Atlas › Modular forms › Weight 2 › Level 506 › 506.2.a › 506.2.a.h

Newform orbit 506.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{29} + 8T_{31}$ , whose minimal polynomial is

$x^{3} - 21x^{2} - 13281x + 692477$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
11	$+1$
23	$-1$

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	3	3	0	−3	−3	−3	9	3	−9	6	15	−12	−3	6	−21	9	24	18	0	12	−6	3	−12	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.456 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, 11, 22, 23, 46, 253, 506]: [0, 1, 2, 5, 3, 8, 26, 76] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (29, 7), (31, 8)]; h of degree 21 = 1 x 21 (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(506, 2)[7]   # 506.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 506:2 1000