# Properties

 Label 648.1.b.a Level $648$ Weight $1$ Character orbit 648.b Self dual yes Analytic conductor $0.323$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -8 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,1,Mod(163,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("648.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 648.b (of order $$2$$, degree $$1$$, 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: $$0.323394128186$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.1259712.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - q^8 $$q - q^{2} + q^{4} - q^{8} + q^{11} + q^{16} + q^{17} - q^{19} - q^{22} + q^{25} - q^{32} - q^{34} + q^{38} + q^{41} - q^{43} + q^{44} + q^{49} - q^{50} + q^{59} + q^{64} - q^{67} + q^{68} - q^{73} - q^{76} - q^{82} - 2 q^{83} + q^{86} - q^{88} - 2 q^{89} - q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^8 + q^11 + q^16 + q^17 - q^19 - q^22 + q^25 - q^32 - q^34 + q^38 + q^41 - q^43 + q^44 + q^49 - q^50 + q^59 + q^64 - q^67 + q^68 - q^73 - q^76 - q^82 - 2 * q^83 + q^86 - q^88 - 2 * q^89 - q^97 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0
−1.00000 0 1.00000 0 0 0 −1.00000 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.1.b.a 1
3.b odd 2 1 648.1.b.b 1
4.b odd 2 1 2592.1.b.a 1
8.b even 2 1 2592.1.b.a 1
8.d odd 2 1 CM 648.1.b.a 1
9.c even 3 2 216.1.p.a 2
9.d odd 6 2 72.1.p.a 2
12.b even 2 1 2592.1.b.b 1
24.f even 2 1 648.1.b.b 1
24.h odd 2 1 2592.1.b.b 1
36.f odd 6 2 864.1.t.a 2
36.h even 6 2 288.1.t.a 2
45.h odd 6 2 1800.1.bk.d 2
45.l even 12 4 1800.1.ba.b 4
63.i even 6 2 3528.1.ce.b 2
63.j odd 6 2 3528.1.ce.a 2
63.n odd 6 2 3528.1.ba.b 2
63.o even 6 2 3528.1.cg.a 2
63.s even 6 2 3528.1.ba.a 2
72.j odd 6 2 288.1.t.a 2
72.l even 6 2 72.1.p.a 2
72.n even 6 2 864.1.t.a 2
72.p odd 6 2 216.1.p.a 2
144.u even 12 4 2304.1.o.c 4
144.w odd 12 4 2304.1.o.c 4
360.bd even 6 2 1800.1.bk.d 2
360.bt odd 12 4 1800.1.ba.b 4
504.u odd 6 2 3528.1.ba.a 2
504.bt even 6 2 3528.1.ce.a 2
504.cm odd 6 2 3528.1.ce.b 2
504.co odd 6 2 3528.1.cg.a 2
504.cy even 6 2 3528.1.ba.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 9.d odd 6 2
72.1.p.a 2 72.l even 6 2
216.1.p.a 2 9.c even 3 2
216.1.p.a 2 72.p odd 6 2
288.1.t.a 2 36.h even 6 2
288.1.t.a 2 72.j odd 6 2
648.1.b.a 1 1.a even 1 1 trivial
648.1.b.a 1 8.d odd 2 1 CM
648.1.b.b 1 3.b odd 2 1
648.1.b.b 1 24.f even 2 1
864.1.t.a 2 36.f odd 6 2
864.1.t.a 2 72.n even 6 2
1800.1.ba.b 4 45.l even 12 4
1800.1.ba.b 4 360.bt odd 12 4
1800.1.bk.d 2 45.h odd 6 2
1800.1.bk.d 2 360.bd even 6 2
2304.1.o.c 4 144.u even 12 4
2304.1.o.c 4 144.w odd 12 4
2592.1.b.a 1 4.b odd 2 1
2592.1.b.a 1 8.b even 2 1
2592.1.b.b 1 12.b even 2 1
2592.1.b.b 1 24.h odd 2 1
3528.1.ba.a 2 63.s even 6 2
3528.1.ba.a 2 504.u odd 6 2
3528.1.ba.b 2 63.n odd 6 2
3528.1.ba.b 2 504.cy even 6 2
3528.1.ce.a 2 63.j odd 6 2
3528.1.ce.a 2 504.bt even 6 2
3528.1.ce.b 2 63.i even 6 2
3528.1.ce.b 2 504.cm odd 6 2
3528.1.cg.a 2 63.o even 6 2
3528.1.cg.a 2 504.co odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 1$$ acting on $$S_{1}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

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