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Properties

Label 36100.469
Modulus $36100$
Conductor $9025$
Order $1710$
Real no
Primitive no
Minimal yes
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(36100, base_ring=CyclotomicField(1710)) M = H._module chi = DirichletCharacter(H, M([0,1539,385]))
 
Copy content gp:[g,chi] = znchar(Mod(469, 36100))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("36100.469");
 

Basic properties

Modulus: \(36100\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(9025\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1710\)
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 \(\chi_{9025}(469,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 36100.fd

\(\chi_{36100}(29,\cdot)\) \(\chi_{36100}(89,\cdot)\) \(\chi_{36100}(109,\cdot)\) \(\chi_{36100}(129,\cdot)\) \(\chi_{36100}(269,\cdot)\) \(\chi_{36100}(409,\cdot)\) \(\chi_{36100}(469,\cdot)\) \(\chi_{36100}(489,\cdot)\) \(\chi_{36100}(509,\cdot)\) \(\chi_{36100}(629,\cdot)\) \(\chi_{36100}(789,\cdot)\) \(\chi_{36100}(869,\cdot)\) \(\chi_{36100}(889,\cdot)\) \(\chi_{36100}(1009,\cdot)\) \(\chi_{36100}(1169,\cdot)\) \(\chi_{36100}(1229,\cdot)\) \(\chi_{36100}(1269,\cdot)\) \(\chi_{36100}(1389,\cdot)\) \(\chi_{36100}(1409,\cdot)\)

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: $\Q(\zeta_{855})$
Fixed field: Number field defined by a degree 1710 polynomial (not computed)

Values on generators

\((18051,5777,21301)\) → \((1,e\left(\frac{9}{10}\right),e\left(\frac{77}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 36100 }(469, a) \) \(-1\)\(1\)\(e\left(\frac{509}{855}\right)\)\(e\left(\frac{31}{114}\right)\)\(e\left(\frac{163}{855}\right)\)\(e\left(\frac{104}{285}\right)\)\(e\left(\frac{148}{855}\right)\)\(e\left(\frac{997}{1710}\right)\)\(e\left(\frac{1483}{1710}\right)\)\(e\left(\frac{1319}{1710}\right)\)\(e\left(\frac{224}{285}\right)\)\(e\left(\frac{1073}{1710}\right)\)
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)
 
\( \chi_{ 36100 }(469,a) \;\) at \(\;a = \) e.g. 2