Properties

Label 1008.2.cz.g
Level $1008$
Weight $2$
Character orbit 1008.cz
Analytic conductor $8.049$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

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

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

\(\operatorname{Tr}(f)(q) = \) \( 24q - 3q^{3} - 3q^{5} + 4q^{7} + 17q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 3q^{3} - 3q^{5} + 4q^{7} + 17q^{9} - 9q^{11} - 3q^{13} - 6q^{15} - 3q^{17} - 4q^{19} + 13q^{21} - 6q^{23} + 15q^{25} + 9q^{27} + 18q^{29} + 34q^{31} - 21q^{33} - 42q^{35} - 3q^{37} + 27q^{39} + 36q^{41} + 24q^{43} + 21q^{45} - 42q^{47} + 30q^{49} - 6q^{51} - 12q^{53} - 30q^{55} - 13q^{57} - 12q^{59} - 3q^{63} + 6q^{69} + 48q^{73} + 36q^{75} - 48q^{77} - 31q^{81} - 48q^{83} - 21q^{85} + 15q^{87} + 39q^{89} + 9q^{91} + 10q^{93} + 3q^{97} + 30q^{99} + O(q^{100}) \)

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.

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 607.12
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\(T_{5}^{24} + \cdots\)
\(T_{11}^{24} + \cdots\)