Atlas › Modular forms › Weight 2 › Level 689 › 689.2.a

Newspace 689.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(689))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
689.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	689a1 rank 1
689.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + q^{3} - 2q^{4} + q^{5} - q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
689.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} + 6q^{4} - q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
689.2.a.d	3	degree 3	$3q + 3q^{2} - q^{3} - 3q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
689.2.a.e	9	degree 9	$9q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} + q^{6} - 13q^{7} + O(q^{8})$	− −	$-1$
689.2.a.f	18	degree 18	$18q + 2q^{2} + q^{3} + 26q^{4} + q^{5} + 5q^{6} + 19q^{7} + O(q^{8})$	− +	$+1$
689.2.a.g	18	degree 18	$18q + 2q^{2} + 5q^{3} + 30q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
689.2.a.a	$x + 7$
689.2.a.b	$x^{2} - x - 11$
689.2.a.c	$x^{2} + 3x + 1$
689.2.a.d	$x^{3} - 48x + 101$
689.2.a.e	$x^{9} + 12x^{8} - 72x^{7} - 1043x^{6} + 1584x^{5} + 28218x^{4} - 10953x^{3} - 235446x^{2} + 3660x + 302525$
689.2.a.f	$x^{18} - 5x^{17} - 362x^{16} + 1681x^{15} + 50080x^{14} - 196399x^{13} - 3576555x^{12} + 11028396x^{11} + 145234150x^{10} - 336976478x^{9} - 3441005218x^{8} + 5943203460x^{7} + 46765949293x^{6} - 61393728369x^{5} - 337180309360x^{4} + 356793961763x^{3} + 1018606803678x^{2} - 948764433015x - 261538730199$
689.2.a.g	$x^{18} - 17x^{17} - 250x^{16} + 5391x^{15} + 19798x^{14} - 692301x^{13} - 141841x^{12} + 46847932x^{11} - 61375822x^{10} - 1826810696x^{9} + 3581022684x^{8} + 42342540288x^{7} - 87928170131x^{6} - 577299076167x^{5} + 1030196372610x^{4} + 4278007755245x^{3} - 5078192176312x^{2} - 13041963871671x + 6287148002125$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(689))$	53
Old forms	8
Cusp forms $S_{2}(\Gamma_0(689))$	61
Eisenstein series $E_{2}(\Gamma_0(689))$	3
Modular forms $M_{2}(\Gamma_0(689))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(689)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.237 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, 13, 53, 689]: [0, 1, 5, 64] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 53 = 1 x 53 (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(689, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 689:2 1000

← 688.2.a · 690.2.a →