Atlas › Modular forms › Weight 2 › Level 661 › 661.2.a › 661.2.a.c

Newform orbit 661.2.a.c

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q + 7q^{2} + 5q^{3} + 33q^{4} + 9q^{5} + 8q^{6} + 10q^{7} + 18q^{8} + 48q^{9} - 10q^{10} + 50q^{11} - 2q^{12} - 11q^{13} + 2q^{14} + 4q^{15} + 33q^{16} + 5q^{17} + 16q^{18} + 25q^{19} - 4q^{20} + q^{21} - 2q^{22} + 23q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} - 12x^{28} - 49x^{27} + 1092x^{26} - 753x^{25} - 39102x^{24} + 94671x^{23} + 721625x^{22} - 2587472x^{21} - 7436949x^{20} + 36277245x^{19} + 42122117x^{18} - 302761304x^{17} - 105115787x^{16} + 1588625986x^{15} - 113354791x^{14} - 5341321031x^{13} + 1437982132x^{12} + 11463346065x^{11} - 3640598996x^{10} - 15281258236x^{9} + 3801570744x^{8} + 11840607958x^{7} - 1050162530x^{6} - 4488524491x^{5} - 492753867x^{4} + 417132388x^{3} + 56508819x^{2} - 3989722x - 145919$

Atkin–Lehner signs

$p$	$w_p$
661	$-1$

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
$\operatorname{tr} a_p$	7	5	9	10	50	−11	5	25	23	35	10	5	25	14	−5	4	75	−2	−14	53	−22	−2	98	15	−45

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 661]: [0, 55] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 6 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(661, 2)[2]   # 661.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 661:2 1000