Properties

Label 3751.42
Modulus $3751$
Conductor $3751$
Order $330$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3751, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([288,253]))
 
pari: [g,chi] = znchar(Mod(42,3751))
 

Basic properties

Modulus: \(3751\)
Conductor: \(3751\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3751.dq

\(\chi_{3751}(42,\cdot)\) \(\chi_{3751}(86,\cdot)\) \(\chi_{3751}(115,\cdot)\) \(\chi_{3751}(158,\cdot)\) \(\chi_{3751}(203,\cdot)\) \(\chi_{3751}(300,\cdot)\) \(\chi_{3751}(322,\cdot)\) \(\chi_{3751}(383,\cdot)\) \(\chi_{3751}(427,\cdot)\) \(\chi_{3751}(456,\cdot)\) \(\chi_{3751}(499,\cdot)\) \(\chi_{3751}(544,\cdot)\) \(\chi_{3751}(641,\cdot)\) \(\chi_{3751}(663,\cdot)\) \(\chi_{3751}(664,\cdot)\) \(\chi_{3751}(724,\cdot)\) \(\chi_{3751}(768,\cdot)\) \(\chi_{3751}(797,\cdot)\) \(\chi_{3751}(840,\cdot)\) \(\chi_{3751}(885,\cdot)\) \(\chi_{3751}(982,\cdot)\) \(\chi_{3751}(1004,\cdot)\) \(\chi_{3751}(1005,\cdot)\) \(\chi_{3751}(1065,\cdot)\) \(\chi_{3751}(1109,\cdot)\) \(\chi_{3751}(1138,\cdot)\) \(\chi_{3751}(1181,\cdot)\) \(\chi_{3751}(1226,\cdot)\) \(\chi_{3751}(1323,\cdot)\) \(\chi_{3751}(1345,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((2543,2421)\) → \((e\left(\frac{48}{55}\right),e\left(\frac{23}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 3751 }(42, a) \) \(-1\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{151}{165}\right)\)\(e\left(\frac{277}{330}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{31}{165}\right)\)\(e\left(\frac{37}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3751 }(42,a) \;\) at \(\;a = \) e.g. 2