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

Newform orbit 703.2.a.f

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 9x^{4} - 15x^{3} - 101x^{2} + 80x + 159$

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	1	−7	−4	−3	−12	5	−16	−11	−2	5	−1	9	13	−27	−19	18	−1	−3	10	22	1	−16	−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)[5]   # 703.2.a.f

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

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