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

Newform orbit 115.4.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 6q^{2} - 3q^{3} + 2q^{4} + 10q^{5} + 9q^{6} + q^{7} + 42q^{8} + 5q^{9} - 30q^{10} - 27q^{11} - 3q^{12} - 15q^{13} - 3q^{14} - 15q^{15} - 142q^{16} - 79q^{17} - 15q^{18} - 71q^{19} + 10q^{20} - 274q^{21} + 81q^{22} - 46q^{23} - 63q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{109})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 9x - 7$

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
$\operatorname{tr} a_p$	−6	−3	10	1	−27	−15	−79	−71	−46	−430	−305	−68	−593	648	382	−464	−18	−7	60	−1029	74	692	−1460	−220	1339

$\operatorname{tr} a_n$ 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)[2]   # 115.4.a.c

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

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