Atlas › Modular forms › Weight 4 › Level 212 › 212.4.a › 212.4.a.b

Newform orbit 212.4.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 2q^{3} + 11q^{5} + 22q^{7} + 74q^{9} - 15q^{11} + 140q^{13} + 234q^{15} + 171q^{17} + 136q^{19} + 212q^{21} + 149q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} - 13x^{7} - 1161x^{6} + 17711x^{5} + 338961x^{4} - 6089687x^{3} - 3057399x^{2} + 253090729x - 664386526$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 212$ )
53	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	2	11	22	−15	140	171	136	149	514	431	314	336	−213	236	424	−901	472	−1280	−2804	1102	−1539	−3242	−689	−205

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.129 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, 4, 53, 106, 212]: [1, 2, 3, 15, 43, 84] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 3 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(212, 4)[1]   # 212.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 212:4 1000