Properties

Label 1224.601
Modulus $1224$
Conductor $153$
Order $48$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,32,45]))
 
pari: [g,chi] = znchar(Mod(601,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(153\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{153}(142,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.db

\(\chi_{1224}(97,\cdot)\) \(\chi_{1224}(193,\cdot)\) \(\chi_{1224}(241,\cdot)\) \(\chi_{1224}(265,\cdot)\) \(\chi_{1224}(313,\cdot)\) \(\chi_{1224}(337,\cdot)\) \(\chi_{1224}(385,\cdot)\) \(\chi_{1224}(481,\cdot)\) \(\chi_{1224}(601,\cdot)\) \(\chi_{1224}(673,\cdot)\) \(\chi_{1224}(745,\cdot)\) \(\chi_{1224}(889,\cdot)\) \(\chi_{1224}(913,\cdot)\) \(\chi_{1224}(1057,\cdot)\) \(\chi_{1224}(1129,\cdot)\) \(\chi_{1224}(1201,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((919,613,137,649)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{15}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1224 }(601, a) \) \(-1\)\(1\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1224 }(601,a) \;\) at \(\;a = \) e.g. 2