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

Newform orbit 105.4.a.g

Trace form

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

$2q + 3q^{2} + 6q^{3} + 9q^{4} - 10q^{5} + 9q^{6} + 14q^{7} + 51q^{8} + 18q^{9} - 15q^{10} + 62q^{11} + 27q^{12} - 6q^{13} + 21q^{14} - 30q^{15} + 25q^{16} + 40q^{17} + 27q^{18} - 122q^{19} - 45q^{20} + 42q^{21} + 52q^{22} + 16q^{23} + 153q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{11}$ , whose minimal polynomial is

$x^{2} - 65x + 1046$

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
$\operatorname{tr} a_p$	3	6	−10	14	62	−6	40	−122	16	352	66	−188	16	−396	−188	982	516	−880	−356	310	326	1832	−680	796	−670

$\operatorname{tr} a_n$ 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)[6]   # 105.4.a.g

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

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