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

Newform orbit 901.2.a.h

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 2q^{2} - 7q^{3} + 10q^{4} - 11q^{5} + 3q^{6} - 7q^{7} - 6q^{8} + 6q^{9} - 4q^{10} - 30q^{11} - 12q^{12} + 8q^{13} - 12q^{14} + 5q^{15} + 24q^{16} + 13q^{17} - 22q^{18} - 7q^{19} - 41q^{20} - 13q^{21} - 10q^{22} - 21q^{23} + O(q^{25})$

Coefficient field

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

$x^{13} + 127x^{12} + 5331x^{11} + 51761x^{10} - 1883563x^{9} - 41182746x^{8} + 89871283x^{7} + 6616475521x^{6} + 15572581604x^{5} - 336907156249x^{4} - 505910983233x^{3} + 7290930210219x^{2} - 10416922731667x + 3377562192879$

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 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
$\operatorname{tr} a_p$	−2	−7	−11	−7	−30	8	13	−7	−21	−14	−13	−16	−6	−9	−13	13	−16	2	−12	−5	−22	−27	−12	−34	−30

$\operatorname{tr} a_n$ 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)[7]   # 901.2.a.h

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

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