Atlas › Modular forms › Weight 2 › Level 693 › 693.2.a

Newspace 693.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(693))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
693.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} - q^{7} + O(q^{8})$	+ +	693a1 rank 0
693.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} - 3q^{5} + q^{7} + O(q^{8})$	− −	693c1 rank 0
693.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - q^{7} + O(q^{8})$	+ −	693b1 rank 1
693.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− −	693d1 rank 0
693.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	− −
693.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	− +
693.2.a.g	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +
693.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{4} + 4q^{5} + 2q^{7} + O(q^{8})$	− −
693.2.a.i	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 3q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	+ −
693.2.a.j	2	$\mathbb{Q}(\sqrt{21})$	$2q + q^{2} + 7q^{4} - 6q^{5} + 2q^{7} + O(q^{8})$	− −
693.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− +
693.2.a.l	3	degree 3	$3q - 2q^{2} + 6q^{4} - 4q^{5} - 3q^{7} + O(q^{8})$	+ −
693.2.a.m	3	degree 3	$3q + 6q^{4} - 3q^{7} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{13} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
693.2.a.a	$x - 315$
693.2.a.b	$x - 190$
693.2.a.c	$x + 298$
693.2.a.d	$x - 135$
693.2.a.e	$x^{2} + 237x + 7381$
693.2.a.f	$x^{2} + 283x + 18821$
693.2.a.g	$x^{2} + 147x + 5139$
693.2.a.h	$x^{2} - 90x + 1945$
693.2.a.i	$x^{2} - 183x - 59959$
693.2.a.j	$x^{2} + 205x - 71525$
693.2.a.k	$x^{2} - 89x + 1919$
693.2.a.l	$x^{3} + 422x^{2} + 58412x + 2660447$
693.2.a.m	$x^{3} - 240x^{2} + 12912x - 140835$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(693))$	24
Old forms	65
Cusp forms $S_{2}(\Gamma_0(693))$	89
Eisenstein series $E_{2}(\Gamma_0(693))$	15
Modular forms $M_{2}(\Gamma_0(693))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(693)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(99))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(231))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.922 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, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693]: [0, 1, 1, 2, 2, 4, 6, 10, 10, 14, 36, 100] (certified at ell = 2147482951)
T = [(2, 1), (5, 2), (17, 3), (19, 4), (23, 5), (13, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 8 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(693, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 693:2 1000

← 692.2.a · 694.2.a →