Atlas › Modular forms › Weight 2 › Level 820 › 820.2.a

Newspace 820.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(820))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
820.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 2q^{5} - q^{7} + O(q^{8})$	− +
820.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q + 2q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	+ −
820.2.a.c	4	degree 4	$4q - 4q^{5} - q^{7} + O(q^{8})$	+ +
820.2.a.d	4	degree 4	$4q + 4q^{5} + q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
820.2.a.a	$x^{2} + 3x + 1$
820.2.a.b	$x^{2} + x - 3$
820.2.a.c	$x^{4} + x^{3} - 49x^{2} + 16x + 256$
820.2.a.d	$x^{4} - x^{3} - 21x^{2} + 56x - 32$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(820))$	12
Old forms	109
Cusp forms $S_{2}(\Gamma_0(820))$	121
Eisenstein series $E_{2}(\Gamma_0(820))$	11
Modular forms $M_{2}(\Gamma_0(820))$	132

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(820)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(164))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(205))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(410))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 5, 10, 20, 41, 82, 164, 205, 410, 820]: [0, 1, 2, 1, 3, 6, 4, 12, 24, 22, 66, 132] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (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(820, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 820:2 1000

← 819.2.a · 821.2.a →