Atlas › Modular forms › Weight 2 › Level 270 › 270.2.a

Newspace 270.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(270))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
270.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + 2q^{7} + O(q^{8})$	+ +	270a1 rank 0
270.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + 2q^{7} + O(q^{8})$	+ −	270d1 rank 0
270.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} + 2q^{7} + O(q^{8})$	− +	270b1 rank 0
270.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− −	270c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 3T_{11} + 5T_{13} + 7T_{17}$ 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 $T$
270.2.a.a	$x - 27$
270.2.a.b	$x - 39$
270.2.a.c	$x - 15$
270.2.a.d	$x + 33$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(270))$	4
Old forms	39
Cusp forms $S_{2}(\Gamma_0(270))$	43
Eisenstein series $E_{2}(\Gamma_0(270))$	23
Modular forms $M_{2}(\Gamma_0(270))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(270)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(135))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.132 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, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270]: [0, 1, 1, 1, 3, 2, 3, 4, 5, 4, 10, 8, 11, 22, 20, 58] (certified at ell = 2147483629)
T = [(7, 1), (11, 3), (13, 5), (17, 7)]; h of degree 4 = 1 x 4 (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(270, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 270:2 1000

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