Atlas › Modular forms › Weight 4 › Level 113 › 113.4.a › 113.4.a.b

Newform orbit 113.4.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 4q^{2} + 18q^{3} + 80q^{4} + 6q^{5} + 5q^{6} + 118q^{7} + 51q^{8} + 178q^{9} + 76q^{10} + 30q^{11} + 164q^{12} + 176q^{13} + 27q^{14} + 170q^{15} + 456q^{16} + 136q^{17} + 133q^{18} + 310q^{19} - 96q^{20} + 158q^{21} + 363q^{22} + 276q^{23} + 252q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 22x^{15} - 172x^{14} + 6232x^{13} + 1358x^{12} - 710592x^{11} + 1624301x^{10} + 41048618x^{9} - 148306138x^{8} - 1221899220x^{7} + 5739235325x^{6} + 15809390778x^{5} - 102632606019x^{4} - 11121966262x^{3} + 652152347414x^{2} - 877473442564x + 224073281402$

Atkin–Lehner signs

$p$	$w_p$
113	$+1$

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
$\operatorname{tr} a_p$	4	18	6	118	30	176	136	310	276	−30	762	344	−218	368	−490	94	172	1602	420	−124	3690	1610	2896	−420	1768

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 113]: [1, 30] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 4 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(113, 4)[1]   # 113.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 113:4 1000