Properties

Label 175.x
Modulus $175$
Conductor $175$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(175, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,10]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,175))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(175\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(8\) \(9\) \(11\) \(12\) \(13\) \(16\)
\(\chi_{175}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{175}(12,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{175}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{175}(33,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{175}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{175}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{175}(52,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{175}(73,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{175}(87,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{175}(103,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{175}(108,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{175}(117,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{175}(122,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{175}(138,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{175}(152,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{175}(173,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{15}\right)\)