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

Newform orbit 899.2.a.f

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q - 2q^{3} + 36q^{4} + 9q^{5} + 2q^{6} + 3q^{7} + 3q^{8} + 47q^{9} + q^{10} + 2q^{11} - 11q^{12} + 13q^{13} - 16q^{14} + 6q^{15} + 70q^{16} + 2q^{17} - 8q^{18} + 22q^{19} + 27q^{20} + 20q^{21} + 27q^{22} - 2q^{23} + 22q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} + 2x^{24} - 104x^{23} - 182x^{22} + 4474x^{21} + 7029x^{20} - 103454x^{19} - 150060x^{18} + 1401750x^{17} + 1906090x^{16} - 11377594x^{15} - 14360794x^{14} + 54423008x^{13} + 60351329x^{12} - 147150478x^{11} - 123107762x^{10} + 215097972x^{9} + 90124661x^{8} - 157629042x^{7} - 5492989x^{6} + 37750569x^{5} - 911334x^{4} - 3733805x^{3} - 64775x^{2} + 131725x + 10575$

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
$\operatorname{tr} a_p$	0	−2	9	3	2	13	2	22	−2	25	−25	54	−20	−3	6	29	1	16	58	32	−14	21	−35	−3	9

$\operatorname{tr} a_n$ 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)[5]   # 899.2.a.f

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

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