Properties

Label 647.g
Modulus $647$
Conductor $647$
Order $323$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(647, base_ring=CyclotomicField(646))
 
M = H._module
 
chi = DirichletCharacter(H, M([158]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(2,647))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(647\)
Conductor: \(647\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(323\)
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: $\Q(\zeta_{323})$
Fixed field: Number field defined by a degree 323 polynomial (not computed)

First 31 of 288 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{647}(2,\cdot)\) \(1\) \(1\) \(e\left(\frac{208}{323}\right)\) \(e\left(\frac{167}{323}\right)\) \(e\left(\frac{93}{323}\right)\) \(e\left(\frac{79}{323}\right)\) \(e\left(\frac{52}{323}\right)\) \(e\left(\frac{243}{323}\right)\) \(e\left(\frac{301}{323}\right)\) \(e\left(\frac{11}{323}\right)\) \(e\left(\frac{287}{323}\right)\) \(e\left(\frac{108}{323}\right)\)
\(\chi_{647}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{167}{323}\right)\) \(e\left(\frac{117}{323}\right)\) \(e\left(\frac{11}{323}\right)\) \(e\left(\frac{183}{323}\right)\) \(e\left(\frac{284}{323}\right)\) \(e\left(\frac{60}{323}\right)\) \(e\left(\frac{178}{323}\right)\) \(e\left(\frac{234}{323}\right)\) \(e\left(\frac{27}{323}\right)\) \(e\left(\frac{242}{323}\right)\)
\(\chi_{647}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{93}{323}\right)\) \(e\left(\frac{11}{323}\right)\) \(e\left(\frac{186}{323}\right)\) \(e\left(\frac{158}{323}\right)\) \(e\left(\frac{104}{323}\right)\) \(e\left(\frac{163}{323}\right)\) \(e\left(\frac{279}{323}\right)\) \(e\left(\frac{22}{323}\right)\) \(e\left(\frac{251}{323}\right)\) \(e\left(\frac{216}{323}\right)\)
\(\chi_{647}(6,\cdot)\) \(1\) \(1\) \(e\left(\frac{52}{323}\right)\) \(e\left(\frac{284}{323}\right)\) \(e\left(\frac{104}{323}\right)\) \(e\left(\frac{262}{323}\right)\) \(e\left(\frac{13}{323}\right)\) \(e\left(\frac{303}{323}\right)\) \(e\left(\frac{156}{323}\right)\) \(e\left(\frac{245}{323}\right)\) \(e\left(\frac{314}{323}\right)\) \(e\left(\frac{27}{323}\right)\)
\(\chi_{647}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{243}{323}\right)\) \(e\left(\frac{60}{323}\right)\) \(e\left(\frac{163}{323}\right)\) \(e\left(\frac{69}{323}\right)\) \(e\left(\frac{303}{323}\right)\) \(e\left(\frac{155}{323}\right)\) \(e\left(\frac{83}{323}\right)\) \(e\left(\frac{120}{323}\right)\) \(e\left(\frac{312}{323}\right)\) \(e\left(\frac{33}{323}\right)\)
\(\chi_{647}(8,\cdot)\) \(1\) \(1\) \(e\left(\frac{301}{323}\right)\) \(e\left(\frac{178}{323}\right)\) \(e\left(\frac{279}{323}\right)\) \(e\left(\frac{237}{323}\right)\) \(e\left(\frac{156}{323}\right)\) \(e\left(\frac{83}{323}\right)\) \(e\left(\frac{257}{323}\right)\) \(e\left(\frac{33}{323}\right)\) \(e\left(\frac{215}{323}\right)\) \(e\left(\frac{1}{323}\right)\)
\(\chi_{647}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{323}\right)\) \(e\left(\frac{234}{323}\right)\) \(e\left(\frac{22}{323}\right)\) \(e\left(\frac{43}{323}\right)\) \(e\left(\frac{245}{323}\right)\) \(e\left(\frac{120}{323}\right)\) \(e\left(\frac{33}{323}\right)\) \(e\left(\frac{145}{323}\right)\) \(e\left(\frac{54}{323}\right)\) \(e\left(\frac{161}{323}\right)\)
