Atlas › Modular forms › Weight 6 › Level 33 › 33.6.a › 33.6.a.e

Newform orbit 33.6.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} + 117q^{6} + 146q^{7} + 39q^{8} + 162q^{9} + 212q^{10} - 242q^{11} + 333q^{12} - 130q^{13} - 74q^{14} + 522q^{15} + 241q^{16} - 728q^{17} + 1053q^{18} - 828q^{19} - 1072q^{20} + 1314q^{21} - 1573q^{22} - 238q^{23} + 351q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 71x + 592$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
11	$+1$

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
$\operatorname{tr} a_p$	13	18	58	146	−242	−130	−728	−828	−238	696	−10480	−1908	36484	9768	43742	−12174	−2788	−25302	−40520	31386	−46780	−16850	79440	−54204	−241568

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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 engine's checks

dim M^+ at levels [1, 3, 11, 33]: [1, 3, 6, 22] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(33, 6)[4]   # 33.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 33:6 1000