Atlas › Modular forms › Weight 2 › Level 703 › 703.2.a › 703.2.a.g

Newform orbit 703.2.a.g

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 3q^{2} - 2q^{3} + 9q^{4} - 7q^{5} - 5q^{6} - 7q^{7} - 12q^{8} - 3q^{9} + 6q^{10} + 4q^{11} - 11q^{12} - 3q^{13} - 16q^{14} + q^{15} + 21q^{16} - 26q^{17} + 7q^{18} - 9q^{19} - 32q^{20} + 2q^{21} - 11q^{22} - 22q^{23} - q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} + 9x^{8} - 66x^{7} - 640x^{6} + 933x^{5} + 10242x^{4} - 9306x^{3} - 49049x^{2} + 30944x + 62944$

Atkin–Lehner signs

$p$	$w_p$
19	$+1$
37	$+1$

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 odd 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
$\operatorname{tr} a_p$	−3	−2	−7	−7	4	−3	−26	−9	−22	−15	10	−9	21	−17	−3	−33	5	−14	−5	−1	−34	2	−21	4	−25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 19, 37, 703]: [0, 2, 3, 64] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (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(703, 2)[6]   # 703.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 703:2 1000