Properties

Label 6815.cw
Modulus $6815$
Conductor $6815$
Order $644$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6815, base_ring=CyclotomicField(644))
 
M = H._module
 
chi = DirichletCharacter(H, M([483,460,70]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(23,6815))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(6815\)
Conductor: \(6815\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(644\)
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: $\Q(\zeta_{644})$
Fixed field: Number field defined by a degree 644 polynomial (not computed)

First 31 of 264 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{6815}(23,\cdot)\) \(1\) \(1\) \(e\left(\frac{271}{644}\right)\) \(e\left(\frac{641}{644}\right)\) \(e\left(\frac{271}{322}\right)\) \(e\left(\frac{67}{161}\right)\) \(e\left(\frac{515}{644}\right)\) \(e\left(\frac{169}{644}\right)\) \(e\left(\frac{319}{322}\right)\) \(e\left(\frac{199}{322}\right)\) \(e\left(\frac{77}{92}\right)\) \(e\left(\frac{195}{644}\right)\)
\(\chi_{6815}(52,\cdot)\) \(1\) \(1\) \(e\left(\frac{229}{644}\right)\) \(e\left(\frac{487}{644}\right)\) \(e\left(\frac{229}{322}\right)\) \(e\left(\frac{18}{161}\right)\) \(e\left(\frac{333}{644}\right)\) \(e\left(\frac{43}{644}\right)\) \(e\left(\frac{165}{322}\right)\) \(e\left(\frac{3}{322}\right)\) \(e\left(\frac{43}{92}\right)\) \(e\left(\frac{545}{644}\right)\)
\(\chi_{6815}(78,\cdot)\) \(1\) \(1\) \(e\left(\frac{503}{644}\right)\) \(e\left(\frac{541}{644}\right)\) \(e\left(\frac{181}{322}\right)\) \(e\left(\frac{100}{161}\right)\) \(e\left(\frac{79}{644}\right)\) \(e\left(\frac{221}{644}\right)\) \(e\left(\frac{219}{322}\right)\) \(e\left(\frac{285}{322}\right)\) \(e\left(\frac{37}{92}\right)\) \(e\left(\frac{255}{644}\right)\)
\(\chi_{6815}(82,\cdot)\) \(1\) \(1\) \(e\left(\frac{289}{644}\right)\) \(e\left(\frac{339}{644}\right)\) \(e\left(\frac{289}{322}\right)\) \(e\left(\frac{157}{161}\right)\) \(e\left(\frac{409}{644}\right)\) \(e\left(\frac{223}{644}\right)\) \(e\left(\frac{17}{322}\right)\) \(e\left(\frac{53}{322}\right)\) \(e\left(\frac{39}{92}\right)\) \(e\left(\frac{505}{644}\right)\)
\(\chi_{6815}(107,\cdot)\) \(1\) \(1\) \(e\left(\frac{265}{644}\right)\) \(e\left(\frac{527}{644}\right)\) \(e\left(\frac{265}{322}\right)\) \(e\left(\frac{37}{161}\right)\) \(e\left(\frac{121}{644}\right)\) \(e\left(\frac{151}{644}\right)\) \(e\left(\frac{205}{322}\right)\) \(e\left(\frac{33}{322}\right)\) \(e\left(\frac{59}{92}\right)\) \(e\left(\frac{521}{644}\right)\)
\(\chi_{6815}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{563}{644}\right)\) \(e\left(\frac{393}{644}\right)\) \(e\left(\frac{241}{322}\right)\) \(e\left(\frac{78}{161}\right)\) \(e\left(\frac{155}{644}\right)\) \(e\left(\frac{401}{644}\right)\) \(e\left(\frac{71}{322}\right)\) \(e\left(\frac{13}{322}\right)\) \(e\left(\frac{33}{92}\right)\) \(e\left(\frac{215}{644}\right)\)
