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

Newspace 212.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(212))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
212.4.a.a	5	degree 5	$5q + 2q^{3} - 9q^{5} - 34q^{7} + O(q^{8})$	+
212.4.a.b	8	degree 8	$8q + 2q^{3} + 11q^{5} + 22q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
212.4.a.a	$x^{5} + 7x^{4} - 158x^{3} - 1502x^{2} - 2043x + 3183$
212.4.a.b	$x^{8} - 13x^{7} - 1161x^{6} + 17711x^{5} + 338961x^{4} - 6089687x^{3} - 3057399x^{2} + 253090729x - 664386526$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(212))$	13
Old forms	65
Cusp forms $S_{4}(\Gamma_0(212))$	78
Eisenstein series $E_{4}(\Gamma_0(212))$	6
Modular forms $M_{4}(\Gamma_0(212))$	84

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(212)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(106))^{\oplus 2}$

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)

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

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

← 211.4.a · 213.4.a →