Properties

Label 650.2.p.a
Level $650$
Weight $2$
Character orbit 650.p
Analytic conductor $5.190$
Analytic rank $0$
Dimension $136$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(181,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.p (of order \(10\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 136 q + 34 q^{4} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q + 34 q^{4} - 30 q^{9} - 4 q^{10} - 12 q^{13} + 8 q^{14} - 34 q^{16} + 4 q^{23} + 40 q^{25} + 4 q^{26} - 12 q^{27} + 16 q^{29} + 10 q^{30} - 18 q^{35} + 30 q^{36} - 24 q^{38} + 32 q^{39} + 4 q^{40} - 12 q^{42} + 24 q^{43} - 104 q^{49} + 84 q^{51} + 12 q^{52} + 4 q^{53} - 60 q^{55} - 8 q^{56} + 12 q^{61} + 18 q^{62} + 34 q^{64} + 26 q^{65} + 16 q^{66} + 84 q^{69} - 88 q^{74} - 214 q^{75} + 56 q^{78} - 8 q^{79} - 74 q^{81} + 160 q^{82} - 164 q^{87} - 16 q^{90} - 40 q^{91} + 16 q^{92} + 48 q^{94} - 4 q^{95}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −0.951057 + 0.309017i −2.77229 + 2.01419i 0.809017 0.587785i −2.17604 + 0.514625i 2.01419 2.77229i 1.58290i −0.587785 + 0.809017i 2.70160 8.31467i 1.91051 1.16187i
181.2 −0.951057 + 0.309017i −2.39624 + 1.74097i 0.809017 0.587785i 2.00896 + 0.981871i 1.74097 2.39624i 0.812549i −0.587785 + 0.809017i 1.78393 5.49037i −2.21405 0.313011i
181.3 −0.951057 + 0.309017i −1.64902 + 1.19809i 0.809017 0.587785i 1.65401 + 1.50474i 1.19809 1.64902i 3.17306i −0.587785 + 0.809017i 0.356819 1.09818i −2.03805 0.919973i
181.4 −0.951057 + 0.309017i −1.63416 + 1.18728i 0.809017 0.587785i −0.970013 2.01471i 1.18728 1.63416i 1.37615i −0.587785 + 0.809017i 0.333771 1.02724i 1.54512 + 1.61636i
181.5 −0.951057 + 0.309017i −1.03852 + 0.754528i 0.809017 0.587785i −1.01800 + 1.99090i 0.754528 1.03852i 0.731554i −0.587785 + 0.809017i −0.417843 + 1.28599i 0.352954 2.20804i
181.6 −0.951057 + 0.309017i −0.802302 + 0.582906i 0.809017 0.587785i 0.515358 + 2.17587i 0.582906 0.802302i 4.67023i −0.587785 + 0.809017i −0.623143 + 1.91784i −1.16251 1.91012i
181.7 −0.951057 + 0.309017i −0.757142 + 0.550096i 0.809017 0.587785i −2.06002 0.869669i 0.550096 0.757142i 2.56855i −0.587785 + 0.809017i −0.656392 + 2.02017i 2.22794 + 0.190524i
181.8 −0.951057 + 0.309017i −0.454733 + 0.330383i 0.809017 0.587785i 2.02575 0.946747i 0.330383 0.454733i 3.59180i −0.587785 + 0.809017i −0.829422 + 2.55270i −1.63404 + 1.52640i
181.9 −0.951057 + 0.309017i −0.271435 + 0.197209i 0.809017 0.587785i −2.19946 + 0.402964i 0.197209 0.271435i 3.44579i −0.587785 + 0.809017i −0.892265 + 2.74611i 1.96729 1.06291i
181.10 −0.951057 + 0.309017i −0.0856483 + 0.0622271i 0.809017 0.587785i 1.86289 1.23679i 0.0622271 0.0856483i 1.22434i −0.587785 + 0.809017i −0.923588 + 2.84251i −1.38952 + 1.75192i
181.11 −0.951057 + 0.309017i 1.08542 0.788606i 0.809017 0.587785i −1.26746 1.84216i −0.788606 + 1.08542i 1.88803i −0.587785 + 0.809017i −0.370807 + 1.14123i 1.77468 + 1.36033i
181.12 −0.951057 + 0.309017i 1.23469 0.897052i 0.809017 0.587785i −2.23531 + 0.0581791i −0.897052 + 1.23469i 2.12559i −0.587785 + 0.809017i −0.207303 + 0.638013i 2.10793 0.746081i
181.13 −0.951057 + 0.309017i 1.37663 1.00018i 0.809017 0.587785i 0.672142 2.13266i −1.00018 + 1.37663i 3.88027i −0.587785 + 0.809017i −0.0322965 + 0.0993983i 0.0197820 + 2.23598i
181.14 −0.951057 + 0.309017i 1.45190 1.05487i 0.809017 0.587785i 0.184771 + 2.22842i −1.05487 + 1.45190i 1.28692i −0.587785 + 0.809017i 0.0682224 0.209967i −0.864348 2.06226i
181.15 −0.951057 + 0.309017i 2.12082 1.54087i 0.809017 0.587785i −1.15648 + 1.91378i −1.54087 + 2.12082i 3.25927i −0.587785 + 0.809017i 1.19656 3.68265i 0.508489 2.17748i
181.16 −0.951057 + 0.309017i 2.13781 1.55321i 0.809017 0.587785i 2.03958 + 0.916581i −1.55321 + 2.13781i 0.170643i −0.587785 + 0.809017i 1.23072 3.78776i −2.22299 0.241456i
181.17 −0.951057 + 0.309017i 2.45421 1.78309i 0.809017 0.587785i 0.943747 2.02715i −1.78309 + 2.45421i 5.11933i −0.587785 + 0.809017i 1.91669 5.89896i −0.271133 + 2.21957i
181.18 0.951057 0.309017i −2.77229 + 2.01419i 0.809017 0.587785i 2.17604 0.514625i −2.01419 + 2.77229i 1.58290i 0.587785 0.809017i 2.70160 8.31467i 1.91051 1.16187i
181.19 0.951057 0.309017i −2.39624 + 1.74097i 0.809017 0.587785i −2.00896 0.981871i −1.74097 + 2.39624i 0.812549i 0.587785 0.809017i 1.78393 5.49037i −2.21405 0.313011i
181.20 0.951057 0.309017i −1.64902 + 1.19809i 0.809017 0.587785i −1.65401 1.50474i −1.19809 + 1.64902i 3.17306i 0.587785 0.809017i 0.356819 1.09818i −2.03805 0.919973i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
25.d even 5 1 inner
325.q even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.p.a 136
13.b even 2 1 inner 650.2.p.a 136
25.d even 5 1 inner 650.2.p.a 136
325.q even 10 1 inner 650.2.p.a 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.p.a 136 1.a even 1 1 trivial
650.2.p.a 136 13.b even 2 1 inner
650.2.p.a 136 25.d even 5 1 inner
650.2.p.a 136 325.q even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(650, [\chi])\).