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Properties

Label 147.3.n.b
Level $147$
Weight $3$
Character orbit 147.n
Analytic conductor $4.005$
Analytic rank $0$
Dimension $408$
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] = [147,3,Mod(2,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 26]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.n (of order \(42\), 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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(408\)
Relative dimension: \(34\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 408 q - 10 q^{3} - 94 q^{4} - 22 q^{6} - 42 q^{7} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 408 q - 10 q^{3} - 94 q^{4} - 22 q^{6} - 42 q^{7} + 42 q^{9} - 18 q^{10} + 17 q^{12} - 20 q^{13} + 2 q^{15} + 54 q^{16} - 31 q^{18} + 18 q^{19} - 42 q^{21} + 88 q^{22} - 44 q^{24} - 164 q^{25} - 31 q^{27} + 196 q^{28} - 9 q^{30} - 20 q^{31} - 64 q^{33} + 12 q^{34} - 250 q^{36} - 910 q^{37} + 7 q^{39} - 1062 q^{40} + 112 q^{42} - 220 q^{43} + 495 q^{45} - 272 q^{46} + 796 q^{48} + 126 q^{49} + 72 q^{51} + 480 q^{52} - 84 q^{54} + 852 q^{55} + 282 q^{57} + 126 q^{58} - 524 q^{60} + 284 q^{61} - 735 q^{63} + 464 q^{64} - 198 q^{66} - 14 q^{67} - 302 q^{69} + 448 q^{70} + 26 q^{72} + 8 q^{73} - 1194 q^{75} + 1100 q^{76} + 900 q^{78} - 56 q^{79} - 1190 q^{81} - 798 q^{82} - 1232 q^{84} + 620 q^{85} - 1064 q^{87} - 118 q^{88} - 1359 q^{90} - 112 q^{91} - 1178 q^{93} - 1124 q^{94} + 756 q^{96} + 316 q^{97} + 296 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.12696 + 3.65352i 1.91870 + 2.30621i −8.77323 − 5.98149i 0.888153 + 5.89251i −10.5881 + 4.41101i 2.20371 + 6.64407i 19.7836 − 15.7769i −1.63717 + 8.84984i −22.5293 − 3.39575i
2.2 −1.08259 + 3.50967i −2.76622 − 1.16106i −7.84083 − 5.34578i 0.298817 + 1.98252i 7.06960 − 8.45156i 0.658817 − 6.96893i 15.7642 − 12.5715i 6.30390 + 6.42346i −7.28148 − 1.09751i
2.3 −1.07223 + 3.47609i 2.92107 − 0.683641i −7.62857 − 5.20107i −1.20098 − 7.96799i −0.755663 + 10.8869i −5.56740 − 4.24312i 14.8827 − 11.8686i 8.06527 − 3.99392i 28.9852 + 4.36881i
2.4 −0.960499 + 3.11386i −1.13772 + 2.77589i −5.46861 − 3.72844i −1.20687 − 8.00703i −7.55096 − 6.20895i 6.76273 − 1.80705i 6.67162 − 5.32044i −6.41117 − 6.31640i 26.0920 + 3.93273i
2.5 −0.932175 + 3.02204i −0.0395900 − 2.99974i −4.95880 − 3.38085i 0.272129 + 1.80546i 9.10223 + 2.67664i −5.23386 + 4.64830i 4.94924 − 3.94689i −8.99687 + 0.237519i −5.70983 − 0.860619i
2.6 −0.871348 + 2.82484i 2.42878 − 1.76098i −3.91553 − 2.66956i 0.555437 + 3.68508i 2.85817 + 8.39534i 6.97873 − 0.545338i 1.70796 − 1.36205i 2.79791 − 8.55404i −10.8938 − 1.64197i
2.7 −0.784867 + 2.54448i −0.369837 + 2.97712i −2.55338 − 1.74087i 0.476597 + 3.16201i −7.28493 − 3.27768i −4.77319 − 5.12022i −1.89371 + 1.51018i −8.72644 − 2.20209i −8.41973 − 1.26907i
2.8 −0.768147 + 2.49027i −2.33630 − 1.88194i −2.30645 − 1.57251i −0.962174 − 6.38361i 6.48117 − 4.37242i 4.04889 + 5.71021i −2.46230 + 1.96362i 1.91660 + 8.79356i 16.6360 + 2.50747i
