# Properties

 Label 176.2.x.a Level $176$ Weight $2$ Character orbit 176.x Analytic conductor $1.405$ Analytic rank $0$ Dimension $176$ 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(19,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 15, 6]))

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

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

chi := DirichletCharacter("176.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.x (of order $$20$$, degree $$8$$, 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: $$176$$ Relative dimension: $$22$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$176 q - 10 q^{2} - 6 q^{3} - 10 q^{4} - 6 q^{5} - 10 q^{6} - 20 q^{7} - 10 q^{8}+O(q^{10})$$ 176 * q - 10 * q^2 - 6 * q^3 - 10 * q^4 - 6 * q^5 - 10 * q^6 - 20 * q^7 - 10 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$176 q - 10 q^{2} - 6 q^{3} - 10 q^{4} - 6 q^{5} - 10 q^{6} - 20 q^{7} - 10 q^{8} - 4 q^{11} - 32 q^{12} - 10 q^{13} - 12 q^{14} + 14 q^{16} - 20 q^{17} + 40 q^{18} - 10 q^{19} - 14 q^{20} - 66 q^{22} - 16 q^{23} - 10 q^{24} + 56 q^{26} - 18 q^{27} - 10 q^{28} - 10 q^{29} - 50 q^{30} - 16 q^{33} - 4 q^{34} - 10 q^{35} - 22 q^{36} - 30 q^{37} + 22 q^{38} - 20 q^{39} - 10 q^{40} - 10 q^{42} - 46 q^{44} - 48 q^{45} - 30 q^{46} + 14 q^{48} + 8 q^{49} - 10 q^{50} - 10 q^{51} - 50 q^{52} + 18 q^{53} + 16 q^{55} + 8 q^{56} + 14 q^{58} + 10 q^{59} - 82 q^{60} - 10 q^{61} - 10 q^{62} + 38 q^{64} + 58 q^{66} - 56 q^{67} - 140 q^{68} - 24 q^{69} + 68 q^{70} + 20 q^{71} + 190 q^{72} + 40 q^{74} + 10 q^{75} - 14 q^{77} - 4 q^{78} + 148 q^{80} + 42 q^{82} - 110 q^{83} + 40 q^{84} - 10 q^{85} - 2 q^{86} + 182 q^{88} + 60 q^{90} + 54 q^{91} + 90 q^{92} - 18 q^{93} + 180 q^{94} + 90 q^{96} - 12 q^{97} - 68 q^{99}+O(q^{100})$$ 176 * q - 10 * q^2 - 6 * q^3 - 10 * q^4 - 6 * q^5 - 10 * q^6 - 20 * q^7 - 10 * q^8 - 4 * q^11 - 32 * q^12 - 10 * q^13 - 12 * q^14 + 14 * q^16 - 20 * q^17 + 40 * q^18 - 10 * q^19 - 14 * q^20 - 66 * q^22 - 16 * q^23 - 10 * q^24 + 56 * q^26 - 18 * q^27 - 10 * q^28 - 10 * q^29 - 50 * q^30 - 16 * q^33 - 4 * q^34 - 10 * q^35 - 22 * q^36 - 30 * q^37 + 22 * q^38 - 20 * q^39 - 10 * q^40 - 10 * q^42 - 46 * q^44 - 48 * q^45 - 30 * q^46 + 14 * q^48 + 8 * q^49 - 10 * q^50 - 10 * q^51 - 50 * q^52 + 18 * q^53 + 16 * q^55 + 8 * q^56 + 14 * q^58 + 10 * q^59 - 82 * q^60 - 10 * q^61 - 10 * q^62 + 38 * q^64 + 58 * q^66 - 56 * q^67 - 140 * q^68 - 24 * q^69 + 68 * q^70 + 20 * q^71 + 190 * q^72 + 40 * q^74 + 10 * q^75 - 14 * q^77 - 4 * q^78 + 148 * q^80 + 42 * q^82 - 110 * q^83 + 40 * q^84 - 10 * q^85 - 2 * q^86 + 182 * q^88 + 60 * q^90 + 54 * q^91 + 90 * q^92 - 18 * q^93 + 180 * q^94 + 90 * q^96 - 12 * q^97 - 68 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.41139 + 0.0892747i −1.05924 0.539708i 1.98406 0.252003i −1.33741 + 0.211825i 1.54318 + 0.667177i 2.96719 + 0.964100i −2.77779 + 0.532802i −0.932658 1.28369i 1.86870 0.418365i
19.2 −1.40401 0.169535i 1.69449 + 0.863383i 1.94252 + 0.476059i 2.58727 0.409783i −2.23271 1.49948i 0.508988 + 0.165380i −2.64661 0.997719i 0.362493 + 0.498929i −3.70204 + 0.136709i
19.3 −1.35193 0.415085i −2.19704 1.11945i 1.65541 + 1.12233i 0.453172 0.0717753i 2.50557 + 2.42537i −3.69158 1.19947i −1.77213 2.20444i 1.81047 + 2.49190i −0.642447 0.0910698i
19.4 −1.18083 + 0.778224i 0.465640 + 0.237255i 0.788734 1.83791i −3.20760 + 0.508035i −0.734481 + 0.0822133i −2.50286 0.813229i 0.498941 + 2.78407i −1.60283 2.20610i 3.39228 3.09614i
19.5 −1.09947 0.889470i 2.08544 + 1.06259i 0.417686 + 1.95590i −3.64392 + 0.577140i −1.34775 3.02322i 3.51004 + 1.14048i 1.28048 2.52198i 1.45663 + 2.00487i 4.51974 + 2.60661i
