↑

Properties

Label 15204.8825
Modulus $15204$
Conductor $3801$
Order $90$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15204, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([0,45,75,77]))
 
Copy content gp:[g,chi] = znchar(Mod(8825, 15204))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15204.8825");
 

Basic properties

Modulus: \(15204\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3801\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{3801}(1223,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 15204.nf

\(\chi_{15204}(185,\cdot)\) \(\chi_{15204}(1361,\cdot)\) \(\chi_{15204}(1445,\cdot)\) \(\chi_{15204}(1865,\cdot)\) \(\chi_{15204}(2453,\cdot)\) \(\chi_{15204}(2525,\cdot)\) \(\chi_{15204}(3545,\cdot)\) \(\chi_{15204}(3785,\cdot)\) \(\chi_{15204}(7481,\cdot)\) \(\chi_{15204}(8165,\cdot)\) \(\chi_{15204}(8405,\cdot)\) \(\chi_{15204}(8825,\cdot)\) \(\chi_{15204}(9161,\cdot)\) \(\chi_{15204}(9761,\cdot)\) \(\chi_{15204}(9917,\cdot)\) \(\chi_{15204}(10169,\cdot)\) \(\chi_{15204}(11093,\cdot)\) \(\chi_{15204}(11177,\cdot)\) \(\chi_{15204}(13541,\cdot)\) \(\chi_{15204}(13793,\cdot)\) \(\chi_{15204}(13949,\cdot)\) \(\chi_{15204}(14129,\cdot)\) \(\chi_{15204}(14285,\cdot)\) \(\chi_{15204}(14465,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((7603,5069,2173,12853)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{77}{90}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 15204 }(8825, a) \) \(1\)\(1\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{41}{45}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 15204 }(8825,a) \;\) at \(\;a = \) e.g. 2