Atlas › Modular forms › Weight 2 › Level 885 › 885.2.a › 885.2.a.h

Newform orbit 885.2.a.h

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{3} - 2x^{2} - 54x + 176$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$+1$
59	$-1$

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	−3	−3	0	3	−5	7	−5	6	−4	−8	7	−2	29	4	5	3	−8	15	−2	−29	−21	30	−5	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.438 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, 3, 5, 15, 59, 177, 295, 885]: [0, 1, 1, 4, 6, 22, 32, 124] (certified at ell = 2147482481)
T = [(2, 1), (7, 3)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(885, 2)[7]   # 885.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 885:2 1000