19.6 −0.961599 + 1.03698i 1.08914 + 0.554944i −0.150656 1.99432i 1.84875 0.292814i −1.62278 + 0.595782i 0.739925 + 0.240416i 2.21294 + 1.76151i −0.885095 1.21823i −1.47412 + 2.19869i
19.7 −0.799521 1.16652i −0.748231 0.381243i −0.721532 + 1.86531i 2.75331 0.436081i 0.153500 + 1.17764i 1.71180 + 0.556197i 2.75280 0.649676i −1.34885 1.85654i −2.71002 2.86313i
19.8 −0.488029 1.32734i −2.46533 1.25615i −1.52365 + 1.29556i −2.25517 + 0.357184i −0.464181 + 3.88537i 2.36438 + 0.768234i 2.46323 + 1.39013i 2.73660 + 3.76660i 1.57469 + 2.81906i
19.9 −0.400688 1.35626i 2.50372 + 1.27571i −1.67890 + 1.08688i 2.08614 0.330411i 0.726986 3.90687i −3.37867 1.09780i 2.14681 + 1.84153i 2.87783 + 3.96099i −1.28401 2.69696i
19.10 −0.275705 + 1.38708i −1.44186 0.734666i −1.84797 0.764848i 0.774751 0.122708i 1.41657 1.79743i −3.19838 1.03922i 1.57040 2.35241i −0.224121 0.308477i −0.0433961 + 1.10847i
19.11 −0.0657923 + 1.41268i 2.65021 + 1.35035i −1.99134 0.185887i −0.419911 + 0.0665073i −2.08198 + 3.65506i −1.07355 0.348817i 0.393614 2.80090i 3.43681 + 4.73036i −0.0663268 0.597576i
19.12 0.317671 1.37807i −0.349980 0.178324i −1.79817 0.875548i −2.89168 + 0.457997i −0.356922 + 0.425650i −2.68173 0.871347i −1.77779 + 2.19987i −1.67267 2.30223i −0.287449 + 4.13044i
19.13 0.524739 + 1.31326i −1.25130 0.637570i −1.44930 + 1.37824i −3.61695 + 0.572869i 0.180688 1.97784i 0.568084 + 0.184582i −2.57048 1.18009i −0.604094 0.831464i −2.65028 4.44939i
19.14 0.546837 1.30421i 1.88744 + 0.961700i −1.40194 1.42638i −0.474891 + 0.0752154i 2.28639 1.93573i 3.46190 + 1.12484i −2.62694 + 1.04843i 0.874219 + 1.20326i −0.161591 + 0.660489i
19.15 0.547313 1.30401i −1.19426 0.608505i −1.40090 1.42741i 3.97256 0.629192i −1.44713 + 1.22428i −0.338364 0.109941i −2.62808 + 1.04555i −0.707382 0.973627i 1.35376 5.52464i
19.16 0.838432 + 1.13887i 1.18196 + 0.602240i −0.594065 + 1.90973i −0.0178621 + 0.00282908i 0.305120 + 1.85104i 0.570788 + 0.185460i −2.67303 + 0.924618i −0.729013 1.00340i −0.0181981 0.0179707i
19.17 0.962542 + 1.03610i −2.82726 1.44056i −0.147025 + 1.99459i 3.09886 0.490811i −1.22879 4.31593i 1.76298 + 0.572825i −2.20812 + 1.76754i 4.15482 + 5.71862i 3.49131 + 2.73831i
19.18 1.19855 0.750654i −2.76720 1.40996i 0.873038 1.79939i −0.955959 + 0.151409i −4.37501 + 0.387305i −1.82777 0.593878i −0.304341 2.81201i 3.90606 + 5.37622i −1.03211 + 0.899065i
19.19 1.22937 0.699026i 0.0932365 + 0.0475063i 1.02272 1.71873i −0.234287 + 0.0371074i 0.147831 0.00677164i 0.905351 + 0.294166i 0.0558734 2.82788i −1.75692 2.41819i −0.262087 + 0.209392i
19.20 1.32736 + 0.487958i 0.357504 + 0.182157i 1.52379 + 1.29540i 2.88474 0.456898i 0.385653 + 0.416236i −4.93701 1.60413i 1.39053 + 2.46301i −1.66873 2.29681i 4.05205 + 0.801162i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.22 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.d odd 10 1 inner
16.f odd 4 1 inner
176.x even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.x.a 176
4.b odd 2 1 704.2.bf.a 176
11.d odd 10 1 inner 176.2.x.a 176
16.e even 4 1 704.2.bf.a 176
16.f odd 4 1 inner 176.2.x.a 176
44.g even 10 1 704.2.bf.a 176
176.u odd 20 1 704.2.bf.a 176
176.x even 20 1 inner 176.2.x.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.x.a 176 1.a even 1 1 trivial
176.2.x.a 176 11.d odd 10 1 inner
176.2.x.a 176 16.f odd 4 1 inner
176.2.x.a 176 176.x even 20 1 inner
704.2.bf.a 176 4.b odd 2 1
704.2.bf.a 176 16.e even 4 1
704.2.bf.a 176 44.g even 10 1
704.2.bf.a 176 176.u odd 20 1

## Hecke kernels

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