\(\chi_{647}(12,\cdot)\) \(1\) \(1\) \(e\left(\frac{260}{323}\right)\) \(e\left(\frac{128}{323}\right)\) \(e\left(\frac{197}{323}\right)\) \(e\left(\frac{18}{323}\right)\) \(e\left(\frac{65}{323}\right)\) \(e\left(\frac{223}{323}\right)\) \(e\left(\frac{134}{323}\right)\) \(e\left(\frac{256}{323}\right)\) \(e\left(\frac{278}{323}\right)\) \(e\left(\frac{135}{323}\right)\)
\(\chi_{647}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{124}{323}\right)\) \(e\left(\frac{230}{323}\right)\) \(e\left(\frac{248}{323}\right)\) \(e\left(\frac{103}{323}\right)\) \(e\left(\frac{31}{323}\right)\) \(e\left(\frac{2}{323}\right)\) \(e\left(\frac{49}{323}\right)\) \(e\left(\frac{137}{323}\right)\) \(e\left(\frac{227}{323}\right)\) \(e\left(\frac{288}{323}\right)\)
\(\chi_{647}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{128}{323}\right)\) \(e\left(\frac{227}{323}\right)\) \(e\left(\frac{256}{323}\right)\) \(e\left(\frac{148}{323}\right)\) \(e\left(\frac{32}{323}\right)\) \(e\left(\frac{75}{323}\right)\) \(e\left(\frac{61}{323}\right)\) \(e\left(\frac{131}{323}\right)\) \(e\left(\frac{276}{323}\right)\) \(e\left(\frac{141}{323}\right)\)
\(\chi_{647}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{186}{323}\right)\) \(e\left(\frac{22}{323}\right)\) \(e\left(\frac{49}{323}\right)\) \(e\left(\frac{316}{323}\right)\) \(e\left(\frac{208}{323}\right)\) \(e\left(\frac{3}{323}\right)\) \(e\left(\frac{235}{323}\right)\) \(e\left(\frac{44}{323}\right)\) \(e\left(\frac{179}{323}\right)\) \(e\left(\frac{109}{323}\right)\)
\(\chi_{647}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{218}{323}\right)\) \(e\left(\frac{321}{323}\right)\) \(e\left(\frac{113}{323}\right)\) \(e\left(\frac{30}{323}\right)\) \(e\left(\frac{216}{323}\right)\) \(e\left(\frac{264}{323}\right)\) \(e\left(\frac{8}{323}\right)\) \(e\left(\frac{319}{323}\right)\) \(e\left(\frac{248}{323}\right)\) \(e\left(\frac{225}{323}\right)\)
\(\chi_{647}(18,\cdot)\) \(1\) \(1\) \(e\left(\frac{219}{323}\right)\) \(e\left(\frac{78}{323}\right)\) \(e\left(\frac{115}{323}\right)\) \(e\left(\frac{122}{323}\right)\) \(e\left(\frac{297}{323}\right)\) \(e\left(\frac{40}{323}\right)\) \(e\left(\frac{11}{323}\right)\) \(e\left(\frac{156}{323}\right)\) \(e\left(\frac{18}{323}\right)\) \(e\left(\frac{269}{323}\right)\)
\(\chi_{647}(21,\cdot)\) \(1\) \(1\) \(e\left(\frac{87}{323}\right)\) \(e\left(\frac{177}{323}\right)\) \(e\left(\frac{174}{323}\right)\) \(e\left(\frac{252}{323}\right)\) \(e\left(\frac{264}{323}\right)\) \(e\left(\frac{215}{323}\right)\) \(e\left(\frac{261}{323}\right)\) \(e\left(\frac{31}{323}\right)\) \(e\left(\frac{16}{323}\right)\) \(e\left(\frac{275}{323}\right)\)
\(\chi_{647}(24,\cdot)\) \(1\) \(1\) \(e\left(\frac{145}{323}\right)\) \(e\left(\frac{295}{323}\right)\) \(e\left(\frac{290}{323}\right)\) \(e\left(\frac{97}{323}\right)\) \(e\left(\frac{117}{323}\right)\) \(e\left(\frac{143}{323}\right)\) \(e\left(\frac{112}{323}\right)\) \(e\left(\frac{267}{323}\right)\) \(e\left(\frac{242}{323}\right)\) \(e\left(\frac{243}{323}\right)\)
