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Properties

Label 574.2.u.a
Level $574$
Weight $2$
Character orbit 574.u
Analytic conductor $4.583$
Analytic rank $0$
Dimension $80$
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] = [574,2,Mod(43,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.u (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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + 20 q^{4} + 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + 20 q^{4} + 16 q^{6} - 16 q^{11} - 4 q^{12} + 20 q^{13} + 36 q^{15} - 20 q^{16} + 4 q^{17} + 20 q^{18} + 28 q^{19} - 20 q^{20} + 16 q^{22} - 8 q^{23} + 4 q^{24} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 36 q^{30} - 12 q^{31} + 4 q^{34} - 20 q^{36} + 12 q^{37} - 32 q^{38} - 80 q^{39} - 8 q^{41} + 16 q^{44} - 108 q^{45} - 20 q^{46} + 8 q^{47} - 16 q^{48} - 20 q^{51} - 40 q^{53} + 16 q^{54} + 60 q^{55} + 52 q^{57} + 12 q^{58} - 12 q^{59} + 24 q^{60} + 100 q^{61} - 16 q^{63} + 20 q^{64} - 20 q^{65} - 36 q^{67} + 36 q^{68} - 32 q^{69} - 12 q^{71} - 20 q^{72} + 20 q^{74} + 40 q^{75} + 12 q^{76} + 40 q^{77} + 16 q^{78} + 16 q^{79} - 20 q^{80} - 104 q^{81} - 12 q^{82} - 16 q^{85} + 12 q^{86} + 140 q^{87} + 24 q^{88} - 32 q^{89} + 8 q^{92} + 8 q^{93} - 12 q^{94} + 12 q^{95} - 4 q^{96} + 4 q^{97} + 20 q^{98} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 0.951057 − 0.309017i −2.33712 + 2.33712i 0.809017 − 0.587785i −1.70120 − 2.34151i −1.50053 + 2.94495i 0.453990 + 0.891007i 0.587785 − 0.809017i − 7.92429i −2.34151 − 1.70120i
43.2 0.951057 − 0.309017i −1.62445 + 1.62445i 0.809017 − 0.587785i −0.764320 − 1.05200i −1.04296 + 2.04693i −0.453990 − 0.891007i 0.587785 − 0.809017i − 2.27768i −1.05200 − 0.764320i
43.3 0.951057 − 0.309017i −1.27895 + 1.27895i 0.809017 − 0.587785i −1.21136 − 1.66730i −0.821136 + 1.61157i −0.453990 − 0.891007i 0.587785 − 0.809017i − 0.271421i −1.66730 − 1.21136i
43.4 0.951057 − 0.309017i −0.506337 + 0.506337i 0.809017 − 0.587785i 1.31445 + 1.80919i −0.325088 + 0.638022i 0.453990 + 0.891007i 0.587785 − 0.809017i 2.48725i 1.80919 + 1.31445i
43.5 0.951057 − 0.309017i 0.0482483 − 0.0482483i 0.809017 − 0.587785i −0.234147 − 0.322276i 0.0309773 − 0.0607964i 0.453990 + 0.891007i 0.587785 − 0.809017i 2.99534i −0.322276 − 0.234147i
43.6 0.951057 − 0.309017i 0.398829 − 0.398829i 0.809017 − 0.587785i 1.92408 + 2.64827i 0.256064 − 0.502554i −0.453990 − 0.891007i 0.587785 − 0.809017i 2.68187i 2.64827 + 1.92408i
43.7 0.951057 − 0.309017i 1.34719 − 1.34719i 0.809017 − 0.587785i −0.232029 − 0.319360i 0.864951 − 1.69756i −0.453990 − 0.891007i 0.587785 − 0.809017i − 0.629857i −0.319360 − 0.232029i
