Atlas › Modular forms › Weight 2 › Level 688 › 688.2.a

Newspace 688.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(688))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
688.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	+	688a1 rank 1
688.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 4q^{5} + O(q^{8})$	−	688c1 rank 1
688.2.a.c	1	$\mathbb{Q}$	$q + 2q^{3} + 4q^{7} + O(q^{8})$	+	688b1 rank 0
688.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{3} + O(q^{8})$	−
688.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - 3q^{5} + O(q^{8})$	−
688.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{5} + 4q^{7} + O(q^{8})$	+
688.2.a.g	2	$\mathbb{Q}(\sqrt{21})$	$2q + q^{3} + 3q^{5} - 4q^{7} + O(q^{8})$	+
688.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	−
688.2.a.i	3	degree 3	$3q - 3q^{3} + q^{5} - 6q^{7} + O(q^{8})$	+
688.2.a.j	5	degree 5	$5q + q^{3} + q^{5} - 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
688.2.a.a	$x + 12$
688.2.a.b	$x + 76$
688.2.a.c	$x - 34$
688.2.a.d	$x^{2} + 72x + 1278$
688.2.a.e	$x^{2} - 18x - 639$
688.2.a.f	$x^{2} - 64x + 1006$
688.2.a.g	$x^{2} - 26x + 85$
688.2.a.h	$x^{2} + 6x + 6$
688.2.a.i	$x^{3} + 10x^{2} - 5759x - 50056$
688.2.a.j	$x^{5} - 52x^{4} - 6137x^{3} + 194734x^{2} + 10274302x - 43530736$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(688))$	21
Old forms	62
Cusp forms $S_{2}(\Gamma_0(688))$	83
Eisenstein series $E_{2}(\Gamma_0(688))$	11
Modular forms $M_{2}(\Gamma_0(688))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(688)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(172))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(344))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.345 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, 16, 43, 86, 172, 344, 688]: [0, 1, 2, 3, 4, 4, 13, 25, 48, 92] (certified at ell = 2147483353)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 21 = 1 x 21 (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(688, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 688:2 1000

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