Atlas › Modular forms › Weight 8 › Level 53 › 53.8.a › 53.8.a.b

Newform orbit 53.8.a.b

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 8q^{2} + 121q^{3} + 1344q^{4} + 126q^{5} + 681q^{6} + 2412q^{7} + 2334q^{8} + 15658q^{9} + 8441q^{10} + 6528q^{11} + 16251q^{12} + 22981q^{13} - 79054q^{14} - 3344q^{15} + 128532q^{16} + 7651q^{17} + 118497q^{18} + 113403q^{19} + 277771q^{20} + 48480q^{21} + 283579q^{22} + 192735q^{23} + 806513q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 129x^{16} - 20539x^{15} + 2757637x^{14} + 174012168x^{13} - 23251490146x^{12} - 817438126242x^{11} + 97446799578438x^{10} + 2418180232782225x^{9} - 208219488393214277x^{8} - 4551323234399349339x^{7} + 202393624152310403885x^{6} + 4606733698182362397550x^{5} - 58555754072980933956304x^{4} - 1499972841208514027083008x^{3} + 2096489278558900188376560x^{2} + 140218548524266872776911200x + 512899227957749250911328000$

Atkin–Lehner signs

$p$	$w_p$
53	$+1$

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
$\operatorname{tr} a_p$	8	121	126	2412	6528	22981	7651	113403	192735	299357	447172	−6399	538390	2219438	918734	−2530909	−4795906	−2049804	6642932	−587339	10248748	2834467	27226231	33826432	32952255

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 53]: [1, 33] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (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(53, 8)[1]   # 53.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 53:8 1000