Atlas › Modular forms › Weight 4 › Level 60 › 60.4.a

Newspace 60.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(60))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
60.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 5q^{5} - 28q^{7} + O(q^{8})$	+ +
60.4.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + 5q^{5} + 32q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 3T_{11}$ 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 $T$
60.4.a.a	$x + 100$
60.4.a.b	$x - 140$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(60))$	2
Old forms	28
Cusp forms $S_{4}(\Gamma_0(60))$	30
Eisenstein series $E_{4}(\Gamma_0(60))$	12
Modular forms $M_{4}(\Gamma_0(60))$	42

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(60)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 2, 3, 3, 5, 7, 9, 8, 12, 22, 42] (certified at ell = 2147483629)
T = [(7, 1), (11, 3)]; h of degree 2 = 1 x 2 (phi(ord eps) x dim S_k^new); 1 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 60:4 1000

← 59.4.a · 61.4.a →