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Properties

Label 2700.269
Modulus $2700$
Conductor $75$
Order $10$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,5,9]))
 
Copy content pari:[g,chi] = znchar(Mod(269,2700))
 

Basic properties

Modulus: \(2700\)
Conductor: \(75\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(10\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{75}(44,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2700.z

\(\chi_{2700}(269,\cdot)\) \(\chi_{2700}(809,\cdot)\) \(\chi_{2700}(1889,\cdot)\) \(\chi_{2700}(2429,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: 10.0.185394287109375.1

Values on generators

\((1351,1001,2377)\) → \((1,-1,e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2700 }(269, a) \) \(-1\)\(1\)\(-1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2700 }(269,a) \;\) at \(\;a = \) e.g. 2