# Properties

 Label 945.2.cs.b Level 945 Weight 2 Character orbit 945.cs Analytic conductor 7.546 Analytic rank 0 Dimension 816 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$945 = 3^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 945.cs (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.54586299101$$ Analytic rank: $$0$$ Dimension: $$816$$ Relative dimension: $$136$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$816q - 24q^{4} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$816q - 24q^{4} - 24q^{9} - 12q^{11} - 6q^{14} + 24q^{21} - 12q^{25} + 12q^{29} + 78q^{30} - 36q^{35} + 12q^{36} - 84q^{39} - 36q^{44} - 12q^{46} + 6q^{49} + 24q^{50} - 60q^{51} + 120q^{56} - 150q^{60} - 444q^{64} + 162q^{65} + 21q^{70} - 36q^{71} - 48q^{74} - 48q^{79} - 48q^{81} + 66q^{84} - 102q^{85} - 192q^{86} - 6q^{91} - 210q^{95} - 216q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
104.1 −2.58909 0.942353i −0.600558 + 1.62460i 4.28329 + 3.59410i 1.90382 + 1.17280i 3.08585 3.64031i −0.535422 + 2.59101i −4.94766 8.56959i −2.27866 1.95133i −3.82398 4.83056i
104.2 −2.58909 0.942353i 0.600558 1.62460i 4.28329 + 3.59410i −1.90382 1.17280i −3.08585 + 3.64031i 2.64462 0.0773642i −4.94766 8.56959i −2.27866 1.95133i 3.82398 + 4.83056i
104.3 −2.53031 0.920958i −1.72154 0.190516i 4.02222 + 3.37504i 0.693777 2.12572i 4.18058 + 2.06753i −0.625891 + 2.57065i −4.37650 7.58031i 2.92741 + 0.655961i −3.71317 + 4.73979i
104.4 −2.53031 0.920958i 1.72154 + 0.190516i 4.02222 + 3.37504i −0.693777 + 2.12572i −4.18058 2.06753i 2.64028 0.169993i −4.37650 7.58031i 2.92741 + 0.655961i 3.71317 4.73979i
104.5 −2.50242 0.910806i −0.792257 + 1.54024i 3.90044 + 3.27286i −0.778079 2.09633i 3.38541 3.13272i 1.23163 2.34160i −4.11657 7.13012i −1.74466 2.44053i 0.0377315 + 5.95457i
104.6 −2.50242 0.910806i 0.792257 1.54024i 3.90044 + 3.27286i 0.778079 + 2.09633i −3.38541 + 3.13272i −2.51990 + 0.806302i −4.11657 7.13012i −1.74466 2.44053i −0.0377315 5.95457i
104.7 −2.39005 0.869908i −0.413843 1.68188i 3.42352 + 2.87268i 1.33788 1.79167i −0.473977 + 4.37980i −2.12702 1.57346i −3.14000 5.43864i −2.65747 + 1.39207i −4.75619 + 3.11835i
104.8 −2.39005 0.869908i 0.413843 + 1.68188i 3.42352 + 2.87268i −1.33788 + 1.79167i 0.473977 4.37980i −1.18021 2.36793i −3.14000 5.43864i −2.65747 + 1.39207i 4.75619 3.11835i
104.9 −2.37527 0.864529i −1.19353 1.25518i 3.36242 + 2.82141i 2.05172 + 0.889078i 1.74982 + 4.01325i 2.60397 0.468317i −3.01977 5.23040i −0.150973 + 2.99620i −4.10475 3.88557i
104.10 −2.37527 0.864529i 1.19353 + 1.25518i 3.36242 + 2.82141i −2.05172 0.889078i −1.74982 4.01325i −0.913378 + 2.48309i −3.01977 5.23040i −0.150973 + 2.99620i 4.10475 + 3.88557i
