Properties

Label 639.2.f.d
Level $639$
Weight $2$
Character orbit 639.f
Analytic conductor $5.102$
Analytic rank $0$
Dimension $24$
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] = [639,2,Mod(199,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.f (of order \(5\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 213)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 7 q^{4} + 2 q^{5} - 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 7 q^{4} + 2 q^{5} - 4 q^{7} - 8 q^{8} - 8 q^{10} + 5 q^{11} - 8 q^{13} - 29 q^{14} + 5 q^{16} + 2 q^{19} + 2 q^{20} - 11 q^{22} - 6 q^{23} - 12 q^{25} - 24 q^{26} - 25 q^{28} - 2 q^{29} - 24 q^{31} + 36 q^{32} + 102 q^{34} - 22 q^{35} + 10 q^{37} + 27 q^{38} - 56 q^{40} - 2 q^{41} - 18 q^{43} - 9 q^{44} - 11 q^{46} + 36 q^{47} - 6 q^{49} - 3 q^{50} - 35 q^{52} - 6 q^{53} - 9 q^{55} + 100 q^{56} + 64 q^{58} - 3 q^{61} + 36 q^{62} + 66 q^{64} + 3 q^{65} - 18 q^{67} - 76 q^{68} + 130 q^{70} + 10 q^{71} - 9 q^{73} - 7 q^{74} - 24 q^{76} - 39 q^{77} - 20 q^{79} - 51 q^{80} + 20 q^{82} - 17 q^{83} + 25 q^{85} + 69 q^{86} + 13 q^{88} + 28 q^{89} - 38 q^{91} + 81 q^{92} - 56 q^{94} - 35 q^{95} - 34 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1 −0.782687 + 2.40886i 0 −3.57199 2.59520i −1.75584 + 1.27569i 0 −1.09908 + 3.38262i 4.94904 3.59569i 0 −1.69869 5.22804i
199.2 −0.561101 + 1.72689i 0 −1.04928 0.762350i −0.100283 + 0.0728600i 0 0.566896 1.74473i −1.03271 + 0.750310i 0 −0.0695523 0.214060i
199.3 −0.221575 + 0.681938i 0 1.20209 + 0.873370i 1.12207 0.815232i 0 0.879183 2.70585i −2.02212 + 1.46916i 0 0.307315 + 0.945817i
199.4 0.0386980 0.119100i 0 1.60535 + 1.16635i −1.70590 + 1.23941i 0 −0.521916 + 1.60629i 0.403662 0.293278i 0 0.0815989 + 0.251135i
199.5 0.472579 1.45445i 0 −0.274058 0.199115i 2.89570 2.10385i 0 −0.0936882 + 0.288343i 2.05534 1.49329i 0 −1.69149 5.20588i
199.6 0.863103 2.65636i 0 −4.69326 3.40985i 0.0442495 0.0321491i 0 1.50467 4.63091i −8.58928 + 6.24048i 0 −0.0472077 0.145290i
289.1 −0.782687 2.40886i 0 −3.57199 + 2.59520i −1.75584 1.27569i 0 −1.09908 3.38262i 4.94904 + 3.59569i 0 −1.69869 + 5.22804i
289.2 −0.561101 1.72689i 0 −1.04928 + 0.762350i −0.100283 0.0728600i 0 0.566896 + 1.74473i −1.03271 0.750310i 0 −0.0695523 + 0.214060i
289.3 −0.221575 0.681938i 0 1.20209 0.873370i 1.12207 + 0.815232i 0 0.879183 + 2.70585i −2.02212 1.46916i 0 0.307315 0.945817i
289.4 0.0386980 + 0.119100i 0 1.60535 1.16635i −1.70590 1.23941i 0 −0.521916 1.60629i 0.403662 + 0.293278i 0 0.0815989 0.251135i
289.5 0.472579 + 1.45445i 0 −0.274058 + 0.199115i 2.89570 + 2.10385i 0 −0.0936882 0.288343i 2.05534 + 1.49329i 0 −1.69149 + 5.20588i
289.6 0.863103 + 2.65636i 0 −4.69326 + 3.40985i 0.0442495 + 0.0321491i 0 1.50467 + 4.63091i −8.58928 6.24048i 0 −0.0472077 + 0.145290i
451.1 −1.95351 + 1.41931i 0 1.18374 3.64316i 1.01092 + 3.11131i 0 −1.65679 + 1.20373i 1.36598 + 4.20407i 0 −6.39076 4.64316i
451.2 −1.86334 + 1.35380i 0 1.02125 3.14307i −0.582378 1.79238i 0 3.38602 2.46009i 0.928686 + 2.85820i 0 3.51169 + 2.55139i
451.3 −1.11950 + 0.813364i 0 −0.0263155 + 0.0809908i −0.789118 2.42866i 0 −2.24566 + 1.63157i −0.891637 2.74418i 0 2.85880 + 2.07704i
451.4 0.145962 0.106048i 0 −0.607975 + 1.87116i 0.911171 + 2.80430i 0 −2.68661 + 1.95194i 0.221196 + 0.680770i 0 0.430386 + 0.312694i
451.5 1.43997 1.04620i 0 0.360951 1.11089i 0.492126 + 1.51461i 0 2.18360 1.58648i 0.457583 + 1.40830i 0 2.29323 + 1.66613i
451.6 2.04140 1.48317i 0 1.34951 4.15337i −0.542726 1.67034i 0 −2.21662 + 1.61047i −1.84574 5.68062i 0 −3.58531 2.60488i
622.1 −1.95351 1.41931i 0 1.18374 + 3.64316i 1.01092 3.11131i 0 −1.65679 1.20373i 1.36598 4.20407i 0 −6.39076 + 4.64316i
622.2 −1.86334 1.35380i 0 1.02125 + 3.14307i −0.582378 + 1.79238i 0 3.38602 + 2.46009i 0.928686 2.85820i 0 3.51169 2.55139i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.f.d 24
3.b odd 2 1 213.2.e.c 24
71.c even 5 1 inner 639.2.f.d 24
213.h odd 10 1 213.2.e.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.2.e.c 24 3.b odd 2 1
213.2.e.c 24 213.h odd 10 1
639.2.f.d 24 1.a even 1 1 trivial
639.2.f.d 24 71.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 3 T_{2}^{23} + 14 T_{2}^{22} + 33 T_{2}^{21} + 120 T_{2}^{20} + 232 T_{2}^{19} + 771 T_{2}^{18} + 1527 T_{2}^{17} + 5570 T_{2}^{16} + 11089 T_{2}^{15} + 24491 T_{2}^{14} + 30579 T_{2}^{13} + 69694 T_{2}^{12} + 93123 T_{2}^{11} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display