Atlas › Modular forms › Weight 2 › Level 903 › 903.2.a › 903.2.a.k

Newform orbit 903.2.a.k

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{8} - 130x^{7} - 5254x^{6} + 720896x^{5} + 3230920x^{4} - 1010417514x^{3} + 11095895862x^{2} + 99667603756x - 1135001887033$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$+1$
43	$-1$

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
$\operatorname{tr} a_p$	2	−8	3	−8	6	9	20	−18	4	6	−13	16	13	8	2	7	32	−17	0	10	24	−2	19	−6	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.697 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, 3, 7, 21, 43, 129, 301, 903]: [0, 1, 1, 4, 4, 16, 30, 120] (certified at ell = 2147483647)
T = [(2, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 12 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(903, 2)[10]   # 903.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 903:2 1000