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

Newform orbit 703.2.a.i

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 3q^{2} + 2q^{3} + 23q^{4} + 9q^{5} - 3q^{6} + 7q^{7} + 6q^{8} + 32q^{9} + 14q^{11} + 3q^{12} + q^{13} + 3q^{14} + 5q^{15} + 9q^{16} + 26q^{17} + 17q^{18} - 18q^{19} + 18q^{20} + 4q^{21} - 9q^{22} + 28q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} - 9x^{17} - 367x^{16} + 3213x^{15} + 53717x^{14} - 446060x^{13} - 4155595x^{12} + 31429778x^{11} + 188234979x^{10} - 1212261984x^{9} - 5194910775x^{8} + 25435760575x^{7} + 86956339588x^{6} - 266405047917x^{5} - 815935875729x^{4} + 1051855386476x^{3} + 3328195169729x^{2} + 198142312108x - 1589648048920$

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 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
$\operatorname{tr} a_p$	3	2	9	7	14	1	26	−18	28	7	4	18	17	−13	23	61	−21	14	13	3	−4	−18	17	−4	15

$\operatorname{tr} a_n$ 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)[8]   # 703.2.a.i

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

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