Atlas › Modular forms › Weight 6 › Level 87 › 87.6.a

Newspace 87.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(87))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
87.6.a.a	4	degree 4	$4q - 3q^{2} + 36q^{3} + 49q^{4} - 136q^{5} - 27q^{6} - 28q^{7} + O(q^{8})$	− −	$-1$
87.6.a.b	5	degree 5	$5q - 9q^{2} - 45q^{3} + 53q^{4} + 14q^{5} + 81q^{6} - 68q^{7} + O(q^{8})$	+ +	$-1$
87.6.a.c	7	degree 7	$7q + 13q^{2} + 63q^{3} + 145q^{4} + 64q^{5} + 117q^{6} + 168q^{7} + O(q^{8})$	− +	$+1$
87.6.a.d	8	degree 8	$8q - q^{2} - 72q^{3} + 149q^{4} + 14q^{5} + 9q^{6} + 128q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
87.6.a.a	$x^{4} + 139x^{3} + 4704x^{2} - 8612x - 1295008$
87.6.a.b	$x^{5} - 5x^{4} - 2990x^{3} - 54620x^{2} + 351880x + 8066816$
87.6.a.c	$x^{7} - 77x^{6} - 14950x^{5} + 1178357x^{4} + 36418217x^{3} - 2773679148x^{2} - 20552192336x + 1221511516352$
87.6.a.d	$x^{8} - 13x^{7} - 20128x^{6} + 91711x^{5} + 126406235x^{4} + 1118232378x^{3} - 241243275704x^{2} - 7031974605920x - 51959124898560$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(87))$	24
Old forms	24
Cusp forms $S_{6}(\Gamma_0(87))$	48
Eisenstein series $E_{6}(\Gamma_0(87))$	4
Modular forms $M_{6}(\Gamma_0(87))$	52

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(87)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 29, 87]: [1, 3, 13, 52] (certified at ell = 2147483549)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(87, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 87:6 1000

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