Atlas › Modular forms › Weight 2 › Level 282 › 282.2.a

Newspace 282.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(282))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
282.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$	282b1 rank 1
282.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + O(q^{8})$	− + +	$+1$	282a1 rank 0
282.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + O(q^{8})$	+ + +	$-1$
282.2.a.d	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$
282.2.a.e	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} + 2q^{5} + 3q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
282.2.a.a	$x + 14$
282.2.a.b	$x - 26$
282.2.a.c	$x^{2} + 122x + 2638$
282.2.a.d	$x^{2} - 68x + 670$
282.2.a.e	$x^{3} + 48x^{2} - 1932x - 92996$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(282))$	9
Old forms	36
Cusp forms $S_{2}(\Gamma_0(282))$	45
Eisenstein series $E_{2}(\Gamma_0(282))$	7
Modular forms $M_{2}(\Gamma_0(282))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(282)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(141))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.224 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, 6, 47, 94, 141, 282]: [0, 1, 1, 3, 5, 14, 18, 52] (certified at ell = 2147483137)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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(282, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 282:2 1000

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