Atlas › Modular forms › Weight 4 › Level 208 › 208.4.a

Newspace 208.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(208))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
208.4.a.a	1	$\mathbb{Q}$	$q - 5q^{3} + 19q^{5} + 3q^{7} + O(q^{8})$	+
208.4.a.b	1	$\mathbb{Q}$	$q - 4q^{3} - 18q^{5} - 20q^{7} + O(q^{8})$	−
208.4.a.c	1	$\mathbb{Q}$	$q - 3q^{3} + 11q^{5} - 19q^{7} + O(q^{8})$	+
208.4.a.d	1	$\mathbb{Q}$	$q - q^{3} - 7q^{5} + 21q^{7} + O(q^{8})$	−
208.4.a.e	1	$\mathbb{Q}$	$q + q^{3} + 17q^{5} + 35q^{7} + O(q^{8})$	−
208.4.a.f	1	$\mathbb{Q}$	$q + 3q^{3} - 13q^{5} + 11q^{7} + O(q^{8})$	+
208.4.a.g	1	$\mathbb{Q}$	$q + 7q^{3} - 7q^{5} + 13q^{7} + O(q^{8})$	−
208.4.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q - 5q^{3} - 3q^{5} + 9q^{7} + O(q^{8})$	+
208.4.a.i	2	$\mathbb{Q}(\sqrt{321})$	$2q + q^{3} - 11q^{5} - q^{7} + O(q^{8})$	+
208.4.a.j	2	$\mathbb{Q}(\sqrt{73})$	$2q + 3q^{3} - 3q^{5} + 25q^{7} + O(q^{8})$	+
208.4.a.k	2	$\mathbb{Q}(\sqrt{217})$	$2q + 3q^{3} + 23q^{5} - 27q^{7} + O(q^{8})$	−
208.4.a.l	3	degree 3	$3q - 8q^{5} - 36q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17} + 9T_{19}$ 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 $T$
208.4.a.a	$x - 1123$
208.4.a.b	$x - 788$
208.4.a.c	$x + 947$
208.4.a.d	$x + 443$
208.4.a.e	$x + 937$
208.4.a.f	$x + 1733$
208.4.a.g	$x - 1789$
208.4.a.h	$x^{2} - 475x + 200$
208.4.a.i	$x^{2} + 1407x + 471720$
208.4.a.j	$x^{2} - 1299x + 404312$
208.4.a.k	$x^{2} - 433x - 211412$
208.4.a.l	$x^{3} + 2212x^{2} + 1475383x + 287831868$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(208))$	18
Old forms	60
Cusp forms $S_{4}(\Gamma_0(208))$	78
Eisenstein series $E_{4}(\Gamma_0(208))$	12
Modular forms $M_{4}(\Gamma_0(208))$	90

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(208)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 5}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(104))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 4, 8, 13, 16, 26, 52, 104, 208]: [1, 2, 3, 5, 5, 8, 13, 24, 46, 88] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7), (19, 9)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(208, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 208:4 1000

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