Atlas › Modular forms › Weight 4 › Level 102 › 102.4.a › 102.4.a.f

Newform orbit 102.4.a.f

Trace form

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

$2q + 4q^{2} + 6q^{3} + 8q^{4} + 3q^{5} + 12q^{6} + 22q^{7} + 16q^{8} + 18q^{9} + 6q^{10} + 3q^{11} + 24q^{12} + 37q^{13} + 44q^{14} + 9q^{15} + 32q^{16} - 34q^{17} + 36q^{18} - 71q^{19} + 12q^{20} + 66q^{21} + 6q^{22} - 111q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{393})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ , whose minimal polynomial is

$x^{2} - 343x - 90944$

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
$\operatorname{tr} a_p$	4	6	3	22	3	37	−34	−71	−111	−66	16	−164	−537	79	−498	−900	−138	316	760	318	1168	1540	606	234	1042

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 2, 5, 6, 16, 20, 58] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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, 4)[5]   # 102.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 102:4 1000