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

Newform orbit 102.6.a.h

Trace form

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

$3q + 12q^{2} + 27q^{3} + 48q^{4} + 53q^{5} + 108q^{6} + 168q^{7} + 192q^{8} + 243q^{9} + 212q^{10} + 365q^{11} + 432q^{12} + 663q^{13} + 672q^{14} + 477q^{15} + 768q^{16} + 867q^{17} + 972q^{18} + 1777q^{19} + 848q^{20} + 1512q^{21} + 1460q^{22} + 573q^{23} + 1728q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{7}$ , whose minimal polynomial is

$x^{3} - 221x^{2} - 3760x + 156572$

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 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	27	53	168	365	663	867	1777	573	1278	−2078	3428	−121	−6521	−11266	−12898	−39618	−17040	−18220	−98124	−47590	−27302	−182354	−199528	91500

$\operatorname{tr} a_n$ 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)[7]   # 102.6.a.h

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

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