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

Newform orbit 965.2.a.f

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + q^{2} + 7q^{3} + 33q^{4} + 22q^{5} + 7q^{6} + 6q^{7} + 29q^{9} + q^{10} + 25q^{11} - q^{12} + 7q^{13} + 6q^{14} + 7q^{15} + 51q^{16} - 5q^{17} - 12q^{18} + 28q^{19} + 33q^{20} - q^{21} - 4q^{22} + 7q^{23} + 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{22} - 22x^{21} - 245x^{20} + 8377x^{19} + 3776x^{18} - 1253036x^{17} + 3970069x^{16} + 95132492x^{15} - 490139586x^{14} - 3975292984x^{13} + 26791295104x^{12} + 90884949948x^{11} - 791828399229x^{10} - 965685744821x^{9} + 12940063235967x^{8} - 371389155035x^{7} - 107904077273163x^{6} + 100362307199160x^{5} + 338075843028277x^{4} - 627614497993563x^{3} + 236313456218277x^{2} + 91746193190482x - 51155939991844$

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
$\operatorname{tr} a_p$	1	7	22	6	25	7	−5	28	7	7	33	9	4	49	−14	−13	29	2	53	52	17	39	−12	2	−5

$\operatorname{tr} a_n$ 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)[5]   # 965.2.a.f

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

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