Atlas › Modular forms › Weight 4 › Level 214 › 214.4.a › 214.4.a.b

Newform orbit 214.4.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 14q^{2} - q^{3} + 28q^{4} + 17q^{5} + 2q^{6} - 28q^{7} - 56q^{8} + 88q^{9} - 34q^{10} + 70q^{11} - 4q^{12} + 32q^{13} + 56q^{14} - 23q^{15} + 112q^{16} + 23q^{17} - 176q^{18} + 16q^{19} + 68q^{20} + 93q^{21} - 140q^{22} + 20q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{7} - 16x^{6} - 518x^{5} + 7052x^{4} + 66501x^{3} - 820180x^{2} - 2448240x + 26803048$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
107	$+1$

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
$\operatorname{tr} a_p$	−14	−1	17	−28	70	32	23	16	20	702	420	657	1452	373	786	1196	597	355	−2838	1165	−1484	−589	323	2438	−4638

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 2, 107, 214]: [1, 2, 28, 83] (certified at ell = 2147482621)
T = [(3, 1), (5, 1)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 5 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(214, 4)[1]   # 214.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 214:4 1000