Atlas › Modular forms › Weight 4 › Level 218 › 218.4.a › 218.4.a.c

Newform orbit 218.4.a.c

Trace form

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

$7q - 14q^{2} + 13q^{3} + 28q^{4} - 12q^{5} - 26q^{6} + 25q^{7} - 56q^{8} + 36q^{9} + 24q^{10} + 37q^{11} + 52q^{12} + 170q^{13} - 50q^{14} + 54q^{15} + 112q^{16} + 25q^{17} - 72q^{18} + 177q^{19} - 48q^{20} + 25q^{21} - 74q^{22} + 20q^{23} - 104q^{24} + O(q^{25})$

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

$x^{7} - x^{6} - 772x^{5} + 1919x^{4} + 197313x^{3} - 821236x^{2} - 16677572x + 101460328$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
109	$+1$

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	13	−12	25	37	170	25	177	20	52	567	931	820	1078	353	612	−415	1056	877	−1164	537	−467	−1908	−4714	−507

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.137 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, 109, 218]: [1, 2, 29, 85] (certified at ell = 2147482801)
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(218, 4)[2]   # 218.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 218:4 1000