Atlas › Modular forms › Weight 6 › Level 57 › 57.6.a

Newspace 57.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(57))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
57.6.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 9q^{3} - 28q^{4} - 98q^{5} - 18q^{6} + 240q^{7} + O(q^{8})$	− +	$+1$
57.6.a.b	1	$\mathbb{Q}$	$q + 11q^{2} + 9q^{3} + 89q^{4} + 6q^{5} + 99q^{6} - 176q^{7} + O(q^{8})$	− +	$+1$
57.6.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - 3q^{2} + 18q^{3} + 17q^{4} - 87q^{5} - 27q^{6} - 251q^{7} + O(q^{8})$	− −	$-1$
57.6.a.d	3	degree 3	$3q - 4q^{2} + 27q^{3} + 84q^{4} + 206q^{5} - 36q^{6} + 186q^{7} + O(q^{8})$	− +	$+1$
57.6.a.e	3	degree 3	$3q + 9q^{2} - 27q^{3} + 21q^{4} - 9q^{5} - 81q^{6} + 141q^{7} + O(q^{8})$	+ −	$+1$
57.6.a.f	4	degree 4	$4q + q^{2} - 36q^{3} + 53q^{4} - 8q^{5} - 9q^{6} - 142q^{7} + O(q^{8})$	+ +	$-1$

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 $T$
57.6.a.a	$x + 100$
57.6.a.b	$x - 17$
57.6.a.c	$x^{2} + 90x + 1413$
57.6.a.d	$x^{3} - 202x^{2} + 12689x - 237650$
57.6.a.e	$x^{3} - 4041x + 98856$
57.6.a.f	$x^{4} + 7x^{3} - 7461x^{2} - 75769x + 7767098$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(57))$	14
Old forms	18
Cusp forms $S_{6}(\Gamma_0(57))$	32
Eisenstein series $E_{6}(\Gamma_0(57))$	4
Modular forms $M_{6}(\Gamma_0(57))$	36

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(57)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 19, 57]: [1, 3, 10, 36] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(57, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 57:6 1000

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