Properties

Label 59.d
Modulus $59$
Conductor $59$
Order $58$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(59, base_ring=CyclotomicField(58))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(2,59))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(59\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{59}(2,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{58}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{51}{58}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{3}{58}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{7}{58}\right)\) \(e\left(\frac{25}{58}\right)\)
\(\chi_{59}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{51}{58}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{49}{58}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{37}{58}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{9}{58}\right)\) \(e\left(\frac{57}{58}\right)\)
\(\chi_{59}(8,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{58}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{37}{58}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{9}{58}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{21}{58}\right)\) \(e\left(\frac{17}{58}\right)\)
\(\chi_{59}(10,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{58}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{9}{58}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{21}{58}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{49}{58}\right)\) \(e\left(\frac{1}{58}\right)\)
\(\chi_{59}(11,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{58}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{57}{58}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{17}{58}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{1}{58}\right)\) \(e\left(\frac{45}{58}\right)\)
\(\chi_{59}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{45}{58}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{33}{58}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{19}{58}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{25}{58}\right)\) \(e\left(\frac{23}{58}\right)\)
\(\chi_{59}(14,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{58}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{41}{58}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{57}{58}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{17}{58}\right)\) \(e\left(\frac{11}{58}\right)\)
\(\chi_{59}(18,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{58}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{47}{58}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{13}{58}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{11}{58}\right)\) \(e\left(\frac{31}{58}\right)\)
\(\chi_{59}(23,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{58}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{11}{58}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{45}{58}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{47}{58}\right)\) \(e\left(\frac{27}{58}\right)\)
\(\chi_{59}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{58}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{35}{58}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{43}{58}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{23}{58}\right)\) \(e\left(\frac{49}{58}\right)\)
\(\chi_{59}(30,\cdot)\) \(-1\) \(1\) \(e\left(\frac{57}{58}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{7}{58}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{55}{58}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{51}{58}\right)\) \(e\left(\frac{33}{58}\right)\)
\(\chi_{59}(31,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{58}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{5}{58}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{31}{58}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{53}{58}\right)\) \(e\left(\frac{7}{58}\right)\)
\(\chi_{59}(32,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{58}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{23}{58}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{15}{58}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{35}{58}\right)\) \(e\left(\frac{9}{58}\right)\)
\(\chi_{59}(33,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{58}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{55}{58}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{51}{58}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{3}{58}\right)\) \(e\left(\frac{19}{58}\right)\)
\(\chi_{59}(34,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{58}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{3}{58}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{7}{58}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{55}{58}\right)\) \(e\left(\frac{39}{58}\right)\)
\(\chi_{59}(37,\cdot)\) \(-1\) \(1\) \(e\left(\frac{55}{58}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{20}{29}\right)\) \(e\left(\frac{21}{58}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{49}{58}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{37}{58}\right)\) \(e\left(\frac{41}{58}\right)\)
\(\chi_{59}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{39}{58}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{17}{58}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{1}{58}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{41}{58}\right)\) \(e\left(\frac{47}{58}\right)\)
\(\chi_{59}(39,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{58}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{31}{58}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{53}{58}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{27}{58}\right)\) \(e\left(\frac{55}{58}\right)\)
\(\chi_{59}(40,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{58}\right)\) \(e\left(\frac{22}{29}\right)\) \(e\left(\frac{9}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{53}{58}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{27}{58}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{5}{58}\right)\) \(e\left(\frac{51}{58}\right)\)
\(\chi_{59}(42,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{58}\right)\) \(e\left(\frac{14}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{39}{58}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{33}{58}\right)\) \(e\left(\frac{28}{29}\right)\) \(e\left(\frac{19}{58}\right)\) \(e\left(\frac{43}{58}\right)\)
\(\chi_{59}(43,\cdot)\) \(-1\) \(1\) \(e\left(\frac{33}{58}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{1}{58}\right)\) \(e\left(\frac{7}{29}\right)\) \(e\left(\frac{41}{58}\right)\) \(e\left(\frac{26}{29}\right)\) \(e\left(\frac{57}{58}\right)\) \(e\left(\frac{13}{58}\right)\)
\(\chi_{59}(44,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{58}\right)\) \(e\left(\frac{8}{29}\right)\) \(e\left(\frac{27}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{43}{58}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{23}{58}\right)\) \(e\left(\frac{16}{29}\right)\) \(e\left(\frac{15}{58}\right)\) \(e\left(\frac{37}{58}\right)\)
\(\chi_{59}(47,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{58}\right)\) \(e\left(\frac{24}{29}\right)\) \(e\left(\frac{23}{29}\right)\) \(e\left(\frac{11}{29}\right)\) \(e\left(\frac{13}{58}\right)\) \(e\left(\frac{4}{29}\right)\) \(e\left(\frac{11}{58}\right)\) \(e\left(\frac{19}{29}\right)\) \(e\left(\frac{45}{58}\right)\) \(e\left(\frac{53}{58}\right)\)
\(\chi_{59}(50,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{58}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{25}{58}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{39}{58}\right)\) \(e\left(\frac{12}{29}\right)\) \(e\left(\frac{33}{58}\right)\) \(e\left(\frac{35}{58}\right)\)
\(\chi_{59}(52,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{58}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{19}{58}\right)\) \(e\left(\frac{17}{29}\right)\) \(e\left(\frac{25}{58}\right)\) \(e\left(\frac{1}{29}\right)\) \(e\left(\frac{39}{58}\right)\) \(e\left(\frac{15}{58}\right)\)
\(\chi_{59}(54,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{58}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{45}{58}\right)\) \(e\left(\frac{25}{29}\right)\) \(e\left(\frac{47}{58}\right)\) \(e\left(\frac{10}{29}\right)\) \(e\left(\frac{13}{58}\right)\) \(e\left(\frac{5}{58}\right)\)
\(\chi_{59}(55,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{58}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{2}{29}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{15}{58}\right)\) \(e\left(\frac{18}{29}\right)\) \(e\left(\frac{35}{58}\right)\) \(e\left(\frac{13}{29}\right)\) \(e\left(\frac{43}{58}\right)\) \(e\left(\frac{21}{58}\right)\)
\(\chi_{59}(56,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{58}\right)\) \(e\left(\frac{3}{29}\right)\) \(e\left(\frac{21}{29}\right)\) \(e\left(\frac{5}{29}\right)\) \(e\left(\frac{27}{58}\right)\) \(e\left(\frac{15}{29}\right)\) \(e\left(\frac{5}{58}\right)\) \(e\left(\frac{6}{29}\right)\) \(e\left(\frac{31}{58}\right)\) \(e\left(\frac{3}{58}\right)\)