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Properties

Label 207.3.f.b
Level $207$
Weight $3$
Character orbit 207.f
Analytic conductor $5.640$
Analytic rank $0$
Dimension $80$
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] = [207,3,Mod(22,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.22");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.f (of order \(6\), degree \(2\), 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: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{2} - 6 q^{3} - 66 q^{4} - 42 q^{6} - 16 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{2} - 6 q^{3} - 66 q^{4} - 42 q^{6} - 16 q^{8} - 30 q^{9} + 30 q^{12} - 2 q^{13} - 74 q^{16} + 6 q^{18} - 154 q^{23} + 288 q^{24} + 344 q^{25} - 544 q^{26} - 180 q^{27} + 52 q^{29} - 32 q^{31} + 30 q^{32} - 108 q^{35} + 462 q^{36} + 42 q^{39} + 178 q^{41} + 112 q^{46} - 86 q^{47} - 660 q^{48} + 482 q^{49} + 46 q^{50} + 354 q^{52} + 246 q^{54} - 240 q^{55} + 332 q^{58} - 296 q^{59} + 380 q^{62} - 1048 q^{64} - 42 q^{69} - 132 q^{70} + 340 q^{71} + 258 q^{72} - 8 q^{73} - 84 q^{75} - 276 q^{77} + 492 q^{78} + 258 q^{81} - 448 q^{82} + 48 q^{85} - 1122 q^{87} - 252 q^{92} - 798 q^{93} - 214 q^{94} + 480 q^{95} + 792 q^{96} + 304 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −1.83872 − 3.18477i 1.13642 − 2.77643i −4.76182 + 8.24771i −6.89902 − 3.98315i −10.9318 + 1.48587i −7.54858 + 4.35817i 20.3129 −6.41711 − 6.31036i 29.2957i
22.2 −1.83872 − 3.18477i 1.13642 − 2.77643i −4.76182 + 8.24771i 6.89902 + 3.98315i −10.9318 + 1.48587i 7.54858 − 4.35817i 20.3129 −6.41711 − 6.31036i − 29.2957i
22.3 −1.72200 − 2.98260i 2.85242 + 0.929370i −3.93059 + 6.80798i −3.40010 − 1.96305i −2.13993 − 10.1080i 9.57295 − 5.52694i 13.2979 7.27254 + 5.30190i 13.5215i
22.4 −1.72200 − 2.98260i 2.85242 + 0.929370i −3.93059 + 6.80798i 3.40010 + 1.96305i −2.13993 − 10.1080i −9.57295 + 5.52694i 13.2979 7.27254 + 5.30190i − 13.5215i
22.5 −1.54722 − 2.67986i −0.177594 + 2.99474i −2.78776 + 4.82855i −0.0557816 − 0.0322055i 8.30026 − 4.15759i −5.56237 + 3.21144i 4.87537 −8.93692 − 1.06370i 0.199316i
22.6 −1.54722 − 2.67986i −0.177594 + 2.99474i −2.78776 + 4.82855i 0.0557816 + 0.0322055i 8.30026 − 4.15759i 5.56237 − 3.21144i 4.87537 −8.93692 − 1.06370i − 0.199316i
22.7 −1.44247 − 2.49843i −2.83232 − 0.988928i −2.16144 + 3.74373i −5.33590 − 3.08068i 1.61476 + 8.50285i −0.866425 + 0.500231i 0.931515 7.04404 + 5.60192i 17.7752i
22.8 −1.44247 − 2.49843i −2.83232 − 0.988928i −2.16144 + 3.74373i 5.33590 + 3.08068i 1.61476 + 8.50285i 0.866425 − 0.500231i 0.931515 7.04404 + 5.60192i − 17.7752i
22.9 −1.11584 − 1.93270i −2.17939 + 2.06162i −0.490207 + 0.849064i −8.45649 − 4.88236i 6.41633 + 1.91165i 3.20583 − 1.85089i −6.73876 0.499457 − 8.98613i 21.7918i
