Atlas › Modular forms › Weight 2 › Level 485 › 485.2.a

Newspace 485.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(485))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
485.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} - q^{5} - q^{7} + O(q^{8})$	+ −	$+1$	485a1 rank 0
485.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - q^{7} + O(q^{8})$	− −	$-1$	485b1 rank 1
485.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} + 6q^{4} - 2q^{5} + 5q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
485.2.a.d	2	$\mathbb{Q}(\sqrt{29})$	$2q + 2q^{2} + q^{3} - 2q^{4} + 2q^{5} + q^{6} + O(q^{8})$	− +	$+1$
485.2.a.e	3	degree 3	$3q - 2q^{2} + 6q^{3} + 8q^{4} + 3q^{5} - 4q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$
485.2.a.f	4	degree 4	$4q - q^{2} - 4q^{3} + q^{4} + 4q^{5} - 4q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
485.2.a.g	6	degree 6	$6q - q^{2} - q^{3} + 7q^{4} + 6q^{5} + 5q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
485.2.a.h	7	degree 7	$7q - q^{2} - 6q^{3} + 5q^{4} - 7q^{5} - 4q^{6} - 10q^{7} + O(q^{8})$	+ +	$-1$
485.2.a.i	7	degree 7	$7q + 2q^{2} + 9q^{3} + 10q^{4} - 7q^{5} - 7q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
485.2.a.a	$x + 2$
485.2.a.b	$x$
485.2.a.c	$x^{2} - x - 11$
485.2.a.d	$x^{2} - 3x - 5$
485.2.a.e	$x^{3} - 4x^{2} - x + 2$
485.2.a.f	$x^{4} + 5x^{3} + 4x^{2} - 8x - 9$
485.2.a.g	$x^{6} + 2x^{5} - 22x^{4} - 9x^{3} + 117x^{2} - 5x - 161$
485.2.a.h	$x^{7} + 7x^{6} + 3x^{5} - 37x^{4} - x^{3} + 36x^{2} - 3x - 4$
485.2.a.i	$x^{7} - 11x^{6} + 36x^{5} - 17x^{4} - 76x^{3} + 54x^{2} + 44x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(485))$	33
Old forms	14
Cusp forms $S_{2}(\Gamma_0(485))$	47
Eisenstein series $E_{2}(\Gamma_0(485))$	3
Modular forms $M_{2}(\Gamma_0(485))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(485)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 5, 97, 485]: [0, 1, 8, 50] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (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(485, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 485:2 1000

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