Atlas › Modular forms › Weight 2 › Level 901 › 901.2.a › 901.2.a.i

Newform orbit 901.2.a.i

Trace form

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

$18q + 4q^{2} + 8q^{3} + 28q^{4} + 7q^{5} + 3q^{6} + 14q^{7} + 32q^{9} + q^{10} + 28q^{11} + 2q^{12} - 12q^{13} + 9q^{14} + 18q^{15} + 32q^{16} + 18q^{17} - 24q^{18} + 10q^{19} + 22q^{20} - 3q^{21} - 10q^{22} + 15q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{18} - 161x^{17} + 7505x^{16} + 113975x^{15} - 19780240x^{14} + 401327667x^{13} + 12156364134x^{12} - 559769382750x^{11} + 1677415594251x^{10} + 229145717550745x^{9} - 3134390249054724x^{8} - 24473238921374068x^{7} + 642145590725431561x^{6} + 163554407036172306x^{5} - 48297444271892832423x^{4} + 11227733841924261235x^{3} + 1499176497920749630739x^{2} + 2929649693357771587611x + 435180186045612510860$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
53	$+1$

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	8	7	14	28	−12	18	10	15	32	−11	−6	13	−10	−10	−18	38	8	−17	12	−10	0	5	−20	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 0.418 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, 17, 53, 901]: [0, 2, 5, 82] (certified at ell = 2147482481)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 69 = 1 x 69 (phi(ord eps) x dim S_k^new); 15 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(901, 2)[8]   # 901.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 901:2 1000