Properties

Label 683.e
Modulus $683$
Conductor $683$
Order $31$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(683, base_ring=CyclotomicField(62))
 
M = H._module
 
chi = DirichletCharacter(H, M([30]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(3,683))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(683\)
Conductor: \(683\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(31\)
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: $\Q(\zeta_{31})$
Fixed field: Number field defined by a degree 31 polynomial

First 30 of 30 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{683}(3,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{21}{31}\right)\) \(1\) \(e\left(\frac{15}{31}\right)\) \(e\left(\frac{21}{31}\right)\) \(e\left(\frac{7}{31}\right)\) \(1\) \(e\left(\frac{11}{31}\right)\) \(e\left(\frac{15}{31}\right)\) \(e\left(\frac{28}{31}\right)\)
\(\chi_{683}(9,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{11}{31}\right)\) \(1\) \(e\left(\frac{30}{31}\right)\) \(e\left(\frac{11}{31}\right)\) \(e\left(\frac{14}{31}\right)\) \(1\) \(e\left(\frac{22}{31}\right)\) \(e\left(\frac{30}{31}\right)\) \(e\left(\frac{25}{31}\right)\)
\(\chi_{683}(27,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{31}\right)\) \(1\) \(e\left(\frac{14}{31}\right)\) \(e\left(\frac{1}{31}\right)\) \(e\left(\frac{21}{31}\right)\) \(1\) \(e\left(\frac{2}{31}\right)\) \(e\left(\frac{14}{31}\right)\) \(e\left(\frac{22}{31}\right)\)
\(\chi_{683}(46,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{31}\right)\) \(1\) \(e\left(\frac{28}{31}\right)\) \(e\left(\frac{2}{31}\right)\) \(e\left(\frac{11}{31}\right)\) \(1\) \(e\left(\frac{4}{31}\right)\) \(e\left(\frac{28}{31}\right)\) \(e\left(\frac{13}{31}\right)\)
\(\chi_{683}(67,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{4}{31}\right)\) \(1\) \(e\left(\frac{25}{31}\right)\) \(e\left(\frac{4}{31}\right)\) \(e\left(\frac{22}{31}\right)\) \(1\) \(e\left(\frac{8}{31}\right)\) \(e\left(\frac{25}{31}\right)\) \(e\left(\frac{26}{31}\right)\)
\(\chi_{683}(76,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{20}{31}\right)\) \(1\) \(e\left(\frac{1}{31}\right)\) \(e\left(\frac{20}{31}\right)\) \(e\left(\frac{17}{31}\right)\) \(1\) \(e\left(\frac{9}{31}\right)\) \(e\left(\frac{1}{31}\right)\) \(e\left(\frac{6}{31}\right)\)
\(\chi_{683}(81,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{22}{31}\right)\) \(1\) \(e\left(\frac{29}{31}\right)\) \(e\left(\frac{22}{31}\right)\) \(e\left(\frac{28}{31}\right)\) \(1\) \(e\left(\frac{13}{31}\right)\) \(e\left(\frac{29}{31}\right)\) \(e\left(\frac{19}{31}\right)\)
\(\chi_{683}(104,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{19}{31}\right)\) \(1\) \(e\left(\frac{18}{31}\right)\) \(e\left(\frac{19}{31}\right)\) \(e\left(\frac{27}{31}\right)\) \(1\) \(e\left(\frac{7}{31}\right)\) \(e\left(\frac{18}{31}\right)\) \(e\left(\frac{15}{31}\right)\)
\(\chi_{683}(138,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{23}{31}\right)\) \(1\) \(e\left(\frac{12}{31}\right)\) \(e\left(\frac{23}{31}\right)\) \(e\left(\frac{18}{31}\right)\) \(1\) \(e\left(\frac{15}{31}\right)\) \(e\left(\frac{12}{31}\right)\) \(e\left(\frac{10}{31}\right)\)
