Properties

Label 177.e
Modulus $177$
Conductor $59$
Order $29$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(177, base_ring=CyclotomicField(58))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,2]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(4,177))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(177\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 59.c
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: $\Q(\zeta_{29})$
Fixed field: \(\Q(\zeta_{59})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{177}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{4}{29}\right)\)
\(\chi_{177}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{7}{29}\right)\)
\(\chi_{177}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{8}{29}\right)\)
\(\chi_{177}(19,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{18}{29}\right)\)
\(\chi_{177}(22,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{23}{29}\right)\)
\(\chi_{177}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{24}{29}\right)\)
\(\chi_{177}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{11}{29}\right)\)
\(\chi_{177}(46,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{3}{29}\right)\)
\(\chi_{177}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{14}{29}\right)\)
\(\chi_{177}(64,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{12}{29}\right)\)
\(\chi_{177}(76,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{22}{29}\right)\)
\(\chi_{177}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{16}{29}\right)\)
\(\chi_{177}(85,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{5}{29}\right)\)
\(\chi_{177}(88,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{27}{29}\right)\)
\(\chi_{177}(94,\cdot)\) \(1\) \(1\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{19}{29}\right)\)
\(\chi_{177}(100,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{28}{29}\right)\)
\(\chi_{177}(112,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{15}{29}\right)\)
\(\chi_{177}(121,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{13}{29}\right)\)
\(\chi_{177}(127,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{26}{29}\right)\)
\(\chi_{177}(130,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{17}{29}\right)\)
\(\chi_{177}(133,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{25}{29}\right)\)
\(\chi_{177}(139,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{20}{29}\right)\)
\(\chi_{177}(145,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{10}{29}\right)\)
\(\chi_{177}(154,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{1}{29}\right)\)
\(\chi_{177}(163,\cdot)\) \(1\) \(1\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{9}{29}\right)\)
\(\chi_{177}(166,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{21}{29}\right)\)
\(\chi_{177}(169,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{6}{29}\right)\)
\(\chi_{177}(175,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{2}{29}\right)\)