Atlas › Modular forms › Weight 2 › Level 642 › 642.2.a › 642.2.a.f

Newform orbit 642.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} - 3q^{8} + 3q^{9} - q^{11} - 3q^{12} - q^{13} + 3q^{14} + 3q^{16} - 9q^{17} - 3q^{18} + 4q^{19} + 3q^{21} + q^{22} - 14q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{3} + 9x^{2} - 13x - 5$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
107	$+1$

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
$\operatorname{tr} a_p$	−3	−3	0	−3	−1	−1	−9	4	−14	−4	−1	−3	−5	−8	−24	−14	−21	1	−19	−22	−2	−13	5	7	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.473 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, 2, 3, 6, 107, 214, 321, 642]: [0, 1, 1, 3, 10, 29, 38, 112] (certified at ell = 2147482621)
T = [(5, 1), (7, 3)]; h of degree 19 = 1 x 19 (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(642, 2)[5]   # 642.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 642:2 1000