Properties

Label 3549.1945
Modulus $3549$
Conductor $1183$
Order $52$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3549, base_ring=CyclotomicField(52))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,26,41]))
 
pari: [g,chi] = znchar(Mod(1945,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(762,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.cu

\(\chi_{3549}(34,\cdot)\) \(\chi_{3549}(265,\cdot)\) \(\chi_{3549}(307,\cdot)\) \(\chi_{3549}(538,\cdot)\) \(\chi_{3549}(580,\cdot)\) \(\chi_{3549}(811,\cdot)\) \(\chi_{3549}(853,\cdot)\) \(\chi_{3549}(1126,\cdot)\) \(\chi_{3549}(1357,\cdot)\) \(\chi_{3549}(1399,\cdot)\) \(\chi_{3549}(1630,\cdot)\) \(\chi_{3549}(1672,\cdot)\) \(\chi_{3549}(1903,\cdot)\) \(\chi_{3549}(1945,\cdot)\) \(\chi_{3549}(2176,\cdot)\) \(\chi_{3549}(2218,\cdot)\) \(\chi_{3549}(2449,\cdot)\) \(\chi_{3549}(2491,\cdot)\) \(\chi_{3549}(2722,\cdot)\) \(\chi_{3549}(2764,\cdot)\) \(\chi_{3549}(2995,\cdot)\) \(\chi_{3549}(3037,\cdot)\) \(\chi_{3549}(3268,\cdot)\) \(\chi_{3549}(3541,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((1184,1522,3382)\) → \((1,-1,e\left(\frac{41}{52}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{41}{52}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{11}{52}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(-i\)\(e\left(\frac{9}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(1945,a) \;\) at \(\;a = \) e.g. 2