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Properties

Label 230.4.g.b
Level $230$
Weight $4$
Character orbit 230.g
Analytic conductor $13.570$
Analytic rank $0$
Dimension $60$
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 = [60,12,-3] 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: \(60\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9} + 60 q^{10} - 51 q^{11} + 120 q^{12} + 184 q^{13} + 20 q^{14} - 15 q^{15} - 96 q^{16} - 334 q^{17} - 138 q^{18}+ \cdots - 10589 q^{99}+O(q^{100}) \) 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 −3.91607 − 8.57500i −3.83797 − 1.12693i −3.27430 − 3.77875i −18.0901 + 5.31172i 13.7433 + 8.83226i −3.32332 + 7.27706i −40.5137 + 46.7553i −8.41254 + 5.40641i
31.2 0.284630 − 1.97964i −1.76914 − 3.87387i −3.83797 − 1.12693i −3.27430 − 3.77875i −8.17242 + 2.39964i −26.5942 − 17.0910i −3.32332 + 7.27706i 5.80424 − 6.69844i −8.41254 + 5.40641i
31.3 0.284630 − 1.97964i −0.871814 − 1.90901i −3.83797 − 1.12693i −3.27430 − 3.77875i −4.02729 + 1.18252i 25.9987 + 16.7083i −3.32332 + 7.27706i 14.7970 − 17.0766i −8.41254 + 5.40641i
31.4 0.284630 − 1.97964i −0.654798 − 1.43381i −3.83797 − 1.12693i −3.27430 − 3.77875i −3.02480 + 0.888162i 7.64693 + 4.91439i −3.32332 + 7.27706i 16.0542 − 18.5275i −8.41254 + 5.40641i
31.5 0.284630 − 1.97964i 2.61500 + 5.72604i −3.83797 − 1.12693i −3.27430 − 3.77875i 12.0798 − 3.54696i −5.58110 − 3.58676i −3.32332 + 7.27706i −8.26814 + 9.54194i −8.41254 + 5.40641i
31.6 0.284630 − 1.97964i 2.99586 + 6.56001i −3.83797 − 1.12693i −3.27430 − 3.77875i 13.8392 − 4.06355i 2.05971 + 1.32369i −3.32332 + 7.27706i −16.3774 + 18.9005i −8.41254 + 5.40641i
41.1 1.91899 + 0.563465i −4.61458 + 5.32551i 3.36501 + 2.16256i −0.711574 + 4.94911i −11.8561 + 7.61942i 10.2472 + 22.4383i 5.23889 + 6.04600i −3.22420 − 22.4248i −4.15415 + 9.09632i
41.2 1.91899 + 0.563465i −2.52693 + 2.91624i 3.36501 + 2.16256i −0.711574 + 4.94911i −6.49235 + 4.17238i −13.4966 − 29.5535i 5.23889 + 6.04600i 1.72346 + 11.9869i −4.15415 + 9.09632i
41.3 1.91899 + 0.563465i 0.460964 − 0.531981i 3.36501 + 2.16256i −0.711574 + 4.94911i 1.18434 − 0.761126i −1.15601 − 2.53131i 5.23889 + 6.04600i 3.77198 + 26.2347i −4.15415 + 9.09632i
41.4 1.91899 + 0.563465i 1.94747 − 2.24751i 3.36501 + 2.16256i −0.711574 + 4.94911i 5.00357 − 3.21560i 8.06153 + 17.6523i 5.23889 + 6.04600i 2.58388 + 17.9713i −4.15415 + 9.09632i
41.5 1.91899 + 0.563465i 3.92901 − 4.53431i 3.36501 + 2.16256i −0.711574 + 4.94911i 10.0946 − 6.48743i −9.52105 − 20.8482i 5.23889 + 6.04600i −1.28042 − 8.90551i −4.15415 + 9.09632i
41.6 1.91899 + 0.563465i 6.53866 − 7.54601i 3.36501 + 2.16256i −0.711574 + 4.94911i 16.7995 − 10.7964i 14.9302 + 32.6926i 5.23889 + 6.04600i −10.3458 − 71.9564i −4.15415 + 9.09632i
71.1 −1.68251 − 1.08128i −1.10005 − 7.65101i 1.66166 + 3.63853i −4.79746 − 1.40866i −6.42205 + 14.0623i −11.9276 + 13.7651i 1.13852 − 7.91857i −31.4215 + 9.22619i 6.54861 + 7.55750i
71.2 −1.68251 − 1.08128i −0.719442 − 5.00383i 1.66166 + 3.63853i −4.79746 − 1.40866i −4.20008 + 9.19689i 9.79868 − 11.3083i 1.13852 − 7.91857i 1.38563 − 0.406859i 6.54861 + 7.55750i
71.3 −1.68251 − 1.08128i −0.259790 − 1.80688i 1.66166 + 3.63853i −4.79746 − 1.40866i −1.51665 + 3.32099i −3.50130 + 4.04072i 1.13852 − 7.91857i 22.7090 − 6.66796i 6.54861 + 7.55750i
71.4 −1.68251 − 1.08128i −0.161442 − 1.12285i 1.66166 + 3.63853i −4.79746 − 1.40866i −0.942491 + 2.06377i 20.3265 − 23.4580i 1.13852 − 7.91857i 24.6716 − 7.24423i 6.54861 + 7.55750i
71.5 −1.68251 − 1.08128i 0.839469 + 5.83864i 1.66166 + 3.63853i −4.79746 − 1.40866i 4.90080 − 10.7312i 5.10412 − 5.89047i 1.13852 − 7.91857i −7.47864 + 2.19593i 6.54861 + 7.55750i
71.6 −1.68251 − 1.08128i 1.10986 + 7.71925i 1.66166 + 3.63853i −4.79746 − 1.40866i 6.47933 − 14.1878i −5.92245 + 6.83487i 1.13852 − 7.91857i −32.4487 + 9.52780i 6.54861 + 7.55750i
81.1 −1.68251 + 1.08128i −1.10005 + 7.65101i 1.66166 − 3.63853i −4.79746 + 1.40866i −6.42205 − 14.0623i −11.9276 − 13.7651i 1.13852 + 7.91857i −31.4215 − 9.22619i 6.54861 − 7.55750i
81.2 −1.68251 + 1.08128i −0.719442 + 5.00383i 1.66166 − 3.63853i −4.79746 + 1.40866i −4.20008 − 9.19689i 9.79868 + 11.3083i 1.13852 + 7.91857i 1.38563 + 0.406859i 6.54861 − 7.55750i
See all 60 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.6
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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.b ✓ 60
23.c even 11 1 inner 230.4.g.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.g.b ✓ 60 1.a even 1 1 trivial
230.4.g.b ✓ 60 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 3 T_{3}^{59} + 51 T_{3}^{58} + 469 T_{3}^{57} + 6348 T_{3}^{56} + 46774 T_{3}^{55} + \cdots + 52\!\cdots\!61 \) acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display