# Properties

 Label 165.3.h.a Level $165$ Weight $3$ Character orbit 165.h Analytic conductor $4.496$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,3,Mod(109,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("165.109");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 165.h (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: no Analytic conductor: $$4.49592436194$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 48 q^{4} + 4 q^{5} - 72 q^{9}+O(q^{10})$$ 24 * q + 48 * q^4 + 4 * q^5 - 72 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 48 q^{4} + 4 q^{5} - 72 q^{9} - 28 q^{11} - 24 q^{14} - 12 q^{15} + 152 q^{16} + 12 q^{20} - 80 q^{26} - 96 q^{31} + 104 q^{34} - 144 q^{36} - 92 q^{44} - 12 q^{45} - 264 q^{49} + 64 q^{55} + 280 q^{56} + 296 q^{59} - 12 q^{60} - 112 q^{64} - 36 q^{66} + 192 q^{69} - 208 q^{70} + 608 q^{71} + 312 q^{75} + 4 q^{80} + 216 q^{81} - 800 q^{86} - 272 q^{89} - 736 q^{91} + 84 q^{99}+O(q^{100})$$ 24 * q + 48 * q^4 + 4 * q^5 - 72 * q^9 - 28 * q^11 - 24 * q^14 - 12 * q^15 + 152 * q^16 + 12 * q^20 - 80 * q^26 - 96 * q^31 + 104 * q^34 - 144 * q^36 - 92 * q^44 - 12 * q^45 - 264 * q^49 + 64 * q^55 + 280 * q^56 + 296 * q^59 - 12 * q^60 - 112 * q^64 - 36 * q^66 + 192 * q^69 - 208 * q^70 + 608 * q^71 + 312 * q^75 + 4 * q^80 + 216 * q^81 - 800 * q^86 - 272 * q^89 - 736 * q^91 + 84 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −3.63542 1.73205i 9.21627 2.02329 + 4.57234i 6.29673i 4.25629 −18.9633 −3.00000 −7.35550 16.6224i
109.2 −3.63542 1.73205i 9.21627 2.02329 4.57234i 6.29673i 4.25629 −18.9633 −3.00000 −7.35550 + 16.6224i
109.3 −3.46254 1.73205i 7.98919 −3.32476 3.73443i 5.99730i −5.79297 −13.8127 −3.00000 11.5121 + 12.9306i
109.4 −3.46254 1.73205i 7.98919 −3.32476 + 3.73443i 5.99730i −5.79297 −13.8127 −3.00000 11.5121 12.9306i
109.5 −2.63268 1.73205i 2.93098 4.42289 2.33196i 4.55993i −0.961295 2.81437 −3.00000 −11.6440 + 6.13929i
109.6 −2.63268 1.73205i 2.93098 4.42289 + 2.33196i 4.55993i −0.961295 2.81437 −3.00000 −11.6440 6.13929i
109.7 −1.49337 1.73205i −1.76983 −1.35352 4.81331i 2.58660i 8.34829 8.61652 −3.00000 2.02132 + 7.18807i
109.8 −1.49337 1.73205i −1.76983 −1.35352 + 4.81331i 2.58660i 8.34829 8.61652 −3.00000 2.02132 7.18807i
109.9 −1.26695 1.73205i −2.39485 −4.75221 + 1.55451i 2.19442i 1.85326 8.10193 −3.00000 6.02080 1.96948i
109.10 −1.26695 1.73205i −2.39485 −4.75221 1.55451i 2.19442i 1.85326 8.10193 −3.00000 6.02080 + 1.96948i
109.11 −0.168030 1.73205i −3.97177 3.98431 + 3.02080i 0.291037i −10.1130 1.33950 −3.00000 −0.669486 0.507586i
109.12 −0.168030 1.73205i −3.97177 3.98431 3.02080i 0.291037i −10.1130 1.33950 −3.00000 −0.669486 + 0.507586i
109.13 0.168030 1.73205i −3.97177 3.98431 + 3.02080i 0.291037i 10.1130 −1.33950 −3.00000 0.669486 + 0.507586i
109.14 0.168030 1.73205i −3.97177 3.98431 3.02080i 0.291037i 10.1130 −1.33950 −3.00000 0.669486 0.507586i
109.15 1.26695 1.73205i −2.39485 −4.75221 + 1.55451i 2.19442i −1.85326 −8.10193 −3.00000 −6.02080 + 1.96948i
109.16 1.26695 1.73205i −2.39485 −4.75221 1.55451i 2.19442i −1.85326 −8.10193 −3.00000 −6.02080 1.96948i
109.17 1.49337 1.73205i −1.76983 −1.35352 4.81331i 2.58660i −8.34829 −8.61652 −3.00000 −2.02132 7.18807i
109.18 1.49337 1.73205i −1.76983 −1.35352 + 4.81331i 2.58660i −8.34829 −8.61652 −3.00000 −2.02132 + 7.18807i
109.19 2.63268 1.73205i 2.93098 4.42289 2.33196i 4.55993i 0.961295 −2.81437 −3.00000 11.6440 6.13929i
109.20 2.63268 1.73205i 2.93098 4.42289 + 2.33196i 4.55993i 0.961295 −2.81437 −3.00000 11.6440 + 6.13929i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.h.a 24
3.b odd 2 1 495.3.h.h 24
5.b even 2 1 inner 165.3.h.a 24
5.c odd 4 2 825.3.b.e 24
11.b odd 2 1 inner 165.3.h.a 24
15.d odd 2 1 495.3.h.h 24
33.d even 2 1 495.3.h.h 24
55.d odd 2 1 inner 165.3.h.a 24
55.e even 4 2 825.3.b.e 24
165.d even 2 1 495.3.h.h 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.h.a 24 1.a even 1 1 trivial
165.3.h.a 24 5.b even 2 1 inner
165.3.h.a 24 11.b odd 2 1 inner
165.3.h.a 24 55.d odd 2 1 inner
495.3.h.h 24 3.b odd 2 1
495.3.h.h 24 15.d odd 2 1
495.3.h.h 24 33.d even 2 1
495.3.h.h 24 165.d even 2 1
825.3.b.e 24 5.c odd 4 2
825.3.b.e 24 55.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(165, [\chi])$$.