Atlas › Modular forms › Weight 2 › Level 256 › 256.2.a

Newspace 256.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(256))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
256.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + O(q^{8})$	$\mathbb{Q}(\sqrt{-8})$	256a1 rank 1
256.2.a.b	1	$\mathbb{Q}$	$q - 4q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	256b1 rank 1
256.2.a.c	1	$\mathbb{Q}$	$q + 4q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	256c1 rank 0
256.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + O(q^{8})$	$\mathbb{Q}(\sqrt{-8})$	256d1 rank 0
256.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$	$\mathbb{Q}(\sqrt{-8})$

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 $T$
256.2.a.a	$x + 2$
256.2.a.b	$x + 12$
256.2.a.c	$x - 12$
256.2.a.d	$x - 2$
256.2.a.e	$x^{2} - 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(256))$	6
Old forms	15
Cusp forms $S_{2}(\Gamma_0(256))$	21
Eisenstein series $E_{2}(\Gamma_0(256))$	23
Modular forms $M_{2}(\Gamma_0(256))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(256)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(128))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.192 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, 4, 8, 16, 32, 64, 128, 256]: [0, 1, 2, 3, 4, 6, 10, 18, 34] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 6 = 1 x 6 (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(256, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 256:2 1000

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