# Properties

 Label 189.2.a.a Level $189$ Weight $2$ Character orbit 189.a Self dual yes Analytic conductor $1.509$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("189.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.a (trivial)

## 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: $$1.50917259820$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} - q^{5} - q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 - q^5 - q^7 $$q - 2 q^{2} + 2 q^{4} - q^{5} - q^{7} + 2 q^{10} - 4 q^{11} - 2 q^{13} + 2 q^{14} - 4 q^{16} + 3 q^{17} - 8 q^{19} - 2 q^{20} + 8 q^{22} - 6 q^{23} - 4 q^{25} + 4 q^{26} - 2 q^{28} - 4 q^{29} + 6 q^{31} + 8 q^{32} - 6 q^{34} + q^{35} - 3 q^{37} + 16 q^{38} + q^{41} + 11 q^{43} - 8 q^{44} + 12 q^{46} + 9 q^{47} + q^{49} + 8 q^{50} - 4 q^{52} + 6 q^{53} + 4 q^{55} + 8 q^{58} - 15 q^{59} + 4 q^{61} - 12 q^{62} - 8 q^{64} + 2 q^{65} - 8 q^{67} + 6 q^{68} - 2 q^{70} - 12 q^{71} + 6 q^{73} + 6 q^{74} - 16 q^{76} + 4 q^{77} - q^{79} + 4 q^{80} - 2 q^{82} - 9 q^{83} - 3 q^{85} - 22 q^{86} + 2 q^{89} + 2 q^{91} - 12 q^{92} - 18 q^{94} + 8 q^{95} + 12 q^{97} - 2 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 - q^5 - q^7 + 2 * q^10 - 4 * q^11 - 2 * q^13 + 2 * q^14 - 4 * q^16 + 3 * q^17 - 8 * q^19 - 2 * q^20 + 8 * q^22 - 6 * q^23 - 4 * q^25 + 4 * q^26 - 2 * q^28 - 4 * q^29 + 6 * q^31 + 8 * q^32 - 6 * q^34 + q^35 - 3 * q^37 + 16 * q^38 + q^41 + 11 * q^43 - 8 * q^44 + 12 * q^46 + 9 * q^47 + q^49 + 8 * q^50 - 4 * q^52 + 6 * q^53 + 4 * q^55 + 8 * q^58 - 15 * q^59 + 4 * q^61 - 12 * q^62 - 8 * q^64 + 2 * q^65 - 8 * q^67 + 6 * q^68 - 2 * q^70 - 12 * q^71 + 6 * q^73 + 6 * q^74 - 16 * q^76 + 4 * q^77 - q^79 + 4 * q^80 - 2 * q^82 - 9 * q^83 - 3 * q^85 - 22 * q^86 + 2 * q^89 + 2 * q^91 - 12 * q^92 - 18 * q^94 + 8 * q^95 + 12 * q^97 - 2 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 0 2.00000 −1.00000 0 −1.00000 0 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.a.a 1
3.b odd 2 1 189.2.a.d yes 1
4.b odd 2 1 3024.2.a.l 1
5.b even 2 1 4725.2.a.s 1
7.b odd 2 1 1323.2.a.b 1
9.c even 3 2 567.2.f.h 2
9.d odd 6 2 567.2.f.a 2
12.b even 2 1 3024.2.a.u 1
15.d odd 2 1 4725.2.a.c 1
21.c even 2 1 1323.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.a 1 1.a even 1 1 trivial
189.2.a.d yes 1 3.b odd 2 1
567.2.f.a 2 9.d odd 6 2
567.2.f.h 2 9.c even 3 2
1323.2.a.b 1 7.b odd 2 1
1323.2.a.r 1 21.c even 2 1
3024.2.a.l 1 4.b odd 2 1
3024.2.a.u 1 12.b even 2 1
4725.2.a.c 1 15.d odd 2 1
4725.2.a.s 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(189))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{5} + 1$$ T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T - 3$$
$19$ $$T + 8$$
$23$ $$T + 6$$
$29$ $$T + 4$$
$31$ $$T - 6$$
$37$ $$T + 3$$
$41$ $$T - 1$$
$43$ $$T - 11$$
$47$ $$T - 9$$
$53$ $$T - 6$$
$59$ $$T + 15$$
$61$ $$T - 4$$
$67$ $$T + 8$$
$71$ $$T + 12$$
$73$ $$T - 6$$
$79$ $$T + 1$$
$83$ $$T + 9$$
$89$ $$T - 2$$
$97$ $$T - 12$$