Atlas › Modular forms › Weight 2 › Level 473 › 473.2.a

Newspace 473.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(473))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
473.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - 2q^{6} + O(q^{8})$	+ +	$-1$	473a1 rank 1
473.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 4q^{3} - q^{4} + 2q^{5} - 2q^{6} + O(q^{8})$	+ +	$-1$
473.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
473.2.a.d	5	degree 5	$5q - 3q^{2} - q^{3} + 7q^{4} - 4q^{5} - q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
473.2.a.e	5	degree 5	$5q + q^{2} - 3q^{3} + 3q^{4} - 6q^{5} + 3q^{6} - 15q^{7} + O(q^{8})$	+ +	$-1$
473.2.a.f	9	degree 9	$9q + 4q^{2} + 5q^{3} + 8q^{4} - 5q^{6} + 19q^{7} + O(q^{8})$	− +	$+1$
473.2.a.g	11	degree 11	$11q - q^{2} + 6q^{3} + 13q^{4} + 3q^{5} + 5q^{6} + 17q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
473.2.a.a	$x + 1$
473.2.a.b	$x^{2} + 3x + 1$
473.2.a.c	$x^{2} + x - 1$
473.2.a.d	$x^{5} + 4x^{4} - 9x^{3} - 22x^{2} + 50x - 23$
473.2.a.e	$x^{5} + 2x^{4} - 19x^{3} + 8x^{2} + 10x - 1$
473.2.a.f	$x^{9} - 9x^{8} + 16x^{7} + 43x^{6} - 93x^{5} - 117x^{4} + 125x^{3} + 189x^{2} + 64x + 4$
473.2.a.g	$x^{11} - 5x^{10} - 35x^{9} + 208x^{8} + 251x^{7} - 2596x^{6} + 1292x^{5} + 10290x^{4} - 12209x^{3} - 8433x^{2} + 14980x - 2712$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(473))$	35
Old forms	8
Cusp forms $S_{2}(\Gamma_0(473))$	43
Eisenstein series $E_{2}(\Gamma_0(473))$	3
Modular forms $M_{2}(\Gamma_0(473))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(473)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 11, 43, 473]: [0, 2, 4, 46] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 4 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(473, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 473:2 1000

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