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Properties

Label 164.3.n.c
Level $164$
Weight $3$
Character orbit 164.n
Analytic conductor $4.469$
Analytic rank $0$
Dimension $304$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,3,Mod(39,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.39");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.n (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: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 10 q^{2} - 22 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 10 q^{2} - 22 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 46 q^{10} - 20 q^{12} - 48 q^{13} - 30 q^{14} + 58 q^{16} - 80 q^{17} + 18 q^{18} - 220 q^{20} - 20 q^{21} - 68 q^{22} - 26 q^{24} + 384 q^{25} + 144 q^{26} - 274 q^{28} + 240 q^{29} - 120 q^{30} - 20 q^{33} + 186 q^{34} + 150 q^{36} - 12 q^{37} - 4 q^{38} - 116 q^{40} + 104 q^{41} + 128 q^{42} + 30 q^{44} + 192 q^{45} - 510 q^{46} + 146 q^{48} - 20 q^{49} - 600 q^{52} + 192 q^{53} + 296 q^{54} + 58 q^{56} - 412 q^{57} - 44 q^{58} - 612 q^{60} - 20 q^{61} - 10 q^{62} - 460 q^{64} - 828 q^{65} + 278 q^{66} - 196 q^{68} - 164 q^{69} - 1244 q^{70} + 44 q^{72} - 510 q^{74} - 906 q^{76} - 20 q^{77} + 570 q^{78} - 110 q^{80} - 864 q^{81} + 68 q^{82} + 1430 q^{84} + 748 q^{85} + 310 q^{86} + 640 q^{88} + 304 q^{89} + 2150 q^{90} + 912 q^{92} - 52 q^{93} + 62 q^{94} + 1380 q^{96} + 304 q^{97} + 124 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −1.99099 − 0.189578i 1.44150 − 1.44150i 3.92812 + 0.754899i 4.69954 + 6.46836i −3.14329 + 2.59673i 1.67033 + 3.27821i −7.67775 − 2.24769i 4.84418i −8.13049 − 13.7694i
39.2 −1.98832 − 0.215877i −2.84245 + 2.84245i 3.90679 + 0.858461i −0.466405 − 0.641951i 6.26531 − 5.03808i 5.33042 + 10.4615i −7.58262 − 2.55028i − 7.15909i 0.788778 + 1.37709i
39.3 −1.97579 + 0.310272i −2.24952 + 2.24952i 3.80746 − 1.22606i 0.190291 + 0.261913i 3.74661 − 5.14253i −3.66571 − 7.19435i −7.14232 + 3.60379i − 1.12068i −0.457239 − 0.458443i
39.4 −1.93755 − 0.495890i 4.18164 − 4.18164i 3.50819 + 1.92162i −1.99856 − 2.75078i −10.1758 + 6.02850i −0.135172 − 0.265289i −5.84437 − 5.46291i − 25.9723i 2.50822 + 6.32084i
39.5 −1.78082 + 0.910323i 2.24952 − 2.24952i 2.34263 − 3.24224i 0.190291 + 0.261913i −1.95820 + 6.05378i 3.66571 + 7.19435i −1.22031 + 7.90638i − 1.12068i −0.577300 − 0.293194i
39.6 −1.75072 − 0.966938i −0.0697249 + 0.0697249i 2.13006 + 3.38568i −0.0379555 − 0.0522413i 0.189489 − 0.0546493i −2.16753 − 4.25401i −0.455403 − 7.98703i 8.99028i 0.0159355 + 0.128161i
39.7 −1.68526 − 1.07698i 0.989033 − 0.989033i 1.68021 + 3.63000i −3.65044 − 5.02440i −2.73195 + 0.601607i −0.740377 − 1.45307i 1.07786 − 7.92706i 7.04363i 0.740747 + 12.3989i
39.8 −1.49932 + 1.32365i −1.44150 + 1.44150i 0.495905 − 3.96914i 4.69954 + 6.46836i 0.253224 − 4.06929i −1.67033 − 3.27821i 4.51023 + 6.60741i 4.84418i −15.6079 − 3.47758i
