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

Newform orbit 214.4.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 10q^{2} - 13q^{3} + 20q^{4} - 22q^{5} - 26q^{6} - 34q^{7} + 40q^{8} + 2q^{9} - 44q^{10} - 98q^{11} - 52q^{12} - 88q^{13} - 68q^{14} - 103q^{15} + 80q^{16} - 187q^{17} + 4q^{18} - 190q^{19} - 88q^{20} - 79q^{21} - 196q^{22} - 297q^{23} - 104q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 35x^{4} + 413x^{3} + 1697x^{2} + 562x - 884$

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 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
$\operatorname{tr} a_p$	10	−13	−22	−34	−98	−88	−187	−190	−297	−134	66	−69	−263	256	−273	472	−614	−9	1711	635	572	360	−479	−1513	146

$\operatorname{tr} a_n$ 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)[0]   # 214.4.a.a

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

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