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Properties

Label 650.2.bd.a
Level $650$
Weight $2$
Character orbit 650.bd
Analytic conductor $5.190$
Analytic rank $0$
Dimension $136$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [650,2,Mod(47,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([17, 5])) 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("650.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.bd (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [136] 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q + 34 q^{4} + 20 q^{9} + 2 q^{13} - 14 q^{15} - 34 q^{16} - 4 q^{17} + 128 q^{18} - 20 q^{19} + 6 q^{21} - 12 q^{23} - 14 q^{26} + 36 q^{27} + 20 q^{29} - 12 q^{30} + 6 q^{31} + 30 q^{33} + 16 q^{34}+ \cdots + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0.951057 + 0.309017i −3.22017 + 0.510024i 0.809017 + 0.587785i 2.15554 − 0.594697i −3.22017 − 0.510024i −2.30965 0.587785 + 0.809017i 7.25618 − 2.35768i 2.23381 + 0.100506i
47.2 0.951057 + 0.309017i −2.50399 + 0.396592i 0.809017 + 0.587785i −1.89180 − 1.19209i −2.50399 − 0.396592i 0.532627 0.587785 + 0.809017i 3.25949 − 1.05907i −1.43083 − 1.71835i
47.3 0.951057 + 0.309017i −2.33269 + 0.369462i 0.809017 + 0.587785i −0.733554 − 2.11232i −2.33269 − 0.369462i −0.211097 0.587785 + 0.809017i 2.45179 − 0.796635i −0.0449084 − 2.23562i
47.4 0.951057 + 0.309017i −2.19119 + 0.347051i 0.809017 + 0.587785i −1.35035 + 1.78229i −2.19119 − 0.347051i −1.67700 0.587785 + 0.809017i 1.82771 − 0.593859i −1.83502 + 1.27778i
47.5 0.951057 + 0.309017i −1.82274 + 0.288693i 0.809017 + 0.587785i 1.50886 + 1.65025i −1.82274 − 0.288693i 1.50402 0.587785 + 0.809017i 0.385856 − 0.125372i 0.925058 + 2.03575i
47.6 0.951057 + 0.309017i −1.50262 + 0.237992i 0.809017 + 0.587785i 2.20865 + 0.349092i −1.50262 − 0.237992i 4.46957 0.587785 + 0.809017i −0.651933 + 0.211826i 1.99268 + 1.01452i
47.7 0.951057 + 0.309017i −0.903505 + 0.143101i 0.809017 + 0.587785i 0.399063 − 2.20017i −0.903505 − 0.143101i −2.83760 0.587785 + 0.809017i −2.05733 + 0.668466i 1.05942 − 1.96917i
47.8 0.951057 + 0.309017i −0.269503 + 0.0426851i 0.809017 + 0.587785i 0.00208248 + 2.23607i −0.269503 − 0.0426851i −2.15888 0.587785 + 0.809017i −2.78236 + 0.904043i −0.689002 + 2.12727i
47.9 0.951057 + 0.309017i 0.252785 − 0.0400371i 0.809017 + 0.587785i 1.91696 − 1.15120i 0.252785 + 0.0400371i 1.39851 0.587785 + 0.809017i −2.79087 + 0.906809i 2.17888 − 0.502478i
47.10 0.951057 + 0.309017i 0.472657 − 0.0748615i 0.809017 + 0.587785i −2.05820 + 0.873968i 0.472657 + 0.0748615i 1.46967 0.587785 + 0.809017i −2.63537 + 0.856283i −2.22753 + 0.195175i
47.11 0.951057 + 0.309017i 0.569849 − 0.0902552i 0.809017 + 0.587785i −2.15998 − 0.578331i 0.569849 + 0.0902552i −4.73060 0.587785 + 0.809017i −2.53659 + 0.824187i −1.87555 − 1.21750i
47.12 0.951057 + 0.309017i 1.15419 − 0.182806i 0.809017 + 0.587785i −0.390455 − 2.20171i 1.15419 + 0.182806i 3.34305 0.587785 + 0.809017i −1.55443 + 0.505066i 0.309022 − 2.21461i
47.13 0.951057 + 0.309017i 1.80881 − 0.286487i 0.809017 + 0.587785i −0.222584 + 2.22496i 1.80881 + 0.286487i 5.10419 0.587785 + 0.809017i 0.336540 − 0.109349i −0.899241 + 2.04728i
47.14 0.951057 + 0.309017i 1.93702 − 0.306794i 0.809017 + 0.587785i 1.97487 + 1.04875i 1.93702 + 0.306794i −1.91770 0.587785 + 0.809017i 0.804764 − 0.261484i 1.55414 + 1.60769i
47.15 0.951057 + 0.309017i 2.55484 − 0.404646i 0.809017 + 0.587785i 1.44258 − 1.70850i 2.55484 + 0.404646i −0.742587 0.587785 + 0.809017i 3.51028 − 1.14056i 1.89993 − 1.17910i
47.16 0.951057 + 0.309017i 2.81652 − 0.446092i 0.809017 + 0.587785i −2.16243 − 0.569102i 2.81652 + 0.446092i 1.59260 0.587785 + 0.809017i 4.88060 − 1.58580i −1.88074 − 1.20948i
47.17 0.951057 + 0.309017i 3.17975 − 0.503622i 0.809017 + 0.587785i −0.639249 + 2.14275i 3.17975 + 0.503622i −2.82913 0.587785 + 0.809017i 7.00398 − 2.27573i −1.27011 + 1.84033i
83.1 0.951057 − 0.309017i −3.22017 − 0.510024i 0.809017 − 0.587785i 2.15554 + 0.594697i −3.22017 + 0.510024i −2.30965 0.587785 − 0.809017i 7.25618 + 2.35768i 2.23381 − 0.100506i
83.2 0.951057 − 0.309017i −2.50399 − 0.396592i 0.809017 − 0.587785i −1.89180 + 1.19209i −2.50399 + 0.396592i 0.532627 0.587785 − 0.809017i 3.25949 + 1.05907i −1.43083 + 1.71835i
83.3 0.951057 − 0.309017i −2.33269 − 0.369462i 0.809017 − 0.587785i −0.733554 + 2.11232i −2.33269 + 0.369462i −0.211097 0.587785 − 0.809017i 2.45179 + 0.796635i −0.0449084 + 2.23562i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.17
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
325.be even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.bd.a yes 136
13.d odd 4 1 650.2.ba.a ✓ 136
25.f odd 20 1 650.2.ba.a ✓ 136
325.be even 20 1 inner 650.2.bd.a yes 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.ba.a ✓ 136 13.d odd 4 1
650.2.ba.a ✓ 136 25.f odd 20 1
650.2.bd.a yes 136 1.a even 1 1 trivial
650.2.bd.a yes 136 325.be even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{136} - 10 T_{3}^{134} - 12 T_{3}^{133} - 139 T_{3}^{132} + 176 T_{3}^{131} + \cdots + 11\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display