Properties

Label 1225.26
Modulus $1225$
Conductor $49$
Order $42$
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([0,17]))
 
pari: [g,chi] = znchar(Mod(26,1225))
 

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 \(\chi_{49}(26,\cdot)\)
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()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((1177,101)\) → \((1,e\left(\frac{17}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{2}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})\)