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

Newform orbit 899.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 3x^{9} - 21x^{8} - 64x^{7} + 105x^{6} + 322x^{5} - 223x^{4} - 575x^{3} + 238x^{2} + 335x - 113$

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 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
$\operatorname{tr} a_p$	0	−3	−7	−2	−9	−13	5	−14	−2	10	10	−34	−16	−13	16	−29	−13	−25	−37	6	−38	−11	−17	−12	−1

$\operatorname{tr} a_n$ 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)[2]   # 899.2.a.c

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

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