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Properties

Label 132300.9973
Modulus $132300$
Conductor $11025$
Order $420$
Real no
Primitive no
Minimal no
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(132300, base_ring=CyclotomicField(420)) M = H._module chi = DirichletCharacter(H, M([0,140,231,170]))
 
Copy content gp:[g,chi] = znchar(Mod(9973, 132300))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("132300.9973");
 

Basic properties

Modulus: \(132300\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11025\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(420\)
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 \(\chi_{11025}(2623,\cdot)\)
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: no
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 132300.bad

\(\chi_{132300}(73,\cdot)\) \(\chi_{132300}(2413,\cdot)\) \(\chi_{132300}(3097,\cdot)\) \(\chi_{132300}(5437,\cdot)\) \(\chi_{132300}(6877,\cdot)\) \(\chi_{132300}(7633,\cdot)\) \(\chi_{132300}(9217,\cdot)\) \(\chi_{132300}(9973,\cdot)\) \(\chi_{132300}(11413,\cdot)\) \(\chi_{132300}(12997,\cdot)\) \(\chi_{132300}(13753,\cdot)\) \(\chi_{132300}(17533,\cdot)\) \(\chi_{132300}(18217,\cdot)\) \(\chi_{132300}(18973,\cdot)\) \(\chi_{132300}(21313,\cdot)\) \(\chi_{132300}(21997,\cdot)\) \(\chi_{132300}(22753,\cdot)\) \(\chi_{132300}(24337,\cdot)\) \(\chi_{132300}(25777,\cdot)\) \(\chi_{132300}(26533,\cdot)\) \(\chi_{132300}(28117,\cdot)\) \(\chi_{132300}(28873,\cdot)\) \(\chi_{132300}(31897,\cdot)\) \(\chi_{132300}(33337,\cdot)\) \(\chi_{132300}(35677,\cdot)\) \(\chi_{132300}(36433,\cdot)\) \(\chi_{132300}(37117,\cdot)\) \(\chi_{132300}(37873,\cdot)\) \(\chi_{132300}(40213,\cdot)\) \(\chi_{132300}(41653,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((66151,122501,15877,54001)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{11}{20}\right),e\left(\frac{17}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 132300 }(9973, a) \) \(1\)\(1\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{199}{420}\right)\)\(e\left(\frac{113}{420}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{41}{420}\right)\)\(e\left(\frac{151}{210}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{379}{420}\right)\)\(e\left(\frac{197}{210}\right)\)\(e\left(\frac{1}{84}\right)\)
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)
 
\( \chi_{ 132300 }(9973,a) \;\) at \(\;a = \) e.g. 2