Atlas › Modular forms › Weight 8 › Level 57 › 57.8.a

Newspace 57.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(57))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
57.8.a.a	4	degree 4	$4q - 7q^{2} + 108q^{3} + 77q^{4} + 32q^{5} - 189q^{6} - 2074q^{7} + O(q^{8})$	− +	$-1$
57.8.a.b	5	degree 5	$5q + q^{2} - 135q^{3} + 553q^{4} + 138q^{5} - 27q^{6} + 670q^{7} + O(q^{8})$	+ +	$+1$
57.8.a.c	6	degree 6	$6q - 15q^{2} - 162q^{3} + 681q^{4} - 921q^{5} + 405q^{6} - 855q^{7} + O(q^{8})$	+ −	$-1$
57.8.a.d	7	degree 7	$7q + 9q^{2} + 189q^{3} + 589q^{4} - 27q^{5} + 243q^{6} + 1889q^{7} + O(q^{8})$	− −	$+1$

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 $T$
57.8.a.a	$x^{4} - 25x^{3} - 113025x^{2} + 3808981x + 224438108$
57.8.a.b	$x^{5} - 139x^{4} - 231425x^{3} + 12583319x^{2} + 14249164912x + 450743069492$
57.8.a.c	$x^{6} + 936x^{5} + 157002x^{4} - 61780950x^{3} - 15822464559x^{2} - 105612333246x + 11168440783056$
57.8.a.d	$x^{7} + 18x^{6} - 440184x^{5} - 12400776x^{4} + 54737490963x^{3} + 3479081791590x^{2} - 1975610049808500x - 184101405051708360$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(57))$	22
Old forms	22
Cusp forms $S_{8}(\Gamma_0(57))$	44
Eisenstein series $E_{8}(\Gamma_0(57))$	4
Modular forms $M_{8}(\Gamma_0(57))$	48

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(57)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 12, 48] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 7 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 57:8 1000

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