Properties

Label 137.e
Modulus $137$
Conductor $137$
Order $17$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(137, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(16,137))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(137\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
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: \(\Q(\zeta_{17})\)
Fixed field: 17.17.15400296222263289476715621650663041.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{137}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(1\) \(e\left(\frac{15}{17}\right)\)
\(\chi_{137}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(1\) \(e\left(\frac{1}{17}\right)\)
\(\chi_{137}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(1\) \(e\left(\frac{4}{17}\right)\)
\(\chi_{137}(50,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(1\) \(e\left(\frac{9}{17}\right)\)
\(\chi_{137}(56,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(1\) \(e\left(\frac{10}{17}\right)\)
\(\chi_{137}(59,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(1\) \(e\left(\frac{5}{17}\right)\)
\(\chi_{137}(60,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(1\) \(e\left(\frac{2}{17}\right)\)
\(\chi_{137}(72,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(1\) \(e\left(\frac{12}{17}\right)\)
\(\chi_{137}(73,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(1\) \(e\left(\frac{14}{17}\right)\)
\(\chi_{137}(74,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(1\) \(e\left(\frac{8}{17}\right)\)
\(\chi_{137}(88,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(1\) \(e\left(\frac{6}{17}\right)\)
\(\chi_{137}(115,\cdot)\) \(1\) \(1\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(1\) \(e\left(\frac{7}{17}\right)\)
\(\chi_{137}(119,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(1\) \(e\left(\frac{13}{17}\right)\)
\(\chi_{137}(122,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(1\) \(e\left(\frac{3}{17}\right)\)
\(\chi_{137}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(1\) \(e\left(\frac{11}{17}\right)\)
\(\chi_{137}(133,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(1\) \(e\left(\frac{16}{17}\right)\)