Atlas › Modular forms › Weight 2 › Level 905 › 905.2.a › 905.2.a.g

Newform orbit 905.2.a.g

Trace form

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

$18q - 3q^{2} - 2q^{3} + 23q^{4} - 18q^{5} + 10q^{6} + 20q^{7} - 12q^{8} + 22q^{9} + 3q^{10} + 5q^{11} - 9q^{12} + 19q^{13} + 5q^{14} + 2q^{15} + 33q^{16} - q^{17} - 18q^{18} + 36q^{19} - 23q^{20} + 6q^{21} + 7q^{22} - 11q^{23} + 36q^{24} + O(q^{25})$

Coefficient field

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

$x^{18} + 5x^{17} - 65x^{16} - 312x^{15} + 1576x^{14} + 7380x^{13} - 17209x^{12} - 83159x^{11} + 76148x^{10} + 461978x^{9} - 25739x^{8} - 1179469x^{7} - 598407x^{6} + 1021867x^{5} + 941791x^{4} - 9744x^{3} - 223152x^{2} - 69780x - 6236$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
181	$-1$

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	−18	20	5	19	−1	36	−11	7	41	15	3	16	9	−17	18	23	0	34	36	40	2	18	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 5, 181, 905]: [0, 1, 15, 92] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 6 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(905, 2)[6]   # 905.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 905:2 1000