Properties

Label 1449.100
Modulus $1449$
Conductor $161$
Order $33$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,22,18]))
 
pari: [g,chi] = znchar(Mod(100,1449))
 

Basic properties

Modulus: \(1449\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(100,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1449.bz

\(\chi_{1449}(100,\cdot)\) \(\chi_{1449}(163,\cdot)\) \(\chi_{1449}(289,\cdot)\) \(\chi_{1449}(361,\cdot)\) \(\chi_{1449}(478,\cdot)\) \(\chi_{1449}(487,\cdot)\) \(\chi_{1449}(541,\cdot)\) \(\chi_{1449}(604,\cdot)\) \(\chi_{1449}(676,\cdot)\) \(\chi_{1449}(739,\cdot)\) \(\chi_{1449}(928,\cdot)\) \(\chi_{1449}(982,\cdot)\) \(\chi_{1449}(991,\cdot)\) \(\chi_{1449}(1108,\cdot)\) \(\chi_{1449}(1117,\cdot)\) \(\chi_{1449}(1297,\cdot)\) \(\chi_{1449}(1306,\cdot)\) \(\chi_{1449}(1360,\cdot)\) \(\chi_{1449}(1369,\cdot)\) \(\chi_{1449}(1432,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: 33.33.277966181338944111003326058293667039541136678070715028736001.1

Values on generators

\((1289,829,442)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1449 }(100, a) \) \(1\)\(1\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{25}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1449 }(100,a) \;\) at \(\;a = \) e.g. 2