Atlas › Modular forms › Weight 2 › Level 687 › 687.2.a

Newspace 687.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(687))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
687.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} - 2q^{4} - q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
687.2.a.b	5	degree 5	$5q - 2q^{2} + 5q^{3} - 3q^{5} - 2q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
687.2.a.c	7	degree 7	$7q - 2q^{2} - 7q^{3} + 14q^{4} - 6q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
687.2.a.d	10	degree 10	$10q + 3q^{2} - 10q^{3} + 9q^{4} + 9q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
687.2.a.e	15	degree 15	$15q + 2q^{2} + 15q^{3} + 20q^{4} + 7q^{5} + 2q^{6} + 12q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
687.2.a.a	$x^{2} + 3x - 9$
687.2.a.b	$x^{5} + 5x^{4} + 5x^{3} - 6x^{2} - 7x + 1$
687.2.a.c	$x^{7} + 8x^{6} + 9x^{5} - 30x^{4} + 4x^{3} + 16x^{2} - 8x + 1$
687.2.a.d	$x^{10} - 12x^{9} + 21x^{8} + 205x^{7} - 662x^{6} - 841x^{5} + 3726x^{4} + 1541x^{3} - 7030x^{2} - 1930x + 3203$
687.2.a.e	$x^{15} - 9x^{14} - 36x^{13} + 557x^{12} - 613x^{11} - 9394x^{10} + 26869x^{9} + 46448x^{8} - 256909x^{7} + 115750x^{6} + 715069x^{5} - 1077996x^{4} + 401490x^{3} - 2937x^{2} - 3035x + 67$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(687))$	39
Old forms	36
Cusp forms $S_{2}(\Gamma_0(687))$	75
Eisenstein series $E_{2}(\Gamma_0(687))$	3
Modular forms $M_{2}(\Gamma_0(687))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(687)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(229))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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, 229, 687]: [0, 1, 19, 78] (certified at ell = 2147478649)
T = [(2, 1), (5, 1)]; h of degree 39 = 1 x 39 (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(687, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 687:2 1000

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