Atlas › Modular forms › Weight 4 › Level 215 › 215.4.a

Newspace 215.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(215))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
215.4.a.a	6	degree 6	$6q - 5q^{2} - 5q^{3} + 7q^{4} + 30q^{5} - 8q^{6} - 34q^{7} + O(q^{8})$	− +	$-1$
215.4.a.b	8	degree 8	$8q - q^{2} - 7q^{3} + 27q^{4} - 40q^{5} + 8q^{6} - 36q^{7} + O(q^{8})$	+ −	$-1$
215.4.a.c	13	degree 13	$13q + 5q^{2} + 5q^{3} + 63q^{4} - 65q^{5} + 8q^{6} + 34q^{7} + O(q^{8})$	+ +	$+1$
215.4.a.d	15	degree 15	$15q + q^{2} + 7q^{3} + 83q^{4} + 75q^{5} + 40q^{6} + 36q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
215.4.a.a	$x^{6} + 10x^{5} - 32x^{4} - 395x^{3} + 238x^{2} + 3155x + 1275$
215.4.a.b	$x^{8} + 8x^{7} - 156x^{6} - 575x^{5} + 8914x^{4} - 5701x^{3} - 93275x^{2} + 80158x + 283886$
215.4.a.c	$x^{13} - 10x^{12} - 293x^{11} + 2120x^{10} + 34051x^{9} - 134852x^{8} - 1933343x^{7} + 2137715x^{6} + 50832604x^{5} + 43788047x^{4} - 465530684x^{3} - 834417867x^{2} + 899266560x + 2049772335$
215.4.a.d	$x^{15} - 8x^{14} - 433x^{13} + 3704x^{12} + 64573x^{11} - 575938x^{10} - 4127695x^{9} + 38095121x^{8} + 117858376x^{7} - 1176500319x^{6} - 1097050040x^{5} + 16796536397x^{4} - 6138229198x^{3} - 88835608271x^{2} + 89878337536x + 53831237154$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(215))$	42
Old forms	22
Cusp forms $S_{4}(\Gamma_0(215))$	64
Eisenstein series $E_{4}(\Gamma_0(215))$	4
Modular forms $M_{4}(\Gamma_0(215))$	68

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(215)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 5, 43, 215]: [1, 3, 12, 68] (certified at ell = 2147483353)
T = [(2, 1), (3, 1)]; h of degree 42 = 1 x 42 (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(215, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 215:4 1000

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