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Properties

Label 520.2.cl.c
Level $520$
Weight $2$
Character orbit 520.cl
Analytic conductor $4.152$
Analytic rank $0$
Dimension $40$
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] = [520,2,Mod(137,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.cl (of order \(12\), 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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{5} - 2 q^{7} + 6 q^{9} - 6 q^{11} - 10 q^{13} + 2 q^{15} + 2 q^{17} - 16 q^{19} - 18 q^{21} - 6 q^{23} + 8 q^{25} - 6 q^{27} - 12 q^{29} - 24 q^{31} + 48 q^{33} + 46 q^{35} + 4 q^{37} + 18 q^{39} + 10 q^{41} - 50 q^{45} - 36 q^{47} - 56 q^{49} + 6 q^{53} + 38 q^{59} + 16 q^{61} - 6 q^{67} - 28 q^{69} + 2 q^{71} + 16 q^{75} - 56 q^{77} + 54 q^{81} + 48 q^{83} - 6 q^{85} + 42 q^{87} - 2 q^{89} + 4 q^{91} - 4 q^{93} + 70 q^{95} + 48 q^{97} - 22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 0 −0.752849 − 2.80967i 0 0.0201667 + 2.23598i 0 −2.00794 − 3.47785i 0 −4.72939 + 2.73051i 0
137.2 0 −0.540233 − 2.01618i 0 2.23586 − 0.0306185i 0 1.81145 + 3.13753i 0 −1.17504 + 0.678410i 0
137.3 0 −0.303411 − 1.13235i 0 −2.02818 + 0.941541i 0 1.06769 + 1.84930i 0 1.40793 − 0.812866i 0
137.4 0 −0.229111 − 0.855055i 0 0.0815584 − 2.23458i 0 −0.656229 − 1.13662i 0 1.91945 − 1.10819i 0
137.5 0 0.0875190 + 0.326625i 0 1.62239 + 1.53879i 0 −1.16158 − 2.01192i 0 2.49905 − 1.44283i 0
137.6 0 0.110419 + 0.412091i 0 1.56300 − 1.59907i 0 0.0499629 + 0.0865384i 0 2.44045 − 1.40899i 0
137.7 0 0.406501 + 1.51708i 0 −1.79762 − 1.32987i 0 1.69514 + 2.93607i 0 0.461785 − 0.266612i 0
137.8 0 0.538263 + 2.00883i 0 −0.533140 + 2.17158i 0 0.278217 + 0.481885i 0 −1.14758 + 0.662554i 0
137.9 0 0.674846 + 2.51856i 0 −1.93237 − 1.12514i 0 −2.51508 − 4.35625i 0 −3.28965 + 1.89928i 0
137.10 0 0.874082 + 3.26212i 0 2.13437 − 0.666679i 0 1.80440 + 3.12530i 0 −7.27931 + 4.20271i 0
297.1 0 −2.96139 − 0.793503i 0 −1.67030 + 1.48664i 0 −0.312185 + 0.540720i 0 5.54212 + 3.19975i 0
297.2 0 −2.75892 − 0.739251i 0 1.51971 − 1.64027i 0 1.19183 − 2.06430i 0 4.46708 + 2.57907i 0
297.3 0 −1.48300 − 0.397369i 0 −0.0347397 − 2.23580i 0 −1.69162 + 2.92997i 0 −0.556683 − 0.321401i 0
297.4 0 −1.00772 − 0.270018i 0 0.732364 + 2.11273i 0 −1.91516 + 3.31716i 0 −1.65548 − 0.955794i 0
297.5 0 −0.670838 − 0.179751i 0 2.12910 + 0.683315i 0 1.90688 − 3.30282i 0 −2.18036 − 1.25883i 0
297.6 0 −0.377813 − 0.101235i 0 −1.11844 + 1.93626i 0 0.730992 − 1.26612i 0 −2.46558 − 1.42350i 0
297.7 0 0.827235 + 0.221657i 0 −2.21171 − 0.329165i 0 0.660322 − 1.14371i 0 −1.96289 − 1.13328i 0
297.8 0 1.77330 + 0.475154i 0 2.15354 + 0.601876i 0 −1.64176 + 2.84361i 0 0.320741 + 0.185180i 0
297.9 0 2.84956 + 0.763536i 0 −2.21915 + 0.274555i 0 −2.47566 + 4.28797i 0 4.93891 + 2.85148i 0
297.10 0 2.94357 + 0.788728i 0 0.353592 + 2.20793i 0 2.18034 − 3.77646i 0 5.44445 + 3.14336i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.cl.c ✓ 40
5.c odd 4 1 520.2.cw.b yes 40
13.f odd 12 1 520.2.cw.b yes 40
65.t even 12 1 inner 520.2.cl.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.cl.c ✓ 40 1.a even 1 1 trivial
520.2.cl.c ✓ 40 65.t even 12 1 inner
520.2.cw.b yes 40 5.c odd 4 1
520.2.cw.b yes 40 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 3 T_{3}^{38} + 14 T_{3}^{37} - 153 T_{3}^{36} - 184 T_{3}^{35} + 566 T_{3}^{34} + \cdots + 1982464 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display