Atlas › Modular forms › Weight 2 › Level 211 › 211.2.a › 211.2.a.d

Newform orbit 211.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - q^{2} - q^{3} + 11q^{4} + 15q^{5} - 5q^{6} - 2q^{7} - 6q^{8} + 14q^{9} - 2q^{10} + 13q^{11} - 5q^{12} - 4q^{13} + 13q^{14} + 2q^{15} + 3q^{16} + 4q^{17} - 21q^{18} - 2q^{19} + 32q^{20} + 9q^{21} - 8q^{22} - 3q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} + 4x^{8} - 176x^{7} - 395x^{6} + 10908x^{5} + 3867x^{4} - 246603x^{3} + 347656x^{2} + 536892x - 786568$

Atkin–Lehner signs

$p$	$w_p$
211	$-1$

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	−1	15	−2	13	−4	4	−2	−3	26	5	5	20	−37	4	13	14	23	−3	19	17	7	6	33	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.166 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 211]: [0, 18] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 3 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(211, 2)[3]   # 211.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 211:2 1000