Properties

Label 7800.17
Modulus $7800$
Conductor $975$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7800, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,30,39,10]))
 
pari: [g,chi] = znchar(Mod(17,7800))
 

Basic properties

Modulus: \(7800\)
Conductor: \(975\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{975}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7800.lz

\(\chi_{7800}(17,\cdot)\) \(\chi_{7800}(953,\cdot)\) \(\chi_{7800}(1577,\cdot)\) \(\chi_{7800}(1817,\cdot)\) \(\chi_{7800}(2513,\cdot)\) \(\chi_{7800}(2753,\cdot)\) \(\chi_{7800}(3137,\cdot)\) \(\chi_{7800}(3377,\cdot)\) \(\chi_{7800}(4073,\cdot)\) \(\chi_{7800}(4313,\cdot)\) \(\chi_{7800}(4697,\cdot)\) \(\chi_{7800}(4937,\cdot)\) \(\chi_{7800}(5633,\cdot)\) \(\chi_{7800}(5873,\cdot)\) \(\chi_{7800}(6497,\cdot)\) \(\chi_{7800}(7433,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1951,3901,5201,7177,4201)\) → \((1,1,-1,e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7800 }(17, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7800 }(17,a) \;\) at \(\;a = \) e.g. 2