Atlas › Modular forms › Weight 4 › Level 111 › 111.4.a

Newspace 111.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(111))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
111.4.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 3q^{3} + 8q^{4} + 2q^{5} - 12q^{6} - 28q^{7} + O(q^{8})$	− +	$-1$
111.4.a.b	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - 7q^{4} - 4q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+ +	$+1$
111.4.a.c	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} - 7q^{4} - 8q^{5} + 3q^{6} - 13q^{7} + O(q^{8})$	− +	$-1$
111.4.a.d	3	degree 3	$3q - 3q^{2} - 9q^{3} + 9q^{4} - 6q^{5} + 9q^{6} + 15q^{7} + O(q^{8})$	+ −	$-1$
111.4.a.e	5	degree 5	$5q + 4q^{2} - 15q^{3} + 36q^{4} + 18q^{5} - 12q^{6} - 12q^{7} + O(q^{8})$	+ +	$+1$
111.4.a.f	7	degree 7	$7q + 3q^{2} + 21q^{3} + 45q^{4} + 14q^{5} + 9q^{6} + 43q^{7} + O(q^{8})$	− −	$+1$

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 $T$
111.4.a.a	$x + 2$
111.4.a.b	$x + 5$
111.4.a.c	$x + 7$
111.4.a.d	$x^{3} + 9x^{2} - 108x - 918$
111.4.a.e	$x^{5} - 22x^{4} - 378x^{3} + 9652x^{2} - 46995x + 2764$
111.4.a.f	$x^{7} - 17x^{6} - 542x^{5} + 10580x^{4} + 25047x^{3} - 936117x^{2} + 82026x + 23245110$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(111))$	18
Old forms	18
Cusp forms $S_{4}(\Gamma_0(111))$	36
Eisenstein series $E_{4}(\Gamma_0(111))$	4
Modular forms $M_{4}(\Gamma_0(111))$	40

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(111)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 2, 11, 40] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 111:4 1000

← 110.4.a · 112.4.a →