Atlas › Modular forms › Weight 4 › Level 219 › 219.4.a

Newspace 219.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(219))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
219.4.a.a	6	degree 6	$6q - 2q^{2} + 18q^{3} + 4q^{4} - 28q^{5} - 6q^{6} - 56q^{7} + O(q^{8})$	− +	$-1$
219.4.a.b	8	degree 8	$8q - 24q^{3} + 24q^{4} - 42q^{5} + 28q^{7} + O(q^{8})$	+ −	$-1$
219.4.a.c	10	degree 10	$10q + 4q^{2} - 30q^{3} + 40q^{4} + 28q^{5} - 12q^{6} - 28q^{7} + O(q^{8})$	+ +	$+1$
219.4.a.d	12	degree 12	$12q + 2q^{2} + 36q^{3} + 68q^{4} + 42q^{5} + 6q^{6} + 84q^{7} + O(q^{8})$	− −	$+1$

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 $T$
219.4.a.a	$x^{6} + 30x^{5} + 221x^{4} - 360x^{3} - 4673x^{2} + 2586x + 14643$
219.4.a.b	$x^{8} + 42x^{7} + 225x^{6} - 7740x^{5} - 53685x^{4} + 539298x^{3} + 2132811x^{2} - 14545008x + 12719592$
219.4.a.c	$x^{10} - 32x^{9} - 409x^{8} + 17264x^{7} + 36326x^{6} - 3053836x^{5} + 811770x^{4} + 211567808x^{3} - 131877759x^{2} - 4766372324x + 3250268695$
219.4.a.d	$x^{12} - 44x^{11} - 189x^{10} + 31472x^{9} - 240374x^{8} - 6982668x^{7} + 91817334x^{6} + 415058256x^{5} - 10110679587x^{4} + 18033718176x^{3} + 320222277207x^{2} - 1671048657864x + 2292185172456$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(219))$	36
Old forms	36
Cusp forms $S_{4}(\Gamma_0(219))$	72
Eisenstein series $E_{4}(\Gamma_0(219))$	4
Modular forms $M_{4}(\Gamma_0(219))$	76

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(219)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.141 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, 73, 219]: [1, 2, 20, 76] (certified at ell = 2147482873)
T = [(2, 1), (5, 1)]; h of degree 36 = 1 x 36 (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(219, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 219:4 1000

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