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Properties

Label 230.4.g.a
Level $230$
Weight $4$
Character orbit 230.g
Analytic conductor $13.570$
Analytic rank $0$
Dimension $50$
Inner twists $2$

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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] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [50,-10,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 10 q^{2} - 5 q^{3} - 20 q^{4} + 25 q^{5} - 10 q^{6} - 19 q^{7} - 40 q^{8} - 126 q^{9} + 50 q^{10} - 61 q^{11} + 156 q^{12} + 312 q^{13} - 16 q^{14} + 25 q^{15} - 80 q^{16} + 114 q^{17} - 252 q^{18}+ \cdots - 10743 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.284630 + 1.97964i −4.01390 − 8.78921i −3.83797 − 1.12693i 3.27430 + 3.77875i 18.5420 − 5.44441i 6.54570 + 4.20667i 3.32332 − 7.27706i −43.4576 + 50.1527i −8.41254 + 5.40641i
31.2 −0.284630 + 1.97964i −2.18029 − 4.77416i −3.83797 − 1.12693i 3.27430 + 3.77875i 10.0717 − 2.95732i −14.2452 − 9.15482i 3.32332 − 7.27706i −0.357709 + 0.412819i −8.41254 + 5.40641i
31.3 −0.284630 + 1.97964i 0.678933 + 1.48666i −3.83797 − 1.12693i 3.27430 + 3.77875i −3.13629 + 0.920899i −9.83778 − 6.32236i 3.32332 − 7.27706i 15.9320 − 18.3866i −8.41254 + 5.40641i
31.4 −0.284630 + 1.97964i 1.00192 + 2.19390i −3.83797 − 1.12693i 3.27430 + 3.77875i −4.62832 + 1.35900i 20.7474 + 13.3335i 3.32332 − 7.27706i 13.8719 − 16.0090i −8.41254 + 5.40641i
31.5 −0.284630 + 1.97964i 2.79412 + 6.11828i −3.83797 − 1.12693i 3.27430 + 3.77875i −12.9073 + 3.78992i −7.74472 − 4.97723i 3.32332 − 7.27706i −11.9450 + 13.7852i −8.41254 + 5.40641i
41.1 −1.91899 − 0.563465i −3.58270 + 4.13466i 3.36501 + 2.16256i 0.711574 − 4.94911i 9.20488 − 5.91562i 0.232124 + 0.508281i −5.23889 − 6.04600i −0.417140 − 2.90127i −4.15415 + 9.09632i
41.2 −1.91899 − 0.563465i −0.118925 + 0.137247i 3.36501 + 2.16256i 0.711574 − 4.94911i 0.305550 − 0.196365i 0.694472 + 1.52068i −5.23889 − 6.04600i 3.83781 + 26.6925i −4.15415 + 9.09632i
41.3 −1.91899 − 0.563465i 1.08156 − 1.24819i 3.36501 + 2.16256i 0.711574 − 4.94911i −2.77882 + 1.78584i 0.565922 + 1.23920i −5.23889 − 6.04600i 3.45430 + 24.0252i −4.15415 + 9.09632i
41.4 −1.91899 − 0.563465i 4.22541 − 4.87638i 3.36501 + 2.16256i 0.711574 − 4.94911i −10.8562 + 6.97683i −15.2702 − 33.4370i −5.23889 − 6.04600i −2.08251 − 14.4842i −4.15415 + 9.09632i
41.5 −1.91899 − 0.563465i 5.38591 − 6.21567i 3.36501 + 2.16256i 0.711574 − 4.94911i −13.8378 + 8.89302i 7.81863 + 17.1204i −5.23889 − 6.04600i −5.78405 − 40.2289i −4.15415 + 9.09632i
71.1 1.68251 + 1.08128i −1.06222 − 7.38793i 1.66166 + 3.63853i 4.79746 + 1.40866i 6.20123 − 13.5788i 2.08726 − 2.40883i −1.13852 + 7.91857i −27.5468 + 8.08848i 6.54861 + 7.55750i
71.2 1.68251 + 1.08128i −0.937040 − 6.51725i 1.66166 + 3.63853i 4.79746 + 1.40866i 5.47041 − 11.9785i −20.6340 + 23.8129i −1.13852 + 7.91857i −15.6903 + 4.60708i 6.54861 + 7.55750i
71.3 1.68251 + 1.08128i 0.0393624 + 0.273772i 1.66166 + 3.63853i 4.79746 + 1.40866i −0.229797 + 0.503185i −2.34264 + 2.70355i −1.13852 + 7.91857i 25.8329 − 7.58523i 6.54861 + 7.55750i
71.4 1.68251 + 1.08128i 0.339971 + 2.36455i 1.66166 + 3.63853i 4.79746 + 1.40866i −1.98474 + 4.34597i 21.3773 − 24.6707i −1.13852 + 7.91857i 20.4308 − 5.99902i 6.54861 + 7.55750i
71.5 1.68251 + 1.08128i 1.08909 + 7.57480i 1.66166 + 3.63853i 4.79746 + 1.40866i −6.35809 + 13.9223i −6.64671 + 7.67071i −1.13852 + 7.91857i −30.2851 + 8.89252i 6.54861 + 7.55750i
81.1 1.68251 − 1.08128i −1.06222 + 7.38793i 1.66166 − 3.63853i 4.79746 − 1.40866i 6.20123 + 13.5788i 2.08726 + 2.40883i −1.13852 − 7.91857i −27.5468 − 8.08848i 6.54861 − 7.55750i
81.2 1.68251 − 1.08128i −0.937040 + 6.51725i 1.66166 − 3.63853i 4.79746 − 1.40866i 5.47041 + 11.9785i −20.6340 − 23.8129i −1.13852 − 7.91857i −15.6903 − 4.60708i 6.54861 − 7.55750i
81.3 1.68251 − 1.08128i 0.0393624 − 0.273772i 1.66166 − 3.63853i 4.79746 − 1.40866i −0.229797 − 0.503185i −2.34264 − 2.70355i −1.13852 − 7.91857i 25.8329 + 7.58523i 6.54861 − 7.55750i
81.4 1.68251 − 1.08128i 0.339971 − 2.36455i 1.66166 − 3.63853i 4.79746 − 1.40866i −1.98474 − 4.34597i 21.3773 + 24.6707i −1.13852 − 7.91857i 20.4308 + 5.99902i 6.54861 − 7.55750i
81.5 1.68251 − 1.08128i 1.08909 − 7.57480i 1.66166 − 3.63853i 4.79746 − 1.40866i −6.35809 − 13.9223i −6.64671 − 7.67071i −1.13852 − 7.91857i −30.2851 − 8.89252i 6.54861 − 7.55750i
See all 50 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.5
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.g.a ✓ 50
23.c even 11 1 inner 230.4.g.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.g.a ✓ 50 1.a even 1 1 trivial
230.4.g.a ✓ 50 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} + 5 T_{3}^{49} + 143 T_{3}^{48} + 535 T_{3}^{47} + 12570 T_{3}^{46} + 51170 T_{3}^{45} + \cdots + 40\!\cdots\!69 \) acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display