Atlas › Modular forms › Weight 2 › Level 965 › 965.2.a › 965.2.a.e

Newform orbit 965.2.a.e

Trace form

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

$20q + 3q^{2} + 11q^{3} + 31q^{4} - 20q^{5} - q^{6} + 10q^{7} + 18q^{8} + 35q^{9} - 3q^{10} - q^{11} + 27q^{12} + 13q^{13} - 14q^{14} - 11q^{15} + 53q^{16} + 9q^{17} + 22q^{18} + 10q^{19} - 31q^{20} + 3q^{21} + 20q^{22} + 25q^{23} - 14q^{24} + O(q^{25})$

Coefficient field

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

$x^{20} - 36x^{19} + 188x^{18} + 7331x^{17} - 88198x^{16} - 453055x^{15} + 10399619x^{14} + 662405x^{13} - 614747707x^{12} + 1157853741x^{11} + 21427118989x^{10} - 59977175395x^{9} - 469376631360x^{8} + 1478261624832x^{7} + 6585238456393x^{6} - 20002732812391x^{5} - 57987369580272x^{4} + 142682387131879x^{3} + 294705589616521x^{2} - 419252764273466x - 664160920628556$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
193	$-1$

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	11	−20	10	−1	13	9	10	25	1	−7	19	−8	79	28	7	−7	0	89	−40	5	15	36	−38	17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 193, 965]: [0, 1, 16, 98] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 9 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(965, 2)[4]   # 965.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 965:2 1000