↑

Properties

Label 141.2.e.a
Level $141$
Weight $2$
Character orbit 141.e
Analytic conductor $1.126$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [141,2,Mod(4,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([0, 36]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 141.e (of order \(23\), degree \(22\), 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: \(1.12589066850\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(4\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{2} - 4 q^{3} - 10 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} - 18 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{2} - 4 q^{3} - 10 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} - 18 q^{8} - 4 q^{9} + 26 q^{10} + 11 q^{11} - 10 q^{12} - 12 q^{13} - 24 q^{14} - 4 q^{15} - 22 q^{16} - 16 q^{17} - 2 q^{18} - 20 q^{19} - 32 q^{20} - 2 q^{21} + 2 q^{22} + 56 q^{23} - 18 q^{24} - 30 q^{25} - 36 q^{26} - 4 q^{27} - 44 q^{28} - 28 q^{29} - 20 q^{30} - 28 q^{31} - 58 q^{32} + 11 q^{33} + 116 q^{34} + 32 q^{35} - 10 q^{36} + q^{37} - 38 q^{38} - 12 q^{39} + 42 q^{40} + 13 q^{41} - 24 q^{42} - 21 q^{43} + 20 q^{44} - 4 q^{45} + 124 q^{46} + 92 q^{47} + 70 q^{48} - 8 q^{49} + 14 q^{50} - 16 q^{51} - 8 q^{52} - 33 q^{53} - 2 q^{54} - 25 q^{55} + 156 q^{56} + 3 q^{57} + 84 q^{58} + 17 q^{59} - 32 q^{60} - 41 q^{61} - 108 q^{62} - 2 q^{63} - 118 q^{64} - 76 q^{65} - 44 q^{66} - 72 q^{67} - 24 q^{68} + 10 q^{69} + 6 q^{70} + 66 q^{71} - 18 q^{72} - 64 q^{73} - 108 q^{74} - 30 q^{75} - 117 q^{76} - 62 q^{77} + 33 q^{78} + 2 q^{79} + 168 q^{80} - 4 q^{81} + 48 q^{82} + 8 q^{83} + 94 q^{84} - 36 q^{85} + 84 q^{86} + 18 q^{87} - 54 q^{88} - 32 q^{89} + 26 q^{90} + 72 q^{91} + 199 q^{92} + 156 q^{93} + 138 q^{94} + 144 q^{95} + 80 q^{96} - 40 q^{97} + 29 q^{98} + 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.19461 − 0.614902i 0.460065 − 0.887885i 2.72938 + 1.65978i 0.484542 − 0.394205i −1.55563 + 1.66567i 2.41488 + 0.331918i −1.85809 − 1.98953i −0.576680 − 0.816970i −1.30578 + 0.567180i
4.2 −0.866341 − 0.242737i 0.460065 − 0.887885i −1.01721 − 0.618581i −2.97424 + 2.41973i −0.614096 + 0.657537i −3.74150 − 0.514258i 1.95929 + 2.09789i −0.576680 − 0.816970i 3.16407 − 1.37435i
4.3 0.0642270 + 0.0179955i 0.460065 − 0.887885i −1.70504 − 1.03686i 1.02138 − 0.830956i 0.0455266 − 0.0487470i 1.49751 + 0.205828i −0.181904 − 0.194771i −0.576680 − 0.816970i 0.0805538 − 0.0349894i
4.4 1.50284 + 0.421075i 0.460065 − 0.887885i 0.372373 + 0.226445i 0.811572 − 0.660263i 1.06527 − 1.14062i −0.0356795 − 0.00490404i −1.66627 − 1.78414i −0.576680 − 0.816970i 1.49768 − 0.650534i
7.1 −0.127689 + 1.86675i 0.962917 − 0.269797i −1.48708 − 0.204394i 0.709454 − 1.00507i 0.380689 + 1.83197i 1.36393 + 1.46041i −0.189942 + 0.914052i 0.854419 − 0.519584i 1.78562 + 1.45271i
7.2 −0.0733426 + 1.07223i 0.962917 − 0.269797i 0.837071 + 0.115053i −2.25609 + 3.19616i 0.218662 + 1.05226i −1.20115 − 1.28612i −0.622080 + 2.99361i 0.854419 − 0.519584i −3.26155 − 2.65347i
7.3 0.0915530 − 1.33846i 0.962917 − 0.269797i 0.198285 + 0.0272536i −0.328000 + 0.464670i −0.272954 − 1.31352i 0.468644 + 0.501795i 0.600539 − 2.88995i 0.854419 − 0.519584i 0.591912 + 0.481556i
