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Properties

Label 259200.baq
Modulus $259200$
Conductor $259200$
Order $4320$
Real no
Primitive yes
Minimal yes
Parity odd

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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(259200, base_ring=CyclotomicField(4320)) M = H._module chi = DirichletCharacter(H, M([0,2025,640,4104])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(13, 259200)) 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("259200.13"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(259200\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(259200\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4320\)
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);
 

Related number fields

Field of values: $\Q(\zeta_{4320})$
Fixed field: Number field defined by a degree 4320 polynomial (not computed)

First 8 of 1152 characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\)
\(\chi_{259200}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{349}{432}\right)\) \(e\left(\frac{4189}{4320}\right)\) \(e\left(\frac{1151}{4320}\right)\) \(e\left(\frac{131}{360}\right)\) \(e\left(\frac{1429}{1440}\right)\) \(e\left(\frac{1387}{2160}\right)\) \(e\left(\frac{163}{4320}\right)\) \(e\left(\frac{169}{540}\right)\) \(e\left(\frac{707}{1440}\right)\) \(e\left(\frac{1543}{2160}\right)\)
\(\chi_{259200}(277,\cdot)\) \(-1\) \(1\) \(e\left(\frac{167}{432}\right)\) \(e\left(\frac{2231}{4320}\right)\) \(e\left(\frac{349}{4320}\right)\) \(e\left(\frac{289}{360}\right)\) \(e\left(\frac{671}{1440}\right)\) \(e\left(\frac{353}{2160}\right)\) \(e\left(\frac{1577}{4320}\right)\) \(e\left(\frac{131}{540}\right)\) \(e\left(\frac{73}{1440}\right)\) \(e\left(\frac{1637}{2160}\right)\)
\(\chi_{259200}(517,\cdot)\) \(-1\) \(1\) \(e\left(\frac{211}{432}\right)\) \(e\left(\frac{3763}{4320}\right)\) \(e\left(\frac{3377}{4320}\right)\) \(e\left(\frac{197}{360}\right)\) \(e\left(\frac{283}{1440}\right)\) \(e\left(\frac{1429}{2160}\right)\) \(e\left(\frac{3661}{4320}\right)\) \(e\left(\frac{463}{540}\right)\) \(e\left(\frac{269}{1440}\right)\) \(e\left(\frac{361}{2160}\right)\)
\(\chi_{259200}(733,\cdot)\) \(-1\) \(1\) \(e\left(\frac{337}{432}\right)\) \(e\left(\frac{433}{4320}\right)\) \(e\left(\frac{1307}{4320}\right)\) \(e\left(\frac{287}{360}\right)\) \(e\left(\frac{553}{1440}\right)\) \(e\left(\frac{1879}{2160}\right)\) \(e\left(\frac{4111}{4320}\right)\) \(e\left(\frac{373}{540}\right)\) \(e\left(\frac{719}{1440}\right)\) \(e\left(\frac{1891}{2160}\right)\)
\(\chi_{259200}(997,\cdot)\) \(-1\) \(1\) \(e\left(\frac{299}{432}\right)\) \(e\left(\frac{2507}{4320}\right)\) \(e\left(\frac{793}{4320}\right)\) \(e\left(\frac{13}{360}\right)\) \(e\left(\frac{947}{1440}\right)\) \(e\left(\frac{1421}{2160}\right)\) \(e\left(\frac{3509}{4320}\right)\) \(e\left(\frac{47}{540}\right)\) \(e\left(\frac{661}{1440}\right)\) \(e\left(\frac{2129}{2160}\right)\)
\(\chi_{259200}(1213,\cdot)\) \(-1\) \(1\) \(e\left(\frac{425}{432}\right)\) \(e\left(\frac{1769}{4320}\right)\) \(e\left(\frac{451}{4320}\right)\) \(e\left(\frac{31}{360}\right)\) \(e\left(\frac{929}{1440}\right)\) \(e\left(\frac{1007}{2160}\right)\) \(e\left(\frac{503}{4320}\right)\) \(e\left(\frac{389}{540}\right)\) \(e\left(\frac{247}{1440}\right)\) \(e\left(\frac{203}{2160}\right)\)
\(\chi_{259200}(1237,\cdot)\) \(-1\) \(1\) \(e\left(\frac{199}{432}\right)\) \(e\left(\frac{2599}{4320}\right)\) \(e\left(\frac{941}{4320}\right)\) \(e\left(\frac{281}{360}\right)\) \(e\left(\frac{559}{1440}\right)\) \(e\left(\frac{1057}{2160}\right)\) \(e\left(\frac{4153}{4320}\right)\) \(e\left(\frac{19}{540}\right)\) \(e\left(\frac{857}{1440}\right)\) \(e\left(\frac{1573}{2160}\right)\)
\(\chi_{259200}(1453,\cdot)\) \(-1\) \(1\) \(e\left(\frac{325}{432}\right)\) \(e\left(\frac{997}{4320}\right)\) \(e\left(\frac{1463}{4320}\right)\) \(e\left(\frac{83}{360}\right)\) \(e\left(\frac{1117}{1440}\right)\) \(e\left(\frac{211}{2160}\right)\) \(e\left(\frac{3739}{4320}\right)\) \(e\left(\frac{37}{540}\right)\) \(e\left(\frac{731}{1440}\right)\) \(e\left(\frac{79}{2160}\right)\)