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Properties

Label 1008.2.cz.g
Level $1008$
Weight $2$
Character orbit 1008.cz
Analytic conductor $8.049$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(367,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.367"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cz (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,-3,0,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{3} - 3 q^{5} + 4 q^{7} + 17 q^{9} - 9 q^{11} - 3 q^{13} - 6 q^{15} - 3 q^{17} - 4 q^{19} + 13 q^{21} - 6 q^{23} + 15 q^{25} + 9 q^{27} + 18 q^{29} + 34 q^{31} - 21 q^{33} - 42 q^{35} - 3 q^{37}+ \cdots + 30 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1 0 −1.72726 + 0.128734i 0 2.43562 + 1.40621i 0 −0.717569 + 2.54659i 0 2.96686 − 0.444713i 0
367.2 0 −1.62154 − 0.608772i 0 1.53539 + 0.886460i 0 2.64391 − 0.0987102i 0 2.25879 + 1.97430i 0
367.3 0 −1.60558 − 0.649697i 0 −2.10605 − 1.21593i 0 0.347128 − 2.62288i 0 2.15579 + 2.08628i 0
367.4 0 −1.42297 + 0.987503i 0 −0.679706 − 0.392428i 0 −2.49091 − 0.891831i 0 1.04968 − 2.81037i 0
367.5 0 −0.908151 + 1.47488i 0 −0.243063 − 0.140332i 0 2.20451 − 1.46292i 0 −1.35052 − 2.67882i 0
367.6 0 −0.706216 − 1.58154i 0 −1.27943 − 0.738680i 0 1.34028 + 2.28115i 0 −2.00252 + 2.23381i 0
367.7 0 −0.597950 − 1.62556i 0 2.47441 + 1.42860i 0 −2.59845 − 0.498066i 0 −2.28491 + 1.94401i 0
367.8 0 1.02195 + 1.39843i 0 −3.67461 − 2.12154i 0 2.04787 − 1.67518i 0 −0.911216 + 2.85827i 0
367.9 0 1.29235 − 1.15318i 0 −2.47996 − 1.43180i 0 1.38867 + 2.25202i 0 0.340334 − 2.98063i 0
367.10 0 1.43691 + 0.967101i 0 1.73177 + 0.999840i 0 2.38101 + 1.15360i 0 1.12943 + 2.77928i 0
367.11 0 1.64174 − 0.551995i 0 2.87107 + 1.65762i 0 −2.01336 + 1.71651i 0 2.39060 − 1.81246i 0
367.12 0 1.69672 + 0.348077i 0 −2.08545 − 1.20403i 0 −2.53309 + 0.763830i 0 2.75769 + 1.18117i 0
607.1 0 −1.72726 − 0.128734i 0 2.43562 − 1.40621i 0 −0.717569 − 2.54659i 0 2.96686 + 0.444713i 0
607.2 0 −1.62154 + 0.608772i 0 1.53539 − 0.886460i 0 2.64391 + 0.0987102i 0 2.25879 − 1.97430i 0
607.3 0 −1.60558 + 0.649697i 0 −2.10605 + 1.21593i 0 0.347128 + 2.62288i 0 2.15579 − 2.08628i 0
607.4 0 −1.42297 − 0.987503i 0 −0.679706 + 0.392428i 0 −2.49091 + 0.891831i 0 1.04968 + 2.81037i 0
607.5 0 −0.908151 − 1.47488i 0 −0.243063 + 0.140332i 0 2.20451 + 1.46292i 0 −1.35052 + 2.67882i 0
607.6 0 −0.706216 + 1.58154i 0 −1.27943 + 0.738680i 0 1.34028 − 2.28115i 0 −2.00252 − 2.23381i 0
607.7 0 −0.597950 + 1.62556i 0 2.47441 − 1.42860i 0 −2.59845 + 0.498066i 0 −2.28491 − 1.94401i 0
607.8 0 1.02195 − 1.39843i 0 −3.67461 + 2.12154i 0 2.04787 + 1.67518i 0 −0.911216 − 2.85827i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.bj even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cz.g yes 24
3.b odd 2 1 3024.2.cz.h 24
4.b odd 2 1 1008.2.cz.h yes 24
7.d odd 6 1 1008.2.bf.g ✓ 24
9.c even 3 1 1008.2.bf.h yes 24
9.d odd 6 1 3024.2.bf.g 24
12.b even 2 1 3024.2.cz.g 24
21.g even 6 1 3024.2.bf.h 24
28.f even 6 1 1008.2.bf.h yes 24
36.f odd 6 1 1008.2.bf.g ✓ 24
36.h even 6 1 3024.2.bf.h 24
63.i even 6 1 3024.2.cz.g 24
63.t odd 6 1 1008.2.cz.h yes 24
84.j odd 6 1 3024.2.bf.g 24
252.r odd 6 1 3024.2.cz.h 24
252.bj even 6 1 inner 1008.2.cz.g yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.g ✓ 24 7.d odd 6 1
1008.2.bf.g ✓ 24 36.f odd 6 1
1008.2.bf.h yes 24 9.c even 3 1
1008.2.bf.h yes 24 28.f even 6 1
1008.2.cz.g yes 24 1.a even 1 1 trivial
1008.2.cz.g yes 24 252.bj even 6 1 inner
1008.2.cz.h yes 24 4.b odd 2 1
1008.2.cz.h yes 24 63.t odd 6 1
3024.2.bf.g 24 9.d odd 6 1
3024.2.bf.g 24 84.j odd 6 1
3024.2.bf.h 24 21.g even 6 1
3024.2.bf.h 24 36.h even 6 1
3024.2.cz.g 24 12.b even 2 1
3024.2.cz.g 24 63.i even 6 1
3024.2.cz.h 24 3.b odd 2 1
3024.2.cz.h 24 252.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{24} + 3 T_{5}^{23} - 33 T_{5}^{22} - 108 T_{5}^{21} + 723 T_{5}^{20} + 2178 T_{5}^{19} + \cdots + 4782969 \) Copy content Toggle raw display
\( T_{11}^{24} + 9 T_{11}^{23} - 36 T_{11}^{22} - 567 T_{11}^{21} + 984 T_{11}^{20} + 20682 T_{11}^{19} + \cdots + 109592116209 \) Copy content Toggle raw display