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Properties

Label 30100.15661
Modulus $30100$
Conductor $7525$
Order $105$
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(30100, base_ring=CyclotomicField(210)) M = H._module chi = DirichletCharacter(H, M([0,168,70,10]))
 
Copy content gp:[g,chi] = znchar(Mod(15661, 30100))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("30100.15661");
 

Basic properties

Modulus: \(30100\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7525\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(105\)
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 \(\chi_{7525}(611,\cdot)\)
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 30100.om

\(\chi_{30100}(81,\cdot)\) \(\chi_{30100}(541,\cdot)\) \(\chi_{30100}(961,\cdot)\) \(\chi_{30100}(2081,\cdot)\) \(\chi_{30100}(2181,\cdot)\) \(\chi_{30100}(2461,\cdot)\) \(\chi_{30100}(3621,\cdot)\) \(\chi_{30100}(3721,\cdot)\) \(\chi_{30100}(4281,\cdot)\) \(\chi_{30100}(5861,\cdot)\) \(\chi_{30100}(6421,\cdot)\) \(\chi_{30100}(6561,\cdot)\) \(\chi_{30100}(6981,\cdot)\) \(\chi_{30100}(7221,\cdot)\) \(\chi_{30100}(8481,\cdot)\) \(\chi_{30100}(9641,\cdot)\) \(\chi_{30100}(9741,\cdot)\) \(\chi_{30100}(11881,\cdot)\) \(\chi_{30100}(12121,\cdot)\) \(\chi_{30100}(12441,\cdot)\) \(\chi_{30100}(12581,\cdot)\) \(\chi_{30100}(13241,\cdot)\) \(\chi_{30100}(14121,\cdot)\) \(\chi_{30100}(14221,\cdot)\) \(\chi_{30100}(15661,\cdot)\) \(\chi_{30100}(15761,\cdot)\) \(\chi_{30100}(16321,\cdot)\) \(\chi_{30100}(18141,\cdot)\) \(\chi_{30100}(18461,\cdot)\) \(\chi_{30100}(19021,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((15051,22877,4301,25201)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 30100 }(15661, a) \) \(1\)\(1\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{101}{105}\right)\)\(e\left(\frac{59}{105}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{58}{105}\right)\)\(e\left(\frac{37}{105}\right)\)
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)
 
\( \chi_{ 30100 }(15661,a) \;\) at \(\;a = \) e.g. 2