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Properties

Label 339575.6687
Modulus $339575$
Conductor $19975$
Order $1840$
Real no
Primitive no
Minimal yes
Parity odd

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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(339575, base_ring=CyclotomicField(1840)) M = H._module chi = DirichletCharacter(H, M([828,1725,440]))
 
Copy content gp:[g,chi] = znchar(Mod(6687, 339575))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("339575.6687");
 

Basic properties

Modulus: \(339575\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(19975\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1840\)
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 \(\chi_{19975}(6687,\cdot)\)
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 339575.jb

\(\chi_{339575}(513,\cdot)\) \(\chi_{339575}(792,\cdot)\) \(\chi_{339575}(998,\cdot)\) \(\chi_{339575}(1603,\cdot)\) \(\chi_{339575}(1958,\cdot)\) \(\chi_{339575}(2088,\cdot)\) \(\chi_{339575}(3048,\cdot)\) \(\chi_{339575}(3403,\cdot)\) \(\chi_{339575}(4978,\cdot)\) \(\chi_{339575}(5162,\cdot)\) \(\chi_{339575}(5277,\cdot)\) \(\chi_{339575}(5333,\cdot)\) \(\chi_{339575}(6423,\cdot)\) \(\chi_{339575}(6572,\cdot)\) \(\chi_{339575}(6687,\cdot)\) \(\chi_{339575}(6778,\cdot)\) \(\chi_{339575}(7738,\cdot)\) \(\chi_{339575}(8052,\cdot)\) \(\chi_{339575}(8223,\cdot)\) \(\chi_{339575}(8828,\cdot)\) \(\chi_{339575}(9462,\cdot)\) \(\chi_{339575}(10942,\cdot)\) \(\chi_{339575}(12073,\cdot)\) \(\chi_{339575}(12203,\cdot)\) \(\chi_{339575}(12352,\cdot)\) \(\chi_{339575}(12387,\cdot)\) \(\chi_{339575}(13797,\cdot)\) \(\chi_{339575}(13947,\cdot)\) \(\chi_{339575}(14003,\cdot)\) \(\chi_{339575}(14608,\cdot)\) ...

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: $\Q(\zeta_{1840})$
Fixed field: Number field defined by a degree 1840 polynomial (not computed)

Values on generators

\((298827,160976,180626)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{15}{16}\right),e\left(\frac{11}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 339575 }(6687, a) \) \(-1\)\(1\)\(e\left(\frac{809}{920}\right)\)\(e\left(\frac{1601}{1840}\right)\)\(e\left(\frac{349}{460}\right)\)\(e\left(\frac{1379}{1840}\right)\)\(e\left(\frac{79}{368}\right)\)\(e\left(\frac{587}{920}\right)\)\(e\left(\frac{681}{920}\right)\)\(e\left(\frac{803}{1840}\right)\)\(e\left(\frac{1157}{1840}\right)\)\(e\left(\frac{107}{115}\right)\)
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)
 
\( \chi_{ 339575 }(6687,a) \;\) at \(\;a = \) e.g. 2