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

Newform orbit 965.2.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} + 22x^{11} + 23x^{10} - 2289x^{9} - 10188x^{8} + 79804x^{7} + 466594x^{6} - 1049250x^{5} - 7389521x^{4} + 3399795x^{3} + 35168341x^{2} + 11303278x - 9529324$

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 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
$\operatorname{tr} a_p$	−1	−7	−12	−6	−1	−9	−3	−14	−11	1	−7	−23	12	−55	−14	−17	13	0	−61	28	−5	−29	−14	10	−7

$\operatorname{tr} a_n$ 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)[3]   # 965.2.a.d

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

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