Atlas › Modular forms › Weight 4 › Level 218 › 218.4.a

Newspace 218.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(218))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
218.4.a.a	5	degree 5	$5q + 10q^{2} - 10q^{3} + 20q^{4} - 28q^{5} - 20q^{6} - 37q^{7} + O(q^{8})$	− +	$-1$
218.4.a.b	6	degree 6	$6q - 12q^{2} - 8q^{3} + 24q^{4} + 13q^{5} + 16q^{6} - 45q^{7} + O(q^{8})$	+ −	$-1$
218.4.a.c	7	degree 7	$7q - 14q^{2} + 13q^{3} + 28q^{4} - 12q^{5} - 26q^{6} + 25q^{7} + O(q^{8})$	+ +	$+1$
218.4.a.d	9	degree 9	$9q + 18q^{2} + 11q^{3} + 36q^{4} + 27q^{5} + 22q^{6} + 61q^{7} + O(q^{8})$	− −	$+1$

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 $T$
218.4.a.a	$x^{5} + 38x^{4} + 493x^{3} + 2620x^{2} + 4772x - 632$
218.4.a.b	$x^{6} - 5x^{5} - 529x^{4} + 1173x^{3} + 6476x^{2} - 18572x + 8648$
218.4.a.c	$x^{7} - x^{6} - 772x^{5} + 1919x^{4} + 197313x^{3} - 821236x^{2} - 16677572x + 101460328$
218.4.a.d	$x^{9} - 38x^{8} - 341x^{7} + 29721x^{6} - 281826x^{5} - 2849883x^{4} + 55928450x^{3} - 244510828x^{2} + 166395344x - 30936912$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(218))$	27
Old forms	54
Cusp forms $S_{4}(\Gamma_0(218))$	81
Eisenstein series $E_{4}(\Gamma_0(218))$	4
Modular forms $M_{4}(\Gamma_0(218))$	85

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(218)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.137 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, 109, 218]: [1, 2, 29, 85] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(218, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 218:4 1000

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