Properties

Label 100.2.l.b
Level $100$
Weight $2$
Character orbit 100.l
Analytic conductor $0.799$
Analytic rank $0$
Dimension $96$
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] = [100,2,Mod(3,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 100.l (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: \(0.798504020213\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 10 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 10 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 20 q^{9} - 10 q^{10} - 10 q^{12} - 20 q^{13} - 10 q^{14} - 14 q^{16} - 20 q^{17} + 20 q^{18} + 10 q^{20} - 12 q^{21} - 10 q^{22} - 20 q^{25} - 12 q^{26} - 10 q^{28} - 20 q^{29} - 10 q^{30} - 50 q^{32} - 20 q^{33} - 60 q^{34} - 10 q^{36} + 40 q^{37} + 20 q^{38} + 40 q^{40} - 28 q^{41} + 90 q^{42} + 60 q^{44} - 20 q^{45} - 6 q^{46} + 120 q^{48} + 80 q^{50} + 80 q^{52} - 40 q^{53} + 120 q^{54} - 6 q^{56} - 20 q^{57} + 60 q^{58} + 90 q^{60} + 12 q^{61} + 40 q^{62} + 20 q^{64} - 100 q^{65} - 30 q^{66} - 10 q^{68} - 20 q^{69} - 10 q^{70} - 40 q^{72} - 20 q^{73} - 20 q^{77} + 20 q^{78} - 10 q^{80} - 36 q^{81} - 50 q^{82} - 90 q^{84} + 100 q^{85} - 6 q^{86} - 130 q^{88} + 160 q^{89} - 160 q^{90} - 110 q^{92} + 60 q^{93} - 170 q^{94} + 14 q^{96} + 180 q^{97} - 130 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41421 0.000293203i −1.30110 + 0.206075i 2.00000 0.000829304i −2.22277 + 0.243473i 1.84010 0.291052i −1.76525 1.76525i −2.82843 0.00175922i −1.20277 + 0.390802i 3.14355 0.343670i
3.2 −1.16222 + 0.805750i 1.77314 0.280838i 0.701533 1.87293i 1.24326 1.85858i −1.83450 + 1.75511i −2.22547 2.22547i 0.693772 + 2.74202i 0.211989 0.0688795i 0.0526065 + 3.16184i
3.3 −1.15706 0.813156i 1.79631 0.284507i 0.677554 + 1.88173i −0.456295 2.18902i −2.30978 1.13149i 2.23536 + 2.23536i 0.746176 2.72823i 0.292613 0.0950758i −1.25205 + 2.90385i
3.4 −1.06026 0.935866i −1.97842 + 0.313350i 0.248311 + 1.98453i 1.82539 + 1.29150i 2.39089 + 1.51930i 1.00369 + 1.00369i 1.59398 2.33650i 0.962775 0.312824i −0.726717 3.07764i
3.5 −0.934812 + 1.06119i 0.173581 0.0274925i −0.252253 1.98403i −0.138125 + 2.23180i −0.133091 + 0.209903i 2.95323 + 2.95323i 2.34124 + 1.58700i −2.82379 + 0.917507i −2.23924 2.23289i
3.6 −0.133925 1.40786i 1.97842 0.313350i −1.96413 + 0.377095i 1.82539 + 1.29150i −0.706112 2.74336i −1.00369 1.00369i 0.793941 + 2.71471i 0.962775 0.312824i 1.57378 2.74285i
3.7 0.0222429 1.41404i −1.79631 + 0.284507i −1.99901 0.0629046i −0.456295 2.18902i 0.362349 + 2.54638i −2.23536 2.23536i −0.133413 + 2.82528i 0.292613 0.0950758i −3.10550 + 0.596529i
3.8 0.189713 + 1.40143i 3.09018 0.489437i −1.92802 + 0.531739i −2.06046 + 0.868611i 1.27216 + 4.23782i −1.45172 1.45172i −1.11097 2.60111i 6.45651 2.09785i −1.60820 2.72281i
3.9 0.831017 1.14430i 1.30110 0.206075i −0.618823 1.90186i −2.22277 + 0.243473i 0.845429 1.66010i 1.76525 + 1.76525i −2.69054 0.872359i −1.20277 + 0.390802i −1.56856 + 2.74584i
3.10 1.02227 + 0.977222i −3.09018 + 0.489437i 0.0900763 + 1.99797i −2.06046 + 0.868611i −3.63729 2.51946i 1.45172 + 1.45172i −1.86038 + 2.13049i 6.45651 2.09785i −2.95518 1.12557i
3.11 1.33500 0.466651i −1.77314 + 0.280838i 1.56447 1.24596i 1.24326 1.85858i −2.23610 + 1.20236i 2.22547 + 2.22547i 1.50715 2.39343i 0.211989 0.0688795i 0.792444 3.06138i
