Atlas › Modular forms › Weight 2 › Level 106 › 106.2.a

Newspace 106.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(106))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
106.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} + O(q^{8})$	+ +	$-1$	106b1 rank 1
106.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$	106d1 rank 0
106.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + 3q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	106a1 rank 0
106.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$	106c1 rank 0

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 $T$
106.2.a.a	$x + 13$
106.2.a.b	$x - 5$
106.2.a.c	$x - 7$
106.2.a.d	$x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(106))$	4
Old forms	8
Cusp forms $S_{2}(\Gamma_0(106))$	12
Eisenstein series $E_{2}(\Gamma_0(106))$	3
Modular forms $M_{2}(\Gamma_0(106))$	15

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 53, 106]: [0, 1, 5, 15] (certified at ell = 2147483053)
T = [(3, 1), (5, 3)]; h of degree 4 = 1 x 4 (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(106, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 106:2 1000

← 105.2.a · 107.2.a →