Atlas › Modular forms › Weight 2 › Level 225 › 225.2.a

Newspace 225.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(225))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
225.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 3q^{7} + O(q^{8})$		225e1 rank 1
225.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + O(q^{8})$		225c1 rank 0
225.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} - 5q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	225a1 rank 1
225.2.a.d	1	$\mathbb{Q}$	$q - 2q^{4} + 5q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	225b1 rank 0
225.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 3q^{7} + O(q^{8})$		225d1 rank 0
225.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-15})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
225.2.a.a	$x + 14$
225.2.a.b	$x - 33$
225.2.a.c	$x + 50$
225.2.a.d	$x - 50$
225.2.a.e	$x + 6$
225.2.a.f	$x^{2} - 5$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(225))$	7
Old forms	12
Cusp forms $S_{2}(\Gamma_0(225))$	19
Eisenstein series $E_{2}(\Gamma_0(225))$	23
Modular forms $M_{2}(\Gamma_0(225))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(225)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 5, 9, 15, 25, 45, 75, 225]: [0, 1, 1, 2, 4, 3, 8, 12, 32] (certified at ell = 2147482921)
T = [(2, 1), (7, 3), (11, 5), (13, 7)]; h of degree 7 = 1 x 7 (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(225, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 225:2 1000

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