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Properties

Label 650.2.ba.a
Level $650$
Weight $2$
Character orbit 650.ba
Analytic conductor $5.190$
Analytic rank $0$
Dimension $136$
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] = [650,2,Mod(73,650)] 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([11, 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("650.73"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.ba (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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 34 q^{2} - 34 q^{4} - 34 q^{8} + 20 q^{9} + 6 q^{13} + 32 q^{15} - 34 q^{16} + 4 q^{17} - 20 q^{19} + 6 q^{21} + 12 q^{23} - 14 q^{26} + 36 q^{27} + 20 q^{29} + 12 q^{30} + 6 q^{31} + 136 q^{32}+ \cdots - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.809017 + 0.587785i −2.99761 + 1.52736i 0.309017 − 0.951057i 0.965394 − 2.01693i 1.52736 − 2.99761i − 4.41301i 0.309017 + 0.951057i 4.88947 − 6.72978i 0.404502 + 2.19918i
73.2 −0.809017 + 0.587785i −2.52731 + 1.28773i 0.309017 − 0.951057i 1.05748 + 1.97021i 1.28773 − 2.52731i 0.476085i 0.309017 + 0.951057i 2.96569 − 4.08193i −2.01358 − 0.972368i
73.3 −0.809017 + 0.587785i −2.39892 + 1.22231i 0.309017 − 0.951057i −1.05123 − 1.97355i 1.22231 − 2.39892i 2.84188i 0.309017 + 0.951057i 2.49742 − 3.43741i 2.01049 + 0.978738i
73.4 −0.809017 + 0.587785i −2.00034 + 1.01922i 0.309017 − 0.951057i −1.78750 + 1.34344i 1.01922 − 2.00034i 0.767561i 0.309017 + 0.951057i 1.19918 − 1.65052i 0.656468 − 2.13753i
73.5 −0.809017 + 0.587785i −1.12262 + 0.572004i 0.309017 − 0.951057i −1.94251 − 1.10755i 0.572004 − 1.12262i 0.338022i 0.309017 + 0.951057i −0.830267 + 1.14277i 2.22252 − 0.245755i
73.6 −0.809017 + 0.587785i −1.06973 + 0.545054i 0.309017 − 0.951057i 2.08999 − 0.794959i 0.545054 − 1.06973i 1.77360i 0.309017 + 0.951057i −0.916121 + 1.26093i −1.22357 + 1.87160i
73.7 −0.809017 + 0.587785i −0.847148 + 0.431643i 0.309017 − 0.951057i 1.80370 − 1.32162i 0.431643 − 0.847148i − 2.12124i 0.309017 + 0.951057i −1.23201 + 1.69572i −0.682395 + 2.12940i
73.8 −0.809017 + 0.587785i −0.294147 + 0.149876i 0.309017 − 0.951057i 0.251296 + 2.22190i 0.149876 − 0.294147i 4.89348i 0.309017 + 0.951057i −1.69930 + 2.33888i −1.50930 − 1.64985i
73.9 −0.809017 + 0.587785i −0.0663642 + 0.0338142i 0.309017 − 0.951057i 1.54212 + 1.61922i 0.0338142 − 0.0663642i − 1.82460i 0.309017 + 0.951057i −1.76009 + 2.42256i −2.19935 − 0.403539i
73.10 −0.809017 + 0.587785i 0.177131 − 0.0902528i 0.309017 − 0.951057i −0.641599 + 2.14204i −0.0902528 + 0.177131i − 4.46227i 0.309017 + 0.951057i −1.74013 + 2.39508i −0.739997 − 2.11007i
73.11 −0.809017 + 0.587785i 0.799872 − 0.407555i 0.309017 − 0.951057i −1.99344 + 1.01301i −0.407555 + 0.799872i − 0.585140i 0.309017 + 0.951057i −1.28966 + 1.77507i 1.01730 − 1.99126i
73.12 −0.809017 + 0.587785i 1.15122 − 0.586577i 0.309017 − 0.951057i −2.09067 − 0.793162i −0.586577 + 1.15122i 1.85103i 0.309017 + 0.951057i −0.782115 + 1.07649i 2.15759 − 0.587183i
73.13 −0.809017 + 0.587785i 1.44850 − 0.738047i 0.309017 − 0.951057i 0.723163 − 2.11590i −0.738047 + 1.44850i 4.34065i 0.309017 + 0.951057i −0.209920 + 0.288930i 0.658644 + 2.13686i
73.14 −0.809017 + 0.587785i 1.86581 − 0.950677i 0.309017 − 0.951057i 0.467110 − 2.18673i −0.950677 + 1.86581i − 2.95692i 0.309017 + 0.951057i 0.814098 − 1.12051i 0.907430 + 2.04367i
73.15 −0.809017 + 0.587785i 2.33973 − 1.19215i 0.309017 − 0.951057i 2.22577 + 0.214378i −1.19215 + 2.33973i 1.94060i 0.309017 + 0.951057i 2.28976 − 3.15159i −1.92669 + 1.13484i
73.16 −0.809017 + 0.587785i 2.56648 − 1.30768i 0.309017 − 0.951057i 0.585893 + 2.15795i −1.30768 + 2.56648i 0.941662i 0.309017 + 0.951057i 3.11340 − 4.28523i −1.74241 − 1.40144i
73.17 −0.809017 + 0.587785i 2.97544 − 1.51606i 0.309017 − 0.951057i −2.20495 − 0.371749i −1.51606 + 2.97544i − 3.80138i 0.309017 + 0.951057i 4.79145 − 6.59486i 2.00235 − 0.995286i
187.1 −0.809017 − 0.587785i −2.99761 − 1.52736i 0.309017 + 0.951057i 0.965394 + 2.01693i 1.52736 + 2.99761i 4.41301i 0.309017 − 0.951057i 4.88947 + 6.72978i 0.404502 − 2.19918i
187.2 −0.809017 − 0.587785i −2.52731 − 1.28773i 0.309017 + 0.951057i 1.05748 − 1.97021i 1.28773 + 2.52731i − 0.476085i 0.309017 − 0.951057i 2.96569 + 4.08193i −2.01358 + 0.972368i
187.3 −0.809017 − 0.587785i −2.39892 − 1.22231i 0.309017 + 0.951057i −1.05123 + 1.97355i 1.22231 + 2.39892i − 2.84188i 0.309017 − 0.951057i 2.49742 + 3.43741i 2.01049 − 0.978738i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.17
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Inner twists

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

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