39.9 −1.48169 + 1.34335i 2.84245 − 2.84245i 0.390821 − 3.98086i −0.466405 − 0.641951i −0.393230 + 8.03005i −5.33042 − 10.4615i 4.76862 + 6.42342i − 7.15909i 1.55343 + 0.324629i
39.10 −1.44283 − 1.38501i −2.92028 + 2.92028i 0.163493 + 3.99666i 3.53020 + 4.85890i 8.25806 − 0.168838i −0.522596 − 1.02565i 5.29952 − 5.99292i − 8.05603i 1.63616 − 11.8999i
39.11 −1.27603 + 1.54005i −4.18164 + 4.18164i −0.743480 − 3.93030i −1.99856 − 2.75078i −1.10401 − 11.7758i 0.135172 + 0.265289i 7.00154 + 3.87019i − 25.9723i 6.78656 + 0.432214i
39.12 −1.11080 − 1.66316i 1.70364 − 1.70364i −1.53223 + 3.69490i 0.728137 + 1.00219i −4.72585 − 0.941024i 5.82261 + 11.4275i 7.84723 − 1.55596i 3.19522i 0.857997 − 2.32425i
39.13 −0.999784 − 1.73218i 3.16452 − 3.16452i −2.00087 + 3.46360i 5.16063 + 7.10300i −8.64534 − 2.31767i −5.60325 − 10.9970i 8.00000 + 0.00299873i − 11.0284i 7.14413 − 16.0406i
39.14 −0.848013 + 1.81132i 0.0697249 − 0.0697249i −2.56175 − 3.07204i −0.0379555 − 0.0522413i 0.0671664 + 0.185422i 2.16753 + 4.25401i 7.73684 − 2.03501i 8.99028i 0.126812 − 0.0244483i
39.15 −0.846420 − 1.81206i −2.77435 + 2.77435i −2.56715 + 3.06753i −4.56566 − 6.28409i 7.37556 + 2.67903i 2.76317 + 5.42303i 7.73145 + 2.05542i − 6.39401i −7.52271 + 13.5922i
39.16 −0.730370 + 1.86187i −0.989033 + 0.989033i −2.93312 − 2.71971i −3.65044 − 5.02440i −1.11909 − 2.56381i 0.740377 + 1.45307i 7.20600 − 3.47470i 7.04363i 12.0209 − 3.12698i
39.17 −0.466285 − 1.94489i −0.457998 + 0.457998i −3.56516 + 1.81374i −1.43199 − 1.97096i 1.10431 + 0.677196i −4.44878 − 8.73121i 5.18990 + 6.08810i 8.58048i −3.16558 + 3.70408i
39.18 −0.353182 + 1.96857i 2.92028 − 2.92028i −3.75052 − 1.39053i 3.53020 + 4.85890i 4.71738 + 6.78016i 0.522596 + 1.02565i 4.06197 − 6.89206i − 8.05603i −10.8119 + 5.23336i
39.19 −0.0663434 − 1.99890i 3.12766 − 3.12766i −3.99120 + 0.265228i −3.20481 − 4.41104i −6.45938 − 6.04438i 0.253371 + 0.497269i 0.794953 + 7.96041i − 10.5646i −8.60460 + 6.69873i
39.20 0.0789243 + 1.99844i −1.70364 + 1.70364i −3.98754 + 0.315451i 0.728137 + 1.00219i −3.53909 − 3.27017i −5.82261 − 11.4275i −0.945126 − 7.94397i 3.19522i −1.94536 + 1.53424i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
41.g even 20 1 inner
164.n odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.3.n.c ✓ 304
4.b odd 2 1 inner 164.3.n.c ✓ 304
41.g even 20 1 inner 164.3.n.c ✓ 304
164.n odd 20 1 inner 164.3.n.c ✓ 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.3.n.c ✓ 304 1.a even 1 1 trivial
164.3.n.c ✓ 304 4.b odd 2 1 inner
164.3.n.c ✓ 304 41.g even 20 1 inner
164.3.n.c ✓ 304 164.n odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(164, [\chi])\):

\( T_{3}^{304} + 19332 T_{3}^{300} + 180511450 T_{3}^{296} + 1084885705048 T_{3}^{292} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
\( T_{5}^{152} + 10 T_{5}^{151} - 521 T_{5}^{150} - 5540 T_{5}^{149} + 146707 T_{5}^{148} + \cdots + 83\!\cdots\!56 \) Copy content Toggle raw display