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Properties

Label 222.3.r.d
Level $222$
Weight $3$
Character orbit 222.r
Analytic conductor $6.049$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [222,3,Mod(13,222)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(222, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("222.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 222.r (of order \(36\), degree \(12\), 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: \(6.04906186880\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{8} - 18 q^{11} + 12 q^{13} + 24 q^{14} + 36 q^{15} - 36 q^{17} - 150 q^{19} - 36 q^{20} - 36 q^{21} - 30 q^{22} + 90 q^{23} + 216 q^{25} + 216 q^{27} - 72 q^{28} + 210 q^{29} + 72 q^{30} - 12 q^{31} - 54 q^{34} + 180 q^{35} - 24 q^{37} - 108 q^{38} + 18 q^{39} - 438 q^{41} - 90 q^{42} - 294 q^{46} + 42 q^{47} - 288 q^{48} + 54 q^{49} + 144 q^{50} - 72 q^{51} - 120 q^{52} + 30 q^{53} - 264 q^{55} + 126 q^{57} - 102 q^{58} + 66 q^{59} + 660 q^{61} - 18 q^{62} - 36 q^{63} + 60 q^{65} + 36 q^{66} - 72 q^{67} + 12 q^{68} + 18 q^{69} + 42 q^{70} - 30 q^{71} + 282 q^{74} + 144 q^{75} - 300 q^{76} + 60 q^{77} - 108 q^{78} + 426 q^{79} + 48 q^{82} - 510 q^{83} + 72 q^{84} - 414 q^{85} + 528 q^{86} + 306 q^{87} + 24 q^{88} - 414 q^{89} - 1098 q^{91} - 132 q^{92} + 90 q^{93} + 528 q^{94} + 6 q^{95} + 966 q^{97} + 108 q^{98} - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0.811160 + 1.15846i 1.70574 − 0.300767i −0.684040 + 1.87939i −9.26361 − 0.810461i 1.73205 + 1.73205i −4.86520 + 4.08239i −2.73205 + 0.732051i 2.81908 − 1.02606i −6.57538 − 11.3889i
13.2 0.811160 + 1.15846i 1.70574 − 0.300767i −0.684040 + 1.87939i −1.66321 − 0.145512i 1.73205 + 1.73205i 7.05340 − 5.91851i −2.73205 + 0.732051i 2.81908 − 1.02606i −1.18056 − 2.04479i
13.3 0.811160 + 1.15846i 1.70574 − 0.300767i −0.684040 + 1.87939i 4.40749 + 0.385605i 1.73205 + 1.73205i −9.56010 + 8.02187i −2.73205 + 0.732051i 2.81908 − 1.02606i 3.12847 + 5.41867i
13.4 0.811160 + 1.15846i 1.70574 − 0.300767i −0.684040 + 1.87939i 9.65635 + 0.844821i 1.73205 + 1.73205i 5.21571 − 4.37650i −2.73205 + 0.732051i 2.81908 − 1.02606i 6.85415 + 11.8717i
19.1 −1.40883 + 0.123257i −1.11334 − 1.32683i 1.96962 − 0.347296i −3.77272 − 8.09062i 1.73205 + 1.73205i −11.2603 − 4.09842i −2.73205 + 0.732051i −0.520945 + 2.95442i 6.31235 + 10.9333i
19.2 −1.40883 + 0.123257i −1.11334 − 1.32683i 1.96962 − 0.347296i −0.248455 − 0.532814i 1.73205 + 1.73205i 2.07268 + 0.754394i −2.73205 + 0.732051i −0.520945 + 2.95442i 0.415705 + 0.720022i
19.3 −1.40883 + 0.123257i −1.11334 − 1.32683i 1.96962 − 0.347296i −0.175672 − 0.376730i 1.73205 + 1.73205i 7.77415 + 2.82956i −2.73205 + 0.732051i −0.520945 + 2.95442i 0.293927 + 0.509096i
