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

Newform orbit 703.2.a.h

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 5q^{2} - 3q^{3} + 15q^{4} + 7q^{5} - 9q^{6} - 7q^{7} + 15q^{8} + 12q^{9} - 2q^{10} - 4q^{11} + 9q^{12} + 10q^{13} + 9q^{14} + 3q^{15} + 19q^{16} + 18q^{17} - q^{18} + 11q^{19} + 26q^{20} - 29q^{21} + 28q^{22} + 4q^{23} - 19q^{24} + O(q^{25})$

Coefficient field

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

$x^{11} + 4x^{10} - 186x^{9} - 1101x^{8} + 8240x^{7} + 66618x^{6} - 20987x^{5} - 941865x^{4} - 1284380x^{3} + 2950189x^{2} + 7102168x + 3697047$

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
$\operatorname{tr} a_p$	5	−3	7	−7	−4	10	18	11	4	14	17	−11	10	3	11	18	6	−8	−25	0	4	−11	3	31	−16

$\operatorname{tr} a_n$ 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)[7]   # 703.2.a.h

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

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