\(\chi_{683}(201,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{25}{31}\right)\) \(1\) \(e\left(\frac{9}{31}\right)\) \(e\left(\frac{25}{31}\right)\) \(e\left(\frac{29}{31}\right)\) \(1\) \(e\left(\frac{19}{31}\right)\) \(e\left(\frac{9}{31}\right)\) \(e\left(\frac{23}{31}\right)\)
\(\chi_{683}(228,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{10}{31}\right)\) \(1\) \(e\left(\frac{16}{31}\right)\) \(e\left(\frac{10}{31}\right)\) \(e\left(\frac{24}{31}\right)\) \(1\) \(e\left(\frac{20}{31}\right)\) \(e\left(\frac{16}{31}\right)\) \(e\left(\frac{3}{31}\right)\)
\(\chi_{683}(243,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{12}{31}\right)\) \(1\) \(e\left(\frac{13}{31}\right)\) \(e\left(\frac{12}{31}\right)\) \(e\left(\frac{4}{31}\right)\) \(1\) \(e\left(\frac{24}{31}\right)\) \(e\left(\frac{13}{31}\right)\) \(e\left(\frac{16}{31}\right)\)
\(\chi_{683}(250,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{14}{31}\right)\) \(1\) \(e\left(\frac{10}{31}\right)\) \(e\left(\frac{14}{31}\right)\) \(e\left(\frac{15}{31}\right)\) \(1\) \(e\left(\frac{28}{31}\right)\) \(e\left(\frac{10}{31}\right)\) \(e\left(\frac{29}{31}\right)\)
\(\chi_{683}(253,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{30}{31}\right)\) \(1\) \(e\left(\frac{17}{31}\right)\) \(e\left(\frac{30}{31}\right)\) \(e\left(\frac{10}{31}\right)\) \(1\) \(e\left(\frac{29}{31}\right)\) \(e\left(\frac{17}{31}\right)\) \(e\left(\frac{9}{31}\right)\)
\(\chi_{683}(311,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{24}{31}\right)\) \(1\) \(e\left(\frac{26}{31}\right)\) \(e\left(\frac{24}{31}\right)\) \(e\left(\frac{8}{31}\right)\) \(1\) \(e\left(\frac{17}{31}\right)\) \(e\left(\frac{26}{31}\right)\) \(e\left(\frac{1}{31}\right)\)
\(\chi_{683}(312,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{9}{31}\right)\) \(1\) \(e\left(\frac{2}{31}\right)\) \(e\left(\frac{9}{31}\right)\) \(e\left(\frac{3}{31}\right)\) \(1\) \(e\left(\frac{18}{31}\right)\) \(e\left(\frac{2}{31}\right)\) \(e\left(\frac{12}{31}\right)\)
\(\chi_{683}(347,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{28}{31}\right)\) \(1\) \(e\left(\frac{20}{31}\right)\) \(e\left(\frac{28}{31}\right)\) \(e\left(\frac{30}{31}\right)\) \(1\) \(e\left(\frac{25}{31}\right)\) \(e\left(\frac{20}{31}\right)\) \(e\left(\frac{27}{31}\right)\)
\(\chi_{683}(350,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{6}{31}\right)\) \(1\) \(e\left(\frac{22}{31}\right)\) \(e\left(\frac{6}{31}\right)\) \(e\left(\frac{2}{31}\right)\) \(1\) \(e\left(\frac{12}{31}\right)\) \(e\left(\frac{22}{31}\right)\) \(e\left(\frac{8}{31}\right)\)
\(\chi_{683}(358,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{18}{31}\right)\) \(1\) \(e\left(\frac{4}{31}\right)\) \(e\left(\frac{18}{31}\right)\) \(e\left(\frac{6}{31}\right)\) \(1\) \(e\left(\frac{5}{31}\right)\) \(e\left(\frac{4}{31}\right)\) \(e\left(\frac{24}{31}\right)\)
