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Properties

Label 47.17.b.b
Level $47$
Weight $17$
Character orbit 47.b
Analytic conductor $76.293$
Analytic rank $0$
Dimension $58$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [47,17,Mod(46,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.46");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 47.b (of order \(2\), degree \(1\), 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: \(76.2925356126\)
Analytic rank: \(0\)
Dimension: \(58\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 58 q - 2 q^{2} - 7912 q^{3} + 1810082 q^{4} + 1079932 q^{6} - 1154132 q^{7} - 30175738 q^{8} + 748683554 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 58 q - 2 q^{2} - 7912 q^{3} + 1810082 q^{4} + 1079932 q^{6} - 1154132 q^{7} - 30175738 q^{8} + 748683554 q^{9} + 3460528164 q^{12} - 11577182096 q^{14} + 69339077058 q^{16} - 4758439220 q^{17} + 102064390450 q^{18} - 22746599936 q^{21} - 806721324556 q^{24} - 2508619152170 q^{25} - 1345644529432 q^{27} - 698878535136 q^{28} + 3461680411430 q^{32} - 13155644143888 q^{34} + 30956868331470 q^{36} - 5195227703024 q^{37} + 161793018765560 q^{42} - 167107917118626 q^{47} + 133081345573756 q^{48} + 55015569300482 q^{49} - 116924163720758 q^{50} - 121523770796876 q^{51} - 132871941935192 q^{53} + 132592717709416 q^{54} - 250863215859996 q^{55} - 346333439502312 q^{56} - 49472195706536 q^{59} - 803950920247640 q^{61} + 673179316773520 q^{63} + 561616823145010 q^{64} + 12\!\cdots\!40 q^{65}+ \cdots - 13\!\cdots\!62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −483.920 10218.9 168642. − 629554.i −4.94514e6 −3.94268e6 −4.98951e7 6.13798e7 3.04653e8i
46.2 −483.920 10218.9 168642. 629554.i −4.94514e6 −3.94268e6 −4.98951e7 6.13798e7 − 3.04653e8i
46.3 −478.811 −880.197 163724. 331340.i 421448. 1.44361e6 −4.70137e7 −4.22720e7 − 1.58649e8i
46.4 −478.811 −880.197 163724. − 331340.i 421448. 1.44361e6 −4.70137e7 −4.22720e7 1.58649e8i
46.5 −430.523 −10115.6 119814. 531121.i 4.35501e6 1.08046e7 −2.33678e7 5.92791e7 − 2.28660e8i
46.6 −430.523 −10115.6 119814. − 531121.i 4.35501e6 1.08046e7 −2.33678e7 5.92791e7 2.28660e8i
46.7 −410.356 −2674.55 102856. − 459309.i 1.09751e6 −1.23203e6 −1.53143e7 −3.58935e7 1.88480e8i
46.8 −410.356 −2674.55 102856. 459309.i 1.09751e6 −1.23203e6 −1.53143e7 −3.58935e7 − 1.88480e8i
46.9 −397.406 7787.59 92395.2 171163.i −3.09483e6 6.63274e6 −1.06740e7 1.75998e7 − 6.80212e7i
46.10 −397.406 7787.59 92395.2 − 171163.i −3.09483e6 6.63274e6 −1.06740e7 1.75998e7 6.80212e7i
46.11 −361.726 4867.85 65309.7 1422.42i −1.76083e6 −1.05988e7 81872.3 −1.93508e7 − 514528.i
46.12 −361.726 4867.85 65309.7 − 1422.42i −1.76083e6 −1.05988e7 81872.3 −1.93508e7 514528.i
46.13 −331.512 −8443.28 44364.3 396261.i 2.79905e6 −2.84989e6 7.01868e6 2.82423e7 − 1.31365e8i
46.14 −331.512 −8443.28 44364.3 − 396261.i 2.79905e6 −2.84989e6 7.01868e6 2.82423e7 1.31365e8i
46.15 −258.183 6897.07 1122.25 723277.i −1.78070e6 2.06292e6 1.66305e7 4.52288e6 − 1.86738e8i
46.16 −258.183 6897.07 1122.25 − 723277.i −1.78070e6 2.06292e6 1.66305e7 4.52288e6 1.86738e8i
46.17 −223.925 −1007.60 −15393.7 73912.4i 225626. 8.62188e6 1.81222e7 −4.20315e7 − 1.65508e7i
46.18 −223.925 −1007.60 −15393.7 − 73912.4i 225626. 8.62188e6 1.81222e7 −4.20315e7 1.65508e7i
46.19 −213.973 −375.596 −19751.5 514990.i 80367.6 −3.09913e6 1.82492e7 −4.29056e7 − 1.10194e8i
46.20 −213.973 −375.596 −19751.5 − 514990.i 80367.6 −3.09913e6 1.82492e7 −4.29056e7 1.10194e8i
See all 58 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.17.b.b ✓ 58
47.b odd 2 1 inner 47.17.b.b ✓ 58
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.17.b.b ✓ 58 1.a even 1 1 trivial
47.17.b.b ✓ 58 47.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{29} + T_{2}^{28} - 1402792 T_{2}^{27} + 3670188 T_{2}^{26} + 868488784956 T_{2}^{25} + \cdots + 37\!\cdots\!60 \) acting on \(S_{17}^{\mathrm{new}}(47, [\chi])\). Copy content Toggle raw display