2.9 −0.655891 + 2.12635i −2.79535 + 1.08903i −0.786196 − 0.536019i 1.20976 + 8.02623i −0.482213 − 6.65818i 5.96035 + 3.67072i −5.30352 + 4.22942i 6.62801 − 6.08847i −17.8600 − 2.69196i
2.10 −0.619640 + 2.00882i 2.15472 + 2.08739i −0.346458 − 0.236211i −0.729124 − 4.83742i −5.52834 + 3.03502i −3.03239 + 6.30909i −5.88513 + 4.69324i 0.285621 + 8.99547i 10.1693 + 1.53278i
2.11 −0.492446 + 1.59647i −2.93645 + 0.614227i 0.998738 + 0.680928i −0.547555 − 3.63279i 0.465447 − 4.99043i −6.93551 + 0.947997i −6.80372 + 5.42578i 8.24545 − 3.60729i 6.06929 + 0.914798i
2.12 −0.461166 + 1.49506i 0.681711 − 2.92152i 1.28242 + 0.874337i −0.798583 − 5.29825i 4.05347 + 2.36650i 1.18892 − 6.89829i −6.79152 + 5.41606i −8.07054 − 3.98326i 8.28949 + 1.24944i
2.13 −0.421309 + 1.36585i 2.63073 + 1.44196i 1.61691 + 1.10239i −0.103834 − 0.688896i −3.07786 + 2.98567i 6.30417 − 3.04261i −6.65697 + 5.30876i 4.84148 + 7.58684i 0.984677 + 0.148416i
2.14 −0.384124 + 1.24530i 2.88815 − 0.811540i 1.90173 + 1.29658i 1.07865 + 7.15635i −0.0987975 + 3.90834i −6.94304 + 0.891213i −6.42066 + 5.12030i 7.68281 − 4.68769i −9.32614 − 1.40569i
2.15 −0.245875 + 0.797108i −2.37464 − 1.83333i 2.73003 + 1.86130i 0.692011 + 4.59120i 2.04523 − 1.44207i −4.59873 − 5.27747i −4.76362 + 3.79886i 2.27781 + 8.70699i −3.82983 − 0.577254i
2.16 −0.0418302 + 0.135610i −0.565930 − 2.94614i 3.28831 + 2.24194i 0.754596 + 5.00642i 0.423199 + 0.0464916i 2.90301 + 6.36966i −0.885394 + 0.706078i −8.35945 + 3.33461i −0.710486 − 0.107088i
2.17 −0.0161938 + 0.0524989i −2.52790 + 1.61546i 3.30246 + 2.25158i −0.837526 − 5.55662i −0.0438736 − 0.158873i 4.49519 − 5.36594i −0.343499 + 0.273932i 3.78058 − 8.16745i 0.305280 + 0.0460135i
2.18 0.0161938 − 0.0524989i 0.754293 + 2.90363i 3.30246 + 2.25158i 0.837526 + 5.55662i 0.164652 + 0.00742112i 4.49519 − 5.36594i 0.343499 − 0.273932i −7.86209 + 4.38037i 0.305280 + 0.0460135i
2.19 0.0418302 − 0.135610i 2.41874 − 1.77474i 3.28831 + 2.24194i −0.754596 − 5.00642i −0.139497 − 0.402243i 2.90301 + 6.36966i 0.885394 − 0.706078i 2.70059 − 8.58527i −0.710486 − 0.107088i
2.20 0.245875 − 0.797108i 2.98771 + 0.271238i 2.73003 + 1.86130i −0.692011 − 4.59120i 0.950811 − 2.31484i −4.59873 − 5.27747i 4.76362 − 3.79886i 8.85286 + 1.62076i −3.82983 − 0.577254i
See next 80 embeddings (of 408 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.g even 21 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.n.b ✓ 408
3.b odd 2 1 inner 147.3.n.b ✓ 408
49.g even 21 1 inner 147.3.n.b ✓ 408
147.n odd 42 1 inner 147.3.n.b ✓ 408
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.n.b ✓ 408 1.a even 1 1 trivial
147.3.n.b ✓ 408 3.b odd 2 1 inner
147.3.n.b ✓ 408 49.g even 21 1 inner
147.3.n.b ✓ 408 147.n odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{408} + 115 T_{2}^{406} + 6129 T_{2}^{404} + 190458 T_{2}^{402} + 3319707 T_{2}^{400} + \cdots + 57\!\cdots\!01 \) acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display