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Properties

Label 125.3.f.c
Level $125$
Weight $3$
Character orbit 125.f
Analytic conductor $3.406$
Analytic rank $0$
Dimension $32$
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] = [125,3,Mod(7,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 125.f (of order \(20\), degree \(8\), not 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: \(3.40600330450\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 25)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{2} + 10 q^{3} - 10 q^{4} - 6 q^{6} + 10 q^{7} + 10 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{2} + 10 q^{3} - 10 q^{4} - 6 q^{6} + 10 q^{7} + 10 q^{8} - 10 q^{9} - 6 q^{11} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 2 q^{16} - 60 q^{17} - 140 q^{18} + 90 q^{19} - 6 q^{21} - 70 q^{22} - 10 q^{23} + 4 q^{26} + 100 q^{27} + 250 q^{28} - 110 q^{29} - 6 q^{31} + 290 q^{32} + 190 q^{33} - 260 q^{34} - 58 q^{36} - 50 q^{37} - 320 q^{38} + 390 q^{39} - 86 q^{41} - 690 q^{42} - 230 q^{43} + 340 q^{44} - 6 q^{46} - 70 q^{47} - 160 q^{48} - 16 q^{51} + 320 q^{52} + 190 q^{53} - 660 q^{54} - 70 q^{56} + 650 q^{57} + 640 q^{58} - 260 q^{59} + 114 q^{61} - 60 q^{62} + 20 q^{63} + 340 q^{64} + 138 q^{66} - 270 q^{67} - 710 q^{68} + 340 q^{69} - 66 q^{71} - 360 q^{72} - 30 q^{73} - 80 q^{76} + 250 q^{77} + 500 q^{78} - 210 q^{79} + 62 q^{81} - 30 q^{82} - 10 q^{84} - 6 q^{86} - 300 q^{87} - 190 q^{88} - 10 q^{89} - 6 q^{91} + 30 q^{92} - 520 q^{93} + 790 q^{94} + 174 q^{96} - 270 q^{97} - 170 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.70026 + 1.37585i 4.42692 − 0.701156i 3.04732 − 4.19427i 0 −10.9892 + 7.98410i 4.77540 + 4.77540i −0.561510 + 3.54523i 10.5465 − 3.42677i 0
7.2 −0.972743 + 0.495637i −0.872241 + 0.138149i −1.65057 + 2.27181i 0 0.779995 − 0.566699i −1.62783 − 1.62783i 1.16272 − 7.34115i −7.81779 + 2.54015i 0
7.3 1.61837 − 0.824603i −3.42034 + 0.541729i −0.411975 + 0.567034i 0 −5.08868 + 3.69715i 8.06323 + 8.06323i −1.33571 + 8.43332i 2.84577 − 0.924645i 0
7.4 2.38234 − 1.21387i 3.57679 − 0.566508i 1.85096 − 2.54762i 0 7.83348 − 5.69136i −6.54971 − 6.54971i −0.355933 + 2.24727i 3.91299 − 1.27141i 0
18.1 −2.70026 − 1.37585i 4.42692 + 0.701156i 3.04732 + 4.19427i 0 −10.9892 − 7.98410i 4.77540 − 4.77540i −0.561510 − 3.54523i 10.5465 + 3.42677i 0
18.2 −0.972743 − 0.495637i −0.872241 − 0.138149i −1.65057 − 2.27181i 0 0.779995 + 0.566699i −1.62783 + 1.62783i 1.16272 + 7.34115i −7.81779 − 2.54015i 0
18.3 1.61837 + 0.824603i −3.42034 − 0.541729i −0.411975 − 0.567034i 0 −5.08868 − 3.69715i 8.06323 − 8.06323i −1.33571 − 8.43332i 2.84577 + 0.924645i 0
18.4 2.38234 + 1.21387i 3.57679 + 0.566508i 1.85096 + 2.54762i 0 7.83348 + 5.69136i −6.54971 + 6.54971i −0.355933 − 2.24727i 3.91299 + 1.27141i 0