43.8 0.951057 − 0.309017i 1.54408 − 1.54408i 0.809017 − 0.587785i −1.84191 − 2.53517i 0.991360 − 1.94565i 0.453990 + 0.891007i 0.587785 − 0.809017i − 1.76836i −2.53517 − 1.84191i
43.9 0.951057 − 0.309017i 1.79942 − 1.79942i 0.809017 − 0.587785i −1.05059 − 1.44602i 1.15530 − 2.26740i −0.453990 − 0.891007i 0.587785 − 0.809017i − 3.47580i −1.44602 − 1.05059i
43.10 0.951057 − 0.309017i 1.89317 − 1.89317i 0.809017 − 0.587785i 1.56096 + 2.14848i 1.21549 − 2.38554i 0.453990 + 0.891007i 0.587785 − 0.809017i − 4.16820i 2.14848 + 1.56096i
169.1 −0.587785 − 0.809017i −2.34137 − 2.34137i −0.309017 + 0.951057i −0.332119 − 0.107912i −0.517986 + 3.27043i 0.156434 + 0.987688i 0.951057 − 0.309017i 7.96404i 0.107912 + 0.332119i
169.2 −0.587785 − 0.809017i −1.38661 − 1.38661i −0.309017 + 0.951057i 1.51658 + 0.492765i −0.306761 + 1.93682i 0.156434 + 0.987688i 0.951057 − 0.309017i 0.845358i −0.492765 − 1.51658i
169.3 −0.587785 − 0.809017i −1.37500 − 1.37500i −0.309017 + 0.951057i 2.70783 + 0.879827i −0.304193 + 1.92060i −0.156434 − 0.987688i 0.951057 − 0.309017i 0.781229i −0.879827 − 2.70783i
169.4 −0.587785 − 0.809017i −0.887212 − 0.887212i −0.309017 + 0.951057i −0.911575 − 0.296189i −0.196279 + 1.23926i −0.156434 − 0.987688i 0.951057 − 0.309017i − 1.42571i 0.296189 + 0.911575i
169.5 −0.587785 − 0.809017i −0.364190 − 0.364190i −0.309017 + 0.951057i −2.56002 − 0.831800i −0.0805703 + 0.508701i −0.156434 − 0.987688i 0.951057 − 0.309017i − 2.73473i 0.831800 + 2.56002i
169.6 −0.587785 − 0.809017i 0.350958 + 0.350958i −0.309017 + 0.951057i −0.961282 − 0.312339i 0.0776431 − 0.490219i 0.156434 + 0.987688i 0.951057 − 0.309017i − 2.75366i 0.312339 + 0.961282i
169.7 −0.587785 − 0.809017i 0.756754 + 0.756754i −0.309017 + 0.951057i 3.33364 + 1.08317i 0.167418 − 1.05704i −0.156434 − 0.987688i 0.951057 − 0.309017i − 1.85465i −1.08317 − 3.33364i
169.8 −0.587785 − 0.809017i 1.52921 + 1.52921i −0.309017 + 0.951057i −3.74452 − 1.21667i 0.338311 − 2.13601i 0.156434 + 0.987688i 0.951057 − 0.309017i 1.67699i 1.21667 + 3.74452i
169.9 −0.587785 − 0.809017i 2.06904 + 2.06904i −0.309017 + 0.951057i 1.41569 + 0.459984i 0.457737 − 2.89004i 0.156434 + 0.987688i 0.951057 − 0.309017i 5.56184i −0.459984 − 1.41569i
169.10 −0.587785 − 0.809017i 2.09088 + 2.09088i −0.309017 + 0.951057i 1.77185 + 0.575708i 0.462568 − 2.92054i −0.156434 − 0.987688i 0.951057 − 0.309017i 5.74352i −0.575708 − 1.77185i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.u.a ✓ 80
41.g even 20 1 inner 574.2.u.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.u.a ✓ 80 1.a even 1 1 trivial
574.2.u.a ✓ 80 41.g even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 4 T_{3}^{79} + 8 T_{3}^{78} - 8 T_{3}^{77} + 566 T_{3}^{76} - 2296 T_{3}^{75} + \cdots + 430336 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display