Atlas › Modular forms › Weight 2 › Level 854 › 854.2.a › 854.2.a.f

Newform orbit 854.2.a.f

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{6} - 54x^{5} - 5462x^{4} + 94509x^{3} + 9949621x^{2} + 133067276x - 58935069$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$-1$
61	$+1$

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
$\operatorname{tr} a_p$	−6	4	3	6	−2	15	−6	12	−9	7	11	9	−6	−8	5	33	10	−6	8	−17	20	11	6	8	22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.436 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, 2, 7, 14, 61, 122, 427, 854]: [0, 1, 1, 4, 5, 17, 42, 128] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 8 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(854, 2)[5]   # 854.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 854:2 1000