\(\chi_{6815}(132,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{644}\right)\) \(e\left(\frac{551}{644}\right)\) \(e\left(\frac{29}{322}\right)\) \(e\left(\frac{145}{161}\right)\) \(e\left(\frac{509}{644}\right)\) \(e\left(\frac{87}{644}\right)\) \(e\left(\frac{229}{322}\right)\) \(e\left(\frac{51}{322}\right)\) \(e\left(\frac{87}{92}\right)\) \(e\left(\frac{249}{644}\right)\)
\(\chi_{6815}(152,\cdot)\) \(1\) \(1\) \(e\left(\frac{269}{644}\right)\) \(e\left(\frac{603}{644}\right)\) \(e\left(\frac{269}{322}\right)\) \(e\left(\frac{57}{161}\right)\) \(e\left(\frac{169}{644}\right)\) \(e\left(\frac{163}{644}\right)\) \(e\left(\frac{281}{322}\right)\) \(e\left(\frac{251}{322}\right)\) \(e\left(\frac{71}{92}\right)\) \(e\left(\frac{89}{644}\right)\)
\(\chi_{6815}(198,\cdot)\) \(1\) \(1\) \(e\left(\frac{303}{644}\right)\) \(e\left(\frac{605}{644}\right)\) \(e\left(\frac{303}{322}\right)\) \(e\left(\frac{66}{161}\right)\) \(e\left(\frac{255}{644}\right)\) \(e\left(\frac{265}{644}\right)\) \(e\left(\frac{283}{322}\right)\) \(e\left(\frac{11}{322}\right)\) \(e\left(\frac{81}{92}\right)\) \(e\left(\frac{603}{644}\right)\)
\(\chi_{6815}(223,\cdot)\) \(1\) \(1\) \(e\left(\frac{335}{644}\right)\) \(e\left(\frac{569}{644}\right)\) \(e\left(\frac{13}{322}\right)\) \(e\left(\frac{65}{161}\right)\) \(e\left(\frac{639}{644}\right)\) \(e\left(\frac{361}{644}\right)\) \(e\left(\frac{247}{322}\right)\) \(e\left(\frac{145}{322}\right)\) \(e\left(\frac{85}{92}\right)\) \(e\left(\frac{367}{644}\right)\)
\(\chi_{6815}(227,\cdot)\) \(1\) \(1\) \(e\left(\frac{429}{644}\right)\) \(e\left(\frac{423}{644}\right)\) \(e\left(\frac{107}{322}\right)\) \(e\left(\frac{52}{161}\right)\) \(e\left(\frac{157}{644}\right)\) \(e\left(\frac{643}{644}\right)\) \(e\left(\frac{101}{322}\right)\) \(e\left(\frac{277}{322}\right)\) \(e\left(\frac{91}{92}\right)\) \(e\left(\frac{197}{644}\right)\)
\(\chi_{6815}(228,\cdot)\) \(1\) \(1\) \(e\left(\frac{543}{644}\right)\) \(e\left(\frac{13}{644}\right)\) \(e\left(\frac{221}{322}\right)\) \(e\left(\frac{139}{161}\right)\) \(e\left(\frac{559}{644}\right)\) \(e\left(\frac{341}{644}\right)\) \(e\left(\frac{13}{322}\right)\) \(e\left(\frac{211}{322}\right)\) \(e\left(\frac{65}{92}\right)\) \(e\left(\frac{443}{644}\right)\)
\(\chi_{6815}(248,\cdot)\) \(1\) \(1\) \(e\left(\frac{127}{644}\right)\) \(e\left(\frac{481}{644}\right)\) \(e\left(\frac{127}{322}\right)\) \(e\left(\frac{152}{161}\right)\) \(e\left(\frac{75}{644}\right)\) \(e\left(\frac{381}{644}\right)\) \(e\left(\frac{159}{322}\right)\) \(e\left(\frac{79}{322}\right)\) \(e\left(\frac{13}{92}\right)\) \(e\left(\frac{291}{644}\right)\)
\(\chi_{6815}(257,\cdot)\) \(1\) \(1\) \(e\left(\frac{389}{644}\right)\) \(e\left(\frac{307}{644}\right)\) \(e\left(\frac{67}{322}\right)\) \(e\left(\frac{13}{161}\right)\) \(e\left(\frac{321}{644}\right)\) \(e\left(\frac{523}{644}\right)\) \(e\left(\frac{307}{322}\right)\) \(e\left(\frac{29}{322}\right)\) \(e\left(\frac{63}{92}\right)\) \(e\left(\frac{9}{644}\right)\)
\(\chi_{6815}(268,\cdot)\) \(1\) \(1\) \(e\left(\frac{479}{644}\right)\) \(e\left(\frac{85}{644}\right)\) \(e\left(\frac{157}{322}\right)\) \(e\left(\frac{141}{161}\right)\) \(e\left(\frac{435}{644}\right)\) \(e\left(\frac{149}{644}\right)\) \(e\left(\frac{85}{322}\right)\) \(e\left(\frac{265}{322}\right)\) \(e\left(\frac{57}{92}\right)\) \(e\left(\frac{271}{644}\right)\)
