Atlas › Modular forms › Weight 2 › Level 470 › 470.2.a

Newspace 470.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(470))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
470.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$	470c1 rank 1
470.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	470a1 rank 1
470.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$	470b1 rank 0
470.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} + q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	470f1 rank 1
470.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	470e1 rank 1
470.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$	470d1 rank 0
470.2.a.g	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{2} + q^{3} + 2q^{4} + 2q^{5} - q^{6} + 8q^{7} + O(q^{8})$	+ − +	$+1$
470.2.a.h	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + O(q^{8})$	+ + −	$+1$
470.2.a.i	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− − −	$+1$
470.2.a.j	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
470.2.a.a	$x + 80$
470.2.a.b	$x + 22$
470.2.a.c	$x - 4$
470.2.a.d	$x + 116$
470.2.a.e	$x + 6$
470.2.a.f	$x + 26$
470.2.a.g	$x^{2} + 21x - 8715$
470.2.a.h	$x^{3} - 141x^{2} + 3795x + 87026$
470.2.a.i	$x^{3} - 11x^{2} - 2157x - 20104$
470.2.a.j	$x^{3} + 39x^{2} - 1899x + 4478$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(470))$	17
Old forms	52
Cusp forms $S_{2}(\Gamma_0(470))$	69
Eisenstein series $E_{2}(\Gamma_0(470))$	7
Modular forms $M_{2}(\Gamma_0(470))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(470)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(235))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.44 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, 5, 10, 47, 94, 235, 470]: [0, 1, 1, 3, 5, 14, 26, 76] (certified at ell = 2147483137)
T = [(3, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 5 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(470, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 470:2 1000

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