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Properties

Label 1005.2.g.a
Level $1005$
Weight $2$
Character orbit 1005.g
Analytic conductor $8.025$
Analytic rank $0$
Dimension $46$
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] = [1005,2,Mod(401,1005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1005.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1005 = 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1005.g (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: \(8.02496540314\)
Analytic rank: \(0\)
Dimension: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 48 q^{4} - 46 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 48 q^{4} - 46 q^{5} + 8 q^{9} - 8 q^{11} + 36 q^{16} + 9 q^{18} - 20 q^{19} - 48 q^{20} - 4 q^{21} + 16 q^{22} - 8 q^{24} + 46 q^{25} + 6 q^{27} + 20 q^{32} - 20 q^{33} + 28 q^{36} - 16 q^{37} - 8 q^{38} + 12 q^{39} + 28 q^{41} - 34 q^{42} + 20 q^{44} - 8 q^{45} + 9 q^{48} - 22 q^{49} - 8 q^{51} - 24 q^{53} - 8 q^{54} + 8 q^{55} + 20 q^{57} - 10 q^{63} + 16 q^{64} - 8 q^{66} - 28 q^{67} + 4 q^{69} + 22 q^{72} + 8 q^{73} + 8 q^{74} - 64 q^{76} + 49 q^{78} - 36 q^{80} + 20 q^{81} - 12 q^{82} - 17 q^{84} + 50 q^{87} + 4 q^{88} - 9 q^{90} + 28 q^{91} + 2 q^{93} + 20 q^{95} - 49 q^{96} - 16 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1 −2.66731 0.285202 + 1.70841i 5.11454 −1.00000 −0.760721 − 4.55685i 0.896523i −8.30744 −2.83732 + 0.974482i 2.66731
401.2 −2.66731 0.285202 − 1.70841i 5.11454 −1.00000 −0.760721 + 4.55685i − 0.896523i −8.30744 −2.83732 − 0.974482i 2.66731
401.3 −2.52072 1.70185 − 0.322009i 4.35404 −1.00000 −4.28990 + 0.811695i − 3.33211i −5.93388 2.79262 − 1.09602i 2.52072
401.4 −2.52072 1.70185 + 0.322009i 4.35404 −1.00000 −4.28990 − 0.811695i 3.33211i −5.93388 2.79262 + 1.09602i 2.52072
401.5 −2.32029 −1.72316 + 0.175226i 3.38375 −1.00000 3.99824 − 0.406576i − 3.13277i −3.21070 2.93859 − 0.603887i 2.32029
401.6 −2.32029 −1.72316 − 0.175226i 3.38375 −1.00000 3.99824 + 0.406576i 3.13277i −3.21070 2.93859 + 0.603887i 2.32029
401.7 −2.19117 −1.11268 − 1.32738i 2.80121 −1.00000 2.43807 + 2.90851i 4.21796i −1.75558 −0.523884 + 2.95390i 2.19117
401.8 −2.19117 −1.11268 + 1.32738i 2.80121 −1.00000 2.43807 − 2.90851i − 4.21796i −1.75558 −0.523884 − 2.95390i 2.19117
401.9 −1.88925 −0.963495 + 1.43933i 1.56928 −1.00000 1.82029 − 2.71926i 2.68731i 0.813737 −1.14336 − 2.77358i 1.88925
401.10 −1.88925 −0.963495 − 1.43933i 1.56928 −1.00000 1.82029 + 2.71926i − 2.68731i 0.813737 −1.14336 + 2.77358i 1.88925
401.11 −1.57525 1.73156 + 0.0410832i 0.481403 −1.00000 −2.72764 − 0.0647161i − 1.96979i 2.39217 2.99662 + 0.142276i 1.57525
401.12 −1.57525 1.73156 − 0.0410832i 0.481403 −1.00000 −2.72764 + 0.0647161i 1.96979i 2.39217 2.99662 − 0.142276i 1.57525
401.13 −1.50390 0.269913 − 1.71089i 0.261705 −1.00000 −0.405922 + 2.57300i − 0.213008i 2.61422 −2.85429 − 0.923585i 1.50390
401.14 −1.50390 0.269913 + 1.71089i 0.261705 −1.00000 −0.405922 − 2.57300i 0.213008i 2.61422 −2.85429 + 0.923585i 1.50390
401.15 −1.46364 0.833877 − 1.51811i 0.142257 −1.00000 −1.22050 + 2.22197i 2.73332i 2.71908 −1.60930 − 2.53183i 1.46364
401.16 −1.46364 0.833877 + 1.51811i 0.142257 −1.00000 −1.22050 − 2.22197i − 2.73332i 2.71908 −1.60930 + 2.53183i 1.46364
401.17 −0.830690 −1.69013 − 0.378750i −1.30995 −1.00000 1.40398 + 0.314624i − 1.89932i 2.74955 2.71310 + 1.28028i 0.830690
401.18 −0.830690 −1.69013 + 0.378750i −1.30995 −1.00000 1.40398 − 0.314624i 1.89932i 2.74955 2.71310 − 1.28028i 0.830690
401.19 −0.415407 −0.406382 + 1.68370i −1.82744 −1.00000 0.168814 − 0.699421i − 3.65614i 1.58994 −2.66971 − 1.36845i 0.415407
401.20 −0.415407 −0.406382 − 1.68370i −1.82744 −1.00000 0.168814 + 0.699421i 3.65614i 1.58994 −2.66971 + 1.36845i 0.415407
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1005.2.g.a ✓ 46
3.b odd 2 1 1005.2.g.b yes 46
67.b odd 2 1 1005.2.g.b yes 46
201.d even 2 1 inner 1005.2.g.a ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1005.2.g.a ✓ 46 1.a even 1 1 trivial
1005.2.g.a ✓ 46 201.d even 2 1 inner
1005.2.g.b yes 46 3.b odd 2 1
1005.2.g.b yes 46 67.b odd 2 1