Atlas › Modular forms › Weight 4 › Level 229 › 229.4.a › 229.4.a.b

Newform orbit 229.4.a.b

Trace form

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

$31q + 11q^{2} + 19q^{3} + 149q^{4} + 30q^{5} + 31q^{6} + 29q^{7} + 135q^{8} + 338q^{9} + 13q^{10} + 369q^{11} + 128q^{12} + 65q^{13} + 237q^{14} + 260q^{15} + 737q^{16} + 185q^{17} + 453q^{18} + 397q^{19} + 352q^{20} + 404q^{21} + 145q^{22} + 299q^{23} + 467q^{24} + O(q^{25})$

Coefficient field

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

$x^{31} - 30x^{30} - 367x^{29} + 18311x^{28} + 3126x^{27} - 4634392x^{26} + 17410725x^{25} + 651824178x^{24} - 3702862859x^{23} - 57454286416x^{22} + 388749609093x^{21} + 3384370938808x^{20} - 24819849617343x^{19} - 138058438425777x^{18} + 1030894208563515x^{17} + 3949657643576784x^{16} - 28505814688491465x^{15} - 78122675881146596x^{14} + 525342919106882527x^{13} + 1015371447468136145x^{12} - 6381828149885594116x^{11} - 7777664063494704445x^{10} + 50066780534268035823x^{9} + 26603354807267517422x^{8} - 241288790204262957948x^{7} + 24563552556077848792x^{6} + 628159922567349794424x^{5} - 402723332826731389824x^{4} - 614151207945499813984x^{3} + 702006389344174588416x^{2} - 87632935406700121728x - 71796309641843151360$

Atkin–Lehner signs

$p$	$w_p$
229	$+1$

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
$\operatorname{tr} a_p$	11	19	30	29	369	65	185	397	299	761	277	410	732	1053	1287	1146	3730	−918	−1250	4534	−1796	605	3665	2682	−3529

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.157 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 engine's checks

dim M^+ at levels [1, 229]: [1, 59] (certified at ell = 2147478649)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(229, 4)[1]   # 229.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 229:4 1000