Properties

Label 4011.428
Modulus $4011$
Conductor $573$
Order $190$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, base_ring=CyclotomicField(190))
 
M = H._module
 
chi = DirichletCharacter(H, M([95,0,178]))
 
pari: [g,chi] = znchar(Mod(428,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(573\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(190\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{573}(428,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.ca

\(\chi_{4011}(8,\cdot)\) \(\chi_{4011}(50,\cdot)\) \(\chi_{4011}(92,\cdot)\) \(\chi_{4011}(134,\cdot)\) \(\chi_{4011}(218,\cdot)\) \(\chi_{4011}(239,\cdot)\) \(\chi_{4011}(281,\cdot)\) \(\chi_{4011}(386,\cdot)\) \(\chi_{4011}(428,\cdot)\) \(\chi_{4011}(449,\cdot)\) \(\chi_{4011}(512,\cdot)\) \(\chi_{4011}(554,\cdot)\) \(\chi_{4011}(575,\cdot)\) \(\chi_{4011}(596,\cdot)\) \(\chi_{4011}(638,\cdot)\) \(\chi_{4011}(659,\cdot)\) \(\chi_{4011}(701,\cdot)\) \(\chi_{4011}(722,\cdot)\) \(\chi_{4011}(743,\cdot)\) \(\chi_{4011}(911,\cdot)\) \(\chi_{4011}(995,\cdot)\) \(\chi_{4011}(1058,\cdot)\) \(\chi_{4011}(1163,\cdot)\) \(\chi_{4011}(1205,\cdot)\) \(\chi_{4011}(1226,\cdot)\) \(\chi_{4011}(1352,\cdot)\) \(\chi_{4011}(1415,\cdot)\) \(\chi_{4011}(1457,\cdot)\) \(\chi_{4011}(1499,\cdot)\) \(\chi_{4011}(1541,\cdot)\) ...

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: $\Q(\zeta_{95})$
Fixed field: Number field defined by a degree 190 polynomial (not computed)

Values on generators

\((2675,2866,2311)\) → \((-1,1,e\left(\frac{89}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(428, a) \) \(-1\)\(1\)\(e\left(\frac{137}{190}\right)\)\(e\left(\frac{42}{95}\right)\)\(e\left(\frac{13}{38}\right)\)\(e\left(\frac{31}{190}\right)\)\(e\left(\frac{6}{95}\right)\)\(e\left(\frac{5}{38}\right)\)\(e\left(\frac{88}{95}\right)\)\(e\left(\frac{84}{95}\right)\)\(e\left(\frac{59}{190}\right)\)\(e\left(\frac{89}{95}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(428,a) \;\) at \(\;a = \) e.g. 2