Atlas › Modular forms › Weight 4 › Level 213 › 213.4.a › 213.4.a.e

Newform orbit 213.4.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 7q^{2} + 30q^{3} + 51q^{4} + 28q^{5} + 21q^{6} + 28q^{7} + 84q^{8} + 90q^{9} + 57q^{10} + 150q^{11} + 153q^{12} + 14q^{13} + 21q^{14} + 84q^{15} + 203q^{16} + 158q^{17} + 63q^{18} + 140q^{19} + 430q^{20} + 84q^{21} - 233q^{22} + 242q^{23} + 252q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} - 35x^{9} - 244x^{8} + 17421x^{7} - 67173x^{6} - 2374754x^{5} + 19226536x^{4} + 65335464x^{3} - 1081168848x^{2} + 3177330496x - 1847260288$

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
$\operatorname{tr} a_p$	7	30	28	28	150	14	158	140	242	266	−244	214	1102	864	824	506	346	−70	800	710	−1054	−1560	1388	1670	−3506

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.141 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, 3, 71, 213]: [1, 2, 19, 74] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 6 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, 4)[4]   # 213.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 213:4 1000