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Properties

Label 200.4.f.d
Level $200$
Weight $4$
Character orbit 200.f
Analytic conductor $11.800$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,4,Mod(149,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.149"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{4} + 18 q^{6} + 216 q^{9} + 224 q^{14} + 338 q^{16} + 570 q^{24} - 376 q^{26} - 528 q^{31} - 930 q^{34} - 1400 q^{36} + 600 q^{39} - 40 q^{41} - 766 q^{44} + 824 q^{46} - 456 q^{49} + 1674 q^{54}+ \cdots + 5470 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −2.79815 − 0.412714i −7.69300 7.65933 + 2.30967i 0 21.5262 + 3.17501i 15.6248i −20.4788 − 9.62394i 32.1823 0
149.2 −2.79815 + 0.412714i −7.69300 7.65933 − 2.30967i 0 21.5262 − 3.17501i − 15.6248i −20.4788 + 9.62394i 32.1823 0
149.3 −2.79756 − 0.416712i 2.73090 7.65270 + 2.33156i 0 −7.63987 − 1.13800i 31.2997i −20.4373 − 9.71165i −19.5422 0
149.4 −2.79756 + 0.416712i 2.73090 7.65270 − 2.33156i 0 −7.63987 + 1.13800i − 31.2997i −20.4373 + 9.71165i −19.5422 0
149.5 −2.15676 − 1.82985i 5.31349 1.30327 + 7.89313i 0 −11.4600 − 9.72291i 15.7169i 11.6324 − 19.4084i 1.23319 0
149.6 −2.15676 + 1.82985i 5.31349 1.30327 − 7.89313i 0 −11.4600 + 9.72291i − 15.7169i 11.6324 + 19.4084i 1.23319 0
149.7 −1.68939 − 2.26847i −1.26108 −2.29192 + 7.66467i 0 2.13045 + 2.86072i − 14.2186i 21.2590 − 7.74946i −25.4097 0
149.8 −1.68939 + 2.26847i −1.26108 −2.29192 − 7.66467i 0 2.13045 − 2.86072i 14.2186i 21.2590 + 7.74946i −25.4097 0
149.9 −0.673381 − 2.74710i −5.16961 −7.09312 + 3.69969i 0 3.48111 + 14.2014i − 7.07059i 14.9398 + 16.9942i −0.275157 0
149.10 −0.673381 + 2.74710i −5.16961 −7.09312 − 3.69969i 0 3.48111 − 14.2014i 7.07059i 14.9398 − 16.9942i −0.275157 0
149.11 −0.367242 − 2.80448i 9.63387 −7.73027 + 2.05985i 0 −3.53796 − 27.0180i 21.1900i 8.61568 + 20.9230i 65.8115 0
149.12 −0.367242 + 2.80448i 9.63387 −7.73027 − 2.05985i 0 −3.53796 + 27.0180i − 21.1900i 8.61568 − 20.9230i 65.8115 0
149.13 0.367242 − 2.80448i −9.63387 −7.73027 − 2.05985i 0 −3.53796 + 27.0180i 21.1900i −8.61568 + 20.9230i 65.8115 0
149.14 0.367242 + 2.80448i −9.63387 −7.73027 + 2.05985i 0 −3.53796 − 27.0180i − 21.1900i −8.61568 − 20.9230i 65.8115 0
149.15 0.673381 − 2.74710i 5.16961 −7.09312 − 3.69969i 0 3.48111 − 14.2014i − 7.07059i −14.9398 + 16.9942i −0.275157 0
149.16 0.673381 + 2.74710i 5.16961 −7.09312 + 3.69969i 0 3.48111 + 14.2014i 7.07059i −14.9398 − 16.9942i −0.275157 0
149.17 1.68939 − 2.26847i 1.26108 −2.29192 − 7.66467i 0 2.13045 − 2.86072i − 14.2186i −21.2590 − 7.74946i −25.4097 0
149.18 1.68939 + 2.26847i 1.26108 −2.29192 + 7.66467i 0 2.13045 + 2.86072i 14.2186i −21.2590 + 7.74946i −25.4097 0
149.19 2.15676 − 1.82985i −5.31349 1.30327 − 7.89313i 0 −11.4600 + 9.72291i 15.7169i −11.6324 − 19.4084i 1.23319 0
149.20 2.15676 + 1.82985i −5.31349 1.30327 + 7.89313i 0 −11.4600 − 9.72291i − 15.7169i −11.6324 + 19.4084i 1.23319 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.4.f.d 24
4.b odd 2 1 800.4.f.d 24
5.b even 2 1 inner 200.4.f.d 24
5.c odd 4 1 200.4.d.c ✓ 12
5.c odd 4 1 200.4.d.d yes 12
8.b even 2 1 inner 200.4.f.d 24
8.d odd 2 1 800.4.f.d 24
20.d odd 2 1 800.4.f.d 24
20.e even 4 1 800.4.d.b 12
20.e even 4 1 800.4.d.c 12
40.e odd 2 1 800.4.f.d 24
40.f even 2 1 inner 200.4.f.d 24
40.i odd 4 1 200.4.d.c ✓ 12
40.i odd 4 1 200.4.d.d yes 12
40.k even 4 1 800.4.d.b 12
40.k even 4 1 800.4.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.d.c ✓ 12 5.c odd 4 1
200.4.d.c ✓ 12 40.i odd 4 1
200.4.d.d yes 12 5.c odd 4 1
200.4.d.d yes 12 40.i odd 4 1
200.4.f.d 24 1.a even 1 1 trivial
200.4.f.d 24 5.b even 2 1 inner
200.4.f.d 24 8.b even 2 1 inner
200.4.f.d 24 40.f even 2 1 inner
800.4.d.b 12 20.e even 4 1
800.4.d.b 12 40.k even 4 1
800.4.d.c 12 20.e even 4 1
800.4.d.c 12 40.k even 4 1
800.4.f.d 24 4.b odd 2 1
800.4.f.d 24 8.d odd 2 1
800.4.f.d 24 20.d odd 2 1
800.4.f.d 24 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 216T_{3}^{10} + 16485T_{3}^{8} - 551120T_{3}^{6} + 8086707T_{3}^{4} - 42440280T_{3}^{2} + 49154791 \) acting on \(S_{4}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display