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

Newform orbit 33.4.a.c

Trace form

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

$2q + q^{2} - 6q^{3} + 33q^{4} - 14q^{5} - 3q^{6} + 24q^{7} + 57q^{8} + 18q^{9} - 104q^{10} - 22q^{11} - 99q^{12} + 30q^{13} - 182q^{14} + 42q^{15} + 201q^{16} + 106q^{17} + 9q^{18} + 50q^{19} - 328q^{20} - 72q^{21} - 11q^{22} + 134q^{23} - 171q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{97})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 13x + 18$

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
$\operatorname{tr} a_p$	1	−6	−14	24	−22	30	106	50	134	−198	360	−328	−782	386	266	−522	−172	−778	−776	630	1296	652	−324	−756	−452

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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)[2]   # 33.4.a.c

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

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