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

Newform orbit 905.2.a.f

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 11x^{16} - 3x^{15} + 408x^{14} - 950x^{13} - 4736x^{12} + 17761x^{11} + 16827x^{10} - 117174x^{9} + 25452x^{8} + 296729x^{7} - 188841x^{6} - 247927x^{5} + 176447x^{4} + 65345x^{3} - 42782x^{2} - 3612x + 1568$

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
$\operatorname{tr} a_p$	4	7	17	22	1	11	8	12	19	−4	20	−1	−15	44	43	12	10	21	35	−35	2	26	30	−24	29

$\operatorname{tr} a_n$ 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)[5]   # 905.2.a.f

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

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