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Properties

Label 815409.23705
Modulus $815409$
Conductor $271803$
Order $1806$
Real no
Primitive no
Minimal yes
Parity odd

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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(815409, base_ring=CyclotomicField(1806)) M = H._module chi = DirichletCharacter(H, M([903,817,55]))
 
Copy content gp:[g,chi] = znchar(Mod(23705, 815409))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("815409.23705");
 

Basic properties

Modulus: \(815409\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(271803\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1806\)
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 \(\chi_{271803}(23705,\cdot)\)
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 815409.ceo

\(\chi_{815409}(3554,\cdot)\) \(\chi_{815409}(4742,\cdot)\) \(\chi_{815409}(5120,\cdot)\) \(\chi_{815409}(5309,\cdot)\) \(\chi_{815409}(7586,\cdot)\) \(\chi_{815409}(8963,\cdot)\) \(\chi_{815409}(10862,\cdot)\) \(\chi_{815409}(11429,\cdot)\) \(\chi_{815409}(12500,\cdot)\) \(\chi_{815409}(14066,\cdot)\) \(\chi_{815409}(18863,\cdot)\) \(\chi_{815409}(22517,\cdot)\) \(\chi_{815409}(23705,\cdot)\) \(\chi_{815409}(24083,\cdot)\) \(\chi_{815409}(24272,\cdot)\) \(\chi_{815409}(26549,\cdot)\) \(\chi_{815409}(27926,\cdot)\) \(\chi_{815409}(29825,\cdot)\) \(\chi_{815409}(30392,\cdot)\) \(\chi_{815409}(31463,\cdot)\) \(\chi_{815409}(33029,\cdot)\) \(\chi_{815409}(36368,\cdot)\) \(\chi_{815409}(37826,\cdot)\) \(\chi_{815409}(41480,\cdot)\) \(\chi_{815409}(42668,\cdot)\) \(\chi_{815409}(43046,\cdot)\) \(\chi_{815409}(43235,\cdot)\) \(\chi_{815409}(45512,\cdot)\) \(\chi_{815409}(46889,\cdot)\) \(\chi_{815409}(48788,\cdot)\) ...

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: $\Q(\zeta_{903})$
Fixed field: Number field defined by a degree 1806 polynomial (not computed)

Values on generators

\((362405,599077,306937)\) → \((-1,e\left(\frac{19}{42}\right),e\left(\frac{55}{1806}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 815409 }(23705, a) \) \(-1\)\(1\)\(e\left(\frac{118}{903}\right)\)\(e\left(\frac{236}{903}\right)\)\(e\left(\frac{299}{602}\right)\)\(e\left(\frac{118}{301}\right)\)\(e\left(\frac{1133}{1806}\right)\)\(e\left(\frac{1801}{1806}\right)\)\(e\left(\frac{155}{258}\right)\)\(e\left(\frac{472}{903}\right)\)\(e\left(\frac{46}{301}\right)\)\(e\left(\frac{5}{7}\right)\)
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)
 
\( \chi_{ 815409 }(23705,a) \;\) at \(\;a = \) e.g. 2