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Properties

Label 660.2.f.a
Level $660$
Weight $2$
Character orbit 660.f
Analytic conductor $5.270$
Analytic rank $0$
Dimension $40$
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] = [660,2,Mod(551,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.551");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.f (of order \(2\), degree \(1\), 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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{4} - 4 q^{6} + 2 q^{10} - 40 q^{11} + 12 q^{12} + 8 q^{13} + 16 q^{14} - 2 q^{16} - 16 q^{18} - 4 q^{21} - 8 q^{23} + 24 q^{24} - 40 q^{25} + 16 q^{26} - 60 q^{32} - 16 q^{34} + 16 q^{36} - 8 q^{37} + 16 q^{39} - 14 q^{40} - 36 q^{42} - 2 q^{44} + 4 q^{45} + 12 q^{46} + 4 q^{48} - 64 q^{49} - 12 q^{51} + 36 q^{52} - 10 q^{54} - 32 q^{56} + 36 q^{58} + 40 q^{59} + 8 q^{61} + 28 q^{62} + 80 q^{63} - 22 q^{64} + 4 q^{66} + 4 q^{69} - 8 q^{71} + 24 q^{73} + 20 q^{74} + 12 q^{76} - 8 q^{78} + 16 q^{80} + 20 q^{81} - 36 q^{82} + 32 q^{83} + 30 q^{84} - 16 q^{85} + 96 q^{86} - 4 q^{90} - 24 q^{92} - 48 q^{93} - 36 q^{94} + 32 q^{96} + 24 q^{97} - 60 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1 −1.41237 − 0.0721426i 0.180100 − 1.72266i 1.98959 + 0.203784i − 1.00000i −0.378646 + 2.42005i 4.09963i −2.79534 − 0.431353i −2.93513 − 0.620503i −0.0721426 + 1.41237i
551.2 −1.41237 + 0.0721426i 0.180100 + 1.72266i 1.98959 − 0.203784i 1.00000i −0.378646 − 2.42005i − 4.09963i −2.79534 + 0.431353i −2.93513 + 0.620503i −0.0721426 − 1.41237i
551.3 −1.37676 − 0.323326i −1.57838 + 0.713237i 1.79092 + 0.890282i 1.00000i 2.40366 − 0.471622i − 0.0535031i −2.17781 − 1.80475i 1.98259 − 2.25152i 0.323326 − 1.37676i
551.4 −1.37676 + 0.323326i −1.57838 − 0.713237i 1.79092 − 0.890282i − 1.00000i 2.40366 + 0.471622i 0.0535031i −2.17781 + 1.80475i 1.98259 + 2.25152i 0.323326 + 1.37676i
551.5 −1.34406 − 0.439893i 1.62490 − 0.599744i 1.61299 + 1.18249i − 1.00000i −2.44779 + 0.0913069i − 2.33680i −1.64778 − 2.29887i 2.28062 − 1.94905i −0.439893 + 1.34406i
551.6 −1.34406 + 0.439893i 1.62490 + 0.599744i 1.61299 − 1.18249i 1.00000i −2.44779 − 0.0913069i 2.33680i −1.64778 + 2.29887i 2.28062 + 1.94905i −0.439893 − 1.34406i
551.7 −1.30467 − 0.545748i 0.0894831 + 1.72974i 1.40432 + 1.42404i − 1.00000i 0.827254 − 2.30557i − 0.212817i −1.05500 − 2.62430i −2.98399 + 0.309565i −0.545748 + 1.30467i
551.8 −1.30467 + 0.545748i 0.0894831 − 1.72974i 1.40432 − 1.42404i 1.00000i 0.827254 + 2.30557i 0.212817i −1.05500 + 2.62430i −2.98399 − 0.309565i −0.545748 − 1.30467i
551.9 −1.01077 − 0.989110i 0.716877 − 1.57673i 0.0433224 + 1.99953i 1.00000i −2.28416 + 0.884648i 4.05235i 1.93397 − 2.06392i −1.97217 − 2.26065i 0.989110 − 1.01077i
551.10 −1.01077 + 0.989110i 0.716877 + 1.57673i 0.0433224 − 1.99953i − 1.00000i −2.28416 − 0.884648i − 4.05235i 1.93397 + 2.06392i −1.97217 + 2.26065i 0.989110 + 1.01077i
551.11 −0.943010 − 1.05391i −1.38758 + 1.03664i −0.221466 + 1.98770i − 1.00000i 2.40103 + 0.484828i 1.23559i 2.30371 − 1.64101i 0.850756 − 2.87684i −1.05391 + 0.943010i
551.12 −0.943010 + 1.05391i −1.38758 − 1.03664i −0.221466 − 1.98770i 1.00000i 2.40103 − 0.484828i − 1.23559i 2.30371 + 1.64101i 0.850756 + 2.87684i −1.05391 − 0.943010i
551.13 −0.609597 − 1.27608i −1.56907 − 0.733496i −1.25678 + 1.55579i 1.00000i 0.0204968 + 2.44940i − 4.94037i 2.75146 + 0.655356i 1.92397 + 2.30182i 1.27608 − 0.609597i
551.14 −0.609597 + 1.27608i −1.56907 + 0.733496i −1.25678 − 1.55579i − 1.00000i 0.0204968 − 2.44940i 4.94037i 2.75146 − 0.655356i 1.92397 − 2.30182i 1.27608 + 0.609597i
551.15 −0.507881 − 1.31987i 1.29820 − 1.14660i −1.48411 + 1.34067i − 1.00000i −2.17269 − 1.13111i − 1.73424i 2.52327 + 1.27794i 0.370623 − 2.97702i −1.31987 + 0.507881i
551.16 −0.507881 + 1.31987i 1.29820 + 1.14660i −1.48411 − 1.34067i 1.00000i −2.17269 + 1.13111i 1.73424i 2.52327 − 1.27794i 0.370623 + 2.97702i −1.31987 − 0.507881i
551.17 −0.353123 − 1.36942i 1.70975 − 0.277048i −1.75061 + 0.967145i 1.00000i −0.983146 − 2.24353i 0.0732801i 1.94261 + 2.05579i 2.84649 − 0.947365i 1.36942 − 0.353123i
551.18 −0.353123 + 1.36942i 1.70975 + 0.277048i −1.75061 − 0.967145i − 1.00000i −0.983146 + 2.24353i − 0.0732801i 1.94261 − 2.05579i 2.84649 + 0.947365i 1.36942 + 0.353123i
551.19 −0.161785 − 1.40493i −0.639661 + 1.60961i −1.94765 + 0.454592i 1.00000i 2.36487 + 0.638269i 3.68355i 0.953770 + 2.66277i −2.18167 − 2.05921i 1.40493 − 0.161785i
551.20 −0.161785 + 1.40493i −0.639661 − 1.60961i −1.94765 − 0.454592i − 1.00000i 2.36487 − 0.638269i − 3.68355i 0.953770 − 2.66277i −2.18167 + 2.05921i 1.40493 + 0.161785i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.f.a ✓ 40
3.b odd 2 1 660.2.f.b yes 40
4.b odd 2 1 660.2.f.b yes 40
12.b even 2 1 inner 660.2.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.f.a ✓ 40 1.a even 1 1 trivial
660.2.f.a ✓ 40 12.b even 2 1 inner
660.2.f.b yes 40 3.b odd 2 1
660.2.f.b yes 40 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{20} + 4 T_{23}^{19} - 230 T_{23}^{18} - 960 T_{23}^{17} + 20752 T_{23}^{16} + 91488 T_{23}^{15} + \cdots + 620232704 \) acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\). Copy content Toggle raw display