Atlas › Modular forms › Weight 2 › Level 926 › 926.2.a › 926.2.a.b

Newform orbit 926.2.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} - 12x^{3} + 19x^{2} - 8x + 1$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
463	$+1$

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
$\operatorname{tr} a_p$	−5	−1	1	−3	−2	−4	−6	−12	5	−2	−21	−8	−1	−6	13	−1	3	−24	−2	−6	−18	−9	20	−18	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 463, 926]: [0, 1, 39, 118] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(926, 2)[1]   # 926.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 926:2 1000