Atlas › Modular forms › Weight 2 › Level 923 › 923.2.a › 923.2.a.d

Newform orbit 923.2.a.d

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 3q^{2} - q^{3} + 23q^{4} + 6q^{5} + 3q^{6} + 11q^{7} + 9q^{8} + 15q^{9} - q^{10} + 9q^{11} + 22q^{12} - 18q^{13} + 5q^{14} + 7q^{15} + 29q^{16} + 3q^{17} + 19q^{18} + 3q^{19} + 16q^{20} + 9q^{21} + 11q^{22} + 10q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} - 2x^{17} - 65x^{16} + 79x^{15} + 1649x^{14} - 701x^{13} - 20430x^{12} - 6325x^{11} + 125207x^{10} + 117488x^{9} - 320524x^{8} - 480680x^{7} + 159569x^{6} + 460727x^{5} - 3245x^{4} - 177487x^{3} - 2006x^{2} + 24905x - 1701$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
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 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
$\operatorname{tr} a_p$	3	−1	6	11	9	−18	3	3	10	2	5	77	19	5	18	44	6	−25	66	18	9	21	−11	−3	47

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.269 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, 13, 71, 923]: [0, 1, 7, 86] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (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(923, 2)[3]   # 923.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 923:2 1000