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Properties

Label 850.2.y.b
Level $850$
Weight $2$
Character orbit 850.y
Analytic conductor $6.787$
Analytic rank $0$
Dimension $192$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [850,2,Mod(21,850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(850, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([12, 15])) 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("850.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.y (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.78728417181\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 48 q^{4} + 2 q^{5} + 4 q^{7} - 12 q^{10} + 14 q^{11} - 4 q^{13} - 6 q^{14} - 48 q^{16} + 12 q^{17} - 240 q^{18} + 18 q^{20} - 16 q^{21} - 16 q^{22} - 14 q^{23} - 30 q^{27} - 4 q^{28} - 20 q^{29}+ \cdots + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.951057 − 0.309017i −3.38553 + 0.536215i 0.809017 + 0.587785i −2.23599 − 0.0191323i 3.38553 + 0.536215i −2.11048 − 2.11048i −0.587785 − 0.809017i 8.32110 − 2.70369i 2.12064 + 0.709154i
21.2 −0.951057 − 0.309017i −3.13489 + 0.496517i 0.809017 + 0.587785i 0.679154 − 2.13043i 3.13489 + 0.496517i 3.29709 + 3.29709i −0.587785 − 0.809017i 6.72781 − 2.18600i −1.30425 + 1.81629i
21.3 −0.951057 − 0.309017i −2.60483 + 0.412565i 0.809017 + 0.587785i 1.87561 + 1.21742i 2.60483 + 0.412565i 1.09101 + 1.09101i −0.587785 − 0.809017i 3.76176 − 1.22227i −1.40761 − 1.73743i
21.4 −0.951057 − 0.309017i −2.50979 + 0.397512i 0.809017 + 0.587785i 1.75939 − 1.38005i 2.50979 + 0.397512i −3.08443 − 3.08443i −0.587785 − 0.809017i 3.28787 − 1.06829i −2.09974 + 0.768828i
21.5 −0.951057 − 0.309017i −2.03867 + 0.322894i 0.809017 + 0.587785i 1.60930 + 1.55247i 2.03867 + 0.322894i 0.957518 + 0.957518i −0.587785 − 0.809017i 1.19876 − 0.389502i −1.05079 − 1.97379i
21.6 −0.951057 − 0.309017i −1.64092 + 0.259896i 0.809017 + 0.587785i −1.95577 − 1.08396i 1.64092 + 0.259896i 1.10434 + 1.10434i −0.587785 − 0.809017i −0.228105 + 0.0741158i 1.52509 + 1.63527i
21.7 −0.951057 − 0.309017i −1.45163 + 0.229915i 0.809017 + 0.587785i 0.0611085 + 2.23523i 1.45163 + 0.229915i −2.53709 − 2.53709i −0.587785 − 0.809017i −0.798807 + 0.259548i 0.632607 − 2.14472i
21.8 −0.951057 − 0.309017i −1.41605 + 0.224280i 0.809017 + 0.587785i −1.93614 + 1.11865i 1.41605 + 0.224280i 2.21924 + 2.21924i −0.587785 − 0.809017i −0.898279 + 0.291868i 2.18706 − 0.465601i
21.9 −0.951057 − 0.309017i −1.40133 + 0.221948i 0.809017 + 0.587785i −0.942582 − 2.02769i 1.40133 + 0.221948i −1.78448 − 1.78448i −0.587785 − 0.809017i −0.938715 + 0.305007i 0.269857 + 2.21972i
21.10 −0.951057 − 0.309017i −1.16195 + 0.184034i 0.809017 + 0.587785i −0.963981 + 2.01761i 1.16195 + 0.184034i −1.79462 − 1.79462i −0.587785 − 0.809017i −1.53692 + 0.499374i 1.54028 − 1.62097i
21.11 −0.951057 − 0.309017i −0.386307 + 0.0611850i 0.809017 + 0.587785i 1.20042 − 1.88653i 0.386307 + 0.0611850i 1.05857 + 1.05857i −0.587785 − 0.809017i −2.70768 + 0.879779i −1.72464 + 1.42325i
21.12 −0.951057 − 0.309017i −0.312473 + 0.0494909i 0.809017 + 0.587785i −0.392105 − 2.20142i 0.312473 + 0.0494909i −0.191763 − 0.191763i −0.587785 − 0.809017i −2.75798 + 0.896122i −0.307362 + 2.21484i
21.13 −0.951057 − 0.309017i 0.107423 − 0.0170141i 0.809017 + 0.587785i 2.14796 − 0.621510i −0.107423 − 0.0170141i 1.02490 + 1.02490i −0.587785 − 0.809017i −2.84192 + 0.923396i −2.23489 − 0.0726643i
21.14 −0.951057 − 0.309017i 0.521990 − 0.0826751i 0.809017 + 0.587785i 0.570985 + 2.16194i −0.521990 − 0.0826751i 3.63271 + 3.63271i −0.587785 − 0.809017i −2.58753 + 0.840740i 0.125036 − 2.23257i
21.15 −0.951057 − 0.309017i 0.572039 − 0.0906020i 0.809017 + 0.587785i 1.99531 + 1.00933i −0.572039 − 0.0906020i −1.92309 − 1.92309i −0.587785 − 0.809017i −2.53415 + 0.823395i −1.58575 − 1.57652i
21.16 −0.951057 − 0.309017i 1.05413 − 0.166958i 0.809017 + 0.587785i −1.06101 + 1.96832i −1.05413 − 0.166958i 0.439231 + 0.439231i −0.587785 − 0.809017i −1.76985 + 0.575058i 1.61732 − 1.54411i
21.17 −0.951057 − 0.309017i 1.46792 − 0.232496i 0.809017 + 0.587785i −2.20958 − 0.343138i −1.46792 − 0.232496i 1.64329 + 1.64329i −0.587785 − 0.809017i −0.752421 + 0.244476i 1.99540 + 1.00914i
21.18 −0.951057 − 0.309017i 1.69400 − 0.268304i 0.809017 + 0.587785i 2.13448 − 0.666335i −1.69400 − 0.268304i −3.09277 − 3.09277i −0.587785 − 0.809017i −0.0555099 + 0.0180363i −2.23592 − 0.0258678i
21.19 −0.951057 − 0.309017i 1.87252 − 0.296578i 0.809017 + 0.587785i −1.97151 − 1.05506i −1.87252 − 0.296578i −1.04620 − 1.04620i −0.587785 − 0.809017i 0.565194 − 0.183643i 1.54899 + 1.61265i
21.20 −0.951057 − 0.309017i 2.50283 − 0.396410i 0.809017 + 0.587785i 2.01962 − 0.959751i −2.50283 − 0.396410i 2.48849 + 2.48849i −0.587785 − 0.809017i 3.25386 − 1.05724i −2.21736 + 0.288680i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner
25.d even 5 1 inner
425.bb even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.y.b ✓ 192
17.c even 4 1 inner 850.2.y.b ✓ 192
25.d even 5 1 inner 850.2.y.b ✓ 192
425.bb even 20 1 inner 850.2.y.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
850.2.y.b ✓ 192 1.a even 1 1 trivial
850.2.y.b ✓ 192 17.c even 4 1 inner
850.2.y.b ✓ 192 25.d even 5 1 inner
850.2.y.b ✓ 192 425.bb even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{192} + 10 T_{3}^{189} - 375 T_{3}^{188} - 116 T_{3}^{187} + 50 T_{3}^{186} + \cdots + 10\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\). Copy content Toggle raw display