Atlas › Modular forms › Weight 2 › Level 213 › 213.2.a › 213.2.a.d

Newform orbit 213.2.a.d

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 2x - 12$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
71	$+1$

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	2	−1	−2	6	−3	6	−4	3	7	4	−1	3	13	−9	5	−6	−24	13	−2	−2	−9	20	6	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 3, 71, 213]: [0, 1, 7, 26] (certified at ell = 2147483171)
T = [(2, 1), (5, 3)]; h of degree 11 = 1 x 11 (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(213, 2)[3]   # 213.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 213:2 1000