Atlas › Modular forms › Weight 2 › Level 208 › 208.2.a

Newspace 208.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(208))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
208.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + q^{7} + O(q^{8})$	−	208a1 rank 1
208.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - 5q^{7} + O(q^{8})$	+	208b1 rank 1
208.2.a.c	1	$\mathbb{Q}$	$q + 2q^{5} + 2q^{7} + O(q^{8})$	+	208c1 rank 0
208.2.a.d	1	$\mathbb{Q}$	$q + 3q^{3} - q^{5} - q^{7} + O(q^{8})$	+	208d1 rank 0
208.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} + 3q^{5} + q^{7} + O(q^{8})$	−

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 $T$
208.2.a.a	$x + 79$
208.2.a.b	$x - 3$
208.2.a.c	$x - 112$
208.2.a.d	$x + 65$
208.2.a.e	$x^{2} + 7x - 332$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(208))$	6
Old forms	17
Cusp forms $S_{2}(\Gamma_0(208))$	23
Eisenstein series $E_{2}(\Gamma_0(208))$	11
Modular forms $M_{2}(\Gamma_0(208))$	34

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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]: [0, 1, 2, 3, 1, 4, 5, 10, 18, 32] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7), (19, 9)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 208:2 1000

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