Properties

Label 804.749
Modulus $804$
Conductor $201$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,33,41]))
 
pari: [g,chi] = znchar(Mod(749,804))
 

Basic properties

Modulus: \(804\)
Conductor: \(201\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{201}(146,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 804.ba

\(\chi_{804}(41,\cdot)\) \(\chi_{804}(101,\cdot)\) \(\chi_{804}(113,\cdot)\) \(\chi_{804}(185,\cdot)\) \(\chi_{804}(197,\cdot)\) \(\chi_{804}(221,\cdot)\) \(\chi_{804}(233,\cdot)\) \(\chi_{804}(245,\cdot)\) \(\chi_{804}(281,\cdot)\) \(\chi_{804}(329,\cdot)\) \(\chi_{804}(353,\cdot)\) \(\chi_{804}(413,\cdot)\) \(\chi_{804}(497,\cdot)\) \(\chi_{804}(593,\cdot)\) \(\chi_{804}(605,\cdot)\) \(\chi_{804}(653,\cdot)\) \(\chi_{804}(677,\cdot)\) \(\chi_{804}(701,\cdot)\) \(\chi_{804}(749,\cdot)\) \(\chi_{804}(785,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((403,269,337)\) → \((1,-1,e\left(\frac{41}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 804 }(749, a) \) \(1\)\(1\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 804 }(749,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 804 }(749,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 804 }(749,·),\chi_{ 804 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 804 }(749,·)) \;\) at \(\; a,b = \) e.g. 1,2