Properties

Modulus 9900
Conductor 275
Order 20
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 9900.hg

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(9900)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,9,4]))
 
pari: [g,chi] = znchar(Mod(37,9900))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 9900
Conductor = 275
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 20
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 9900.hg
Orbit index = 189

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{9900}(37,\cdot)\) \(\chi_{9900}(1153,\cdot)\) \(\chi_{9900}(1477,\cdot)\) \(\chi_{9900}(1873,\cdot)\) \(\chi_{9900}(2413,\cdot)\) \(\chi_{9900}(4933,\cdot)\) \(\chi_{9900}(6517,\cdot)\) \(\chi_{9900}(6697,\cdot)\)

Values on generators

\((4951,5501,2377,4501)\) → \((1,1,e\left(\frac{9}{20}\right),e\left(\frac{1}{5}\right))\)

Values

-117131719232931374143
\(-1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(-i\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(-i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)