Atlas › Modular forms › Weight 2 › Level 535 › 535.2.a › 535.2.a.b

Newform orbit 535.2.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 6x^{7} - 4x^{6} - 48x^{5} + 13x^{4} + 82x^{3} - 19x^{2} - 10x + 2$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
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
$\operatorname{tr} a_p$	−4	−2	−8	−3	−9	−8	−17	1	−20	2	2	−12	−25	−5	−11	−34	2	7	−9	−6	−13	25	−17	−27	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 5, 107, 535]: [0, 1, 10, 56] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 4 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(535, 2)[1]   # 535.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 535:2 1000