Atlas › Modular forms › Weight 4 › Level 115 › 115.4.a › 115.4.a.d

Newform orbit 115.4.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 5q^{2} - 6q^{3} + 33q^{4} - 25q^{5} - 36q^{6} - 15q^{7} - 102q^{8} + 79q^{9} + 25q^{10} - 153q^{11} - 30q^{12} + 28q^{13} + 69q^{14} + 30q^{15} + 145q^{16} - 341q^{17} - 409q^{18} + 3q^{19} - 165q^{20} - 212q^{21} + 11q^{22} + 115q^{23} - 243q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 11x^{4} - 47x^{3} - 89x^{2} + 437x - 361$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
23	$-1$

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 odd 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	−6	−25	−15	−153	28	−341	3	115	−583	662	−172	344	−230	−337	−942	−1166	499	−972	−14	−229	−88	−72	−90	−1765

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 5, 23, 115]: [1, 3, 7, 38] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 3 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(115, 4)[3]   # 115.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 115:4 1000