Atlas › Modular forms › Weight 6 › Level 11 › 11.6.a › 11.6.a.b

Newform orbit 11.6.a.b

Trace form

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

$3q + 34q^{3} + 84q^{4} + 24q^{5} - 206q^{6} + 84q^{7} - 564q^{8} - 7q^{9} - 414q^{10} + 363q^{11} + 992q^{12} + 486q^{13} - 1020q^{14} + 1654q^{15} + 1992q^{16} + 1086q^{17} - 3706q^{18} + 1380q^{19} - 3480q^{20} - 908q^{21} - 3066q^{23} - 11748q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 34x^{2} + 333x - 918$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	0	34	24	84	363	486	1086	1380	−3066	−3426	−4098	17724	5994	−26208	−17232	50586	−3738	18486	−47754	39282	15426	125148	−143928	−106824	9684

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 11]: [1, 6] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 4 = 1 x 4 (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(11, 6)[1]   # 11.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 11:6 1000