Atlas › Modular forms › Weight 6 › Level 93 › 93.6.a

Newspace 93.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(93))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
93.6.a.a	4	degree 4	$4q - 3q^{2} + 36q^{3} + 49q^{4} - 42q^{5} - 27q^{6} - 284q^{7} + O(q^{8})$	− −	$-1$
93.6.a.b	5	degree 5	$5q + 9q^{2} - 45q^{3} + 53q^{4} + 36q^{5} - 81q^{6} + 108q^{7} + O(q^{8})$	+ −	$+1$
93.6.a.c	7	degree 7	$7q - 3q^{2} - 63q^{3} + 101q^{4} - 64q^{5} + 27q^{6} - 88q^{7} + O(q^{8})$	+ +	$-1$
93.6.a.d	8	degree 8	$8q + 9q^{2} + 72q^{3} + 193q^{4} + 58q^{5} + 81q^{6} + 304q^{7} + O(q^{8})$	− +	$+1$

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 $T$
93.6.a.a	$x^{4} + 45x^{3} - 4243x^{2} - 32409x + 1703974$
93.6.a.b	$x^{5} - 45x^{4} - 8887x^{3} + 230877x^{2} + 11151382x - 62206416$
93.6.a.c	$x^{7} + 67x^{6} - 7453x^{5} - 362048x^{4} + 20876815x^{3} + 357836695x^{2} - 23583329299x + 233824598470$
93.6.a.d	$x^{8} - 67x^{7} - 20785x^{6} + 1261398x^{5} + 133446413x^{4} - 6744283937x^{3} - 256853159373x^{2} + 7359438520174x + 89950798027792$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(93))$	24
Old forms	28
Cusp forms $S_{6}(\Gamma_0(93))$	52
Eisenstein series $E_{6}(\Gamma_0(93))$	4
Modular forms $M_{6}(\Gamma_0(93))$	56

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(93)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 31, 93]: [1, 3, 15, 56] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(93, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 93:6 1000

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