Properties

Label 667.626
Modulus $667$
Conductor $667$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(667, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,33]))
 
pari: [g,chi] = znchar(Mod(626,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 667.q

\(\chi_{667}(17,\cdot)\) \(\chi_{667}(99,\cdot)\) \(\chi_{667}(157,\cdot)\) \(\chi_{667}(191,\cdot)\) \(\chi_{667}(244,\cdot)\) \(\chi_{667}(249,\cdot)\) \(\chi_{667}(273,\cdot)\) \(\chi_{667}(336,\cdot)\) \(\chi_{667}(360,\cdot)\) \(\chi_{667}(365,\cdot)\) \(\chi_{667}(389,\cdot)\) \(\chi_{667}(447,\cdot)\) \(\chi_{667}(452,\cdot)\) \(\chi_{667}(481,\cdot)\) \(\chi_{667}(534,\cdot)\) \(\chi_{667}(539,\cdot)\) \(\chi_{667}(563,\cdot)\) \(\chi_{667}(592,\cdot)\) \(\chi_{667}(626,\cdot)\) \(\chi_{667}(655,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1

Values on generators

\((465,553)\) → \((e\left(\frac{1}{22}\right),-i)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{7}{44}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(626,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{667}(626,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(626,r) e\left(\frac{2r}{667}\right) = 25.4415399785+-4.441626225i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(626,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{667}(626,\cdot),\chi_{667}(1,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(626,r) \chi_{667}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(626,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{667}(626,·)) = \sum_{r \in \Z/667\Z} \chi_{667}(626,r) e\left(\frac{1 r + 2 r^{-1}}{667}\right) = 13.6616486722+-7.459824561i \)