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

Newform orbit 33.10.a.a

Trace form

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

$3q - 51q^{2} + 243q^{3} + 1425q^{4} - 1362q^{5} - 4131q^{6} - 15090q^{7} - 40959q^{8} + 19683q^{9} + 72684q^{10} + 43923q^{11} + 115425q^{12} + 62418q^{13} - 80922q^{14} - 110322q^{15} + 384513q^{16} + 118368q^{17} - 334611q^{18} - 118302q^{19} - 2527860q^{20} - 1222290q^{21} - 746691q^{22} - 3437880q^{23} - 3317679q^{24} + O(q^{25})$

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

$x^{3} + 1413x^{2} - 1055154x + 104055784$

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
$\operatorname{tr} a_p$	−51	243	−1362	−15090	43923	62418	118368	−118302	−3437880	−2155140	−14433072	−27826266	28560876	1421574	−97407648	−93477846	61659492	123400494	−164540244	−212273712	277517946	361627182	−1135330320	−59444730	−798933318

$\operatorname{tr} a_n$ 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)[0]   # 33.10.a.a

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

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