Atlas › Modular forms › Weight 2 › Level 285 › 285.2.a

Newspace 285.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(285))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
285.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	285a1 rank 1
285.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$	285b1 rank 1
285.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	285c1 rank 0
285.2.a.d	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{3} + 10q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	+ − +	$+1$
285.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	− − −	$+1$
285.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ + −	$+1$
285.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
285.2.a.a	$x + 7$
285.2.a.b	$x + 5$
285.2.a.c	$x - 13$
285.2.a.d	$x^{2} + 6x - 19$
285.2.a.e	$x^{2} + 6x - 39$
285.2.a.f	$x^{2} - 14x + 17$
285.2.a.g	$x^{2} - 2x - 7$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(285))$	11
Old forms	26
Cusp forms $S_{2}(\Gamma_0(285))$	37
Eisenstein series $E_{2}(\Gamma_0(285))$	7
Modular forms $M_{2}(\Gamma_0(285))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(285)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(95))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 3, 5, 15, 19, 57, 95, 285]: [0, 1, 1, 4, 2, 8, 12, 44] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 11 = 1 x 11 (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(285, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 285:2 1000

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