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Properties

Label 21315.20849
Modulus $21315$
Conductor $21315$
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(21315, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,42,74,45]))
 
Copy content gp:[g,chi] = znchar(Mod(20849, 21315))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("21315.20849");
 

Basic properties

Modulus: \(21315\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(21315\)
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 21315.yc

\(\chi_{21315}(89,\cdot)\) \(\chi_{21315}(164,\cdot)\) \(\chi_{21315}(479,\cdot)\) \(\chi_{21315}(1529,\cdot)\) \(\chi_{21315}(1634,\cdot)\) \(\chi_{21315}(4079,\cdot)\) \(\chi_{21315}(4289,\cdot)\) \(\chi_{21315}(4709,\cdot)\) \(\chi_{21315}(5549,\cdot)\) \(\chi_{21315}(6464,\cdot)\) \(\chi_{21315}(8279,\cdot)\) \(\chi_{21315}(8354,\cdot)\) \(\chi_{21315}(10274,\cdot)\) \(\chi_{21315}(10484,\cdot)\) \(\chi_{21315}(12584,\cdot)\) \(\chi_{21315}(13184,\cdot)\) \(\chi_{21315}(15494,\cdot)\) \(\chi_{21315}(17519,\cdot)\) \(\chi_{21315}(17624,\cdot)\) \(\chi_{21315}(19064,\cdot)\) \(\chi_{21315}(19589,\cdot)\) \(\chi_{21315}(19694,\cdot)\) \(\chi_{21315}(20144,\cdot)\) \(\chi_{21315}(20849,\cdot)\)

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

\((7106,8527,6961,2206)\) → \((-1,-1,e\left(\frac{37}{42}\right),e\left(\frac{15}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 21315 }(20849, a) \) \(-1\)\(1\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{4}{21}\right)\)
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)
 
\( \chi_{ 21315 }(20849,a) \;\) at \(\;a = \) e.g. 2