Atlas › Modular forms › Weight 4 › Level 207 › 207.4.a

Newspace 207.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(207))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
207.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} - 4q^{4} + 6q^{5} - 8q^{7} + O(q^{8})$	+
207.4.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 8q^{5} - 28q^{7} + O(q^{8})$	+
207.4.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{2} + 2q^{4} + 26q^{5} - 10q^{7} + O(q^{8})$	−
207.4.a.d	4	degree 4	$4q - 4q^{2} + 26q^{4} - 4q^{5} - 14q^{7} + O(q^{8})$	+
207.4.a.e	4	degree 4	$4q - 2q^{2} + 20q^{4} - 14q^{5} + 16q^{7} + O(q^{8})$	−
207.4.a.f	4	degree 4	$4q + 2q^{2} + 26q^{4} + 2q^{5} + 32q^{7} + O(q^{8})$	−
207.4.a.g	5	degree 5	$5q - 4q^{2} + 16q^{4} - 20q^{5} - 10q^{7} + O(q^{8})$	−
207.4.a.h	5	degree 5	$5q + 4q^{2} + 16q^{4} + 20q^{5} - 10q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
207.4.a.a	$x - 20$
207.4.a.b	$x^{2} + 22x - 847$
207.4.a.c	$x^{2} - 82x + 1661$
207.4.a.d	$x^{4} + 16x^{3} - 1491x^{2} - 42164x - 268738$
207.4.a.e	$x^{4} + 44x^{3} - 1926x^{2} - 119953x - 1498594$
207.4.a.f	$x^{4} - 8x^{3} - 4335x^{2} + 30400x + 1191450$
207.4.a.g	$x^{5} + 64x^{4} - 2018x^{3} - 107506x^{2} + 847533x + 20047582$
207.4.a.h	$x^{5} - 64x^{4} - 2018x^{3} + 107506x^{2} + 847533x - 20047582$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(207))$	27
Old forms	41
Cusp forms $S_{4}(\Gamma_0(207))$	68
Eisenstein series $E_{4}(\Gamma_0(207))$	8
Modular forms $M_{4}(\Gamma_0(207))$	76

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(207)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.15 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, 9, 23, 69, 207]: [1, 2, 4, 7, 26, 74] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 27 = 1 x 27 (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(207, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 207:4 1000

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