Atlas › Modular forms › Weight 2 › Level 707 › 707.2.a › 707.2.a.g

Newform orbit 707.2.a.g

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 5q^{2} + q^{3} + 19q^{4} + 10q^{5} + q^{6} - 15q^{7} + 12q^{8} + 18q^{9} - q^{10} + 8q^{11} + 2q^{12} + 31q^{13} - 5q^{14} - 8q^{15} + 27q^{16} + 15q^{17} + 27q^{18} - 10q^{19} + 24q^{20} - q^{21} + 14q^{22} + 18q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 6x^{14} - 39x^{13} + 248x^{12} + 514x^{11} - 3423x^{10} - 3308x^{9} + 19851x^{8} + 12193x^{7} - 47809x^{6} - 18726x^{5} + 48209x^{4} + 10198x^{3} - 17201x^{2} - 786x + 146$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
101	$-1$

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
$\operatorname{tr} a_p$	5	1	10	−15	8	31	15	−10	18	18	11	47	15	21	13	11	−13	17	1	17	40	15	−7	−12	35

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.319 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, 7, 101, 707]: [0, 1, 9, 70] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 5 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(707, 2)[6]   # 707.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 707:2 1000