# Properties

 Label 588.1.p.a Level $588$ Weight $1$ Character orbit 588.p Analytic conductor $0.293$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,1,Mod(557,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("588.557");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 588.p (of order $$6$$, 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: $$0.293450227428$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.588.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.1075850221824.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^3 - z * q^9 $$q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} + q^{13} - \zeta_{6} q^{19} + \zeta_{6}^{2} q^{25} - q^{27} + \zeta_{6}^{2} q^{31} + \zeta_{6} q^{37} - \zeta_{6}^{2} q^{39} - q^{43} - q^{57} + \zeta_{6} q^{61} - \zeta_{6}^{2} q^{67} + \zeta_{6}^{2} q^{73} + \zeta_{6} q^{75} + \zeta_{6} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{93} - q^{97} +O(q^{100})$$ q - z^2 * q^3 - z * q^9 + q^13 - z * q^19 + z^2 * q^25 - q^27 + z^2 * q^31 + z * q^37 - z^2 * q^39 - q^43 - q^57 + z * q^61 - z^2 * q^67 + z^2 * q^73 + z * q^75 + z * q^79 + z^2 * q^81 + z * q^93 - q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{9}+O(q^{10})$$ 2 * q + q^3 - q^9 $$2 q + q^{3} - q^{9} + 2 q^{13} - q^{19} - q^{25} - 2 q^{27} - q^{31} + q^{37} + q^{39} - 2 q^{43} - 2 q^{57} + 2 q^{61} + q^{67} - q^{73} + q^{75} + q^{79} - q^{81} + q^{93} - 4 q^{97}+O(q^{100})$$ 2 * q + q^3 - q^9 + 2 * q^13 - q^19 - q^25 - 2 * q^27 - q^31 + q^37 + q^39 - 2 * q^43 - 2 * q^57 + 2 * q^61 + q^67 - q^73 + q^75 + q^79 - q^81 + q^93 - 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
569.1 0 0.500000 + 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.1.p.a 2
3.b odd 2 1 CM 588.1.p.a 2
4.b odd 2 1 2352.1.bn.a 2
7.b odd 2 1 84.1.p.a 2
7.c even 3 1 588.1.c.a 1
7.c even 3 1 inner 588.1.p.a 2
7.d odd 6 1 84.1.p.a 2
7.d odd 6 1 588.1.c.b 1
12.b even 2 1 2352.1.bn.a 2
21.c even 2 1 84.1.p.a 2
21.g even 6 1 84.1.p.a 2
21.g even 6 1 588.1.c.b 1
21.h odd 6 1 588.1.c.a 1
21.h odd 6 1 inner 588.1.p.a 2
28.d even 2 1 336.1.bn.a 2
28.f even 6 1 336.1.bn.a 2
28.f even 6 1 2352.1.d.a 1
28.g odd 6 1 2352.1.d.b 1
28.g odd 6 1 2352.1.bn.a 2
35.c odd 2 1 2100.1.bn.c 2
35.f even 4 2 2100.1.bh.a 4
35.i odd 6 1 2100.1.bn.c 2
35.k even 12 2 2100.1.bh.a 4
56.e even 2 1 1344.1.bn.a 2
56.h odd 2 1 1344.1.bn.b 2
56.j odd 6 1 1344.1.bn.b 2
56.m even 6 1 1344.1.bn.a 2
63.i even 6 1 2268.1.m.a 2
63.k odd 6 1 2268.1.bh.b 2
63.l odd 6 1 2268.1.m.a 2
63.l odd 6 1 2268.1.bh.b 2
63.o even 6 1 2268.1.m.a 2
63.o even 6 1 2268.1.bh.b 2
63.s even 6 1 2268.1.bh.b 2
63.t odd 6 1 2268.1.m.a 2
84.h odd 2 1 336.1.bn.a 2
84.j odd 6 1 336.1.bn.a 2
84.j odd 6 1 2352.1.d.a 1
84.n even 6 1 2352.1.d.b 1
84.n even 6 1 2352.1.bn.a 2
105.g even 2 1 2100.1.bn.c 2
105.k odd 4 2 2100.1.bh.a 4
105.p even 6 1 2100.1.bn.c 2
105.w odd 12 2 2100.1.bh.a 4
168.e odd 2 1 1344.1.bn.a 2
168.i even 2 1 1344.1.bn.b 2
168.ba even 6 1 1344.1.bn.b 2
168.be odd 6 1 1344.1.bn.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 7.b odd 2 1
84.1.p.a 2 7.d odd 6 1
84.1.p.a 2 21.c even 2 1
84.1.p.a 2 21.g even 6 1
336.1.bn.a 2 28.d even 2 1
336.1.bn.a 2 28.f even 6 1
336.1.bn.a 2 84.h odd 2 1
336.1.bn.a 2 84.j odd 6 1
588.1.c.a 1 7.c even 3 1
588.1.c.a 1 21.h odd 6 1
588.1.c.b 1 7.d odd 6 1
588.1.c.b 1 21.g even 6 1
588.1.p.a 2 1.a even 1 1 trivial
588.1.p.a 2 3.b odd 2 1 CM
588.1.p.a 2 7.c even 3 1 inner
588.1.p.a 2 21.h odd 6 1 inner
1344.1.bn.a 2 56.e even 2 1
1344.1.bn.a 2 56.m even 6 1
1344.1.bn.a 2 168.e odd 2 1
1344.1.bn.a 2 168.be odd 6 1
1344.1.bn.b 2 56.h odd 2 1
1344.1.bn.b 2 56.j odd 6 1
1344.1.bn.b 2 168.i even 2 1
1344.1.bn.b 2 168.ba even 6 1
2100.1.bh.a 4 35.f even 4 2
2100.1.bh.a 4 35.k even 12 2
2100.1.bh.a 4 105.k odd 4 2
2100.1.bh.a 4 105.w odd 12 2
2100.1.bn.c 2 35.c odd 2 1
2100.1.bn.c 2 35.i odd 6 1
2100.1.bn.c 2 105.g even 2 1
2100.1.bn.c 2 105.p even 6 1
2268.1.m.a 2 63.i even 6 1
2268.1.m.a 2 63.l odd 6 1
2268.1.m.a 2 63.o even 6 1
2268.1.m.a 2 63.t odd 6 1
2268.1.bh.b 2 63.k odd 6 1
2268.1.bh.b 2 63.l odd 6 1
2268.1.bh.b 2 63.o even 6 1
2268.1.bh.b 2 63.s even 6 1
2352.1.d.a 1 28.f even 6 1
2352.1.d.a 1 84.j odd 6 1
2352.1.d.b 1 28.g odd 6 1
2352.1.d.b 1 84.n even 6 1
2352.1.bn.a 2 4.b odd 2 1
2352.1.bn.a 2 12.b even 2 1
2352.1.bn.a 2 28.g odd 6 1
2352.1.bn.a 2 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + T + 1$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - T + 1$$
$71$ $$T^{2}$$
$73$ $$T^{2} + T + 1$$
$79$ $$T^{2} - T + 1$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 2)^{2}$$