Atlas › Modular forms › Weight 2 › Level 189 › 189.2.a

Newspace 189.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(189))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
189.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - q^{5} - q^{7} + O(q^{8})$	+	189a1 rank 1
189.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} - 3q^{5} + q^{7} + O(q^{8})$	−	189b1 rank 1
189.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} + 3q^{5} + q^{7} + O(q^{8})$	−	189c1 rank 0
189.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + q^{5} - q^{7} + O(q^{8})$	+	189d1 rank 0
189.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} + 2q^{7} + O(q^{8})$	−
189.2.a.f	2	$\mathbb{Q}(\sqrt{7})$	$2q + 10q^{4} - 2q^{7} + O(q^{8})$	+

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 $T$
189.2.a.a	$x + 5$
189.2.a.b	$x + 9$
189.2.a.c	$x - 9$
189.2.a.d	$x - 5$
189.2.a.e	$x^{2} - 48$
189.2.a.f	$x^{2} - 28$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(189))$	8
Old forms	11
Cusp forms $S_{2}(\Gamma_0(189))$	19
Eisenstein series $E_{2}(\Gamma_0(189))$	11
Modular forms $M_{2}(\Gamma_0(189))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(189)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 7, 9, 21, 27, 63, 189]: [0, 1, 1, 2, 4, 4, 10, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(189, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 189:2 1000

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