Atlas › Modular forms › Weight 2 › Level 555 › 555.2.a

Newspace 555.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(555))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
555.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} - 2q^{7} + O(q^{8})$	− + +	$+1$	555a1 rank 0
555.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− − −	$+1$	555b1 rank 0
555.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 2q^{3} + 3q^{4} + 2q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$
555.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + q^{6} - 6q^{7} + O(q^{8})$	+ − −	$-1$
555.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
555.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - 2q^{5} - q^{6} + O(q^{8})$	+ + +	$-1$
555.2.a.g	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 2q^{3} + 3q^{4} - 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
555.2.a.h	3	degree 3	$3q + q^{2} - 3q^{3} + 5q^{4} + 3q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$
555.2.a.i	3	degree 3	$3q + 2q^{2} + 3q^{3} + 6q^{4} + 3q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− − −	$+1$
555.2.a.j	5	degree 5	$5q - 3q^{2} - 5q^{3} + 7q^{4} - 5q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
555.2.a.a	$x + 2$
555.2.a.b	$x - 2$
555.2.a.c	$x^{2} + 7x + 11$
555.2.a.d	$x^{2} + 7x + 9$
555.2.a.e	$x^{2} + 3x + 1$
555.2.a.f	$x^{2} - x - 1$
555.2.a.g	$x^{2} - 3x - 1$
555.2.a.h	$x^{3} - 5x^{2} + x + 14$
555.2.a.i	$x^{3} + 2x^{2} - 18x + 13$
555.2.a.j	$x^{5} + x^{4} - 46x^{3} - 17x^{2} + 535x - 146$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(555))$	23
Old forms	50
Cusp forms $S_{2}(\Gamma_0(555))$	73
Eisenstein series $E_{2}(\Gamma_0(555))$	7
Modular forms $M_{2}(\Gamma_0(555))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(555)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(185))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.235 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, 37, 111, 185, 555]: [0, 1, 1, 4, 3, 14, 20, 80] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 23 = 1 x 23 (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(555, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 555:2 1000

← 554.2.a · 556.2.a →