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Properties

Label 2256.409
Modulus $2256$
Conductor $376$
Order $46$
Real no
Primitive no
Minimal no
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(2256, base_ring=CyclotomicField(46)) M = H._module chi = DirichletCharacter(H, M([0,23,0,27]))
 
Copy content gp:[g,chi] = znchar(Mod(409, 2256))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2256.409");
 

Basic properties

Modulus: \(2256\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(376\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(46\)
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 \(\chi_{376}(221,\cdot)\)
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: no
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 2256.z

\(\chi_{2256}(73,\cdot)\) \(\chi_{2256}(217,\cdot)\) \(\chi_{2256}(265,\cdot)\) \(\chi_{2256}(313,\cdot)\) \(\chi_{2256}(409,\cdot)\) \(\chi_{2256}(505,\cdot)\) \(\chi_{2256}(649,\cdot)\) \(\chi_{2256}(697,\cdot)\) \(\chi_{2256}(745,\cdot)\) \(\chi_{2256}(793,\cdot)\) \(\chi_{2256}(889,\cdot)\) \(\chi_{2256}(937,\cdot)\) \(\chi_{2256}(985,\cdot)\) \(\chi_{2256}(1321,\cdot)\) \(\chi_{2256}(1561,\cdot)\) \(\chi_{2256}(1609,\cdot)\) \(\chi_{2256}(1705,\cdot)\) \(\chi_{2256}(1801,\cdot)\) \(\chi_{2256}(1993,\cdot)\) \(\chi_{2256}(2041,\cdot)\) \(\chi_{2256}(2137,\cdot)\) \(\chi_{2256}(2185,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: 46.0.1036270912454247829848884601705469565826954223470060599867947317935733537595789906053570110685184.1

Values on generators

\((847,565,1505,193)\) → \((1,-1,1,e\left(\frac{27}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2256 }(409, a) \) \(-1\)\(1\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{22}{23}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{43}{46}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{1}{23}\right)\)\(e\left(\frac{35}{46}\right)\)
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)
 
\( \chi_{ 2256 }(409,a) \;\) at \(\;a = \) e.g. 2