Atlas › Modular forms › Weight 6 › Level 60 › 60.6.a

Newspace 60.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(60))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
60.6.a.a	1	$\mathbb{Q}$	$q - 9q^{3} - 25q^{5} + 44q^{7} + O(q^{8})$	+ +
60.6.a.b	1	$\mathbb{Q}$	$q - 9q^{3} + 25q^{5} - 16q^{7} + O(q^{8})$	+ −
60.6.a.c	1	$\mathbb{Q}$	$q + 9q^{3} - 25q^{5} - 244q^{7} + O(q^{8})$	− +
60.6.a.d	1	$\mathbb{Q}$	$q + 9q^{3} + 25q^{5} + 56q^{7} + O(q^{8})$	− −

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 $T$
60.6.a.a	$x - 260$
60.6.a.b	$x + 580$
60.6.a.c	$x + 388$
60.6.a.d	$x - 212$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(60))$	4
Old forms	50
Cusp forms $S_{6}(\Gamma_0(60))$	54
Eisenstein series $E_{6}(\Gamma_0(60))$	12
Modular forms $M_{6}(\Gamma_0(60))$	66

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(60)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.148 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, 2, 3, 4, 3, 7, 9, 13, 12, 18, 34, 66] (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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 60:6 1000

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