Atlas › Modular forms › Weight 2 › Level 217 › 217.2.a › 217.2.a.a

Newform orbit 217.2.a.a

Trace form

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

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

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

$x^{3} + 6x^{2} + 3x - 1$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
31	$-1$

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	−3	−6	3	0	−3	−12	−3	−12	9	3	6	−21	−3	−21	9	−3	−3	6	−21	−3	−12	−3	−3	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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, 7, 31, 217]: [0, 1, 3, 22] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(217, 2)[0]   # 217.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 217:2 1000