# Properties

 Label 176.2.i.a Level $176$ Weight $2$ Character orbit 176.i Analytic conductor $1.405$ Analytic rank $0$ Dimension $44$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(43,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("176.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.i (of order $$4$$, degree $$2$$, 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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 4 q^{3} - 4 q^{5}+O(q^{10})$$ 44 * q - 4 * q^3 - 4 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 4 q^{3} - 4 q^{5} - 6 q^{11} + 12 q^{12} - 8 q^{14} - 24 q^{16} + 4 q^{20} - 4 q^{22} - 24 q^{23} - 16 q^{26} + 8 q^{27} - 4 q^{33} - 16 q^{34} + 12 q^{36} - 20 q^{37} - 32 q^{38} + 36 q^{44} + 28 q^{45} - 24 q^{48} - 28 q^{49} + 12 q^{53} - 36 q^{55} + 72 q^{56} - 24 q^{58} - 20 q^{59} - 68 q^{60} + 72 q^{64} - 68 q^{66} + 36 q^{67} - 16 q^{69} - 8 q^{70} - 40 q^{71} + 60 q^{75} + 4 q^{77} + 104 q^{78} + 32 q^{80} - 20 q^{81} + 48 q^{82} + 72 q^{86} - 32 q^{88} + 56 q^{91} - 60 q^{92} + 8 q^{93} - 8 q^{97} - 42 q^{99}+O(q^{100})$$ 44 * q - 4 * q^3 - 4 * q^5 - 6 * q^11 + 12 * q^12 - 8 * q^14 - 24 * q^16 + 4 * q^20 - 4 * q^22 - 24 * q^23 - 16 * q^26 + 8 * q^27 - 4 * q^33 - 16 * q^34 + 12 * q^36 - 20 * q^37 - 32 * q^38 + 36 * q^44 + 28 * q^45 - 24 * q^48 - 28 * q^49 + 12 * q^53 - 36 * q^55 + 72 * q^56 - 24 * q^58 - 20 * q^59 - 68 * q^60 + 72 * q^64 - 68 * q^66 + 36 * q^67 - 16 * q^69 - 8 * q^70 - 40 * q^71 + 60 * q^75 + 4 * q^77 + 104 * q^78 + 32 * q^80 - 20 * q^81 + 48 * q^82 + 72 * q^86 - 32 * q^88 + 56 * q^91 - 60 * q^92 + 8 * q^93 - 8 * q^97 - 42 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.39613 0.225455i 1.15889 + 1.15889i 1.89834 + 0.629528i −2.51141 2.51141i −1.35668 1.87924i 4.37614i −2.50839 1.30689i 0.313934i 2.94003 + 4.07245i
43.2 −1.38786 0.271723i −1.64229 1.64229i 1.85233 + 0.754228i 1.97065 + 1.97065i 1.83303 + 2.72552i 1.30445i −2.36585 1.55009i 2.39422i −2.19952 3.27046i
43.3 −1.36446 + 0.371814i −0.766690 0.766690i 1.72351 1.01465i −0.946416 0.946416i 1.33118 + 0.761052i 0.840189i −1.97440 + 2.02528i 1.82437i 1.64324 + 0.939458i
43.4 −1.13279 0.846636i 1.22947 + 1.22947i 0.566413 + 1.91812i −0.282901 0.282901i −0.351814 2.43365i 4.20092i 0.982323 2.65237i 0.0232059i 0.0809523 + 0.559981i
43.5 −1.04390 + 0.954084i 0.108192 + 0.108192i 0.179446 1.99193i 2.69023 + 2.69023i −0.216165 0.00971706i 3.17010i 1.71315 + 2.25058i 2.97659i −5.37502 0.241618i
43.6 −0.889668 1.09931i −1.82463 1.82463i −0.416982 + 1.95605i −2.57632 2.57632i −0.382527 + 3.62916i 2.30642i 2.52129 1.28184i 3.65855i −0.540116 + 5.12426i
43.7 −0.888482 + 1.10027i 2.21950 + 2.21950i −0.421198 1.95515i −0.858137 0.858137i −4.41404 + 0.470067i 1.49353i 2.52542 + 1.27368i 6.85234i 1.70662 0.181745i
43.8 −0.826559 1.14752i −0.429854 0.429854i −0.633599 + 1.89699i 0.703721 + 0.703721i −0.137966 + 0.848565i 3.48768i 2.70053 0.840904i 2.63045i 0.225866 1.38920i
