# Properties

 Label 176.2.w.a Level $176$ Weight $2$ Character orbit 176.w 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(5,176)]

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([0, 5, 8]))

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

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

chi := DirichletCharacter("176.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.w (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.

 $$\operatorname{Tr}(f)(q) =$$ $$176 q - 6 q^{2} - 6 q^{3} - 10 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{8}+O(q^{10})$$ 176 * q - 6 * q^2 - 6 * q^3 - 10 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$176 q - 6 q^{2} - 6 q^{3} - 10 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{8} - 16 q^{10} - 12 q^{11} - 6 q^{13} - 12 q^{15} + 14 q^{16} - 12 q^{17} - 44 q^{18} - 6 q^{19} + 2 q^{20} - 28 q^{21} + 50 q^{22} - 38 q^{24} - 68 q^{26} - 18 q^{27} - 46 q^{28} - 22 q^{29} + 26 q^{30} - 12 q^{31} - 16 q^{32} - 16 q^{33} + 12 q^{34} - 26 q^{35} - 22 q^{36} + 18 q^{37} - 34 q^{38} + 14 q^{40} - 10 q^{42} - 40 q^{43} + 2 q^{44} - 24 q^{45} + 38 q^{46} - 12 q^{47} - 26 q^{48} + 8 q^{49} - 62 q^{50} + 6 q^{51} + 74 q^{52} - 30 q^{53} - 52 q^{54} - 96 q^{56} - 26 q^{58} + 10 q^{59} + 118 q^{60} - 6 q^{61} - 42 q^{62} - 28 q^{63} - 106 q^{64} - 32 q^{65} + 6 q^{66} + 24 q^{67} + 116 q^{68} + 12 q^{69} + 52 q^{70} - 98 q^{72} + 96 q^{74} - 46 q^{75} + 112 q^{76} - 14 q^{77} + 44 q^{78} - 52 q^{79} - 28 q^{80} + 66 q^{82} + 54 q^{83} + 120 q^{84} + 14 q^{85} + 86 q^{86} + 142 q^{88} + 228 q^{90} - 122 q^{91} + 146 q^{92} + 6 q^{93} + 56 q^{94} + 52 q^{95} + 86 q^{96} - 12 q^{97} + 140 q^{98} + 92 q^{99}+O(q^{100})$$ 176 * q - 6 * q^2 - 6 * q^3 - 10 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^8 - 16 * q^10 - 12 * q^11 - 6 * q^13 - 12 * q^15 + 14 * q^16 - 12 * q^17 - 44 * q^18 - 6 * q^19 + 2 * q^20 - 28 * q^21 + 50 * q^22 - 38 * q^24 - 68 * q^26 - 18 * q^27 - 46 * q^28 - 22 * q^29 + 26 * q^30 - 12 * q^31 - 16 * q^32 - 16 * q^33 + 12 * q^34 - 26 * q^35 - 22 * q^36 + 18 * q^37 - 34 * q^38 + 14 * q^40 - 10 * q^42 - 40 * q^43 + 2 * q^44 - 24 * q^45 + 38 * q^46 - 12 * q^47 - 26 * q^48 + 8 * q^49 - 62 * q^50 + 6 * q^51 + 74 * q^52 - 30 * q^53 - 52 * q^54 - 96 * q^56 - 26 * q^58 + 10 * q^59 + 118 * q^60 - 6 * q^61 - 42 * q^62 - 28 * q^63 - 106 * q^64 - 32 * q^65 + 6 * q^66 + 24 * q^67 + 116 * q^68 + 12 * q^69 + 52 * q^70 - 98 * q^72 + 96 * q^74 - 46 * q^75 + 112 * q^76 - 14 * q^77 + 44 * q^78 - 52 * q^79 - 28 * q^80 + 66 * q^82 + 54 * q^83 + 120 * q^84 + 14 * q^85 + 86 * q^86 + 142 * q^88 + 228 * q^90 - 122 * q^91 + 146 * q^92 + 6 * q^93 + 56 * q^94 + 52 * q^95 + 86 * q^96 - 12 * q^97 + 140 * q^98 + 92 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.39423 + 0.236897i 3.10742 0.492167i 1.88776 0.660577i −1.27797 + 0.651156i −4.21587 + 1.42233i 2.30623 + 3.17425i −2.47549 + 1.36820i 6.56066 2.13169i 1.62752 1.21061i
5.2 −1.37854 + 0.315659i 0.901291 0.142750i 1.80072 0.870295i 3.45027 1.75800i −1.19740 + 0.481287i −1.81367 2.49631i −2.20764 + 1.76815i −2.06122 + 0.669732i −4.20139 + 3.51258i
5.3 −1.37053 0.348781i −0.417657 + 0.0661504i 1.75670 + 0.956029i −0.736011 + 0.375016i 0.595484 + 0.0550098i 1.33653 + 1.83957i −2.07417 1.92297i −2.68311 + 0.871795i 1.13952 0.257265i
5.4 −1.34050 + 0.450612i −3.05021 + 0.483105i 1.59390 1.20809i −2.67423 + 1.36259i 3.87112 2.02206i −0.413323 0.568890i −1.59224 + 2.33768i 6.21720 2.02009i 2.97081 3.03159i
5.5 −1.17638 0.784944i −1.93030 + 0.305729i 0.767727 + 1.84678i 1.24626 0.635003i 2.51074 + 1.15552i −1.09820 1.51155i 0.546481 2.77513i 0.779411 0.253246i −1.96452 0.231243i