32.1 −1.80600 − 0.286042i 0.665351 + 1.30583i −0.624420 − 0.202886i 0 −0.828102 − 2.54863i −3.62927 − 3.62927i 7.58652 + 3.86553i 4.02758 − 5.54349i 0
32.2 −0.287585 − 0.0455490i −1.72787 − 3.39113i −3.72360 − 1.20987i 0 0.342446 + 1.05394i 2.38950 + 2.38950i 2.05348 + 1.04630i −3.22416 + 4.43767i 0
32.3 1.86717 + 0.295731i 2.19472 + 4.30737i −0.405347 − 0.131705i 0 2.82409 + 8.69166i 3.57009 + 3.57009i −7.45551 − 3.79877i −8.44662 + 11.6258i 0
32.4 3.57427 + 0.566108i −1.61679 − 3.17313i 8.65068 + 2.81078i 0 −3.98250 − 12.2569i 0.574149 + 0.574149i 16.4311 + 8.37205i −2.16466 + 2.97940i 0
43.1 −1.80600 + 0.286042i 0.665351 − 1.30583i −0.624420 + 0.202886i 0 −0.828102 + 2.54863i −3.62927 + 3.62927i 7.58652 − 3.86553i 4.02758 + 5.54349i 0
43.2 −0.287585 + 0.0455490i −1.72787 + 3.39113i −3.72360 + 1.20987i 0 0.342446 − 1.05394i 2.38950 − 2.38950i 2.05348 − 1.04630i −3.22416 − 4.43767i 0
43.3 1.86717 − 0.295731i 2.19472 − 4.30737i −0.405347 + 0.131705i 0 2.82409 − 8.69166i 3.57009 − 3.57009i −7.45551 + 3.79877i −8.44662 − 11.6258i 0
43.4 3.57427 − 0.566108i −1.61679 + 3.17313i 8.65068 − 2.81078i 0 −3.98250 + 12.2569i 0.574149 − 0.574149i 16.4311 − 8.37205i −2.16466 − 2.97940i 0
82.1 −0.463000 − 2.92327i 0.866921 + 0.441718i −4.52691 + 1.47088i 0 0.889877 − 2.73876i −4.44588 − 4.44588i 1.02103 + 2.00389i −4.73363 − 6.51528i 0
82.2 −0.0933465 − 0.589367i −0.210730 − 0.107372i 3.46559 − 1.12604i 0 −0.0436108 + 0.134220i 7.64532 + 7.64532i −2.07076 − 4.06409i −5.25719 − 7.23590i 0
82.3 0.312579 + 1.97355i 4.02069 + 2.04864i 0.00704800 − 0.00229003i 0 −2.78631 + 8.57538i −3.91191 − 3.91191i 3.63528 + 7.13464i 6.67894 + 9.19277i 0
82.4 0.513943 + 3.24491i −2.81033 − 1.43193i −6.46108 + 2.09933i 0 3.20215 − 9.85519i −7.51823 − 7.51823i −4.16668 − 8.17758i 0.557438 + 0.767248i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.3.f.c 32
5.b even 2 1 25.3.f.a ✓ 32
5.c odd 4 1 125.3.f.a 32
5.c odd 4 1 125.3.f.b 32
15.d odd 2 1 225.3.r.a 32
20.d odd 2 1 400.3.bg.c 32
25.d even 5 1 125.3.f.b 32
25.e even 10 1 125.3.f.a 32
25.f odd 20 1 25.3.f.a ✓ 32
25.f odd 20 1 inner 125.3.f.c 32
75.l even 20 1 225.3.r.a 32
100.l even 20 1 400.3.bg.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.3.f.a ✓ 32 5.b even 2 1
25.3.f.a ✓ 32 25.f odd 20 1
125.3.f.a 32 5.c odd 4 1
125.3.f.a 32 25.e even 10 1
125.3.f.b 32 5.c odd 4 1
125.3.f.b 32 25.d even 5 1
125.3.f.c 32 1.a even 1 1 trivial
125.3.f.c 32 25.f odd 20 1 inner
225.3.r.a 32 15.d odd 2 1
225.3.r.a 32 75.l even 20 1
400.3.bg.c 32 20.d odd 2 1
400.3.bg.c 32 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 10 T_{2}^{31} + 55 T_{2}^{30} - 220 T_{2}^{29} + 612 T_{2}^{28} - 1380 T_{2}^{27} + \cdots + 12952801 \) acting on \(S_{3}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display