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Properties

Label 2300.29
Modulus $2300$
Conductor $575$
Order $110$
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
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(110)) M = H._module chi = DirichletCharacter(H, M([0,11,90]))
 
Copy content gp:[g,chi] = znchar(Mod(29, 2300))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2300.29");
 

Basic properties

Modulus: \(2300\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(575\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(110\)
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 \(\chi_{575}(29,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2300.bo

\(\chi_{2300}(9,\cdot)\) \(\chi_{2300}(29,\cdot)\) \(\chi_{2300}(169,\cdot)\) \(\chi_{2300}(209,\cdot)\) \(\chi_{2300}(269,\cdot)\) \(\chi_{2300}(289,\cdot)\) \(\chi_{2300}(409,\cdot)\) \(\chi_{2300}(469,\cdot)\) \(\chi_{2300}(489,\cdot)\) \(\chi_{2300}(509,\cdot)\) \(\chi_{2300}(629,\cdot)\) \(\chi_{2300}(669,\cdot)\) \(\chi_{2300}(729,\cdot)\) \(\chi_{2300}(809,\cdot)\) \(\chi_{2300}(869,\cdot)\) \(\chi_{2300}(909,\cdot)\) \(\chi_{2300}(929,\cdot)\) \(\chi_{2300}(969,\cdot)\) \(\chi_{2300}(1089,\cdot)\) \(\chi_{2300}(1129,\cdot)\) \(\chi_{2300}(1189,\cdot)\) \(\chi_{2300}(1209,\cdot)\) \(\chi_{2300}(1269,\cdot)\) \(\chi_{2300}(1329,\cdot)\) \(\chi_{2300}(1369,\cdot)\) \(\chi_{2300}(1389,\cdot)\) \(\chi_{2300}(1409,\cdot)\) \(\chi_{2300}(1429,\cdot)\) \(\chi_{2300}(1589,\cdot)\) \(\chi_{2300}(1669,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((1151,277,1201)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 2300 }(29, a) \) \(1\)\(1\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{3}{110}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{41}{110}\right)\)\(e\left(\frac{51}{55}\right)\)
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)
 
\( \chi_{ 2300 }(29,a) \;\) at \(\;a = \) e.g. 2