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Properties

Label 520.2.cw.c
Level $520$
Weight $2$
Character orbit 520.cw
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(33,520)]
 
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, 9, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.cw (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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{5} - 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{5} - 6 q^{7} - 12 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 12 q^{17} - 16 q^{19} + 12 q^{21} - 2 q^{23} - 14 q^{25} + 12 q^{27} - 6 q^{29} + 8 q^{31} - 18 q^{33} + 16 q^{35} - 6 q^{37} + 22 q^{39} - 28 q^{41} + 34 q^{45} + 10 q^{49} + 4 q^{53} - 6 q^{55} - 56 q^{57} + 14 q^{59} + 6 q^{61} + 4 q^{63} - 14 q^{65} - 18 q^{69} - 6 q^{71} + 20 q^{73} + 4 q^{75} - 18 q^{77} - 32 q^{81} + 12 q^{85} - 32 q^{89} - 10 q^{91} + 78 q^{93} - 4 q^{95} - 22 q^{97} + 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −2.82289 + 0.756390i 0 1.25943 + 1.84766i 0 −2.78706 + 1.60911i 0 4.79848 − 2.77040i 0
33.2 0 −1.66343 + 0.445714i 0 −2.03017 + 0.937240i 0 0.210852 − 0.121735i 0 −0.0297478 + 0.0171749i 0
33.3 0 −1.43449 + 0.384371i 0 2.22522 − 0.220024i 0 3.94645 − 2.27848i 0 −0.688053 + 0.397248i 0
33.4 0 −0.767373 + 0.205617i 0 −0.863314 − 2.06269i 0 −0.550523 + 0.317845i 0 −2.05149 + 1.18443i 0
33.5 0 −0.767069 + 0.205535i 0 1.60900 − 1.55278i 0 −3.56559 + 2.05860i 0 −2.05193 + 1.18468i 0
33.6 0 −0.411629 + 0.110296i 0 0.171871 + 2.22945i 0 2.47635 − 1.42972i 0 −2.44080 + 1.40920i 0
33.7 0 1.50726 − 0.403870i 0 −1.23594 − 1.86345i 0 0.930587 − 0.537275i 0 −0.489344 + 0.282523i 0
33.8 0 1.57781 − 0.422773i 0 −1.83333 + 1.28020i 0 −3.81702 + 2.20376i 0 −0.287326 + 0.165888i 0
33.9 0 2.28568 − 0.612445i 0 1.30982 + 1.81229i 0 −0.197669 + 0.114124i 0 2.25116 − 1.29971i 0
33.10 0 2.49612 − 0.668834i 0 1.61947 − 1.54186i 0 1.85363 − 1.07020i 0 3.18521 − 1.83898i 0
97.1 0 −0.822389 − 3.06920i 0 −2.16091 − 0.574856i 0 0.688436 − 0.397469i 0 −6.14557 + 3.54815i 0
97.2 0 −0.593094 − 2.21346i 0 0.325118 + 2.21231i 0 −3.90923 + 2.25700i 0 −1.94956 + 1.12558i 0
97.3 0 −0.468904 − 1.74997i 0 1.19080 − 1.89262i 0 −0.158594 + 0.0915643i 0 −0.244465 + 0.141142i 0
97.4 0 −0.265717 − 0.991668i 0 2.03906 + 0.917731i 0 2.54441 − 1.46902i 0 1.68528 − 0.972995i 0
97.5 0 −0.0975914 − 0.364216i 0 −2.08087 + 0.818519i 0 1.58063 − 0.912577i 0 2.47495 − 1.42891i 0
97.6 0 0.00838687 + 0.0313002i 0 −1.62738 − 1.53350i 0 −3.24800 + 1.87523i 0 2.59717 − 1.49947i 0
97.7 0 0.473146 + 1.76580i 0 0.783605 + 2.09427i 0 2.03695 − 1.17604i 0 −0.296121 + 0.170966i 0
97.8 0 0.477788 + 1.78313i 0 −0.693207 − 2.12590i 0 2.37027 − 1.36848i 0 −0.353196 + 0.203918i 0
97.9 0 0.515085 + 1.92232i 0 −0.976246 + 2.01170i 0 −2.00806 + 1.15935i 0 −0.831939 + 0.480320i 0
97.10 0 0.773290 + 2.88596i 0 1.96798 − 1.06162i 0 −1.39681 + 0.806450i 0 −5.13269 + 2.96336i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 6 T_{3}^{38} - 4 T_{3}^{37} - 91 T_{3}^{36} + 16 T_{3}^{35} - 610 T_{3}^{34} + 204 T_{3}^{33} + \cdots + 10816 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display