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Properties

Label 20016.7003
Modulus $20016$
Conductor $2224$
Order $276$
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(20016, base_ring=CyclotomicField(276)) M = H._module chi = DirichletCharacter(H, M([138,69,0,94]))
 
Copy content gp:[g,chi] = znchar(Mod(7003, 20016))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20016.7003");
 

Basic properties

Modulus: \(20016\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2224\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(276\)
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 \(\chi_{2224}(331,\cdot)\)
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 20016.iu

\(\chi_{20016}(19,\cdot)\) \(\chi_{20016}(379,\cdot)\) \(\chi_{20016}(667,\cdot)\) \(\chi_{20016}(1099,\cdot)\) \(\chi_{20016}(1531,\cdot)\) \(\chi_{20016}(1819,\cdot)\) \(\chi_{20016}(1963,\cdot)\) \(\chi_{20016}(2107,\cdot)\) \(\chi_{20016}(2395,\cdot)\) \(\chi_{20016}(2467,\cdot)\) \(\chi_{20016}(2611,\cdot)\) \(\chi_{20016}(2755,\cdot)\) \(\chi_{20016}(2899,\cdot)\) \(\chi_{20016}(3331,\cdot)\) \(\chi_{20016}(3547,\cdot)\) \(\chi_{20016}(3907,\cdot)\) \(\chi_{20016}(4123,\cdot)\) \(\chi_{20016}(4411,\cdot)\) \(\chi_{20016}(4627,\cdot)\) \(\chi_{20016}(4915,\cdot)\) \(\chi_{20016}(5275,\cdot)\) \(\chi_{20016}(5491,\cdot)\) \(\chi_{20016}(5563,\cdot)\) \(\chi_{20016}(5923,\cdot)\) \(\chi_{20016}(5995,\cdot)\) \(\chi_{20016}(6067,\cdot)\) \(\chi_{20016}(6643,\cdot)\) \(\chi_{20016}(6787,\cdot)\) \(\chi_{20016}(7003,\cdot)\) \(\chi_{20016}(7147,\cdot)\) ...

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: $\Q(\zeta_{276})$
Fixed field: Number field defined by a degree 276 polynomial (not computed)

Values on generators

\((12511,15013,2225,4033)\) → \((-1,i,1,e\left(\frac{47}{138}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 20016 }(7003, a) \) \(1\)\(1\)\(e\left(\frac{149}{276}\right)\)\(e\left(\frac{2}{69}\right)\)\(e\left(\frac{175}{276}\right)\)\(e\left(\frac{151}{276}\right)\)\(e\left(\frac{61}{138}\right)\)\(e\left(\frac{7}{276}\right)\)\(e\left(\frac{9}{46}\right)\)\(e\left(\frac{11}{138}\right)\)\(e\left(\frac{211}{276}\right)\)\(e\left(\frac{79}{138}\right)\)
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)
 
\( \chi_{ 20016 }(7003,a) \;\) at \(\;a = \) e.g. 2