Atlas › Modular forms › Weight 4 › Level 214 › 214.4.a

Newspace 214.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(214))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
214.4.a.a	5	degree 5	$5q + 10q^{2} - 13q^{3} + 20q^{4} - 22q^{5} - 26q^{6} - 34q^{7} + O(q^{8})$	− +	$-1$
214.4.a.b	7	degree 7	$7q - 14q^{2} - q^{3} + 28q^{4} + 17q^{5} + 2q^{6} - 28q^{7} + O(q^{8})$	+ +	$+1$
214.4.a.c	7	degree 7	$7q - 14q^{2} + 2q^{3} + 28q^{4} - 28q^{5} - 4q^{6} + O(q^{8})$	+ −	$-1$
214.4.a.d	8	degree 8	$8q + 16q^{2} + 14q^{3} + 32q^{4} + 23q^{5} + 28q^{6} + 50q^{7} + O(q^{8})$	− −	$+1$

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 $T$
214.4.a.a	$x^{5} + 35x^{4} + 413x^{3} + 1697x^{2} + 562x - 884$
214.4.a.b	$x^{7} - 16x^{6} - 518x^{5} + 7052x^{4} + 66501x^{3} - 820180x^{2} - 2448240x + 26803048$
214.4.a.c	$x^{7} + 26x^{6} - 272x^{5} - 8073x^{4} + 9609x^{3} + 573683x^{2} + 1542014x - 1391660$
214.4.a.d	$x^{8} - 37x^{7} - 202x^{6} + 20159x^{5} - 160264x^{4} - 1367688x^{3} + 15815061x^{2} + 18487350x - 335982492$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(214))$	27
Old forms	52
Cusp forms $S_{4}(\Gamma_0(214))$	79
Eisenstein series $E_{4}(\Gamma_0(214))$	4
Modular forms $M_{4}(\Gamma_0(214))$	83

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(214)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 107, 214]: [1, 2, 28, 83] (certified at ell = 2147482621)
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(214, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 214:4 1000

← 213.4.a · 215.4.a →