Atlas › Modular forms › Weight 2 › Level 563 › 563.2.a › 563.2.a.e

Newform orbit 563.2.a.e

Trace form

The 31 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$31q + 5q^{2} + 8q^{3} + 43q^{4} + 13q^{5} - 4q^{6} + 17q^{7} + 6q^{8} + 43q^{9} - 3q^{10} + 6q^{11} + 17q^{12} + 40q^{13} - 8q^{14} + 2q^{15} + 59q^{16} + 18q^{17} + 13q^{18} + 14q^{19} - 4q^{20} + 2q^{21} + 16q^{22} + 15q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 31. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{31} - 29x^{30} - 232x^{29} + 13940x^{28} - 29071x^{27} - 2854522x^{26} + 17436564x^{25} + 323072810x^{24} - 2947680439x^{23} - 21641390352x^{22} + 277811464813x^{21} + 811522697989x^{20} - 16679883834991x^{19} - 9432300449469x^{18} + 667675215353705x^{17} - 603005215280691x^{16} - 17961687979930377x^{15} + 34631974898035001x^{14} + 318074977860994150x^{13} - 869108179103888194x^{12} - 3502162576795864405x^{11} + 12224028134824904082x^{10} + 21134994752699241653x^{9} - 95471435144729741208x^{8} - 49876001792401340687x^{7} + 373450074946241515600x^{6} - 17215836078819647498x^{5} - 671219715827607257056x^{4} + 179329507139615961430x^{3} + 491100370305546682708x^{2} - 137018612980874512408x - 96271559248689353032$

Atkin–Lehner signs

$p$	$w_p$
563	$-1$

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
$\operatorname{tr} a_p$	5	8	13	17	6	40	18	14	15	15	5	94	−2	50	1	50	−13	41	29	−19	29	15	2	−21	25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 563]: [0, 48] (certified at ell = 2147482367)
T = [(2, 1), (3, 3)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 7 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(563, 2)[4]   # 563.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 563:2 1000