Atlas › Modular forms › Weight 2 › Level 723 › 723.2.a › 723.2.a.g

Newform orbit 723.2.a.g

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + q^{2} + 13q^{3} + 23q^{4} + 6q^{5} + q^{6} + 9q^{7} + 12q^{8} + 13q^{9} - 7q^{10} - 7q^{11} + 23q^{12} + 24q^{13} - 8q^{14} + 6q^{15} + 27q^{16} + q^{18} + 7q^{19} + 13q^{20} + 9q^{21} - 8q^{22} - 9q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{13} - 231x^{12} + 14765x^{11} + 194102x^{10} - 47427643x^{9} + 964463455x^{8} + 24002195436x^{7} - 601737896830x^{6} - 6878329783763x^{5} + 90003161575363x^{4} + 1275862536974915x^{3} + 3791120417983516x^{2} - 563201831489955x - 3667891028810539$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
241	$+1$

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
$\operatorname{tr} a_p$	1	13	6	9	−7	24	0	7	−9	−3	29	25	0	13	−1	25	−7	18	24	−28	36	28	15	30	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 241, 723]: [0, 1, 20, 82] (certified at ell = 2147482801)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 10 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(723, 2)[6]   # 723.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 723:2 1000