22.10 −1.11584 − 1.93270i −2.17939 + 2.06162i −0.490207 + 0.849064i 8.45649 + 4.88236i 6.41633 + 1.91165i −3.20583 + 1.85089i −6.73876 0.499457 − 8.98613i − 21.7918i
22.11 −1.01122 − 1.75148i 2.68387 − 1.34047i −0.0451293 + 0.0781662i −5.25503 − 3.03399i −5.06179 − 3.34524i 1.25968 − 0.727274i −7.90721 5.40627 − 7.19529i 12.2721i
22.12 −1.01122 − 1.75148i 2.68387 − 1.34047i −0.0451293 + 0.0781662i 5.25503 + 3.03399i −5.06179 − 3.34524i −1.25968 + 0.727274i −7.90721 5.40627 − 7.19529i − 12.2721i
22.13 −0.812658 − 1.40757i 1.87004 + 2.34583i 0.679174 − 1.17636i −4.73020 − 2.73098i 1.78221 − 4.53857i −7.20109 + 4.15755i −8.70901 −2.00587 + 8.77362i 8.87742i
22.14 −0.812658 − 1.40757i 1.87004 + 2.34583i 0.679174 − 1.17636i 4.73020 + 2.73098i 1.78221 − 4.53857i 7.20109 − 4.15755i −8.70901 −2.00587 + 8.77362i − 8.87742i
22.15 −0.740154 − 1.28198i 0.0693864 − 2.99920i 0.904345 − 1.56637i −2.25399 − 1.30134i −3.89628 + 2.13091i 10.4541 − 6.03568i −8.59865 −8.99037 − 0.416207i 3.85277i
22.16 −0.740154 − 1.28198i 0.0693864 − 2.99920i 0.904345 − 1.56637i 2.25399 + 1.30134i −3.89628 + 2.13091i −10.4541 + 6.03568i −8.59865 −8.99037 − 0.416207i − 3.85277i
22.17 −0.421673 − 0.730359i −2.99746 − 0.123520i 1.64438 − 2.84816i −0.289393 − 0.167081i 1.17373 + 2.24130i −9.32328 + 5.38280i −6.14695 8.96949 + 0.740494i 0.281814i
22.18 −0.421673 − 0.730359i −2.99746 − 0.123520i 1.64438 − 2.84816i 0.289393 + 0.167081i 1.17373 + 2.24130i 9.32328 − 5.38280i −6.14695 8.96949 + 0.740494i − 0.281814i
22.19 0.0530243 + 0.0918409i 2.98824 + 0.265327i 1.99438 − 3.45436i −2.63064 − 1.51880i 0.134082 + 0.288512i −3.92812 + 2.26790i 0.847197 8.85920 + 1.58572i − 0.322133i
22.20 0.0530243 + 0.0918409i 2.98824 + 0.265327i 1.99438 − 3.45436i 2.63064 + 1.51880i 0.134082 + 0.288512i 3.92812 − 2.26790i 0.847197 8.85920 + 1.58572i 0.322133i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
23.b odd 2 1 inner
207.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.3.f.b ✓ 80
3.b odd 2 1 621.3.f.b 80
9.c even 3 1 inner 207.3.f.b ✓ 80
9.d odd 6 1 621.3.f.b 80
23.b odd 2 1 inner 207.3.f.b ✓ 80
69.c even 2 1 621.3.f.b 80
207.f odd 6 1 inner 207.3.f.b ✓ 80
207.g even 6 1 621.3.f.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.3.f.b ✓ 80 1.a even 1 1 trivial
207.3.f.b ✓ 80 9.c even 3 1 inner
207.3.f.b ✓ 80 23.b odd 2 1 inner
207.3.f.b ✓ 80 207.f odd 6 1 inner
621.3.f.b 80 3.b odd 2 1
621.3.f.b 80 9.d odd 6 1
621.3.f.b 80 69.c even 2 1
621.3.f.b 80 207.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + T_{2}^{39} + 57 T_{2}^{38} + 54 T_{2}^{37} + 1890 T_{2}^{36} + 1722 T_{2}^{35} + \cdots + 97634161 \) acting on \(S_{3}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display