\(\chi_{6815}(297,\cdot)\) \(1\) \(1\) \(e\left(\frac{577}{644}\right)\) \(e\left(\frac{15}{644}\right)\) \(e\left(\frac{255}{322}\right)\) \(e\left(\frac{148}{161}\right)\) \(e\left(\frac{1}{644}\right)\) \(e\left(\frac{443}{644}\right)\) \(e\left(\frac{15}{322}\right)\) \(e\left(\frac{293}{322}\right)\) \(e\left(\frac{75}{92}\right)\) \(e\left(\frac{313}{644}\right)\)
\(\chi_{6815}(313,\cdot)\) \(1\) \(1\) \(e\left(\frac{411}{644}\right)\) \(e\left(\frac{81}{644}\right)\) \(e\left(\frac{89}{322}\right)\) \(e\left(\frac{123}{161}\right)\) \(e\left(\frac{263}{644}\right)\) \(e\left(\frac{589}{644}\right)\) \(e\left(\frac{81}{322}\right)\) \(e\left(\frac{101}{322}\right)\) \(e\left(\frac{37}{92}\right)\) \(e\left(\frac{531}{644}\right)\)
\(\chi_{6815}(342,\cdot)\) \(1\) \(1\) \(e\left(\frac{173}{644}\right)\) \(e\left(\frac{67}{644}\right)\) \(e\left(\frac{173}{322}\right)\) \(e\left(\frac{60}{161}\right)\) \(e\left(\frac{305}{644}\right)\) \(e\left(\frac{519}{644}\right)\) \(e\left(\frac{67}{322}\right)\) \(e\left(\frac{171}{322}\right)\) \(e\left(\frac{59}{92}\right)\) \(e\left(\frac{153}{644}\right)\)
\(\chi_{6815}(368,\cdot)\) \(1\) \(1\) \(e\left(\frac{475}{644}\right)\) \(e\left(\frac{9}{644}\right)\) \(e\left(\frac{153}{322}\right)\) \(e\left(\frac{121}{161}\right)\) \(e\left(\frac{387}{644}\right)\) \(e\left(\frac{137}{644}\right)\) \(e\left(\frac{9}{322}\right)\) \(e\left(\frac{47}{322}\right)\) \(e\left(\frac{45}{92}\right)\) \(e\left(\frac{59}{644}\right)\)
\(\chi_{6815}(372,\cdot)\) \(1\) \(1\) \(e\left(\frac{401}{644}\right)\) \(e\left(\frac{535}{644}\right)\) \(e\left(\frac{79}{322}\right)\) \(e\left(\frac{73}{161}\right)\) \(e\left(\frac{465}{644}\right)\) \(e\left(\frac{559}{644}\right)\) \(e\left(\frac{213}{322}\right)\) \(e\left(\frac{39}{322}\right)\) \(e\left(\frac{7}{92}\right)\) \(e\left(\frac{1}{644}\right)\)
\(\chi_{6815}(373,\cdot)\) \(1\) \(1\) \(e\left(\frac{95}{644}\right)\) \(e\left(\frac{517}{644}\right)\) \(e\left(\frac{95}{322}\right)\) \(e\left(\frac{153}{161}\right)\) \(e\left(\frac{335}{644}\right)\) \(e\left(\frac{285}{644}\right)\) \(e\left(\frac{195}{322}\right)\) \(e\left(\frac{267}{322}\right)\) \(e\left(\frac{9}{92}\right)\) \(e\left(\frac{527}{644}\right)\)
\(\chi_{6815}(402,\cdot)\) \(1\) \(1\) \(e\left(\frac{109}{644}\right)\) \(e\left(\frac{139}{644}\right)\) \(e\left(\frac{109}{322}\right)\) \(e\left(\frac{62}{161}\right)\) \(e\left(\frac{181}{644}\right)\) \(e\left(\frac{327}{644}\right)\) \(e\left(\frac{139}{322}\right)\) \(e\left(\frac{225}{322}\right)\) \(e\left(\frac{51}{92}\right)\) \(e\left(\frac{625}{644}\right)\)
\(\chi_{6815}(442,\cdot)\) \(1\) \(1\) \(e\left(\frac{185}{644}\right)\) \(e\left(\frac{295}{644}\right)\) \(e\left(\frac{185}{322}\right)\) \(e\left(\frac{120}{161}\right)\) \(e\left(\frac{449}{644}\right)\) \(e\left(\frac{555}{644}\right)\) \(e\left(\frac{295}{322}\right)\) \(e\left(\frac{181}{322}\right)\) \(e\left(\frac{3}{92}\right)\) \(e\left(\frac{145}{644}\right)\)