5.6 −1.02618 0.973113i 2.68580 0.425388i 0.106100 + 1.99718i 0.102082 0.0520132i −3.17007 2.17706i −2.36506 3.25523i 1.83461 2.15272i 4.17938 1.35796i −0.155369 0.0459620i
5.7 −0.779948 + 1.17970i 1.33374 0.211244i −0.783362 1.84020i 0.141837 0.0722696i −0.791046 + 1.73817i 1.45574 + 2.00365i 2.78186 + 0.511133i −1.11893 + 0.363561i −0.0253694 + 0.223691i
5.8 −0.592330 + 1.28419i −2.54744 + 0.403475i −1.29829 1.52133i 2.98748 1.52220i 0.990787 3.51039i −1.07945 1.48574i 2.72269 0.766123i 3.47349 1.12860i 0.185217 + 4.73814i
5.9 −0.565804 1.29610i 0.310032 0.0491042i −1.35973 + 1.46667i −3.81933 + 1.94605i −0.239061 0.374048i 1.09214 + 1.50320i 2.67029 + 0.932494i −2.75946 + 0.896603i 4.68326 + 3.84914i
5.10 −0.520421 1.31498i 1.22266 0.193650i −1.45832 + 1.36868i 2.51383 1.28086i −0.890940 1.50698i 2.24956 + 3.09626i 2.55873 + 1.20537i −1.39578 + 0.453517i −2.99255 2.63904i
5.11 −0.287066 + 1.38477i −0.925602 + 0.146601i −1.83519 0.795041i −1.96526 + 1.00135i 0.0626996 1.32383i 0.432076 + 0.594702i 1.62777 2.31309i −2.01792 + 0.655663i −0.822483 3.00889i
5.12 0.0503613 + 1.41332i 2.39469 0.379282i −1.99493 + 0.142353i 2.15797 1.09954i 0.656645 + 3.36536i −0.678287 0.933582i −0.301657 2.81230i 2.73753 0.889476i 1.66268 + 2.99452i
5.13 0.0542058 1.41317i −3.21042 + 0.508480i −1.99412 0.153205i 0.266677 0.135879i 0.544548 + 4.56444i 1.68453 + 2.31856i −0.324598 + 2.80974i 7.19505 2.33782i −0.177565 0.384227i
5.14 0.165330 1.40452i −0.757844 + 0.120031i −1.94533 0.464417i −0.814347 + 0.414930i 0.0432907 + 1.08425i −2.91240 4.00857i −0.973904 + 2.65547i −2.29325 + 0.745122i 0.448140 + 1.21236i
5.15 0.664023 + 1.24863i −1.67818 + 0.265797i −1.11815 + 1.65824i −1.78176 + 0.907854i −1.44623 1.91892i −2.03350 2.79887i −2.81300 0.295040i −0.107544 + 0.0349431i −2.31670 1.62192i
5.16 0.760498 + 1.19233i −1.17460 + 0.186039i −0.843285 + 1.81352i 2.66815 1.35949i −1.11510 1.25903i 2.54995 + 3.50970i −2.80363 + 0.373710i −1.50808 + 0.490006i 3.65008 + 2.14742i
5.17 0.818556 1.15324i 0.562832 0.0891439i −0.659933 1.88799i 1.83674 0.935866i 0.357905 0.722051i 0.677755 + 0.932849i −2.71750 0.784359i −2.54434 + 0.826705i 0.424195 2.88426i
5.18 0.994957 + 1.00502i 2.09875 0.332409i −0.0201206 + 1.99990i −1.30316 + 0.663991i 2.42224 + 1.77855i −0.801284 1.10287i −2.02995 + 1.96959i 1.44108 0.468236i −1.96391 0.649052i
5.19 1.11969 0.863883i 2.35196 0.372513i 0.507411 1.93456i −3.24121 + 1.65148i 2.31165 2.44892i 0.349519 + 0.481071i −1.10309 2.60445i 2.53977 0.825220i −2.20247 + 4.64917i
5.20 1.29627 0.565399i −2.20058 + 0.348538i 1.36065 1.46582i 2.16222 1.10171i −2.65549 + 1.69601i −0.296642 0.408292i 0.934999 2.66942i 1.86792 0.606923i 2.17993 2.65063i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.22 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.c even 5 1 inner
16.e even 4 1 inner
176.w 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.w.a 176
4.b odd 2 1 704.2.be.a 176
11.c even 5 1 inner 176.2.w.a 176
16.e even 4 1 inner 176.2.w.a 176
16.f odd 4 1 704.2.be.a 176
44.h odd 10 1 704.2.be.a 176
176.v odd 20 1 704.2.be.a 176
176.w even 20 1 inner 176.2.w.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.w.a 176 1.a even 1 1 trivial
176.2.w.a 176 11.c even 5 1 inner
176.2.w.a 176 16.e even 4 1 inner
176.2.w.a 176 176.w even 20 1 inner
704.2.be.a 176 4.b odd 2 1
704.2.be.a 176 16.f odd 4 1
704.2.be.a 176 44.h odd 10 1
704.2.be.a 176 176.v odd 20 1

## Hecke kernels

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