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

Newform orbit 554.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} - 23x^{8} + 4x^{7} + 3282x^{6} - 18449x^{5} - 88146x^{4} + 671467x^{3} + 1172635x^{2} - 7086631x - 13646052$

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
$\operatorname{tr} a_p$	9	−1	8	8	0	7	12	15	2	0	7	2	9	−7	−7	9	−21	22	−28	−16	8	−11	−16	−13	21

$\operatorname{tr} a_n$ 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)[3]   # 554.2.a.d

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

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