Atlas › Modular forms › Weight 2 › Level 557 › 557.2.a › 557.2.a.d

Newform orbit 557.2.a.d

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + q^{2} + 16q^{3} + 29q^{4} + 4q^{5} - q^{6} + 14q^{7} + 9q^{8} + 32q^{9} + 10q^{11} + 24q^{12} + 30q^{13} - 10q^{14} + 9q^{15} + 31q^{16} + q^{18} + 36q^{19} - 8q^{20} - 12q^{21} + 18q^{22} + 4q^{23} + q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 17x^{25} + 50x^{24} + 670x^{23} - 4482x^{22} - 4800x^{21} + 102201x^{20} - 116862x^{19} - 1077287x^{18} + 2491481x^{17} + 5855330x^{16} - 20595678x^{15} - 15583224x^{14} + 92797134x^{13} + 10622593x^{12} - 247238093x^{11} + 40044973x^{10} + 395051638x^{9} - 92298701x^{8} - 367371116x^{7} + 61706003x^{6} + 180478376x^{5} - 6166086x^{4} - 35270479x^{3} - 3023906x^{2} + 534244x + 47636$

Atkin–Lehner signs

$p$	$w_p$
557	$-1$

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
$\operatorname{tr} a_p$	1	16	4	14	10	30	0	36	4	−10	36	9	−7	39	19	−7	36	16	99	−14	15	2	32	−5	49

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 557]: [0, 47] (certified at ell = 2147475497)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 5 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(557, 2)[3]   # 557.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 557:2 1000