# Properties

 Label 1088.1.p.a Level $1088$ Weight $1$ Character orbit 1088.p Analytic conductor $0.543$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,1,Mod(191,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1088.191");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1088 = 2^{6} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1088.p (of order $$4$$, degree $$2$$, 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: no Analytic conductor: $$0.542982733745$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.19652.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.321978368.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{5} - i q^{9} +O(q^{10})$$ q + (-z + 1) * q^5 - z * q^9 $$q + ( - i + 1) q^{5} - i q^{9} - q^{17} - i q^{25} + (i - 1) q^{29} + ( - i + 1) q^{37} + (i + 1) q^{41} + ( - i - 1) q^{45} + i q^{49} + (i + 1) q^{61} + ( - i + 1) q^{73} - q^{81} + (i - 1) q^{85} + (i - 1) q^{97} +O(q^{100})$$ q + (-z + 1) * q^5 - z * q^9 - q^17 - z * q^25 + (z - 1) * q^29 + (-z + 1) * q^37 + (z + 1) * q^41 + (-z - 1) * q^45 + z * q^49 + (z + 1) * q^61 + (-z + 1) * q^73 - q^81 + (z - 1) * q^85 + (z - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} - 2 q^{17} - 2 q^{29} + 2 q^{37} + 2 q^{41} - 2 q^{45} + 2 q^{61} + 2 q^{73} - 2 q^{81} - 2 q^{85} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^17 - 2 * q^29 + 2 * q^37 + 2 * q^41 - 2 * q^45 + 2 * q^61 + 2 * q^73 - 2 * q^81 - 2 * q^85 - 2 * q^97

## Character values

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

 $$n$$ $$69$$ $$511$$ $$513$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 1.00000i − 1.00000i
0 0 0 1.00000 1.00000i 0 0 0 1.00000i 0
319.1 0 0 0 1.00000 + 1.00000i 0 0 0 1.00000i 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.c even 4 1 inner
68.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.1.p.a 2
4.b odd 2 1 CM 1088.1.p.a 2
8.b even 2 1 68.1.f.a 2
8.d odd 2 1 68.1.f.a 2
17.c even 4 1 inner 1088.1.p.a 2
24.f even 2 1 612.1.l.a 2
24.h odd 2 1 612.1.l.a 2
40.e odd 2 1 1700.1.p.a 2
40.f even 2 1 1700.1.p.a 2
40.i odd 4 1 1700.1.n.a 2
40.i odd 4 1 1700.1.n.b 2
40.k even 4 1 1700.1.n.a 2
40.k even 4 1 1700.1.n.b 2
56.e even 2 1 3332.1.m.b 2
56.h odd 2 1 3332.1.m.b 2
56.j odd 6 2 3332.1.bc.b 4
56.k odd 6 2 3332.1.bc.c 4
56.m even 6 2 3332.1.bc.b 4
56.p even 6 2 3332.1.bc.c 4
68.f odd 4 1 inner 1088.1.p.a 2
136.e odd 2 1 1156.1.f.b 2
136.h even 2 1 1156.1.f.b 2
136.i even 4 1 68.1.f.a 2
136.i even 4 1 1156.1.f.b 2
136.j odd 4 1 68.1.f.a 2
136.j odd 4 1 1156.1.f.b 2
136.o even 8 2 1156.1.c.b 2
136.o even 8 2 1156.1.d.a 2
136.p odd 8 2 1156.1.c.b 2
136.p odd 8 2 1156.1.d.a 2
136.q odd 16 8 1156.1.g.b 8
136.s even 16 8 1156.1.g.b 8
408.q even 4 1 612.1.l.a 2
408.t odd 4 1 612.1.l.a 2
680.s odd 4 1 1700.1.n.a 2
680.t even 4 1 1700.1.n.b 2
680.bc odd 4 1 1700.1.p.a 2
680.be even 4 1 1700.1.p.a 2
680.bk odd 4 1 1700.1.n.b 2
680.bl even 4 1 1700.1.n.a 2
952.v odd 4 1 3332.1.m.b 2
952.x even 4 1 3332.1.m.b 2
952.bw even 12 2 3332.1.bc.c 4
952.by odd 12 2 3332.1.bc.c 4
952.cb odd 12 2 3332.1.bc.b 4
952.cd even 12 2 3332.1.bc.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 8.b even 2 1
68.1.f.a 2 8.d odd 2 1
68.1.f.a 2 136.i even 4 1
68.1.f.a 2 136.j odd 4 1
612.1.l.a 2 24.f even 2 1
612.1.l.a 2 24.h odd 2 1
612.1.l.a 2 408.q even 4 1
612.1.l.a 2 408.t odd 4 1
1088.1.p.a 2 1.a even 1 1 trivial
1088.1.p.a 2 4.b odd 2 1 CM
1088.1.p.a 2 17.c even 4 1 inner
1088.1.p.a 2 68.f odd 4 1 inner
1156.1.c.b 2 136.o even 8 2
1156.1.c.b 2 136.p odd 8 2
1156.1.d.a 2 136.o even 8 2
1156.1.d.a 2 136.p odd 8 2
1156.1.f.b 2 136.e odd 2 1
1156.1.f.b 2 136.h even 2 1
1156.1.f.b 2 136.i even 4 1
1156.1.f.b 2 136.j odd 4 1
1156.1.g.b 8 136.q odd 16 8
1156.1.g.b 8 136.s even 16 8
1700.1.n.a 2 40.i odd 4 1
1700.1.n.a 2 40.k even 4 1
1700.1.n.a 2 680.s odd 4 1
1700.1.n.a 2 680.bl even 4 1
1700.1.n.b 2 40.i odd 4 1
1700.1.n.b 2 40.k even 4 1
1700.1.n.b 2 680.t even 4 1
1700.1.n.b 2 680.bk odd 4 1
1700.1.p.a 2 40.e odd 2 1
1700.1.p.a 2 40.f even 2 1
1700.1.p.a 2 680.bc odd 4 1
1700.1.p.a 2 680.be even 4 1
3332.1.m.b 2 56.e even 2 1
3332.1.m.b 2 56.h odd 2 1
3332.1.m.b 2 952.v odd 4 1
3332.1.m.b 2 952.x even 4 1
3332.1.bc.b 4 56.j odd 6 2
3332.1.bc.b 4 56.m even 6 2
3332.1.bc.b 4 952.cb odd 12 2
3332.1.bc.b 4 952.cd even 12 2
3332.1.bc.c 4 56.k odd 6 2
3332.1.bc.c 4 56.p even 6 2
3332.1.bc.c 4 952.bw even 12 2
3332.1.bc.c 4 952.by odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1088, [\chi])$$.

## Hecke characteristic polynomials

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