Atlas › Modular forms › Weight 6 › Level 111 › 111.6.a

Newspace 111.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(111))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
111.6.a.a	5	degree 5	$5q - 3q^{2} + 45q^{3} + 49q^{4} - 64q^{5} - 27q^{6} - 216q^{7} + O(q^{8})$	− −	$-1$
111.6.a.b	6	degree 6	$6q - 9q^{2} - 54q^{3} + 53q^{4} - 58q^{5} + 81q^{6} + 176q^{7} + O(q^{8})$	+ +	$-1$
111.6.a.c	9	degree 9	$9q + 3q^{2} - 81q^{3} + 133q^{4} + 42q^{5} - 27q^{6} - 20q^{7} + O(q^{8})$	+ −	$+1$
111.6.a.d	10	degree 10	$10q + 9q^{2} + 90q^{3} + 225q^{4} + 36q^{5} + 81q^{6} + 372q^{7} + O(q^{8})$	− +	$+1$

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 $T$
111.6.a.a	$x^{5} + 67x^{4} - 3980x^{3} - 275284x^{2} + 3998052x + 279930168$
111.6.a.b	$x^{6} + 67x^{5} - 5984x^{4} - 147812x^{3} + 15676940x^{2} - 319365284x + 1938558872$
111.6.a.c	$x^{9} - 45x^{8} - 15980x^{7} + 437466x^{6} + 85744671x^{5} - 1214042481x^{4} - 154228330600x^{3} + 2010156194988x^{2} + 74276944596324x - 1140432891935112$
111.6.a.d	$x^{10} - 45x^{9} - 22504x^{8} + 980598x^{7} + 166248617x^{6} - 6396565335x^{5} - 466754022884x^{4} + 12404835142572x^{3} + 486689462409316x^{2} - 3981043549079628x - 91888937281465720$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(111))$	30
Old forms	32
Cusp forms $S_{6}(\Gamma_0(111))$	62
Eisenstein series $E_{6}(\Gamma_0(111))$	4
Modular forms $M_{6}(\Gamma_0(111))$	66

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(111)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 37, 111]: [1, 3, 17, 66] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 30 = 1 x 30 (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(111, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 111:6 1000

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