\(\chi_{683}(367,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{27}{31}\right)\) \(1\) \(e\left(\frac{6}{31}\right)\) \(e\left(\frac{27}{31}\right)\) \(e\left(\frac{9}{31}\right)\) \(1\) \(e\left(\frac{23}{31}\right)\) \(e\left(\frac{6}{31}\right)\) \(e\left(\frac{5}{31}\right)\)
\(\chi_{683}(391,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{8}{31}\right)\) \(1\) \(e\left(\frac{19}{31}\right)\) \(e\left(\frac{8}{31}\right)\) \(e\left(\frac{13}{31}\right)\) \(1\) \(e\left(\frac{16}{31}\right)\) \(e\left(\frac{19}{31}\right)\) \(e\left(\frac{21}{31}\right)\)
\(\chi_{683}(414,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{13}{31}\right)\) \(1\) \(e\left(\frac{27}{31}\right)\) \(e\left(\frac{13}{31}\right)\) \(e\left(\frac{25}{31}\right)\) \(1\) \(e\left(\frac{26}{31}\right)\) \(e\left(\frac{27}{31}\right)\) \(e\left(\frac{7}{31}\right)\)
\(\chi_{683}(418,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{17}{31}\right)\) \(1\) \(e\left(\frac{21}{31}\right)\) \(e\left(\frac{17}{31}\right)\) \(e\left(\frac{16}{31}\right)\) \(1\) \(e\left(\frac{3}{31}\right)\) \(e\left(\frac{21}{31}\right)\) \(e\left(\frac{2}{31}\right)\)
\(\chi_{683}(443,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{5}{31}\right)\) \(1\) \(e\left(\frac{8}{31}\right)\) \(e\left(\frac{5}{31}\right)\) \(e\left(\frac{12}{31}\right)\) \(1\) \(e\left(\frac{10}{31}\right)\) \(e\left(\frac{8}{31}\right)\) \(e\left(\frac{17}{31}\right)\)
\(\chi_{683}(490,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{29}{31}\right)\) \(1\) \(e\left(\frac{3}{31}\right)\) \(e\left(\frac{29}{31}\right)\) \(e\left(\frac{20}{31}\right)\) \(1\) \(e\left(\frac{27}{31}\right)\) \(e\left(\frac{3}{31}\right)\) \(e\left(\frac{18}{31}\right)\)
\(\chi_{683}(559,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{3}{31}\right)\) \(1\) \(e\left(\frac{11}{31}\right)\) \(e\left(\frac{3}{31}\right)\) \(e\left(\frac{1}{31}\right)\) \(1\) \(e\left(\frac{6}{31}\right)\) \(e\left(\frac{11}{31}\right)\) \(e\left(\frac{4}{31}\right)\)
\(\chi_{683}(571,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{7}{31}\right)\) \(1\) \(e\left(\frac{5}{31}\right)\) \(e\left(\frac{7}{31}\right)\) \(e\left(\frac{23}{31}\right)\) \(1\) \(e\left(\frac{14}{31}\right)\) \(e\left(\frac{5}{31}\right)\) \(e\left(\frac{30}{31}\right)\)
\(\chi_{683}(572,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{16}{31}\right)\) \(1\) \(e\left(\frac{7}{31}\right)\) \(e\left(\frac{16}{31}\right)\) \(e\left(\frac{26}{31}\right)\) \(1\) \(e\left(\frac{1}{31}\right)\) \(e\left(\frac{7}{31}\right)\) \(e\left(\frac{11}{31}\right)\)
\(\chi_{683}(603,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{15}{31}\right)\) \(1\) \(e\left(\frac{24}{31}\right)\) \(e\left(\frac{15}{31}\right)\) \(e\left(\frac{5}{31}\right)\) \(1\) \(e\left(\frac{30}{31}\right)\) \(e\left(\frac{24}{31}\right)\) \(e\left(\frac{20}{31}\right)\)
\(\chi_{683}(646,\cdot)\) \(1\) \(1\) \(1\) \(e\left(\frac{26}{31}\right)\) \(1\) \(e\left(\frac{23}{31}\right)\) \(e\left(\frac{26}{31}\right)\) \(e\left(\frac{19}{31}\right)\) \(1\) \(e\left(\frac{21}{31}\right)\) \(e\left(\frac{23}{31}\right)\) \(e\left(\frac{14}{31}\right)\)