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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 $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(104,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11, 9, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(7.54586299101\)
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.

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

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 104.136
Significant digits:
Format:

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 \( T_{2}^{408} + 6 T_{2}^{406} + 9 T_{2}^{404} + 2955 T_{2}^{402} + 17352 T_{2}^{400} + \cdots + 33\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display