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

Newform orbit 214.4.a.c

Trace form

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

$7q - 14q^{2} + 2q^{3} + 28q^{4} - 28q^{5} - 4q^{6} - 56q^{8} + 61q^{9} + 56q^{10} - 51q^{11} + 8q^{12} - 59q^{13} - 83q^{15} + 112q^{16} - 147q^{17} - 122q^{18} - 79q^{19} - 112q^{20} - 285q^{21} + 102q^{22} - 26q^{23} - 16q^{24} + O(q^{25})$

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

$x^{7} + 26x^{6} - 272x^{5} - 8073x^{4} + 9609x^{3} + 573683x^{2} + 1542014x - 1391660$

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
$\operatorname{tr} a_p$	−14	2	−28	0	−51	−59	−147	−79	−26	−725	−464	−236	−1162	−8	−1210	−895	−1380	−622	3005	−547	−34	43	−408	90	952

$\operatorname{tr} a_n$ 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)[2]   # 214.4.a.c

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

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