# Properties

 Label 160.2.z.a Level $160$ Weight $2$ Character orbit 160.z Analytic conductor $1.278$ Analytic rank $0$ Dimension $88$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 160.z (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.27760643234$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{9} - 4q^{10} - 8q^{11} - 24q^{14} - 8q^{16} - 8q^{19} - 4q^{20} - 8q^{21} - 16q^{24} - 4q^{25} + 32q^{26} - 8q^{29} - 52q^{30} - 64q^{31} - 24q^{34} + 20q^{35} + 72q^{36} - 8q^{39} - 56q^{40} - 8q^{41} - 8q^{44} - 16q^{45} - 8q^{46} - 44q^{50} - 48q^{51} + 24q^{54} + 28q^{55} - 56q^{56} + 24q^{59} + 32q^{60} + 24q^{61} + 64q^{64} - 8q^{65} - 8q^{66} - 40q^{69} + 80q^{70} - 40q^{71} + 128q^{74} + 28q^{75} - 8q^{76} + 48q^{80} + 200q^{84} - 24q^{85} + 24q^{86} - 8q^{89} + 80q^{90} - 8q^{91} + 120q^{94} - 8q^{95} - 56q^{96} - 48q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.41382 + 0.0334026i 1.02536 2.47544i 1.99777 0.0944503i 1.80031 + 1.32623i −1.36699 + 3.53408i 2.81608 + 2.81608i −2.82133 + 0.200266i −2.95514 2.95514i −2.58962 1.81491i
29.2 −1.41308 0.0566552i −0.529266 + 1.27776i 1.99358 + 0.160116i 1.47606 1.67965i 0.820286 1.77559i −0.242153 0.242153i −2.80801 0.339204i 0.768771 + 0.768771i −2.18095 + 2.28986i
29.3 −1.31115 + 0.529979i 0.763124 1.84234i 1.43825 1.38977i −1.81451 1.30674i −0.0241693 + 2.82004i −2.19624 2.19624i −1.14921 + 2.58444i −0.690555 0.690555i 3.07164 + 0.751682i
29.4 −1.20513 0.740042i 0.0462717 0.111710i 0.904676 + 1.78369i −2.14654 + 0.626377i −0.138433 + 0.100382i 2.44906 + 2.44906i 0.229754 2.81908i 2.11098 + 2.11098i 3.05041 + 0.833667i
29.5 −1.08863 + 0.902710i −0.624514 + 1.50771i 0.370229 1.96543i 1.41683 + 1.72991i −0.681161 2.20509i 0.611546 + 0.611546i 1.37117 + 2.47384i 0.238149 + 0.238149i −3.10401 0.604247i
29.6 −0.771562 1.18520i −1.12821 + 2.72375i −0.809383 + 1.82891i −1.18407 1.89683i 4.09867 0.764387i −1.51399 1.51399i 2.79210 0.451838i −4.02463 4.02463i −1.33454 + 2.86688i
29.7 −0.685946 + 1.23672i 0.0724046 0.174800i −1.05896 1.69665i −0.320762 2.21294i 0.166513 + 0.209448i 2.49757 + 2.49757i 2.82467 0.145826i 2.09601 + 2.09601i 2.95682 + 1.12127i
29.8 −0.578672 1.29040i −0.245584 + 0.592892i −1.33028 + 1.49344i 1.89351 + 1.18938i 0.907181 0.0261881i −0.502274 0.502274i 2.69693 + 0.852380i 1.83011 + 1.83011i 0.439066 3.13165i
29.9 −0.425344 + 1.34873i 0.872653 2.10677i −1.63817 1.14735i 2.19410 + 0.431197i 2.47030 + 2.07308i −3.02408 3.02408i 2.24425 1.72143i −1.55564 1.55564i −1.51482 + 2.77585i
29.10 −0.381706 1.36173i 1.23949 2.99239i −1.70860 + 1.03956i −1.80901 + 1.31434i −4.54794 0.545632i −1.10874 1.10874i 2.06778 + 1.92984i −5.29676 5.29676i 2.48028 + 1.96169i
