Atlas › Modular forms › Weight 4 › Level 61 › 61.4.a › 61.4.a.b

Newform orbit 61.4.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 3q^{2} + 11q^{3} + 53q^{4} + 24q^{5} + 7q^{6} + q^{7} + 39q^{8} + 120q^{9} + 29q^{10} + 150q^{11} + 115q^{12} - 2q^{13} + 174q^{14} + 166q^{15} + 45q^{16} + 32q^{17} - 618q^{18} - 25q^{19} - 89q^{20} - 148q^{21} - 637q^{22} + 187q^{23} - 753q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 14x^{8} - 153x^{7} + 3069x^{6} - 3700x^{5} - 144155x^{4} + 772756x^{3} - 945860x^{2} - 772044x + 95616$

Atkin–Lehner signs

$p$	$w_p$
61	$+1$

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	11	24	1	150	−2	32	−25	187	209	85	57	67	848	832	1011	2014	−549	582	1975	−2175	−1232	−1171	936	−6277

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.095 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, 61]: [1, 17] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (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(61, 4)[1]   # 61.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 61:4 1000