Atlas › Modular forms › Weight 2 › Level 280 › 280.2.a

Newspace 280.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(280))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
280.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} + q^{7} + O(q^{8})$	− −	280b1 rank 1
280.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - q^{7} + O(q^{8})$	+ +	280a1 rank 1
280.2.a.c	2	$\mathbb{Q}(\sqrt{33})$	$2q - q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +
280.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{11}$ 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 $T$
280.2.a.a	$x + 18$
280.2.a.b	$x + 16$
280.2.a.c	$x^{2} - 20x - 32$
280.2.a.d	$x^{2} + 2x - 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(280))$	6
Old forms	35
Cusp forms $S_{2}(\Gamma_0(280))$	41
Eisenstein series $E_{2}(\Gamma_0(280))$	15
Modular forms $M_{2}(\Gamma_0(280))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(280)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(70))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(140))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.348 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, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280]: [0, 1, 2, 1, 1, 3, 3, 4, 6, 7, 6, 10, 12, 16, 30, 56] (certified at ell = 2147483629)
T = [(3, 1), (11, 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(280, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 280:2 1000

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