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Properties

Label 69696.1609
Modulus $69696$
Conductor $34848$
Order $1320$
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(69696, base_ring=CyclotomicField(1320)) M = H._module chi = DirichletCharacter(H, M([0,495,880,816]))
 
Copy content gp:[g,chi] = znchar(Mod(1609, 69696))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("69696.1609");
 

Basic properties

Modulus: \(69696\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(34848\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1320\)
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 \(\chi_{34848}(23389,\cdot)\)
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 69696.lt

\(\chi_{69696}(25,\cdot)\) \(\chi_{69696}(169,\cdot)\) \(\chi_{69696}(313,\cdot)\) \(\chi_{69696}(553,\cdot)\) \(\chi_{69696}(697,\cdot)\) \(\chi_{69696}(841,\cdot)\) \(\chi_{69696}(889,\cdot)\) \(\chi_{69696}(1417,\cdot)\) \(\chi_{69696}(1609,\cdot)\) \(\chi_{69696}(1753,\cdot)\) \(\chi_{69696}(1897,\cdot)\) \(\chi_{69696}(2137,\cdot)\) \(\chi_{69696}(2281,\cdot)\) \(\chi_{69696}(2425,\cdot)\) \(\chi_{69696}(2473,\cdot)\) \(\chi_{69696}(3001,\cdot)\) \(\chi_{69696}(3193,\cdot)\) \(\chi_{69696}(3337,\cdot)\) \(\chi_{69696}(3481,\cdot)\) \(\chi_{69696}(3721,\cdot)\) \(\chi_{69696}(3865,\cdot)\) \(\chi_{69696}(4009,\cdot)\) \(\chi_{69696}(4057,\cdot)\) \(\chi_{69696}(4585,\cdot)\) \(\chi_{69696}(4777,\cdot)\) \(\chi_{69696}(5065,\cdot)\) \(\chi_{69696}(5305,\cdot)\) \(\chi_{69696}(5449,\cdot)\) \(\chi_{69696}(5641,\cdot)\) \(\chi_{69696}(6169,\cdot)\) ...

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: $\Q(\zeta_{1320})$
Fixed field: Number field defined by a degree 1320 polynomial (not computed)

Values on generators

\((67519,4357,54209,14401)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{2}{3}\right),e\left(\frac{34}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 69696 }(1609, a) \) \(1\)\(1\)\(e\left(\frac{599}{1320}\right)\)\(e\left(\frac{491}{660}\right)\)\(e\left(\frac{521}{1320}\right)\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{411}{440}\right)\)\(e\left(\frac{113}{132}\right)\)\(e\left(\frac{599}{660}\right)\)\(e\left(\frac{397}{1320}\right)\)\(e\left(\frac{82}{165}\right)\)\(e\left(\frac{87}{440}\right)\)
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)
 
\( \chi_{ 69696 }(1609,a) \;\) at \(\;a = \) e.g. 2