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

Newform orbit 213.2.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{4} + 6x^{3} - 89x^{2} + 106x + 76$

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
$\operatorname{tr} a_p$	3	−4	−3	6	2	5	−8	8	−1	−5	2	19	−19	25	7	−5	10	2	35	4	2	−1	18	−16	−1

$\operatorname{tr} a_n$ 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)[4]   # 213.2.a.e

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

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