Properties

Label 1225.24
Modulus $1225$
Conductor $245$
Order $42$
Real no
Primitive no
Minimal no
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([21,37]))
 
pari: [g,chi] = znchar(Mod(24,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bf

\(\chi_{1225}(24,\cdot)\) \(\chi_{1225}(124,\cdot)\) \(\chi_{1225}(199,\cdot)\) \(\chi_{1225}(299,\cdot)\) \(\chi_{1225}(474,\cdot)\) \(\chi_{1225}(549,\cdot)\) \(\chi_{1225}(649,\cdot)\) \(\chi_{1225}(724,\cdot)\) \(\chi_{1225}(824,\cdot)\) \(\chi_{1225}(899,\cdot)\) \(\chi_{1225}(1074,\cdot)\) \(\chi_{1225}(1174,\cdot)\)

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

\((1177,101)\) → \((-1,e\left(\frac{37}{42}\right))\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.56353276529596271503862578540802938668269419115433656434196014026165008544921875.1