Properties

Label 4056.db
Modulus $4056$
Conductor $4056$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,39,39,20]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(29,4056))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(4056\)
Conductor: \(4056\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{4056}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{73}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{29}{39}\right)\)
\(\chi_{4056}(269,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{17}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{34}{39}\right)\)
\(\chi_{4056}(341,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{43}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{8}{39}\right)\)
\(\chi_{4056}(581,\cdot)\) \(-1\) \(1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{59}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{1}{39}\right)\)
\(\chi_{4056}(893,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{23}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{7}{39}\right)\)
\(\chi_{4056}(965,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{61}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{5}{39}\right)\)
\(\chi_{4056}(1277,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{31}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{23}{39}\right)\)
\(\chi_{4056}(1517,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{29}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{19}{39}\right)\)
\(\chi_{4056}(1589,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{1}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{2}{39}\right)\)
\(\chi_{4056}(1829,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{71}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{25}{39}\right)\)
\(\chi_{4056}(1901,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{49}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{20}{39}\right)\)
\(\chi_{4056}(2141,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{35}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{31}{39}\right)\)
\(\chi_{4056}(2213,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{19}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{38}{39}\right)\)
\(\chi_{4056}(2453,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{77}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{37}{39}\right)\)
\(\chi_{4056}(2525,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{67}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{17}{39}\right)\)
\(\chi_{4056}(2765,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{41}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{4}{39}\right)\)
\(\chi_{4056}(2837,\cdot)\) \(-1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{37}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{35}{39}\right)\)
\(\chi_{4056}(3077,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{5}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{10}{39}\right)\)
\(\chi_{4056}(3149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{7}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{14}{39}\right)\)
\(\chi_{4056}(3389,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{47}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{16}{39}\right)\)
\(\chi_{4056}(3461,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{55}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{32}{39}\right)\)
\(\chi_{4056}(3701,\cdot)\) \(-1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{11}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{22}{39}\right)\)
\(\chi_{4056}(3773,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{25}{78}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{11}{39}\right)\)
\(\chi_{4056}(4013,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{53}{78}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{28}{39}\right)\)