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Properties

Label 94815.78707
Modulus $94815$
Conductor $94815$
Order $84$
Real no
Primitive yes
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(94815, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([14,21,66,76]))
 
Copy content gp:[g,chi] = znchar(Mod(78707, 94815))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("94815.78707");
 

Basic properties

Modulus: \(94815\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(94815\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: yes
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 94815.dpm

\(\chi_{94815}(272,\cdot)\) \(\chi_{94815}(10688,\cdot)\) \(\chi_{94815}(17618,\cdot)\) \(\chi_{94815}(21818,\cdot)\) \(\chi_{94815}(22658,\cdot)\) \(\chi_{94815}(25178,\cdot)\) \(\chi_{94815}(26417,\cdot)\) \(\chi_{94815}(27803,\cdot)\) \(\chi_{94815}(28307,\cdot)\) \(\chi_{94815}(32087,\cdot)\) \(\chi_{94815}(38198,\cdot)\) \(\chi_{94815}(40802,\cdot)\) \(\chi_{94815}(53717,\cdot)\) \(\chi_{94815}(64343,\cdot)\) \(\chi_{94815}(66233,\cdot)\) \(\chi_{94815}(67577,\cdot)\) \(\chi_{94815}(70013,\cdot)\) \(\chi_{94815}(74507,\cdot)\) \(\chi_{94815}(78707,\cdot)\) \(\chi_{94815}(78728,\cdot)\) \(\chi_{94815}(79547,\cdot)\) \(\chi_{94815}(82067,\cdot)\) \(\chi_{94815}(84692,\cdot)\) \(\chi_{94815}(91643,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((21071,37927,87076,79381)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{11}{14}\right),e\left(\frac{19}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 94815 }(78707, a) \) \(-1\)\(1\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{11}{12}\right)\)
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)
 
\( \chi_{ 94815 }(78707,a) \;\) at \(\;a = \) e.g. 2