Properties

Label 1694.23
Modulus $1694$
Conductor $847$
Order $33$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1694, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,42]))
 
pari: [g,chi] = znchar(Mod(23,1694))
 

Basic properties

Modulus: \(1694\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1694.u

\(\chi_{1694}(23,\cdot)\) \(\chi_{1694}(67,\cdot)\) \(\chi_{1694}(177,\cdot)\) \(\chi_{1694}(221,\cdot)\) \(\chi_{1694}(331,\cdot)\) \(\chi_{1694}(375,\cdot)\) \(\chi_{1694}(529,\cdot)\) \(\chi_{1694}(639,\cdot)\) \(\chi_{1694}(683,\cdot)\) \(\chi_{1694}(793,\cdot)\) \(\chi_{1694}(837,\cdot)\) \(\chi_{1694}(947,\cdot)\) \(\chi_{1694}(991,\cdot)\) \(\chi_{1694}(1101,\cdot)\) \(\chi_{1694}(1145,\cdot)\) \(\chi_{1694}(1255,\cdot)\) \(\chi_{1694}(1299,\cdot)\) \(\chi_{1694}(1409,\cdot)\) \(\chi_{1694}(1563,\cdot)\) \(\chi_{1694}(1607,\cdot)\)

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 33 polynomial

Values on generators

\((969,365)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 1694 }(23, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1694 }(23,a) \;\) at \(\;a = \) e.g. 2