Atlas › Modular forms › Weight 6 › Level 49 › 49.6.a › 49.6.a.d

Newform orbit 49.6.a.d

Trace form

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

$2q + 2q^{2} - 8q^{3} + 12q^{4} - 38q^{5} - 82q^{6} + 96q^{8} - 380q^{9} - 778q^{10} + 424q^{11} - 196q^{12} - 924q^{13} + 892q^{15} - 2064q^{16} - 2346q^{17} + 212q^{18} - 360q^{19} - 1708q^{20} + 2126q^{22} - 12q^{23} + 1392q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{37})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 22x - 27$

Atkin–Lehner signs

$p$	$w_p$
7	— ( $7^2 \mid 49$ )

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
$\operatorname{tr} a_p$	2	−8	−38	0	424	−924	−2346	−360	−12	−7052	3548	11090	3500	−12680	−22956	3042	−65808	−42486	42312	−2208	−50506	9004	−104328	−26666	209132

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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, 7, 49]: [1, 5, 25] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 14 = 1 x 14 (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(49, 6)[3]   # 49.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 49:6 1000