Properties

Label 1225.303
Modulus $1225$
Conductor $1225$
Order $420$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([147,20]))
 
pari: [g,chi] = znchar(Mod(303,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
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 1225.bv

\(\chi_{1225}(2,\cdot)\) \(\chi_{1225}(23,\cdot)\) \(\chi_{1225}(37,\cdot)\) \(\chi_{1225}(53,\cdot)\) \(\chi_{1225}(58,\cdot)\) \(\chi_{1225}(72,\cdot)\) \(\chi_{1225}(88,\cdot)\) \(\chi_{1225}(102,\cdot)\) \(\chi_{1225}(123,\cdot)\) \(\chi_{1225}(137,\cdot)\) \(\chi_{1225}(142,\cdot)\) \(\chi_{1225}(158,\cdot)\) \(\chi_{1225}(163,\cdot)\) \(\chi_{1225}(172,\cdot)\) \(\chi_{1225}(198,\cdot)\) \(\chi_{1225}(212,\cdot)\) \(\chi_{1225}(228,\cdot)\) \(\chi_{1225}(233,\cdot)\) \(\chi_{1225}(242,\cdot)\) \(\chi_{1225}(247,\cdot)\) \(\chi_{1225}(277,\cdot)\) \(\chi_{1225}(298,\cdot)\) \(\chi_{1225}(303,\cdot)\) \(\chi_{1225}(317,\cdot)\) \(\chi_{1225}(333,\cdot)\) \(\chi_{1225}(338,\cdot)\) \(\chi_{1225}(347,\cdot)\) \(\chi_{1225}(352,\cdot)\) \(\chi_{1225}(387,\cdot)\) \(\chi_{1225}(403,\cdot)\) ...

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

\((1177,101)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{1}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(303, a) \) \(-1\)\(1\)\(e\left(\frac{247}{420}\right)\)\(e\left(\frac{209}{420}\right)\)\(e\left(\frac{37}{210}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{107}{140}\right)\)\(e\left(\frac{209}{210}\right)\)\(e\left(\frac{53}{105}\right)\)\(e\left(\frac{283}{420}\right)\)\(e\left(\frac{31}{140}\right)\)\(e\left(\frac{37}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(303,a) \;\) at \(\;a = \) e.g. 2