Atlas › Modular forms › Weight 2 › Level 212 › 212.2.a

Newspace 212.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(212))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
212.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	−	212a1 rank 1
212.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + O(q^{8})$	+	212b1 rank 0
212.2.a.c	3	degree 3	$3q - 3q^{3} + 6q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
212.2.a.a	$x + 7$
212.2.a.b	$x - 8$
212.2.a.c	$x^{3} + 3x^{2} - 93x - 457$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(212))$	5
Old forms	20
Cusp forms $S_{2}(\Gamma_0(212))$	25
Eisenstein series $E_{2}(\Gamma_0(212))$	5
Modular forms $M_{2}(\Gamma_0(212))$	30

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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]: [0, 1, 2, 5, 15, 30] (certified at ell = 2147483053)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 212:2 1000

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