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

Newform orbit 535.2.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} - 7x^{8} - 8x^{7} + 111x^{6} - 21x^{5} - 477x^{4} + 117x^{3} + 386x^{2} - 134x + 3$

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 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
$\operatorname{tr} a_p$	5	2	−9	3	9	4	21	−1	18	5	−10	8	11	9	4	22	6	−49	11	30	15	−9	30	37	19

$\operatorname{tr} a_n$ 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)[2]   # 535.2.a.c

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

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