Atlas › Modular forms › Weight 8 › Level 60 › 60.8.a

Newspace 60.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(60))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
60.8.a.a	1	$\mathbb{Q}$	$q - 27q^{3} - 125q^{5} + 1028q^{7} + O(q^{8})$	+ +
60.8.a.b	1	$\mathbb{Q}$	$q - 27q^{3} + 125q^{5} - 832q^{7} + O(q^{8})$	+ −
60.8.a.c	1	$\mathbb{Q}$	$q + 27q^{3} - 125q^{5} + 92q^{7} + O(q^{8})$	− +
60.8.a.d	1	$\mathbb{Q}$	$q + 27q^{3} + 125q^{5} - 1408q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + T_{11}$ 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 $T$
60.8.a.a	$x - 4124$
60.8.a.b	$x - 2324$
60.8.a.c	$x - 3548$
60.8.a.d	$x + 5452$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(60))$	4
Old forms	74
Cusp forms $S_{8}(\Gamma_0(60))$	78
Eisenstein series $E_{8}(\Gamma_0(60))$	12
Modular forms $M_{8}(\Gamma_0(60))$	90

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]: [1, 3, 3, 5, 5, 9, 13, 17, 16, 24, 46, 90] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(60, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 60:8 1000

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