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

Newform orbit 903.2.a.g

Trace form

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

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

Coefficient field

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

$x^{5} + 151x^{4} - 10948x^{3} - 1951532x^{2} - 39113013x + 64488989$

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 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	−5	−5	5	−4	−7	−12	8	−10	−2	−15	6	−7	5	−24	−7	−12	−9	0	4	−32	14	−15	−24	−17

$\operatorname{tr} a_n$ 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)[6]   # 903.2.a.g

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

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