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Properties

Label 9906.1649
Modulus $9906$
Conductor $4953$
Order $84$
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(9906, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,49,6]))
 
Copy content gp:[g,chi] = znchar(Mod(1649, 9906))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9906.1649");
 

Basic properties

Modulus: \(9906\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4953\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 \(\chi_{4953}(1649,\cdot)\)
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 9906.fw

\(\chi_{9906}(119,\cdot)\) \(\chi_{9906}(1619,\cdot)\) \(\chi_{9906}(1649,\cdot)\) \(\chi_{9906}(2381,\cdot)\) \(\chi_{9906}(2411,\cdot)\) \(\chi_{9906}(2663,\cdot)\) \(\chi_{9906}(3365,\cdot)\) \(\chi_{9906}(3413,\cdot)\) \(\chi_{9906}(3425,\cdot)\) \(\chi_{9906}(4127,\cdot)\) \(\chi_{9906}(4175,\cdot)\) \(\chi_{9906}(4691,\cdot)\) \(\chi_{9906}(5453,\cdot)\) \(\chi_{9906}(6221,\cdot)\) \(\chi_{9906}(6953,\cdot)\) \(\chi_{9906}(6983,\cdot)\) \(\chi_{9906}(7235,\cdot)\) \(\chi_{9906}(7715,\cdot)\) \(\chi_{9906}(7937,\cdot)\) \(\chi_{9906}(7997,\cdot)\) \(\chi_{9906}(8699,\cdot)\) \(\chi_{9906}(8747,\cdot)\) \(\chi_{9906}(9263,\cdot)\) \(\chi_{9906}(9509,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((6605,3811,8893)\) → \((-1,e\left(\frac{7}{12}\right),e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 9906 }(1649, a) \) \(-1\)\(1\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{53}{84}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{25}{42}\right)\)
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)
 
\( \chi_{ 9906 }(1649,a) \;\) at \(\;a = \) e.g. 2