Atlas › Modular forms › Weight 2 › Level 610 › 610.2.a

Newspace 610.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(610))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
610.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + O(q^{8})$	+ + −	$+1$	610a1 rank 0
610.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + O(q^{8})$	+ − −	$-1$	610b1 rank 1
610.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + q^{5} + 2q^{6} + O(q^{8})$	− − −	$+1$	610c1 rank 0
610.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
610.2.a.e	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} + 3q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
610.2.a.f	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} - 3q^{5} + 2q^{6} + O(q^{8})$	− + +	$+1$
610.2.a.g	4	degree 4	$4q - 4q^{2} - q^{3} + 4q^{4} - 4q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ + +	$-1$
610.2.a.h	4	degree 4	$4q + 4q^{2} - q^{3} + 4q^{4} + 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
610.2.a.a	$x - 17$
610.2.a.b	$x + 34$
610.2.a.c	$x - 14$
610.2.a.d	$x^{2} + 68x + 551$
610.2.a.e	$x^{3} - 5x^{2} - 845x + 9977$
610.2.a.f	$x^{3} - 57x^{2} - 256x + 25468$
610.2.a.g	$x^{4} + 6x^{3} - 1007x^{2} - 3408x + 239796$
610.2.a.h	$x^{4} - 9x^{3} - 779x^{2} - 4595x + 3502$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(610))$	19
Old forms	70
Cusp forms $S_{2}(\Gamma_0(610))$	89
Eisenstein series $E_{2}(\Gamma_0(610))$	7
Modular forms $M_{2}(\Gamma_0(610))$	96

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(610)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(305))^{\oplus 2}$

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, 2, 5, 10, 61, 122, 305, 610]: [0, 1, 1, 3, 5, 17, 32, 96] (certified at ell = 2147482921)
T = [(3, 1), (7, 3), (11, 5), (13, 7)]; h of degree 19 = 1 x 19 (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(610, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 610:2 1000

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