Atlas › Modular forms › Weight 2 › Level 969 › 969.2.a › 969.2.a.j

Newform orbit 969.2.a.j

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + q^{2} - 8q^{3} + 11q^{4} - 4q^{5} - q^{6} + 9q^{7} + 3q^{8} + 8q^{9} + 9q^{10} + 8q^{11} - 11q^{12} + 2q^{13} - 9q^{14} + 4q^{15} + 25q^{16} + 8q^{17} + q^{18} - 8q^{19} - 26q^{20} - 9q^{21} - q^{22} + 10q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{8} + 11x^{7} - 241x^{6} - 2455x^{5} + 18615x^{4} + 175758x^{3} - 383943x^{2} - 4142656x - 4330966$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
17	$-1$
19	$+1$

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	−8	−4	9	8	2	8	−8	10	−3	4	5	−9	25	27	−21	−2	11	4	22	8	21	3	3	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.341 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, 3, 17, 19, 51, 57, 323, 969]: [0, 1, 2, 2, 8, 8, 32, 124] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 7 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(969, 2)[9]   # 969.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 969:2 1000