19.4 −1.40883 + 0.123257i −1.11334 − 1.32683i 1.96962 − 0.347296i 3.48873 + 7.48160i 1.73205 + 1.73205i −4.40147 − 1.60200i −2.73205 + 0.732051i −0.520945 + 2.95442i −5.83719 − 10.1103i
55.1 −0.123257 − 1.40883i −1.11334 − 1.32683i −1.96962 + 0.347296i −5.84940 + 2.72762i −1.73205 + 1.73205i 11.2015 + 4.07701i 0.732051 + 2.73205i −0.520945 + 2.95442i 4.56373 + 7.90462i
55.2 −0.123257 − 1.40883i −1.11334 − 1.32683i −1.96962 + 0.347296i −3.71431 + 1.73201i −1.73205 + 1.73205i −6.38586 − 2.32426i 0.732051 + 2.73205i −0.520945 + 2.95442i 2.89792 + 5.01935i
55.3 −0.123257 − 1.40883i −1.11334 − 1.32683i −1.96962 + 0.347296i 1.21143 − 0.564901i −1.73205 + 1.73205i −1.35377 − 0.492731i 0.732051 + 2.73205i −0.520945 + 2.95442i −0.945168 − 1.63708i
55.4 −0.123257 − 1.40883i −1.11334 − 1.32683i −1.96962 + 0.347296i 6.83371 − 3.18661i −1.73205 + 1.73205i 8.33856 + 3.03499i 0.732051 + 2.73205i −0.520945 + 2.95442i −5.33170 − 9.23477i
61.1 −1.15846 + 0.811160i 1.70574 − 0.300767i 0.684040 − 1.87939i −0.501528 + 5.73249i −1.73205 + 1.73205i 6.79113 − 5.69844i 0.732051 + 2.73205i 2.81908 − 1.02606i −4.06897 − 7.04766i
61.2 −1.15846 + 0.811160i 1.70574 − 0.300767i 0.684040 − 1.87939i −0.165245 + 1.88876i −1.73205 + 1.73205i −8.55835 + 7.18131i 0.732051 + 2.73205i 2.81908 − 1.02606i −1.34066 − 2.32209i
61.3 −1.15846 + 0.811160i 1.70574 − 0.300767i 0.684040 − 1.87939i 0.290180 − 3.31677i −1.73205 + 1.73205i −5.36367 + 4.50065i 0.732051 + 2.73205i 2.81908 − 1.02606i 2.35427 + 4.07771i
61.4 −1.15846 + 0.811160i 1.70574 − 0.300767i 0.684040 − 1.87939i 0.651048 − 7.44151i −1.73205 + 1.73205i 2.81141 − 2.35906i 0.732051 + 2.73205i 2.81908 − 1.02606i 5.28204 + 9.14876i
79.1 0.597672 − 1.28171i −0.592396 + 1.62760i −1.28558 − 1.53209i −1.98168 + 2.83013i 1.73205 + 1.73205i 0.768630 + 4.35912i −2.73205 + 0.732051i −2.29813 − 1.92836i 2.44302 + 4.23144i
79.2 0.597672 − 1.28171i −0.592396 + 1.62760i −1.28558 − 1.53209i −0.430089 + 0.614231i 1.73205 + 1.73205i −1.95581 − 11.0919i −2.73205 + 0.732051i −2.29813 − 1.92836i 0.530215 + 0.918359i
79.3 0.597672 − 1.28171i −0.592396 + 1.62760i −1.28558 − 1.53209i −0.231340 + 0.330388i 1.73205 + 1.73205i 0.882896 + 5.00715i −2.73205 + 0.732051i −2.29813 − 1.92836i 0.285196 + 0.493975i
79.4 0.597672 − 1.28171i −0.592396 + 1.62760i −1.28558 − 1.53209i 5.41036 − 7.72679i 1.73205 + 1.73205i 0.481205 + 2.72905i −2.73205 + 0.732051i −2.29813 − 1.92836i −6.66991 − 11.5526i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 222.3.r.d ✓ 48
37.i odd 36 1 inner 222.3.r.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.3.r.d ✓ 48 1.a even 1 1 trivial
222.3.r.d ✓ 48 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} - 108 T_{5}^{46} + 270 T_{5}^{45} + 4887 T_{5}^{44} - 37548 T_{5}^{43} + \cdots + 81\!\cdots\!56 \) acting on \(S_{3}^{\mathrm{new}}(222, [\chi])\). Copy content Toggle raw display