Properties

Label 7350.73
Modulus $7350$
Conductor $1225$
Order $420$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, base_ring=CyclotomicField(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,231,370]))
 
pari: [g,chi] = znchar(Mod(73,7350))
 

Basic properties

Modulus: \(7350\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1225}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7350.dp

\(\chi_{7350}(73,\cdot)\) \(\chi_{7350}(103,\cdot)\) \(\chi_{7350}(187,\cdot)\) \(\chi_{7350}(283,\cdot)\) \(\chi_{7350}(367,\cdot)\) \(\chi_{7350}(397,\cdot)\) \(\chi_{7350}(523,\cdot)\) \(\chi_{7350}(577,\cdot)\) \(\chi_{7350}(703,\cdot)\) \(\chi_{7350}(733,\cdot)\) \(\chi_{7350}(787,\cdot)\) \(\chi_{7350}(817,\cdot)\) \(\chi_{7350}(997,\cdot)\) \(\chi_{7350}(1027,\cdot)\) \(\chi_{7350}(1123,\cdot)\) \(\chi_{7350}(1153,\cdot)\) \(\chi_{7350}(1237,\cdot)\) \(\chi_{7350}(1333,\cdot)\) \(\chi_{7350}(1363,\cdot)\) \(\chi_{7350}(1417,\cdot)\) \(\chi_{7350}(1447,\cdot)\) \(\chi_{7350}(1573,\cdot)\) \(\chi_{7350}(1627,\cdot)\) \(\chi_{7350}(1753,\cdot)\) \(\chi_{7350}(1837,\cdot)\) \(\chi_{7350}(1867,\cdot)\) \(\chi_{7350}(1963,\cdot)\) \(\chi_{7350}(2047,\cdot)\) \(\chi_{7350}(2173,\cdot)\) \(\chi_{7350}(2203,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((4901,1177,2551)\) → \((1,e\left(\frac{11}{20}\right),e\left(\frac{37}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7350 }(73, a) \) \(1\)\(1\)\(e\left(\frac{4}{105}\right)\)\(e\left(\frac{73}{140}\right)\)\(e\left(\frac{73}{420}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{221}{420}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{59}{420}\right)\)\(e\left(\frac{29}{70}\right)\)\(e\left(\frac{15}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7350 }(73,a) \;\) at \(\;a = \) e.g. 2