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Properties

Label 8820.fs
Modulus $8820$
Conductor $2205$
Order $42$
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 orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,28,21,38])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(709, 8820)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8820.709"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(8820\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2205\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(42\)
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 2205.db
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);
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\) \(43\)
\(\chi_{8820}(709,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{25}{42}\right)\)
\(\chi_{8820}(1969,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{8820}(2209,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{8820}(3229,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{13}{42}\right)\)
\(\chi_{8820}(3469,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{8820}(4729,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{42}\right)\)
\(\chi_{8820}(5749,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{42}\right)\)
\(\chi_{8820}(5989,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{8820}(7009,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{8820}(7249,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{41}{42}\right)\)
\(\chi_{8820}(8269,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{8820}(8509,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{17}{42}\right)\)