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

Newform orbit 33.10.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 3q^{2} - 243q^{3} - 315q^{4} + 2388q^{5} - 243q^{6} - 228q^{7} + 4809q^{8} + 19683q^{9} + 35922q^{10} - 43923q^{11} + 25515q^{12} + 11976q^{13} - 330744q^{14} - 193428q^{15} - 323055q^{16} - 819972q^{17} + 19683q^{18} - 801270q^{19} - 299982q^{20} + 18468q^{21} - 43923q^{22} - 1773114q^{23} - 389529q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} - 2391x^{2} + 1293552x + 146425460$

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
$\operatorname{tr} a_p$	3	−243	2388	−228	−43923	11976	−819972	−801270	−1773114	−330840	−3669432	5656350	19715148	−8977926	30124242	127081992	−148154568	−269196312	−670193220	−167751066	−487291734	−277052424	−938860824	88531542	−10345650

$\operatorname{tr} a_n$ 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)[1]   # 33.10.a.b

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

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