Properties

Label 2075.4
Modulus $2075$
Conductor $2075$
Order $410$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2075, base_ring=CyclotomicField(410))
 
M = H._module
 
chi = DirichletCharacter(H, M([41,10]))
 
pari: [g,chi] = znchar(Mod(4,2075))
 

Basic properties

Modulus: \(2075\)
Conductor: \(2075\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(410\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2075.t

\(\chi_{2075}(4,\cdot)\) \(\chi_{2075}(9,\cdot)\) \(\chi_{2075}(29,\cdot)\) \(\chi_{2075}(44,\cdot)\) \(\chi_{2075}(59,\cdot)\) \(\chi_{2075}(64,\cdot)\) \(\chi_{2075}(69,\cdot)\) \(\chi_{2075}(94,\cdot)\) \(\chi_{2075}(104,\cdot)\) \(\chi_{2075}(109,\cdot)\) \(\chi_{2075}(114,\cdot)\) \(\chi_{2075}(119,\cdot)\) \(\chi_{2075}(134,\cdot)\) \(\chi_{2075}(144,\cdot)\) \(\chi_{2075}(164,\cdot)\) \(\chi_{2075}(169,\cdot)\) \(\chi_{2075}(189,\cdot)\) \(\chi_{2075}(194,\cdot)\) \(\chi_{2075}(204,\cdot)\) \(\chi_{2075}(214,\cdot)\) \(\chi_{2075}(229,\cdot)\) \(\chi_{2075}(234,\cdot)\) \(\chi_{2075}(244,\cdot)\) \(\chi_{2075}(259,\cdot)\) \(\chi_{2075}(279,\cdot)\) \(\chi_{2075}(289,\cdot)\) \(\chi_{2075}(314,\cdot)\) \(\chi_{2075}(319,\cdot)\) \(\chi_{2075}(339,\cdot)\) \(\chi_{2075}(344,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{205})$
Fixed field: Number field defined by a degree 410 polynomial (not computed)

Values on generators

\((1827,251)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{41}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2075 }(4, a) \) \(1\)\(1\)\(e\left(\frac{51}{410}\right)\)\(e\left(\frac{187}{410}\right)\)\(e\left(\frac{51}{205}\right)\)\(e\left(\frac{119}{205}\right)\)\(e\left(\frac{57}{82}\right)\)\(e\left(\frac{153}{410}\right)\)\(e\left(\frac{187}{205}\right)\)\(e\left(\frac{38}{205}\right)\)\(e\left(\frac{289}{410}\right)\)\(e\left(\frac{319}{410}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2075 }(4,a) \;\) at \(\;a = \) e.g. 2