Atlas › Modular forms › Weight 4 › Level 105 › 105.4.a › 105.4.a.f

Newform orbit 105.4.a.f

Trace form

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

$2q + q^{2} + 6q^{3} + 17q^{4} + 10q^{5} + 3q^{6} - 14q^{7} + 33q^{8} + 18q^{9} + 5q^{10} - 22q^{11} + 51q^{12} - 22q^{13} - 7q^{14} + 30q^{15} - 87q^{16} + 116q^{17} + 9q^{18} + 102q^{19} + 85q^{20} - 42q^{21} - 76q^{22} + 260q^{23} + 99q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{65})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{11}$ , whose minimal polynomial is

$x^{2} + 21x + 94$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$-1$
7	$+1$

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
$\operatorname{tr} a_p$	1	6	10	−14	−22	−22	116	102	260	−196	150	−96	−176	−344	560	326	−844	−204	−104	1670	−386	−888	928	588	522

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 3, 5, 7, 15, 21, 35, 105]: [1, 2, 3, 3, 8, 10, 14, 52] (certified at ell = 2147483629)
T = [(2, 1), (11, 1)]; h of degree 12 = 1 x 12 (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(105, 4)[5]   # 105.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 105:4 1000