Properties

Label 1224.97
Modulus $1224$
Conductor $153$
Order $48$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1224)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,32,39]))
 
pari: [g,chi] = znchar(Mod(97,1224))
 

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 \(\chi_{153}(97,\cdot)\)
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()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((919,613,137,649)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{13}{16}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{31}{48}\right)\)\(1\)
value at e.g. 2

Related number fields

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