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Properties

Label 9020.1101
Modulus $9020$
Conductor $41$
Order $40$
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(9020, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([0,0,0,21]))
 
Copy content gp:[g,chi] = znchar(Mod(1101, 9020))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9020.1101");
 

Basic properties

Modulus: \(9020\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(41\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(40\)
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 \(\chi_{41}(35,\cdot)\)
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 9020.op

\(\chi_{9020}(1101,\cdot)\) \(\chi_{9020}(1541,\cdot)\) \(\chi_{9020}(1981,\cdot)\) \(\chi_{9020}(2201,\cdot)\) \(\chi_{9020}(2641,\cdot)\) \(\chi_{9020}(3081,\cdot)\) \(\chi_{9020}(4621,\cdot)\) \(\chi_{9020}(5501,\cdot)\) \(\chi_{9020}(5721,\cdot)\) \(\chi_{9020}(6161,\cdot)\) \(\chi_{9020}(6381,\cdot)\) \(\chi_{9020}(6821,\cdot)\) \(\chi_{9020}(7041,\cdot)\) \(\chi_{9020}(7481,\cdot)\) \(\chi_{9020}(7701,\cdot)\) \(\chi_{9020}(8581,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((4511,7217,1641,3081)\) → \((1,1,1,e\left(\frac{21}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 9020 }(1101, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{19}{40}\right)\)\(-i\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{27}{40}\right)\)
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)
 
\( \chi_{ 9020 }(1101,a) \;\) at \(\;a = \) e.g. 2