43.9 −0.670735 + 1.24504i −0.259076 0.259076i −1.10023 1.67018i −1.45525 1.45525i 0.496330 0.148788i 0.413174i 2.81739 0.249580i 2.86576i 2.78792 0.835750i
43.10 −0.378371 + 1.36266i −2.13264 2.13264i −1.71367 1.03118i 0.987102 + 0.987102i 3.71298 2.09913i 3.85199i 2.05355 1.94498i 6.09629i −1.71857 + 0.971591i
43.11 −0.181161 1.40256i 1.33912 + 1.33912i −1.93436 + 0.508180i 1.27873 + 1.27873i 1.63560 2.12080i 0.148220i 1.06319 + 2.62100i 0.586494i 1.56185 2.02516i
43.12 0.181161 + 1.40256i 1.33912 + 1.33912i −1.93436 + 0.508180i 1.27873 + 1.27873i −1.63560 + 2.12080i 0.148220i −1.06319 2.62100i 0.586494i −1.56185 + 2.02516i
43.13 0.378371 1.36266i −2.13264 2.13264i −1.71367 1.03118i 0.987102 + 0.987102i −3.71298 + 2.09913i 3.85199i −2.05355 + 1.94498i 6.09629i 1.71857 0.971591i
43.14 0.670735 1.24504i −0.259076 0.259076i −1.10023 1.67018i −1.45525 1.45525i −0.496330 + 0.148788i 0.413174i −2.81739 + 0.249580i 2.86576i −2.78792 + 0.835750i
43.15 0.826559 + 1.14752i −0.429854 0.429854i −0.633599 + 1.89699i 0.703721 + 0.703721i 0.137966 0.848565i 3.48768i −2.70053 + 0.840904i 2.63045i −0.225866 + 1.38920i
43.16 0.888482 1.10027i 2.21950 + 2.21950i −0.421198 1.95515i −0.858137 0.858137i 4.41404 0.470067i 1.49353i −2.52542 1.27368i 6.85234i −1.70662 + 0.181745i
43.17 0.889668 + 1.09931i −1.82463 1.82463i −0.416982 + 1.95605i −2.57632 2.57632i 0.382527 3.62916i 2.30642i −2.52129 + 1.28184i 3.65855i 0.540116 5.12426i
43.18 1.04390 0.954084i 0.108192 + 0.108192i 0.179446 1.99193i 2.69023 + 2.69023i 0.216165 + 0.00971706i 3.17010i −1.71315 2.25058i 2.97659i 5.37502 + 0.241618i
43.19 1.13279 + 0.846636i 1.22947 + 1.22947i 0.566413 + 1.91812i −0.282901 0.282901i 0.351814 + 2.43365i 4.20092i −0.982323 + 2.65237i 0.0232059i −0.0809523 0.559981i
43.20 1.36446 0.371814i −0.766690 0.766690i 1.72351 1.01465i −0.946416 0.946416i −1.33118 0.761052i 0.840189i 1.97440 2.02528i 1.82437i −1.64324 0.939458i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
16.f odd 4 1 inner
176.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.i.a 44
4.b odd 2 1 704.2.i.a 44
8.b even 2 1 1408.2.i.b 44
8.d odd 2 1 1408.2.i.a 44
11.b odd 2 1 inner 176.2.i.a 44
16.e even 4 1 704.2.i.a 44
16.e even 4 1 1408.2.i.a 44
16.f odd 4 1 inner 176.2.i.a 44
16.f odd 4 1 1408.2.i.b 44
44.c even 2 1 704.2.i.a 44
88.b odd 2 1 1408.2.i.b 44
88.g even 2 1 1408.2.i.a 44
176.i even 4 1 inner 176.2.i.a 44
176.i even 4 1 1408.2.i.b 44
176.l odd 4 1 704.2.i.a 44
176.l odd 4 1 1408.2.i.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.i.a 44 1.a even 1 1 trivial
176.2.i.a 44 11.b odd 2 1 inner
176.2.i.a 44 16.f odd 4 1 inner
176.2.i.a 44 176.i even 4 1 inner
704.2.i.a 44 4.b odd 2 1
704.2.i.a 44 16.e even 4 1
704.2.i.a 44 44.c even 2 1
704.2.i.a 44 176.l odd 4 1
1408.2.i.a 44 8.d odd 2 1
1408.2.i.a 44 16.e even 4 1
1408.2.i.a 44 88.g even 2 1
1408.2.i.a 44 176.l odd 4 1
1408.2.i.b 44 8.b even 2 1
1408.2.i.b 44 16.f odd 4 1
1408.2.i.b 44 88.b odd 2 1
1408.2.i.b 44 176.i even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(176, [\chi])$$.