# Properties

 Label 1088.1.g.a Level $1088$ Weight $1$ Character orbit 1088.g Self dual yes Analytic conductor $0.543$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -68, 17 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,1,Mod(1087,1088)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1088.1087");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1088 = 2^{6} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1088.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.542982733745$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.4352.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{9}+O(q^{10})$$ q - q^9 $$q - q^{9} + 2 q^{13} + q^{17} + q^{25} - q^{49} - 2 q^{53} + q^{81} - 2 q^{89}+O(q^{100})$$ q - q^9 + 2 * q^13 + q^17 + q^25 - q^49 - 2 * q^53 + q^81 - 2 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$511$$ $$513$$ $$\chi(n)$$ $$0$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1087.1
 0
0 0 0 0 0 0 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.1.g.a 1
4.b odd 2 1 CM 1088.1.g.a 1
8.b even 2 1 68.1.d.a 1
8.d odd 2 1 68.1.d.a 1
17.b even 2 1 RM 1088.1.g.a 1
24.f even 2 1 612.1.e.a 1
24.h odd 2 1 612.1.e.a 1
40.e odd 2 1 1700.1.h.d 1
40.f even 2 1 1700.1.h.d 1
40.i odd 4 2 1700.1.d.b 2
40.k even 4 2 1700.1.d.b 2
56.e even 2 1 3332.1.g.a 1
56.h odd 2 1 3332.1.g.a 1
56.j odd 6 2 3332.1.o.d 2
56.k odd 6 2 3332.1.o.c 2
56.m even 6 2 3332.1.o.d 2
56.p even 6 2 3332.1.o.c 2
68.d odd 2 1 CM 1088.1.g.a 1
136.e odd 2 1 68.1.d.a 1
136.h even 2 1 68.1.d.a 1
136.i even 4 2 1156.1.c.a 1
136.j odd 4 2 1156.1.c.a 1
136.o even 8 4 1156.1.f.a 2
136.p odd 8 4 1156.1.f.a 2
136.q odd 16 8 1156.1.g.a 4
136.s even 16 8 1156.1.g.a 4
408.b odd 2 1 612.1.e.a 1
408.h even 2 1 612.1.e.a 1
680.h even 2 1 1700.1.h.d 1
680.k odd 2 1 1700.1.h.d 1
680.u even 4 2 1700.1.d.b 2
680.bi odd 4 2 1700.1.d.b 2
952.e odd 2 1 3332.1.g.a 1
952.k even 2 1 3332.1.g.a 1
952.z even 6 2 3332.1.o.c 2
952.bf even 6 2 3332.1.o.d 2
952.bi odd 6 2 3332.1.o.c 2
952.bl odd 6 2 3332.1.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 8.b even 2 1
68.1.d.a 1 8.d odd 2 1
68.1.d.a 1 136.e odd 2 1
68.1.d.a 1 136.h even 2 1
612.1.e.a 1 24.f even 2 1
612.1.e.a 1 24.h odd 2 1
612.1.e.a 1 408.b odd 2 1
612.1.e.a 1 408.h even 2 1
1088.1.g.a 1 1.a even 1 1 trivial
1088.1.g.a 1 4.b odd 2 1 CM
1088.1.g.a 1 17.b even 2 1 RM
1088.1.g.a 1 68.d odd 2 1 CM
1156.1.c.a 1 136.i even 4 2
1156.1.c.a 1 136.j odd 4 2
1156.1.f.a 2 136.o even 8 4
1156.1.f.a 2 136.p odd 8 4
1156.1.g.a 4 136.q odd 16 8
1156.1.g.a 4 136.s even 16 8
1700.1.d.b 2 40.i odd 4 2
1700.1.d.b 2 40.k even 4 2
1700.1.d.b 2 680.u even 4 2
1700.1.d.b 2 680.bi odd 4 2
1700.1.h.d 1 40.e odd 2 1
1700.1.h.d 1 40.f even 2 1
1700.1.h.d 1 680.h even 2 1
1700.1.h.d 1 680.k odd 2 1
3332.1.g.a 1 56.e even 2 1
3332.1.g.a 1 56.h odd 2 1
3332.1.g.a 1 952.e odd 2 1
3332.1.g.a 1 952.k even 2 1
3332.1.o.c 2 56.k odd 6 2
3332.1.o.c 2 56.p even 6 2
3332.1.o.c 2 952.z even 6 2
3332.1.o.c 2 952.bi odd 6 2
3332.1.o.d 2 56.j odd 6 2
3332.1.o.d 2 56.m even 6 2
3332.1.o.d 2 952.bf even 6 2
3332.1.o.d 2 952.bl odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(1088, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T - 1$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T + 2$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 2$$
$97$ $$T$$