Atlas › Modular forms › Weight 4 › Level 33 › 33.4.a › 33.4.a.a

Newform orbit 33.4.a.a

$q$ -expansion

$q - 5q^{2} + 3q^{3} + 17q^{4} - 14q^{5} - 15q^{6} - 32q^{7} - 45q^{8} + 9q^{9} + 70q^{10} - 11q^{11} + 51q^{12} - 38q^{13} + 160q^{14} - 42q^{15} + 89q^{16} - 2q^{17} - 45q^{18} + 72q^{19} - 238q^{20} - 96q^{21} + 55q^{22} + 68q^{23} - 135q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 19$

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 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
$\operatorname{tr} a_p$	−5	3	−14	−32	−11	−38	−2	72	68	−54	−152	174	94	−528	−340	−438	20	570	−460	−1092	562	−16	372	−966	−526

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 33, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 2, 4, 14] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 33.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 33:4 1000