Atlas › Modular forms › Weight 2 › Level 994 › 994.2.a › 994.2.a.n

Newform orbit 994.2.a.n

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{4} + 79x^{3} - 3902x^{2} - 222084x + 6962314$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$
71	$+1$

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
$\operatorname{tr} a_p$	−4	−1	−4	−4	−1	−1	−6	−12	3	−3	−6	3	−27	8	−12	4	−32	7	20	−4	4	−18	−11	−40	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.571 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, 7, 14, 71, 142, 497, 994]: [0, 1, 1, 4, 7, 20, 50, 148] (certified at ell = 2147482681)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 35 = 1 x 35 (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(994, 2)[13]   # 994.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 994:2 1000