Atlas › Modular forms › Weight 2 › Level 554 › 554.2.a › 554.2.a.c

Newform orbit 554.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{8} + 3x^{7} - 256x^{6} - 392x^{5} + 21579x^{4} + 9862x^{3} - 653091x^{2} + 113591x + 4500503$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
277	$-1$

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
$\operatorname{tr} a_p$	−8	3	−2	8	5	4	−4	21	−2	1	24	8	6	22	−11	−13	17	22	18	−2	32	15	−6	2	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.215 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, 277, 554]: [0, 1, 23, 71] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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(554, 2)[2]   # 554.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 554:2 1000