^{15} + 512q^{16} - 578q^{17} - 648q^{18} + 5q^{19} - 848q^{20} - 1170q^{21} - 460q^{22} - 3111q^{23} + 1152q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{769})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{7}$ , whose minimal polynomial is

$x^{2} - 77x - 7938$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
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	−18	−53	130	115	925	−578	5	−3111	−10518	−7576	−11208	−14765	−14341	−16790	−7568	13134	55420	10800	−30522	−51492	69236	−23782	−27290	−2594

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.327 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, 3, 6, 17, 34, 51, 102]: [1, 2, 3, 7, 8, 24, 32, 94] (certified at ell = 2147483489)
T = [(5, 1), (7, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 4 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(102, 6)[5]   # 102.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 102:6 1000