Atlas › Modular forms › Weight 6 › Level 61 › 61.6.a › 61.6.a.a

Newform orbit 61.6.a.a

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 9q^{2} - 46q^{3} + 117q^{4} - 129q^{5} - 11q^{6} - 87q^{7} - 417q^{8} + 513q^{9} - 715q^{10} - 1737q^{11} - 2009q^{12} - 177q^{13} - 2124q^{14} - 2738q^{15} - 3371q^{16} - 2908q^{17} - 10020q^{18} - 6600q^{19} - 11369q^{20} - 7258q^{21} - 3213q^{22} - 4205q^{23} - 3489q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 55x^{10} - 304x^{9} - 44770x^{8} - 52246x^{7} + 11693356x^{6} + 17185157x^{5} - 1119071457x^{4} - 1870384936x^{3} + 32738322800x^{2} + 98548688400x + 17474904000$

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 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
$\operatorname{tr} a_p$	−9	−46	−129	−87	−1737	−177	−2908	−6600	−4205	−6136	2152	−13958	−16055	4300	−26588	−26004	−79439	−40931	−78191	−183128	−87715	−219221	−89104	−45168	96304

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.232 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, 27] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 5 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, 6)[0]   # 61.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 61:6 1000