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Properties

Label 34545.937
Modulus $34545$
Conductor $11515$
Order $644$
Real no
Primitive no
Minimal yes
Parity odd

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(34545, base_ring=CyclotomicField(644)) M = H._module chi = DirichletCharacter(H, M([0,161,414,602]))
 
Copy content gp:[g,chi] = znchar(Mod(937, 34545))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("34545.937");
 

Basic properties

Modulus: \(34545\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11515\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(644\)
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 \(\chi_{11515}(937,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 34545.gi

\(\chi_{34545}(13,\cdot)\) \(\chi_{34545}(223,\cdot)\) \(\chi_{34545}(433,\cdot)\) \(\chi_{34545}(622,\cdot)\) \(\chi_{34545}(727,\cdot)\) \(\chi_{34545}(748,\cdot)\) \(\chi_{34545}(937,\cdot)\) \(\chi_{34545}(1063,\cdot)\) \(\chi_{34545}(1147,\cdot)\) \(\chi_{34545}(1168,\cdot)\) \(\chi_{34545}(1252,\cdot)\) \(\chi_{34545}(1357,\cdot)\) \(\chi_{34545}(1378,\cdot)\) \(\chi_{34545}(1462,\cdot)\) \(\chi_{34545}(1483,\cdot)\) \(\chi_{34545}(1777,\cdot)\) \(\chi_{34545}(1903,\cdot)\) \(\chi_{34545}(1987,\cdot)\) \(\chi_{34545}(2113,\cdot)\) \(\chi_{34545}(2197,\cdot)\) \(\chi_{34545}(2323,\cdot)\) \(\chi_{34545}(2407,\cdot)\) \(\chi_{34545}(2428,\cdot)\) \(\chi_{34545}(2722,\cdot)\) \(\chi_{34545}(2953,\cdot)\) \(\chi_{34545}(3142,\cdot)\) \(\chi_{34545}(3352,\cdot)\) \(\chi_{34545}(3457,\cdot)\) \(\chi_{34545}(3583,\cdot)\) \(\chi_{34545}(3688,\cdot)\) ...

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: $\Q(\zeta_{644})$
Fixed field: Number field defined by a degree 644 polynomial (not computed)

Values on generators

\((11516,27637,9166,32341)\) → \((1,i,e\left(\frac{9}{14}\right),e\left(\frac{43}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 34545 }(937, a) \) \(-1\)\(1\)\(e\left(\frac{509}{644}\right)\)\(e\left(\frac{187}{322}\right)\)\(e\left(\frac{239}{644}\right)\)\(e\left(\frac{83}{322}\right)\)\(e\left(\frac{159}{644}\right)\)\(e\left(\frac{26}{161}\right)\)\(e\left(\frac{179}{644}\right)\)\(e\left(\frac{3}{46}\right)\)\(e\left(\frac{31}{644}\right)\)\(e\left(\frac{549}{644}\right)\)
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)
 
\( \chi_{ 34545 }(937,a) \;\) at \(\;a = \) e.g. 2