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

Newform orbit 901.2.a.j

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 5q^{2} + 3q^{3} + 25q^{4} + 7q^{5} - 3q^{6} - 7q^{7} + 9q^{8} + 31q^{9} - 2q^{10} + 34q^{11} + 8q^{12} + 16q^{14} + 7q^{15} + 27q^{16} - 20q^{17} + 17q^{18} + 13q^{19} - 13q^{20} + 3q^{21} + 8q^{22} + 23q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{20} - 8180x^{18} + 24748x^{17} + 25612023x^{16} - 126996051x^{15} - 40169086461x^{14} + 247096688416x^{13} + 34713106677991x^{12} - 229761039254882x^{11} - 17289591886410447x^{10} + 110111148863132538x^{9} + 5045858848412706555x^{8} - 27405371524654192909x^{7} - 851840260341117931681x^{6} + 3322625274467773254990x^{5} + 77439464481633864244658x^{4} - 151076151204958485708878x^{3} - 2949027906650298113444927x^{2} - 561515747783174683266980x + 18468603393910316271284$

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
$\operatorname{tr} a_p$	5	3	7	−7	34	0	−20	13	23	16	1	−6	16	13	29	20	40	−10	20	41	−16	15	20	34	−36

$\operatorname{tr} a_n$ 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)[9]   # 901.2.a.j

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

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