Properties

Label 1352.bj
Modulus $1352$
Conductor $1352$
Order $52$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,26,3]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(5,1352))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1352\)
Conductor: \(1352\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(15\) \(17\) \(19\) \(21\) \(23\)
\(\chi_{1352}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{35}{52}\right)\) \(e\left(\frac{11}{26}\right)\) \(i\) \(e\left(\frac{43}{52}\right)\) \(-1\)
\(\chi_{1352}(21,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{43}{52}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{49}{52}\right)\) \(e\left(\frac{5}{26}\right)\) \(-i\) \(e\left(\frac{29}{52}\right)\) \(-1\)
\(\chi_{1352}(109,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{41}{52}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{31}{52}\right)\) \(e\left(\frac{9}{26}\right)\) \(i\) \(e\left(\frac{47}{52}\right)\) \(-1\)
\(\chi_{1352}(125,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{3}{52}\right)\) \(e\left(\frac{27}{52}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(-i\) \(e\left(\frac{25}{52}\right)\) \(-1\)
\(\chi_{1352}(213,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{29}{52}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{43}{52}\right)\) \(e\left(\frac{27}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(i\) \(e\left(\frac{51}{52}\right)\) \(-1\)
\(\chi_{1352}(229,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{31}{52}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{9}{26}\right)\) \(-i\) \(e\left(\frac{21}{52}\right)\) \(-1\)
\(\chi_{1352}(317,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{49}{52}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{27}{52}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{5}{26}\right)\) \(i\) \(e\left(\frac{3}{52}\right)\) \(-1\)
\(\chi_{1352}(333,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{27}{52}\right)\) \(e\left(\frac{35}{52}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{49}{52}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{11}{26}\right)\) \(-i\) \(e\left(\frac{17}{52}\right)\) \(-1\)
\(\chi_{1352}(421,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{45}{52}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{11}{52}\right)\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{3}{26}\right)\) \(i\) \(e\left(\frac{7}{52}\right)\) \(-1\)
\(\chi_{1352}(525,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{45}{52}\right)\) \(e\left(\frac{41}{52}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{1}{26}\right)\) \(i\) \(e\left(\frac{11}{52}\right)\) \(-1\)
\(\chi_{1352}(541,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{43}{52}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{29}{52}\right)\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{15}{26}\right)\) \(-i\) \(e\left(\frac{9}{52}\right)\) \(-1\)
\(\chi_{1352}(629,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{31}{52}\right)\) \(e\left(\frac{11}{52}\right)\) \(e\left(\frac{25}{26}\right)\) \(i\) \(e\left(\frac{15}{52}\right)\) \(-1\)
\(\chi_{1352}(645,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{11}{52}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{45}{52}\right)\) \(e\left(\frac{21}{52}\right)\) \(e\left(\frac{17}{26}\right)\) \(-i\) \(e\left(\frac{5}{52}\right)\) \(-1\)
\(\chi_{1352}(733,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{21}{52}\right)\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{23}{26}\right)\) \(i\) \(e\left(\frac{19}{52}\right)\) \(-1\)
\(\chi_{1352}(749,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{25}{52}\right)\) \(e\left(\frac{19}{26}\right)\) \(-i\) \(e\left(\frac{1}{52}\right)\) \(-1\)
\(\chi_{1352}(837,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{29}{52}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{3}{52}\right)\) \(e\left(\frac{21}{26}\right)\) \(i\) \(e\left(\frac{23}{52}\right)\) \(-1\)
\(\chi_{1352}(853,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{35}{52}\right)\) \(e\left(\frac{3}{52}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{25}{52}\right)\) \(e\left(\frac{29}{52}\right)\) \(e\left(\frac{21}{26}\right)\) \(-i\) \(e\left(\frac{49}{52}\right)\) \(-1\)
\(\chi_{1352}(941,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{49}{52}\right)\) \(e\left(\frac{25}{52}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{35}{52}\right)\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{19}{26}\right)\) \(i\) \(e\left(\frac{27}{52}\right)\) \(-1\)
\(\chi_{1352}(957,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{41}{52}\right)\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{23}{26}\right)\) \(-i\) \(e\left(\frac{45}{52}\right)\) \(-1\)
\(\chi_{1352}(1045,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{21}{52}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{17}{26}\right)\) \(i\) \(e\left(\frac{31}{52}\right)\) \(-1\)
\(\chi_{1352}(1061,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{11}{52}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{25}{26}\right)\) \(-i\) \(e\left(\frac{41}{52}\right)\) \(-1\)
\(\chi_{1352}(1149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{25}{52}\right)\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{3}{52}\right)\) \(e\left(\frac{43}{52}\right)\) \(e\left(\frac{15}{26}\right)\) \(i\) \(e\left(\frac{35}{52}\right)\) \(-1\)
\(\chi_{1352}(1165,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{21}{52}\right)\) \(e\left(\frac{41}{52}\right)\) \(e\left(\frac{1}{26}\right)\) \(-i\) \(e\left(\frac{37}{52}\right)\) \(-1\)
\(\chi_{1352}(1269,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{31}{52}\right)\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{45}{52}\right)\) \(e\left(\frac{3}{26}\right)\) \(-i\) \(e\left(\frac{33}{52}\right)\) \(-1\)