Atlas › Modular forms › Weight 2 › Level 533 › 533.2.a › 533.2.a.e

Newform orbit 533.2.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{13} - 10x^{12} + 2x^{11} + 272x^{10} - 769x^{9} - 1302x^{8} + 7205x^{7} - 2055x^{6} - 20182x^{5} + 15170x^{4} + 21447x^{3} - 18261x^{2} - 7616x + 6272$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
41	$+1$

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
$\operatorname{tr} a_p$	0	10	0	10	2	13	9	22	4	1	1	8	−13	36	12	1	−18	12	27	10	24	29	−13	15	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 13, 41, 533]: [0, 1, 4, 50] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(533, 2)[4]   # 533.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 533:2 1000