\(\chi_{6815}(458,\cdot)\) \(1\) \(1\) \(e\left(\frac{243}{644}\right)\) \(e\left(\frac{109}{644}\right)\) \(e\left(\frac{243}{322}\right)\) \(e\left(\frac{88}{161}\right)\) \(e\left(\frac{179}{644}\right)\) \(e\left(\frac{85}{644}\right)\) \(e\left(\frac{109}{322}\right)\) \(e\left(\frac{283}{322}\right)\) \(e\left(\frac{85}{92}\right)\) \(e\left(\frac{643}{644}\right)\)
\(\chi_{6815}(513,\cdot)\) \(1\) \(1\) \(e\left(\frac{447}{644}\right)\) \(e\left(\frac{121}{644}\right)\) \(e\left(\frac{125}{322}\right)\) \(e\left(\frac{142}{161}\right)\) \(e\left(\frac{51}{644}\right)\) \(e\left(\frac{53}{644}\right)\) \(e\left(\frac{121}{322}\right)\) \(e\left(\frac{131}{322}\right)\) \(e\left(\frac{53}{92}\right)\) \(e\left(\frac{507}{644}\right)\)
\(\chi_{6815}(547,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{644}\right)\) \(e\left(\frac{363}{644}\right)\) \(e\left(\frac{53}{322}\right)\) \(e\left(\frac{104}{161}\right)\) \(e\left(\frac{153}{644}\right)\) \(e\left(\frac{159}{644}\right)\) \(e\left(\frac{41}{322}\right)\) \(e\left(\frac{71}{322}\right)\) \(e\left(\frac{67}{92}\right)\) \(e\left(\frac{233}{644}\right)\)
\(\chi_{6815}(558,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{644}\right)\) \(e\left(\frac{589}{644}\right)\) \(e\left(\frac{31}{322}\right)\) \(e\left(\frac{155}{161}\right)\) \(e\left(\frac{211}{644}\right)\) \(e\left(\frac{93}{644}\right)\) \(e\left(\frac{267}{322}\right)\) \(e\left(\frac{321}{322}\right)\) \(e\left(\frac{1}{92}\right)\) \(e\left(\frac{355}{644}\right)\)
\(\chi_{6815}(587,\cdot)\) \(1\) \(1\) \(e\left(\frac{409}{644}\right)\) \(e\left(\frac{43}{644}\right)\) \(e\left(\frac{87}{322}\right)\) \(e\left(\frac{113}{161}\right)\) \(e\left(\frac{561}{644}\right)\) \(e\left(\frac{583}{644}\right)\) \(e\left(\frac{43}{322}\right)\) \(e\left(\frac{153}{322}\right)\) \(e\left(\frac{31}{92}\right)\) \(e\left(\frac{425}{644}\right)\)
\(\chi_{6815}(603,\cdot)\) \(1\) \(1\) \(e\left(\frac{383}{644}\right)\) \(e\left(\frac{193}{644}\right)\) \(e\left(\frac{61}{322}\right)\) \(e\left(\frac{144}{161}\right)\) \(e\left(\frac{571}{644}\right)\) \(e\left(\frac{505}{644}\right)\) \(e\left(\frac{193}{322}\right)\) \(e\left(\frac{185}{322}\right)\) \(e\left(\frac{45}{92}\right)\) \(e\left(\frac{335}{644}\right)\)
\(\chi_{6815}(633,\cdot)\) \(1\) \(1\) \(e\left(\frac{527}{644}\right)\) \(e\left(\frac{353}{644}\right)\) \(e\left(\frac{205}{322}\right)\) \(e\left(\frac{59}{161}\right)\) \(e\left(\frac{367}{644}\right)\) \(e\left(\frac{293}{644}\right)\) \(e\left(\frac{31}{322}\right)\) \(e\left(\frac{305}{322}\right)\) \(e\left(\frac{17}{92}\right)\) \(e\left(\frac{239}{644}\right)\)
\(\chi_{6815}(663,\cdot)\) \(1\) \(1\) \(e\left(\frac{459}{644}\right)\) \(e\left(\frac{349}{644}\right)\) \(e\left(\frac{137}{322}\right)\) \(e\left(\frac{41}{161}\right)\) \(e\left(\frac{195}{644}\right)\) \(e\left(\frac{89}{644}\right)\) \(e\left(\frac{27}{322}\right)\) \(e\left(\frac{141}{322}\right)\) \(e\left(\frac{89}{92}\right)\) \(e\left(\frac{499}{644}\right)\)