29.11 −0.258628 + 1.39036i −0.644653 + 1.55633i −1.86622 0.719173i −2.20768 + 0.355198i −1.99714 1.29881i −1.63067 1.63067i 1.48257 2.40873i 0.114733 + 0.114733i 0.0771112 3.16134i
29.12 0.258628 1.39036i 0.644653 1.55633i −1.86622 0.719173i 1.81223 1.30990i −1.99714 1.29881i 1.63067 + 1.63067i −1.48257 + 2.40873i 0.114733 + 0.114733i −1.35255 2.85843i
29.13 0.381706 + 1.36173i −1.23949 + 2.99239i −1.70860 + 1.03956i 2.20854 0.349785i −4.54794 0.545632i 1.10874 + 1.10874i −2.06778 1.92984i −5.29676 5.29676i 1.31932 + 2.87391i
29.14 0.425344 1.34873i −0.872653 + 2.10677i −1.63817 1.14735i −1.24656 + 1.85636i 2.47030 + 2.07308i 3.02408 + 3.02408i −2.24425 + 1.72143i −1.55564 1.55564i 1.97353 + 2.47087i
29.15 0.578672 + 1.29040i 0.245584 0.592892i −1.33028 + 1.49344i −0.497888 + 2.17993i 0.907181 0.0261881i 0.502274 + 0.502274i −2.69693 0.852380i 1.83011 + 1.83011i −3.10110 + 0.618990i
29.16 0.685946 1.23672i −0.0724046 + 0.174800i −1.05896 1.69665i −1.33797 1.79160i 0.166513 + 0.209448i −2.49757 2.49757i −2.82467 + 0.145826i 2.09601 + 2.09601i −3.13349 + 0.425759i
29.17 0.771562 + 1.18520i 1.12821 2.72375i −0.809383 + 1.82891i −0.504002 2.17853i 4.09867 0.764387i 1.51399 + 1.51399i −2.79210 + 0.451838i −4.02463 4.02463i 2.19312 2.27821i
29.18 1.08863 0.902710i 0.624514 1.50771i 0.370229 1.96543i 0.221383 + 2.22508i −0.681161 2.20509i −0.611546 0.611546i −1.37117 2.47384i 0.238149 + 0.238149i 2.24961 + 2.22245i
29.19 1.20513 + 0.740042i −0.0462717 + 0.111710i 0.904676 + 1.78369i 1.96075 1.07492i −0.138433 + 0.100382i −2.44906 2.44906i −0.229754 + 2.81908i 2.11098 + 2.11098i 3.15845 + 0.155618i
29.20 1.31115 0.529979i −0.763124 + 1.84234i 1.43825 1.38977i 0.359048 2.20705i −0.0241693 + 2.82004i 2.19624 + 2.19624i 1.14921 2.58444i −0.690555 0.690555i −0.698924 3.08407i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
32.g even 8 1 inner
160.z even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.2.z.a 88
4.b odd 2 1 640.2.z.a 88
5.b even 2 1 inner 160.2.z.a 88
5.c odd 4 2 800.2.y.f 88
20.d odd 2 1 640.2.z.a 88
32.g even 8 1 inner 160.2.z.a 88
32.h odd 8 1 640.2.z.a 88
160.v odd 8 1 800.2.y.f 88
160.y odd 8 1 640.2.z.a 88
160.z even 8 1 inner 160.2.z.a 88
160.bb odd 8 1 800.2.y.f 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.z.a 88 1.a even 1 1 trivial
160.2.z.a 88 5.b even 2 1 inner
160.2.z.a 88 32.g even 8 1 inner
160.2.z.a 88 160.z even 8 1 inner
640.2.z.a 88 4.b odd 2 1
640.2.z.a 88 20.d odd 2 1
640.2.z.a 88 32.h odd 8 1
640.2.z.a 88 160.y odd 8 1
800.2.y.f 88 5.c odd 4 2
800.2.y.f 88 160.v odd 8 1
800.2.y.f 88 160.bb odd 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(160, [\chi])$$.