Properties

Label 1225.976
Modulus $1225$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,31]))
 
pari: [g,chi] = znchar(Mod(976,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bg

\(\chi_{1225}(26,\cdot)\) \(\chi_{1225}(101,\cdot)\) \(\chi_{1225}(201,\cdot)\) \(\chi_{1225}(376,\cdot)\) \(\chi_{1225}(451,\cdot)\) \(\chi_{1225}(551,\cdot)\) \(\chi_{1225}(626,\cdot)\) \(\chi_{1225}(726,\cdot)\) \(\chi_{1225}(801,\cdot)\) \(\chi_{1225}(976,\cdot)\) \(\chi_{1225}(1076,\cdot)\) \(\chi_{1225}(1151,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((1177,101)\) → \((1,e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(976, a) \) \(-1\)\(1\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{16}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(976,a) \;\) at \(\;a = \) e.g. 2