3.12 1.40799 0.132526i −0.173581 + 0.0274925i 1.96487 0.373191i −0.138125 + 2.23180i −0.240757 + 0.0617133i −2.95323 2.95323i 2.71707 0.785847i −2.82379 + 0.917507i 0.101293 + 3.16065i
23.1 −1.41083 0.0976968i 0.0867473 + 0.170251i 1.98091 + 0.275668i −1.18066 + 1.89896i −0.105753 0.248671i 1.89427 + 1.89427i −2.76781 0.582451i 1.74190 2.39751i 1.85124 2.56377i
23.2 −1.35937 + 0.390031i 1.41506 + 2.77721i 1.69575 1.06039i 1.36636 1.77005i −3.00678 3.22332i 0.467533 + 0.467533i −1.89156 + 2.10285i −3.94714 + 5.43277i −1.16701 + 2.93906i
23.3 −1.06970 + 0.925063i −0.458745 0.900337i 0.288515 1.97908i −1.53117 1.62957i 1.32359 + 0.538722i −2.42430 2.42430i 1.52215 + 2.38392i 1.16320 1.60100i 3.14535 + 0.326722i
23.4 −0.776413 1.18202i −1.25043 2.45410i −0.794365 + 1.83548i −2.21556 0.302124i −1.92996 + 3.38344i 1.05978 + 1.05978i 2.78634 0.486130i −2.69570 + 3.71031i 1.36307 + 2.85342i
23.5 −0.643767 + 1.25919i −0.700193 1.37421i −1.17113 1.62125i 2.14271 + 0.639373i 2.18115 + 0.00299155i 3.31824 + 3.31824i 2.79540 0.430965i 0.365182 0.502631i −2.18450 + 2.28647i
23.6 −0.481662 1.32966i 0.956383 + 1.87701i −1.53600 + 1.28090i 0.727343 + 2.11447i 2.03513 2.17575i 0.0806175 + 0.0806175i 2.44299 + 1.42541i −0.845131 + 1.16322i 2.46119 1.98558i
23.7 0.223147 + 1.39650i 0.700193 + 1.37421i −1.90041 + 0.623249i 2.14271 + 0.639373i −1.76283 + 1.28447i −3.31824 3.31824i −1.29444 2.51484i 0.365182 0.502631i −0.414743 + 3.13496i
23.8 0.731485 + 1.21034i 0.458745 + 0.900337i −0.929861 + 1.77069i −1.53117 1.62957i −0.754152 + 1.21382i 2.42430 + 2.42430i −2.82333 + 0.169786i 1.16320 1.60100i 0.852314 3.04525i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.2.l.b 96
3.b odd 2 1 900.2.bj.d 96
4.b odd 2 1 inner 100.2.l.b 96
5.b even 2 1 500.2.l.f 96
5.c odd 4 1 500.2.l.d 96
5.c odd 4 1 500.2.l.e 96
12.b even 2 1 900.2.bj.d 96
20.d odd 2 1 500.2.l.f 96
20.e even 4 1 500.2.l.d 96
20.e even 4 1 500.2.l.e 96
25.d even 5 1 500.2.l.e 96
25.e even 10 1 500.2.l.d 96
25.f odd 20 1 inner 100.2.l.b 96
25.f odd 20 1 500.2.l.f 96
75.l even 20 1 900.2.bj.d 96
100.h odd 10 1 500.2.l.d 96
100.j odd 10 1 500.2.l.e 96
100.l even 20 1 inner 100.2.l.b 96
100.l even 20 1 500.2.l.f 96
300.u odd 20 1 900.2.bj.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.l.b 96 1.a even 1 1 trivial
100.2.l.b 96 4.b odd 2 1 inner
100.2.l.b 96 25.f odd 20 1 inner
100.2.l.b 96 100.l even 20 1 inner
500.2.l.d 96 5.c odd 4 1
500.2.l.d 96 20.e even 4 1
500.2.l.d 96 25.e even 10 1
500.2.l.d 96 100.h odd 10 1
500.2.l.e 96 5.c odd 4 1
500.2.l.e 96 20.e even 4 1
500.2.l.e 96 25.d even 5 1
500.2.l.e 96 100.j odd 10 1
500.2.l.f 96 5.b even 2 1
500.2.l.f 96 20.d odd 2 1
500.2.l.f 96 25.f odd 20 1
500.2.l.f 96 100.l even 20 1
900.2.bj.d 96 3.b odd 2 1
900.2.bj.d 96 12.b even 2 1
900.2.bj.d 96 75.l even 20 1
900.2.bj.d 96 300.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} + 10 T_{3}^{94} - 103 T_{3}^{92} - 1600 T_{3}^{90} + 7218 T_{3}^{88} + 167890 T_{3}^{86} - 499696 T_{3}^{84} - 13339180 T_{3}^{82} + 61015935 T_{3}^{80} + 1192607560 T_{3}^{78} + \cdots + 75\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display