↑

Properties

Label 2852.65
Modulus $2852$
Conductor $713$
Order $330$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2852, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([0,225,11]))
 
Copy content gp:[g,chi] = znchar(Mod(65, 2852))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2852.65");
 

Basic properties

Modulus: \(2852\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(713\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(330\)
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 \(\chi_{713}(65,\cdot)\)
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 2852.cj

\(\chi_{2852}(17,\cdot)\) \(\chi_{2852}(21,\cdot)\) \(\chi_{2852}(53,\cdot)\) \(\chi_{2852}(65,\cdot)\) \(\chi_{2852}(145,\cdot)\) \(\chi_{2852}(189,\cdot)\) \(\chi_{2852}(241,\cdot)\) \(\chi_{2852}(313,\cdot)\) \(\chi_{2852}(365,\cdot)\) \(\chi_{2852}(385,\cdot)\) \(\chi_{2852}(389,\cdot)\) \(\chi_{2852}(425,\cdot)\) \(\chi_{2852}(477,\cdot)\) \(\chi_{2852}(513,\cdot)\) \(\chi_{2852}(517,\cdot)\) \(\chi_{2852}(549,\cdot)\) \(\chi_{2852}(569,\cdot)\) \(\chi_{2852}(613,\cdot)\) \(\chi_{2852}(641,\cdot)\) \(\chi_{2852}(757,\cdot)\) \(\chi_{2852}(797,\cdot)\) \(\chi_{2852}(849,\cdot)\) \(\chi_{2852}(861,\cdot)\) \(\chi_{2852}(881,\cdot)\) \(\chi_{2852}(885,\cdot)\) \(\chi_{2852}(889,\cdot)\) \(\chi_{2852}(941,\cdot)\) \(\chi_{2852}(973,\cdot)\) \(\chi_{2852}(985,\cdot)\) \(\chi_{2852}(1009,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((1427,373,2669)\) → \((1,e\left(\frac{15}{22}\right),e\left(\frac{1}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2852 }(65, a) \) \(1\)\(1\)\(e\left(\frac{311}{330}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{293}{330}\right)\)\(e\left(\frac{146}{165}\right)\)\(e\left(\frac{149}{165}\right)\)\(e\left(\frac{301}{330}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{1}{165}\right)\)\(e\left(\frac{119}{330}\right)\)\(e\left(\frac{137}{165}\right)\)
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)
 
\( \chi_{ 2852 }(65,a) \;\) at \(\;a = \) e.g. 2