Atlas › Modular forms › Weight 2 › Level 651 › 651.2.a › 651.2.a.j

Newform orbit 651.2.a.j

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{3} + 53x^{2} - 1517x - 60233$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$-1$
31	$-1$

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
$\operatorname{tr} a_p$	−1	−3	−4	3	−8	0	0	4	−2	−20	3	−12	0	−8	−6	−10	−4	−10	−8	−2	4	−6	10	−18	22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 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, 3, 7, 21, 31, 93, 217, 651]: [0, 1, 1, 4, 3, 12, 22, 88] (certified at ell = 2147482951)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 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(651, 2)[9]   # 651.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 651:2 1000