7.4 0.188187 − 2.75119i 0.962917 − 0.269797i −5.55229 − 0.763144i −0.264175 + 0.374251i −0.561055 − 2.69995i −1.88351 − 2.01674i −2.02232 + 9.73192i 0.854419 − 0.519584i 0.979922 + 0.797226i
16.1 −1.93176 − 1.17473i −0.576680 − 0.816970i 1.43157 + 2.76281i −0.528434 + 2.54296i 0.154289 + 2.25563i 3.27366 + 0.917237i 0.171519 − 2.50752i −0.334880 + 0.942261i 4.00810 − 4.29162i
16.2 −0.777337 − 0.472709i −0.576680 − 0.816970i −0.539331 − 1.04086i −0.0594173 + 0.285932i 0.0620858 + 0.907663i −4.32537 − 1.21191i −0.196954 + 2.87937i −0.334880 + 0.942261i 0.181350 − 0.194178i
16.3 0.745699 + 0.453470i −0.576680 − 0.816970i −0.569697 − 1.09947i 0.310532 − 1.49436i −0.0595590 − 0.870721i 1.83936 + 0.515364i 0.192870 − 2.81965i −0.334880 + 0.942261i 0.909210 − 0.973526i
16.4 2.31107 + 1.40539i −0.576680 − 0.816970i 2.44578 + 4.72014i 0.547965 − 2.63695i −0.184585 − 2.69854i −2.69555 − 0.755256i −0.612114 + 8.94879i −0.334880 + 0.942261i 4.97234 − 5.32407i
25.1 −1.67843 + 1.36551i −0.917211 + 0.398401i 0.545617 − 2.62565i −1.68831 − 0.232053i 0.995458 − 1.92115i −0.639665 − 1.79984i 0.678649 + 1.30973i 0.682553 − 0.730836i 3.15058 − 1.91591i
25.2 0.340790 − 0.277253i −0.917211 + 0.398401i −0.367644 + 1.76920i −4.10077 − 0.563638i −0.202118 + 0.390071i 1.59336 + 4.48330i 0.769463 + 1.48500i 0.682553 − 0.730836i −1.55377 + 0.944869i
25.3 0.933048 − 0.759091i −0.917211 + 0.398401i −0.112553 + 0.541633i 2.91946 + 0.401270i −0.553379 + 1.06797i −0.217945 − 0.613238i 1.41289 + 2.72675i 0.682553 − 0.730836i 3.02860 − 1.84173i
25.4 1.94157 − 1.57958i −0.917211 + 0.398401i 0.867693 − 4.17557i −0.464144 − 0.0637952i −1.15152 + 2.22233i −0.349518 − 0.983449i −2.60793 − 5.03307i 0.682553 − 0.730836i −1.00194 + 0.609292i
28.1 −0.649679 + 1.82802i 0.203456 − 0.979084i −1.36816 − 1.11308i 0.115911 + 1.69456i 1.65761 + 1.00801i 1.06964 + 1.51534i −0.391621 + 0.238150i −0.917211 − 0.398401i −3.17300 − 0.889033i
28.2 −0.125131 + 0.352085i 0.203456 − 0.979084i 1.44312 + 1.17406i −0.281344 − 4.11311i 0.319262 + 0.194148i −0.733515 − 1.03915i −1.23247 + 0.749483i −0.917211 − 0.398401i 1.48337 + 0.415621i
28.3 0.197557 − 0.555872i 0.203456 − 0.979084i 1.28146 + 1.04254i 0.0695492 + 1.01677i −0.504051 − 0.306520i 1.50889 + 2.13760i 1.84078 − 1.11941i −0.917211 − 0.398401i 0.578936 + 0.162210i
28.4 0.622959 − 1.75284i 0.203456 − 0.979084i −1.13294 − 0.921716i 0.0845846 + 1.23658i −1.58943 − 0.966555i −2.37563 − 3.36550i 0.857467 − 0.521437i −0.917211 − 0.398401i 2.22022 + 0.622077i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.c even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.2.e.a ✓ 88
3.b odd 2 1 423.2.i.c 88
47.c even 23 1 inner 141.2.e.a ✓ 88
47.c even 23 1 6627.2.a.bg 44
47.d odd 46 1 6627.2.a.bf 44
141.h odd 46 1 423.2.i.c 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.e.a ✓ 88 1.a even 1 1 trivial
141.2.e.a ✓ 88 47.c even 23 1 inner
423.2.i.c 88 3.b odd 2 1
423.2.i.c 88 141.h odd 46 1
6627.2.a.bf 44 47.d odd 46 1
6627.2.a.bg 44 47.c even 23 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 2 T_{2}^{87} + 11 T_{2}^{86} + 28 T_{2}^{85} + 105 T_{2}^{84} + 294 T_{2}^{83} + \cdots + 2209 \) acting on \(S_{2}^{\mathrm{new}}(141, [\chi])\). Copy content Toggle raw display