\(\chi_{647}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{158}{323}\right)\) \(e\left(\frac{43}{323}\right)\) \(e\left(\frac{316}{323}\right)\) \(e\left(\frac{1}{323}\right)\) \(e\left(\frac{201}{323}\right)\) \(e\left(\frac{138}{323}\right)\) \(e\left(\frac{151}{323}\right)\) \(e\left(\frac{86}{323}\right)\) \(e\left(\frac{159}{323}\right)\) \(e\left(\frac{169}{323}\right)\)
\(\chi_{647}(26,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{323}\right)\) \(e\left(\frac{74}{323}\right)\) \(e\left(\frac{18}{323}\right)\) \(e\left(\frac{182}{323}\right)\) \(e\left(\frac{83}{323}\right)\) \(e\left(\frac{245}{323}\right)\) \(e\left(\frac{27}{323}\right)\) \(e\left(\frac{148}{323}\right)\) \(e\left(\frac{191}{323}\right)\) \(e\left(\frac{73}{323}\right)\)
\(\chi_{647}(27,\cdot)\) \(1\) \(1\) \(e\left(\frac{178}{323}\right)\) \(e\left(\frac{28}{323}\right)\) \(e\left(\frac{33}{323}\right)\) \(e\left(\frac{226}{323}\right)\) \(e\left(\frac{206}{323}\right)\) \(e\left(\frac{180}{323}\right)\) \(e\left(\frac{211}{323}\right)\) \(e\left(\frac{56}{323}\right)\) \(e\left(\frac{81}{323}\right)\) \(e\left(\frac{80}{323}\right)\)
\(\chi_{647}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{323}\right)\) \(e\left(\frac{71}{323}\right)\) \(e\left(\frac{26}{323}\right)\) \(e\left(\frac{227}{323}\right)\) \(e\left(\frac{84}{323}\right)\) \(e\left(\frac{318}{323}\right)\) \(e\left(\frac{39}{323}\right)\) \(e\left(\frac{142}{323}\right)\) \(e\left(\frac{240}{323}\right)\) \(e\left(\frac{249}{323}\right)\)
\(\chi_{647}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{18}{323}\right)\) \(e\left(\frac{148}{323}\right)\) \(e\left(\frac{36}{323}\right)\) \(e\left(\frac{41}{323}\right)\) \(e\left(\frac{166}{323}\right)\) \(e\left(\frac{167}{323}\right)\) \(e\left(\frac{54}{323}\right)\) \(e\left(\frac{296}{323}\right)\) \(e\left(\frac{59}{323}\right)\) \(e\left(\frac{146}{323}\right)\)
\(\chi_{647}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{323}\right)\) \(e\left(\frac{157}{323}\right)\) \(e\left(\frac{12}{323}\right)\) \(e\left(\frac{229}{323}\right)\) \(e\left(\frac{163}{323}\right)\) \(e\left(\frac{271}{323}\right)\) \(e\left(\frac{18}{323}\right)\) \(e\left(\frac{314}{323}\right)\) \(e\left(\frac{235}{323}\right)\) \(e\left(\frac{264}{323}\right)\)
\(\chi_{647}(32,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{323}\right)\) \(e\left(\frac{189}{323}\right)\) \(e\left(\frac{142}{323}\right)\) \(e\left(\frac{72}{323}\right)\) \(e\left(\frac{260}{323}\right)\) \(e\left(\frac{246}{323}\right)\) \(e\left(\frac{213}{323}\right)\) \(e\left(\frac{55}{323}\right)\) \(e\left(\frac{143}{323}\right)\) \(e\left(\frac{217}{323}\right)\)
\(\chi_{647}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{103}{323}\right)\) \(e\left(\frac{165}{323}\right)\) \(e\left(\frac{206}{323}\right)\) \(e\left(\frac{109}{323}\right)\) \(e\left(\frac{268}{323}\right)\) \(e\left(\frac{184}{323}\right)\) \(e\left(\frac{309}{323}\right)\) \(e\left(\frac{7}{323}\right)\) \(e\left(\frac{212}{323}\right)\) \(e\left(\frac{10}{323}\right)\)
