Atlas › Modular forms › Weight 2 › Level 129 › 129.2.a

Newspace 129.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(129))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
129.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +	$-1$	129a1 rank 1
129.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + 2q^{5} + q^{6} + O(q^{8})$	− +	$+1$	129b1 rank 0
129.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
129.2.a.d	3	degree 3	$3q - 2q^{2} + 3q^{3} + 8q^{4} - 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
129.2.a.a	$x + 6$
129.2.a.b	$x - 7$
129.2.a.c	$x^{2} - 8x + 8$
129.2.a.d	$x^{3} + 14x^{2} + 40x + 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(129))$	7
Old forms	6
Cusp forms $S_{2}(\Gamma_0(129))$	13
Eisenstein series $E_{2}(\Gamma_0(129))$	3
Modular forms $M_{2}(\Gamma_0(129))$	16

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(129)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.171 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, 3, 43, 129]: [0, 1, 4, 16] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 7 = 1 x 7 (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(129, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 129:2 1000

← 128.2.a · 130.2.a →