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Properties

Label 667.2.f.b
Level $667$
Weight $2$
Character orbit 667.f
Analytic conductor $5.326$
Analytic rank $0$
Dimension $104$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [667,2,Mod(505,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.f (of order \(4\), degree \(2\), 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: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 8 q^{2} - 4 q^{3} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 8 q^{2} - 4 q^{3} - 16 q^{8} + 40 q^{12} - 104 q^{16} + 72 q^{18} - 16 q^{23} + 120 q^{24} + 168 q^{25} - 44 q^{26} - 52 q^{27} - 28 q^{29} - 16 q^{31} - 60 q^{32} - 72 q^{36} - 24 q^{39} - 28 q^{41} - 8 q^{46} - 28 q^{47} - 20 q^{48} - 208 q^{49} - 40 q^{50} + 24 q^{52} + 176 q^{54} + 32 q^{55} - 4 q^{58} + 24 q^{59} - 32 q^{69} + 40 q^{70} + 68 q^{72} - 28 q^{73} - 44 q^{75} + 20 q^{77} - 8 q^{78} + 128 q^{81} + 12 q^{85} - 136 q^{87} + 152 q^{94} - 88 q^{95} + 152 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1 −1.85268 − 1.85268i 2.22010 + 2.22010i 4.86484i −2.69142 − 8.22628i − 4.74123i 5.30762 − 5.30762i 6.85773i 4.98633 + 4.98633i
505.2 −1.85268 − 1.85268i 2.22010 + 2.22010i 4.86484i 2.69142 − 8.22628i 4.74123i 5.30762 − 5.30762i 6.85773i −4.98633 − 4.98633i
505.3 −1.79430 − 1.79430i −0.986996 − 0.986996i 4.43899i −1.82916 3.54192i − 4.14184i 4.37627 − 4.37627i − 1.05168i 3.28206 + 3.28206i
505.4 −1.79430 − 1.79430i −0.986996 − 0.986996i 4.43899i 1.82916 3.54192i 4.14184i 4.37627 − 4.37627i − 1.05168i −3.28206 − 3.28206i
505.5 −1.69080 − 1.69080i 0.716342 + 0.716342i 3.71762i −4.05425 − 2.42239i 3.99852i 2.90416 − 2.90416i − 1.97371i 6.85494 + 6.85494i
505.6 −1.69080 − 1.69080i 0.716342 + 0.716342i 3.71762i 4.05425 − 2.42239i − 3.99852i 2.90416 − 2.90416i − 1.97371i −6.85494 − 6.85494i
505.7 −1.59667 − 1.59667i −1.46615 − 1.46615i 3.09869i −3.85012 4.68191i − 0.0116925i 1.75424 − 1.75424i 1.29921i 6.14736 + 6.14736i
505.8 −1.59667 − 1.59667i −1.46615 − 1.46615i 3.09869i 3.85012 4.68191i 0.0116925i 1.75424 − 1.75424i 1.29921i −6.14736 − 6.14736i
505.9 −1.32483 − 1.32483i −0.264952 − 0.264952i 1.51032i −0.867389 0.702030i 1.12883i −0.648736 + 0.648736i − 2.85960i 1.14914 + 1.14914i
505.10 −1.32483 − 1.32483i −0.264952 − 0.264952i 1.51032i 0.867389 0.702030i − 1.12883i −0.648736 + 0.648736i − 2.85960i −1.14914 − 1.14914i
505.11 −1.25222 − 1.25222i −1.92455 − 1.92455i 1.13609i −0.621736 4.81990i 3.83416i −1.08181 + 1.08181i 4.40780i 0.778548 + 0.778548i
505.12 −1.25222 − 1.25222i −1.92455 − 1.92455i 1.13609i 0.621736 4.81990i − 3.83416i −1.08181 + 1.08181i 4.40780i −0.778548 − 0.778548i
505.13 −1.18141 − 1.18141i 1.53580 + 1.53580i 0.791450i −1.53719 − 3.62880i − 1.71504i −1.42779 + 1.42779i 1.71734i 1.81605 + 1.81605i
505.14 −1.18141 − 1.18141i 1.53580 + 1.53580i 0.791450i 1.53719 − 3.62880i 1.71504i −1.42779 + 1.42779i 1.71734i −1.81605 − 1.81605i
505.15 −0.969473 − 0.969473i 0.413572 + 0.413572i − 0.120245i −3.11311 − 0.801894i − 4.13594i −2.05552 + 2.05552i − 2.65792i 3.01808 + 3.01808i
505.16 −0.969473 − 0.969473i 0.413572 + 0.413572i − 0.120245i 3.11311 − 0.801894i 4.13594i −2.05552 + 2.05552i − 2.65792i −3.01808 − 3.01808i
505.17 −0.964257 − 0.964257i 2.08716 + 2.08716i − 0.140419i −2.04542 − 4.02512i 2.97120i −2.06391 + 2.06391i 5.71249i 1.97231 + 1.97231i
505.18 −0.964257 − 0.964257i 2.08716 + 2.08716i − 0.140419i 2.04542 − 4.02512i − 2.97120i −2.06391 + 2.06391i 5.71249i −1.97231 − 1.97231i
505.19 −0.623219 − 0.623219i −0.544367 − 0.544367i − 1.22320i −2.10514 0.678519i 2.01641i −2.00876 + 2.00876i − 2.40733i 1.31197 + 1.31197i
505.20 −0.623219 − 0.623219i −0.544367 − 0.544367i − 1.22320i 2.10514 0.678519i − 2.01641i −2.00876 + 2.00876i − 2.40733i −1.31197 − 1.31197i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner
29.c odd 4 1 inner
667.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 667.2.f.b ✓ 104
23.b odd 2 1 inner 667.2.f.b ✓ 104
29.c odd 4 1 inner 667.2.f.b ✓ 104
667.f even 4 1 inner 667.2.f.b ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.f.b ✓ 104 1.a even 1 1 trivial
667.2.f.b ✓ 104 23.b odd 2 1 inner
667.2.f.b ✓ 104 29.c odd 4 1 inner
667.2.f.b ✓ 104 667.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 4 T_{2}^{51} + 8 T_{2}^{50} + 8 T_{2}^{49} + 183 T_{2}^{48} + 722 T_{2}^{47} + 1456 T_{2}^{46} + \cdots + 5476 \) acting on \(S_{2}^{\mathrm{new}}(667, [\chi])\). Copy content Toggle raw display