Atlas › Modular forms › Weight 2 › Level 735 › 735.2.a › 735.2.a.g

Newform orbit 735.2.a.g

Trace form

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

$2q - 2q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} - 12q^{8} + 2q^{9} - 2q^{10} - 2q^{11} - 4q^{12} + 8q^{13} - 2q^{15} + 16q^{16} + 10q^{17} - 2q^{18} + 2q^{19} + 4q^{20} - 4q^{22} - 6q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{2} - 120x + 3408$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$-1$
7	— ( $7^2 \mid 735$ )

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	−2	2	0	−2	8	10	2	−6	2	6	4	2	−4	4	4	10	8	−12	2	8	6	6	6	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.693 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 5, 7, 15, 21, 35, 49, 105, 147, 245, 735]: [0, 1, 1, 1, 4, 4, 6, 5, 20, 20, 30, 116] (certified at ell = 2147483353)
T = [(2, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 8 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(735, 2)[6]   # 735.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 735:2 1000