Atlas › Modular forms › Weight 2 › Level 702 › 702.2.a

Newspace 702.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(702))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
702.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} - q^{7} + O(q^{8})$	+ −	702e1 rank 0
702.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + q^{7} + O(q^{8})$	+ +	702b1 rank 1
702.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ +	702f1 rank 0
702.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{7} + O(q^{8})$	+ −	702h1 rank 1
702.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 5q^{7} + O(q^{8})$	+ +	702a1 rank 1
702.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} - 5q^{7} + O(q^{8})$	+ −	702d1 rank 0
702.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+ −	702c1 rank 0
702.2.a.h	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 4q^{5} + q^{7} + O(q^{8})$	+ +	702g1 rank 0
702.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 4q^{5} + q^{7} + O(q^{8})$	− +	702m1 rank 1
702.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - 5q^{7} + O(q^{8})$	− −	702k1 rank 1
702.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− −	702o1 rank 0
702.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 5q^{7} + O(q^{8})$	− +	702l1 rank 1
702.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{7} + O(q^{8})$	− −	702n1 rank 0
702.2.a.n	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− +	702i1 rank 0
702.2.a.o	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− +	702j1 rank 0
702.2.a.p	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - q^{7} + O(q^{8})$	− −	702p1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = 8751T_{5} + 288T_{7} + 4429T_{11} + 5132T_{17} + 663T_{19} + 393T_{23} + 4651T_{29} + 7220T_{31}$ 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 $T$
702.2.a.a	$x - 56296$
702.2.a.b	$x + 55057$
702.2.a.c	$x - 68942$
702.2.a.d	$x + 22073$
702.2.a.e	$x + 5824$
702.2.a.f	$x - 22291$
702.2.a.g	$x + 102434$
702.2.a.h	$x - 22931$
702.2.a.i	$x + 102511$
702.2.a.j	$x + 120947$
702.2.a.k	$x + 30380$
702.2.a.l	$x - 9428$
702.2.a.m	$x - 96349$
702.2.a.n	$x + 16567$
702.2.a.o	$x - 73784$
702.2.a.p	$x - 17980$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(702))$	16
Old forms	99
Cusp forms $S_{2}(\Gamma_0(702))$	115
Eisenstein series $E_{2}(\Gamma_0(702))$	23
Modular forms $M_{2}(\Gamma_0(702))$	138

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(702)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(117))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(234))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(351))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.408 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, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702]: [0, 1, 1, 3, 2, 1, 5, 5, 4, 6, 11, 18, 16, 46, 44, 130] (certified at ell = 2147483629)
T = [(5, 8751), (7, 288), (11, 4429), (17, 5132), (19, 663), (23, 393), (29, 4651), (31, 7220)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(702, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 702:2 1000

← 701.2.a · 703.2.a →