Atlas › Modular forms › Weight 4 › Level 227 › 227.4.a › 227.4.a.b

Newform orbit 227.4.a.b

Trace form

The 33 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$33q + 7q^{2} + 13q^{3} + 165q^{4} + 49q^{5} + 21q^{6} + 96q^{7} + 108q^{8} + 394q^{9} + 58q^{10} + 55q^{11} + 144q^{12} + 387q^{13} + 84q^{14} + 176q^{15} + 829q^{16} + 164q^{17} + 324q^{18} + 225q^{19} + 240q^{20} + 348q^{21} + 339q^{22} + 414q^{23} + 354q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 33. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{33} - 20x^{32} - 678x^{31} + 14993x^{30} + 188895x^{29} - 4796542x^{28} - 28364577x^{27} + 869795717x^{26} + 2540041902x^{25} - 100247306125x^{24} - 143008162630x^{23} + 7791862995958x^{22} + 5472203568515x^{21} - 421924390958862x^{20} - 182612527516801x^{19} + 16170919540638221x^{18} + 7363397201102890x^{17} - 440017904041735647x^{16} - 295763809896581842x^{15} + 8421140031441099306x^{14} + 8585927113770564095x^{13} - 110500741723726129170x^{12} - 161748791818106771743x^{11} + 941859217409384237969x^{10} + 1907879121854725244841x^{9} - 4599307175007621847013x^{8} - 13397968233192976913270x^{7} + 7885591523463543985609x^{6} + 49620733419536028847276x^{5} + 25163916982268905589726x^{4} - 65028447220428447145898x^{3} - 93961344913851903459680x^{2} - 43307004919780448080896x - 5701841631591785466048$

Atkin–Lehner signs

$p$	$w_p$
227	$+1$

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
$\operatorname{tr} a_p$	7	13	49	96	55	387	164	225	414	409	474	2495	356	1237	862	2001	132	2161	2161	−2254	1528	474	−635	718	6514

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.155 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, 227]: [1, 58] (certified at ell = 2147479573)
T = [(2, 1), (3, 1)]; h of degree 56 = 1 x 56 (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(227, 4)[1]   # 227.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 227:4 1000