Properties

Label 1225.32
Modulus $1225$
Conductor $245$
Order $84$
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(1225)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,8]))
 
pari: [g,chi] = znchar(Mod(32,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bm

\(\chi_{1225}(32,\cdot)\) \(\chi_{1225}(93,\cdot)\) \(\chi_{1225}(107,\cdot)\) \(\chi_{1225}(193,\cdot)\) \(\chi_{1225}(207,\cdot)\) \(\chi_{1225}(268,\cdot)\) \(\chi_{1225}(282,\cdot)\) \(\chi_{1225}(368,\cdot)\) \(\chi_{1225}(382,\cdot)\) \(\chi_{1225}(443,\cdot)\) \(\chi_{1225}(457,\cdot)\) \(\chi_{1225}(543,\cdot)\) \(\chi_{1225}(632,\cdot)\) \(\chi_{1225}(718,\cdot)\) \(\chi_{1225}(732,\cdot)\) \(\chi_{1225}(793,\cdot)\) \(\chi_{1225}(807,\cdot)\) \(\chi_{1225}(893,\cdot)\) \(\chi_{1225}(907,\cdot)\) \(\chi_{1225}(968,\cdot)\) \(\chi_{1225}(982,\cdot)\) \(\chi_{1225}(1068,\cdot)\) \(\chi_{1225}(1082,\cdot)\) \(\chi_{1225}(1143,\cdot)\)

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

\((1177,101)\) → \((i,e\left(\frac{2}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{19}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial