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Properties

Label 59904.19807
Modulus $59904$
Conductor $7488$
Order $48$
Real no
Primitive no
Minimal no
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(59904, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,33,32,12]))
 
Copy content gp:[g,chi] = znchar(Mod(19807, 59904))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("59904.19807");
 

Basic properties

Modulus: \(59904\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7488\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(48\)
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 \(\chi_{7488}(3427,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 59904.np

\(\chi_{59904}(31,\cdot)\) \(\chi_{59904}(4831,\cdot)\) \(\chi_{59904}(9823,\cdot)\) \(\chi_{59904}(10015,\cdot)\) \(\chi_{59904}(15007,\cdot)\) \(\chi_{59904}(19807,\cdot)\) \(\chi_{59904}(24799,\cdot)\) \(\chi_{59904}(24991,\cdot)\) \(\chi_{59904}(29983,\cdot)\) \(\chi_{59904}(34783,\cdot)\) \(\chi_{59904}(39775,\cdot)\) \(\chi_{59904}(39967,\cdot)\) \(\chi_{59904}(44959,\cdot)\) \(\chi_{59904}(49759,\cdot)\) \(\chi_{59904}(54751,\cdot)\) \(\chi_{59904}(54943,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((8191,43525,13313,50689)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{2}{3}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 59904 }(19807, a) \) \(1\)\(1\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{17}{48}\right)\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{16}\right)\)
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)
 
\( \chi_{ 59904 }(19807,a) \;\) at \(\;a = \) e.g. 2