Atlas › Modular forms › Weight 6 › Level 94 › 94.6.a

Newspace 94.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(94))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
94.6.a.a	4	degree 4	$4q - 16q^{2} + 6q^{3} + 64q^{4} - 68q^{5} - 24q^{6} + 88q^{7} + O(q^{8})$	+ +	$-1$
94.6.a.b	4	degree 4	$4q + 16q^{2} - 22q^{3} + 64q^{4} - 68q^{5} - 88q^{6} - 304q^{7} + O(q^{8})$	− −	$-1$
94.6.a.c	6	degree 6	$6q - 24q^{2} + 6q^{3} + 96q^{4} + 32q^{5} - 24q^{6} - 108q^{7} + O(q^{8})$	+ −	$+1$
94.6.a.d	6	degree 6	$6q + 24q^{2} + 14q^{3} + 96q^{4} + 82q^{5} + 56q^{6} + 88q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
94.6.a.a	$x^{4} + 62x^{3} - 2577x^{2} - 102766x + 2468941$
94.6.a.b	$x^{4} + 90x^{3} - 437x^{2} - 56134x - 252655$
94.6.a.c	$x^{6} - 38x^{5} - 16903x^{4} + 496934x^{3} + 90318235x^{2} - 1506714180x - 156065911650$
94.6.a.d	$x^{6} - 96x^{5} - 13793x^{4} + 1578592x^{3} + 1611333x^{2} - 2326917702x - 14535472860$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(94))$	20
Old forms	38
Cusp forms $S_{6}(\Gamma_0(94))$	58
Eisenstein series $E_{6}(\Gamma_0(94))$	4
Modular forms $M_{6}(\Gamma_0(94))$	62

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(94)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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, 47, 94]: [1, 2, 21, 62] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(94, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 94:6 1000

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