Atlas › Modular forms › Weight 10 › Level 33 › 33.10.a › 33.10.a.c

Newform orbit 33.10.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 3q^{2} + 324q^{3} + 709q^{4} + 2388q^{5} - 243q^{6} + 5372q^{7} + 20151q^{8} + 26244q^{9} + 25380q^{10} - 58564q^{11} + 57429q^{12} - 8836q^{13} - 114618q^{14} + 193428q^{15} + 97681q^{16} + 706392q^{17} - 19683q^{18} + 984704q^{19} + 1063608q^{20} + 435132q^{21} + 43923q^{22} + 3635892q^{23} + 1632231q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 2385x^{3} - 3246402x^{2} + 10250147072x - 5302309177728$

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
$\operatorname{tr} a_p$	−3	324	2388	5372	−58564	−8836	706392	984704	3635892	7517448	5910752	−7163752	−596832	−14430328	−6723588	−69704508	13672392	−88290604	−402240496	474595140	425419592	538408868	783087552	747115224	814209296

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.327 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, 4, 10, 38] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 6 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, 10)[2]   # 33.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 33:10 1000