\(\chi_{647}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{104}{323}\right)\) \(e\left(\frac{245}{323}\right)\) \(e\left(\frac{208}{323}\right)\) \(e\left(\frac{201}{323}\right)\) \(e\left(\frac{26}{323}\right)\) \(e\left(\frac{283}{323}\right)\) \(e\left(\frac{312}{323}\right)\) \(e\left(\frac{167}{323}\right)\) \(e\left(\frac{305}{323}\right)\) \(e\left(\frac{54}{323}\right)\)
\(\chi_{647}(39,\cdot)\) \(1\) \(1\) \(e\left(\frac{291}{323}\right)\) \(e\left(\frac{24}{323}\right)\) \(e\left(\frac{259}{323}\right)\) \(e\left(\frac{286}{323}\right)\) \(e\left(\frac{315}{323}\right)\) \(e\left(\frac{62}{323}\right)\) \(e\left(\frac{227}{323}\right)\) \(e\left(\frac{48}{323}\right)\) \(e\left(\frac{254}{323}\right)\) \(e\left(\frac{207}{323}\right)\)
\(\chi_{647}(41,\cdot)\) \(1\) \(1\) \(e\left(\frac{148}{323}\right)\) \(e\left(\frac{212}{323}\right)\) \(e\left(\frac{296}{323}\right)\) \(e\left(\frac{50}{323}\right)\) \(e\left(\frac{37}{323}\right)\) \(e\left(\frac{117}{323}\right)\) \(e\left(\frac{121}{323}\right)\) \(e\left(\frac{101}{323}\right)\) \(e\left(\frac{198}{323}\right)\) \(e\left(\frac{52}{323}\right)\)
\(\chi_{647}(42,\cdot)\) \(1\) \(1\) \(e\left(\frac{295}{323}\right)\) \(e\left(\frac{21}{323}\right)\) \(e\left(\frac{267}{323}\right)\) \(e\left(\frac{8}{323}\right)\) \(e\left(\frac{316}{323}\right)\) \(e\left(\frac{135}{323}\right)\) \(e\left(\frac{239}{323}\right)\) \(e\left(\frac{42}{323}\right)\) \(e\left(\frac{303}{323}\right)\) \(e\left(\frac{60}{323}\right)\)
\(\chi_{647}(48,\cdot)\) \(1\) \(1\) \(e\left(\frac{30}{323}\right)\) \(e\left(\frac{139}{323}\right)\) \(e\left(\frac{60}{323}\right)\) \(e\left(\frac{176}{323}\right)\) \(e\left(\frac{169}{323}\right)\) \(e\left(\frac{63}{323}\right)\) \(e\left(\frac{90}{323}\right)\) \(e\left(\frac{278}{323}\right)\) \(e\left(\frac{206}{323}\right)\) \(e\left(\frac{28}{323}\right)\)
\(\chi_{647}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{163}{323}\right)\) \(e\left(\frac{120}{323}\right)\) \(e\left(\frac{3}{323}\right)\) \(e\left(\frac{138}{323}\right)\) \(e\left(\frac{283}{323}\right)\) \(e\left(\frac{310}{323}\right)\) \(e\left(\frac{166}{323}\right)\) \(e\left(\frac{240}{323}\right)\) \(e\left(\frac{301}{323}\right)\) \(e\left(\frac{66}{323}\right)\)
\(\chi_{647}(50,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{323}\right)\) \(e\left(\frac{210}{323}\right)\) \(e\left(\frac{86}{323}\right)\) \(e\left(\frac{80}{323}\right)\) \(e\left(\frac{253}{323}\right)\) \(e\left(\frac{58}{323}\right)\) \(e\left(\frac{129}{323}\right)\) \(e\left(\frac{97}{323}\right)\) \(e\left(\frac{123}{323}\right)\) \(e\left(\frac{277}{323}\right)\)
\(\chi_{647}(51,\cdot)\) \(1\) \(1\) \(e\left(\frac{62}{323}\right)\) \(e\left(\frac{115}{323}\right)\) \(e\left(\frac{124}{323}\right)\) \(e\left(\frac{213}{323}\right)\) \(e\left(\frac{177}{323}\right)\) \(e\left(\frac{1}{323}\right)\) \(e\left(\frac{186}{323}\right)\) \(e\left(\frac{230}{323}\right)\) \(e\left(\frac{275}{323}\right)\) \(e\left(\frac{144}{323}\right)\)