Atlas › Modular forms › Weight 6 › Level 104 › 104.6.a › 104.6.a.b

Newform orbit 104.6.a.b

Trace form

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

$2q + 23q^{3} - 47q^{5} - 95q^{7} - 53q^{9} - 696q^{11} - 338q^{13} - 1383q^{15} - 285q^{17} - 1860q^{19} - 587q^{21} + 684q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{337})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 24x - 1204$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 104$ )
13	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	23	−47	−95	−696	−338	−285	−1860	684	−1004	−1026	−14427	−14246	−3347	11789	−3134	20296	34966	67364	116491	−27240	11552	13786	−18896	37104

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 4, 8, 13, 26, 52, 104]: [1, 2, 4, 7, 7, 19, 38, 74] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(104, 6)[1]   # 104.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 104:6 1000