Atlas › Modular forms › Weight 2 › Level 866 › 866.2.a › 866.2.a.d

Newform orbit 866.2.a.d

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{13} - 47x^{12} - 4829x^{11} + 285083x^{10} + 3450662x^{9} - 362834512x^{8} - 36090644x^{7} + 192931678792x^{6} - 452459144960x^{5} - 49063689550264x^{4} + 94383148640912x^{3} + 5645972976137472x^{2} - 4123803377868832x - 212480998294164832$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
433	$-1$

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
$\operatorname{tr} a_p$	−13	2	0	9	0	5	−11	18	8	−6	31	23	−1	9	19	−11	14	12	27	41	10	50	−11	8	20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.422 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, 2, 433, 866]: [0, 1, 36, 110] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(866, 2)[3]   # 866.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 866:2 1000