104.11 −2.30468 0.838835i −1.61471 + 0.626663i 3.07581 + 2.58091i −1.65639 + 1.50213i 4.24706 0.0897813i 2.42327 + 1.06196i −2.47122 4.28027i 2.21459 2.02376i 5.07748 2.07249i
104.12 −2.30468 0.838835i 1.61471 0.626663i 3.07581 + 2.58091i 1.65639 1.50213i −4.24706 + 0.0897813i 0.625030 + 2.57086i −2.47122 4.28027i 2.21459 2.02376i −5.07748 + 2.07249i
104.13 −2.27225 0.827033i −1.72657 + 0.137621i 2.94706 + 2.47288i 1.34694 + 1.78487i 4.03703 + 1.11522i −1.82568 1.91491i −2.23324 3.86809i 2.96212 0.475226i −1.58445 5.16963i
104.14 −2.27225 0.827033i 1.72657 0.137621i 2.94706 + 2.47288i −1.34694 1.78487i −4.03703 1.11522i −1.56879 2.13047i −2.23324 3.86809i 2.96212 0.475226i 1.58445 + 5.16963i
104.15 −2.19076 0.797371i −0.522216 1.65145i 2.63153 + 2.20812i −2.22823 0.187052i −0.172770 + 4.03433i −0.716428 + 2.54691i −1.67300 2.89773i −2.45458 + 1.72483i 4.73236 + 2.18651i
104.16 −2.19076 0.797371i 0.522216 + 1.65145i 2.63153 + 2.20812i 2.22823 + 0.187052i 0.172770 4.03433i 2.63262 0.263278i −1.67300 2.89773i −2.45458 + 1.72483i −4.73236 2.18651i
104.17 −2.05378 0.747516i −1.57383 + 0.723218i 2.12716 + 1.78490i 1.84479 1.26363i 3.77293 0.308866i −0.111990 2.64338i −0.848890 1.47032i 1.95391 2.27645i −4.73338 + 1.21621i
104.18 −2.05378 0.747516i 1.57383 0.723218i 2.12716 + 1.78490i −1.84479 + 1.26363i −3.77293 + 0.308866i −2.58377 0.569307i −0.848890 1.47032i 1.95391 2.27645i 4.73338 1.21621i
104.19 −1.96750 0.716110i −1.00171 + 1.41301i 1.82614 + 1.53231i −1.23328 + 1.86521i 2.98272 2.06275i −2.38678 + 1.14161i −0.401851 0.696026i −0.993173 2.83083i 3.76217 2.78664i
104.20 −1.96750 0.716110i 1.00171 1.41301i 1.82614 + 1.53231i 1.23328 1.86521i −2.98272 + 2.06275i 1.53873 2.15228i −0.401851 0.696026i −0.993173 2.83083i −3.76217 + 2.78664i
See next 80 embeddings (of 816 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 839.136 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
7.b odd 2 1 inner
27.f odd 18 1 inner
35.c odd 2 1 inner
135.n odd 18 1 inner
189.be even 18 1 inner
945.cs even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.cs.b 816
5.b even 2 1 inner 945.2.cs.b 816
7.b odd 2 1 inner 945.2.cs.b 816
27.f odd 18 1 inner 945.2.cs.b 816
35.c odd 2 1 inner 945.2.cs.b 816
135.n odd 18 1 inner 945.2.cs.b 816
189.be even 18 1 inner 945.2.cs.b 816
945.cs even 18 1 inner 945.2.cs.b 816

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cs.b 816 1.a even 1 1 trivial
945.2.cs.b 816 5.b even 2 1 inner
945.2.cs.b 816 7.b odd 2 1 inner
945.2.cs.b 816 27.f odd 18 1 inner
945.2.cs.b 816 35.c odd 2 1 inner
945.2.cs.b 816 135.n odd 18 1 inner
945.2.cs.b 816 189.be even 18 1 inner
945.2.cs.b 816 945.cs even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{408} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(945, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database