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

Newform orbit 923.2.a.e

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 3q^{2} + 6q^{3} + 35q^{4} + 6q^{5} - 7q^{6} + 7q^{7} + 9q^{8} + 34q^{9} + 15q^{10} - 3q^{11} + 8q^{12} + 26q^{13} + 7q^{14} + 3q^{15} + 45q^{16} + 9q^{17} + 4q^{18} - 3q^{19} + 3q^{20} + 5q^{21} + 37q^{22} + 4q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 9x^{25} - 52x^{24} + 654x^{23} + 635x^{22} - 19430x^{21} + 12300x^{20} + 305673x^{19} - 444313x^{18} - 2765065x^{17} + 5492212x^{16} + 14751618x^{15} - 35632024x^{14} - 46795977x^{13} + 132809607x^{12} + 90212119x^{11} - 293887222x^{10} - 114827725x^{9} + 388760461x^{8} + 114960573x^{7} - 299714258x^{6} - 95681372x^{5} + 121336227x^{4} + 51487607x^{3} - 17239872x^{2} - 11729116x - 1561960$

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	6	6	7	−3	26	9	−3	4	19	12	75	−22	34	−5	36	−9	53	23	−26	−5	34	17	−13	19

$\operatorname{tr} a_n$ 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)[4]   # 923.2.a.e

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

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