Atlas › Modular forms › Weight 2 › Level 205 › 205.2.a

Newspace 205.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(205))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
205.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	205a1 rank 1
205.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} - q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$	205b1 rank 0
205.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} + q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	205c1 rank 0
205.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - 6q^{3} + 3q^{4} + 2q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
205.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
205.2.a.f	3	degree 3	$3q + 2q^{3} + 2q^{4} + 3q^{5} - 5q^{6} + q^{7} + O(q^{8})$	− +	$+1$
205.2.a.g	3	degree 3	$3q + 2q^{2} + 2q^{3} + 6q^{4} - 3q^{5} + 9q^{6} - 9q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
205.2.a.a	$x + 1$
205.2.a.b	$x - 1$
205.2.a.c	$x - 3$
205.2.a.d	$x^{2} + 7x + 9$
205.2.a.e	$x^{2} + 3x + 1$
205.2.a.f	$x^{3} - 2x^{2} - 4x + 7$
205.2.a.g	$x^{3} - 4x^{2} - 14x + 49$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(205))$	13
Old forms	6
Cusp forms $S_{2}(\Gamma_0(205))$	19
Eisenstein series $E_{2}(\Gamma_0(205))$	3
Modular forms $M_{2}(\Gamma_0(205))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(205)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 5, 41, 205]: [0, 1, 4, 22] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (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(205, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 205:2 1000

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