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Properties

Label 650.2.bd.b
Level $650$
Weight $2$
Character orbit 650.bd
Analytic conductor $5.190$
Analytic rank $0$
Dimension $144$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(47,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{4} - 2 q^{5} - 20 q^{9} + 12 q^{13} - 26 q^{15} - 36 q^{16} + 6 q^{17} - 152 q^{18} - 20 q^{19} - 8 q^{20} - 6 q^{21} - 12 q^{23} - 2 q^{25} + 14 q^{26} - 24 q^{27} + 20 q^{29} - 12 q^{30}+ \cdots - 40 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −0.951057 − 0.309017i −3.19611 + 0.506214i 0.809017 + 0.587785i −1.82170 − 1.29670i 3.19611 + 0.506214i −0.716524 −0.587785 − 0.809017i 7.10571 − 2.30878i 1.33184 + 1.79617i
47.2 −0.951057 − 0.309017i −3.05796 + 0.484334i 0.809017 + 0.587785i 2.07976 + 0.821347i 3.05796 + 0.484334i 4.76595 −0.587785 − 0.809017i 6.26339 − 2.03510i −1.72416 − 1.42383i
47.3 −0.951057 − 0.309017i −2.47594 + 0.392151i 0.809017 + 0.587785i 1.61827 − 1.54312i 2.47594 + 0.392151i −4.37981 −0.587785 − 0.809017i 3.12334 − 1.01483i −2.01591 + 0.967522i
47.4 −0.951057 − 0.309017i −2.37227 + 0.375730i 0.809017 + 0.587785i −1.27427 + 1.83746i 2.37227 + 0.375730i −1.25399 −0.587785 − 0.809017i 2.63331 − 0.855615i 1.77970 − 1.35375i
47.5 −0.951057 − 0.309017i −1.89075 + 0.299466i 0.809017 + 0.587785i −1.55473 + 1.60711i 1.89075 + 0.299466i 3.88967 −0.587785 − 0.809017i 0.632103 − 0.205383i 1.97526 − 1.04801i
47.6 −0.951057 − 0.309017i −0.814873 + 0.129063i 0.809017 + 0.587785i 1.17956 − 1.89964i 0.814873 + 0.129063i 1.70233 −0.587785 − 0.809017i −2.20581 + 0.716711i −1.70885 + 1.44216i
47.7 −0.951057 − 0.309017i −0.682762 + 0.108139i 0.809017 + 0.587785i 1.56813 + 1.59404i 0.682762 + 0.108139i −0.786146 −0.587785 − 0.809017i −2.39870 + 0.779385i −0.998793 − 2.00060i
47.8 −0.951057 − 0.309017i −0.354023 + 0.0560717i 0.809017 + 0.587785i −2.03397 + 0.928954i 0.354023 + 0.0560717i −4.40012 −0.587785 − 0.809017i −2.73098 + 0.887350i 2.22149 − 0.254956i
47.9 −0.951057 − 0.309017i −0.149353 + 0.0236552i 0.809017 + 0.587785i 1.08666 + 1.95427i 0.149353 + 0.0236552i −0.0164037 −0.587785 − 0.809017i −2.83142 + 0.919985i −0.429576 − 2.19442i
47.10 −0.951057 − 0.309017i 0.0938992 − 0.0148722i 0.809017 + 0.587785i −2.03697 − 0.922360i −0.0938992 − 0.0148722i 2.91546 −0.587785 − 0.809017i −2.84457 + 0.924258i 1.65225 + 1.50668i
47.11 −0.951057 − 0.309017i 0.803650 − 0.127286i 0.809017 + 0.587785i −0.844706 − 2.07038i −0.803650 − 0.127286i −2.97396 −0.587785 − 0.809017i −2.22352 + 0.722465i 0.163580 + 2.23008i
47.12 −0.951057 − 0.309017i 1.05015 − 0.166328i 0.809017 + 0.587785i 2.17886 + 0.502556i −1.05015 − 0.166328i −3.12577 −0.587785 − 0.809017i −1.77801 + 0.577711i −1.91692 − 1.15126i
47.13 −0.951057 − 0.309017i 1.12765 − 0.178602i 0.809017 + 0.587785i 2.04543 − 0.903437i −1.12765 − 0.178602i 4.78449 −0.587785 − 0.809017i −1.61348 + 0.524252i −2.22450 + 0.227146i
47.14 −0.951057 − 0.309017i 1.43307 − 0.226976i 0.809017 + 0.587785i −0.382747 − 2.20307i −1.43307 − 0.226976i −0.847356 −0.587785 − 0.809017i −0.851002 + 0.276507i −0.316772 + 2.21352i
47.15 −0.951057 − 0.309017i 2.10003 − 0.332612i 0.809017 + 0.587785i −1.68507 + 1.46988i −2.10003 − 0.332612i −3.12290 −0.587785 − 0.809017i 1.44632 − 0.469939i 2.05681 − 0.877226i
47.16 −0.951057 − 0.309017i 2.21976 − 0.351576i 0.809017 + 0.587785i 0.321222 + 2.21288i −2.21976 − 0.351576i 3.16185 −0.587785 − 0.809017i 1.95058 − 0.633781i 0.378315 − 2.20383i
47.17 −0.951057 − 0.309017i 2.88573 − 0.457055i 0.809017 + 0.587785i −2.17063 + 0.536990i −2.88573 − 0.457055i 1.86528 −0.587785 − 0.809017i 5.26539 − 1.71083i 2.23033 + 0.160054i
47.18 −0.951057 − 0.309017i 3.28011 − 0.519518i 0.809017 + 0.587785i 0.917882 − 2.03899i −3.28011 − 0.519518i −1.46206 −0.587785 − 0.809017i 7.63602 − 2.48109i −1.50304 + 1.65556i
83.1 −0.951057 + 0.309017i −3.19611 − 0.506214i 0.809017 − 0.587785i −1.82170 + 1.29670i 3.19611 − 0.506214i −0.716524 −0.587785 + 0.809017i 7.10571 + 2.30878i 1.33184 − 1.79617i
83.2 −0.951057 + 0.309017i −3.05796 − 0.484334i 0.809017 − 0.587785i 2.07976 − 0.821347i 3.05796 − 0.484334i 4.76595 −0.587785 + 0.809017i 6.26339 + 2.03510i −1.72416 + 1.42383i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.18
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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.b yes 144
13.d odd 4 1 650.2.ba.b ✓ 144
25.f odd 20 1 650.2.ba.b ✓ 144
325.be even 20 1 inner 650.2.bd.b yes 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.ba.b ✓ 144 13.d odd 4 1
650.2.ba.b ✓ 144 25.f odd 20 1
650.2.bd.b yes 144 1.a even 1 1 trivial
650.2.bd.b yes 144 325.be even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} + 10 T_{3}^{142} + 8 T_{3}^{141} - 251 T_{3}^{140} + 136 T_{3}^{139} - 3048 T_{3}^{138} + \cdots + 60\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display