Atlas › Modular forms › Weight 2 › Level 899 › 899.2.a › 899.2.a.e

Newform orbit 899.2.a.e

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q + 2q^{2} + 5q^{3} + 32q^{4} + 3q^{5} + 4q^{6} + 4q^{7} + 6q^{8} + 26q^{9} + 11q^{11} + 3q^{12} + 23q^{13} + 3q^{14} + 16q^{15} + 38q^{16} + 5q^{17} + 14q^{18} + 8q^{19} - 10q^{20} + 39q^{21} - 7q^{22} - 2q^{23} - 30q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} - 7x^{22} - 66x^{21} + 519x^{20} + 1611x^{19} - 15295x^{18} - 18004x^{17} + 235873x^{16} + 80277x^{15} - 2087070x^{14} + 101023x^{13} + 10904937x^{12} - 2092410x^{11} - 33510686x^{10} + 5789342x^{9} + 58616424x^{8} - 2102136x^{7} - 54131013x^{6} - 8576251x^{5} + 21126139x^{4} + 7332479x^{3} - 1377034x^{2} - 710444x - 40433$

Atkin–Lehner signs

$p$	$w_p$
29	$+1$
31	$-1$

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
$\operatorname{tr} a_p$	2	5	3	4	11	23	5	8	−2	−23	23	50	18	3	−12	23	−3	47	−7	16	23	49	5	40	31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 29, 31, 899]: [0, 3, 3, 82] (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(899, 2)[4]   # 899.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 899:2 1000