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Properties

Label 9405.469
Modulus $9405$
Conductor $1045$
Order $90$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9405, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([0,45,63,25]))
 
Copy content gp:[g,chi] = znchar(Mod(469, 9405))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9405.469");
 

Basic properties

Modulus: \(9405\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1045\)
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 \(\chi_{1045}(469,\cdot)\)
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 9405.ne

\(\chi_{9405}(469,\cdot)\) \(\chi_{9405}(964,\cdot)\) \(\chi_{9405}(1009,\cdot)\) \(\chi_{9405}(1459,\cdot)\) \(\chi_{9405}(1504,\cdot)\) \(\chi_{9405}(2404,\cdot)\) \(\chi_{9405}(2719,\cdot)\) \(\chi_{9405}(3214,\cdot)\) \(\chi_{9405}(3889,\cdot)\) \(\chi_{9405}(4384,\cdot)\) \(\chi_{9405}(4429,\cdot)\) \(\chi_{9405}(4879,\cdot)\) \(\chi_{9405}(4924,\cdot)\) \(\chi_{9405}(4969,\cdot)\) \(\chi_{9405}(6454,\cdot)\) \(\chi_{9405}(6679,\cdot)\) \(\chi_{9405}(6949,\cdot)\) \(\chi_{9405}(7444,\cdot)\) \(\chi_{9405}(7849,\cdot)\) \(\chi_{9405}(8164,\cdot)\) \(\chi_{9405}(8344,\cdot)\) \(\chi_{9405}(8389,\cdot)\) \(\chi_{9405}(8659,\cdot)\) \(\chi_{9405}(9154,\cdot)\)

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

\((1046,1882,5986,496)\) → \((1,-1,e\left(\frac{7}{10}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(23\)\(26\)
\( \chi_{ 9405 }(469, a) \) \(1\)\(1\)\(e\left(\frac{43}{90}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{15}\right)\)
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)
 
\( \chi_{ 9405 }(469,a) \;\) at \(\;a = \) e.g. 2