LMFDB - Dirichlet character orbit 13688.do

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13688, base_ring=CyclotomicField(812)) M = H._module chi = DirichletCharacter(H, M([406,406,725,350])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(11, 13688)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13688.11"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$13688$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$13688$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$812$	comment:Order sage:chi.multiplicative_order() gp:charorder(g,chi) magma:Order(chi);
Real:	no	comment:Whether the character is real sage:chi.multiplicative_order() <= 2 gp:charorder(g,chi) <= 2 magma:Order(chi) le 2;
Primitive:	yes	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{812})$	comment:Field of values of chi sage:CyclotomicField(chi.multiplicative_order()) gp:nfinit(polcyclo(charorder(g,chi))) magma:CyclotomicField(Order(chi));
Fixed field:	Number field defined by a degree 812 polynomial (not computed)	comment:Fixed field sage:chi.fixed_field()

First 31 of 336 characters in Galois orbit

Character	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{13688}(11,\cdot)$	$-1$	$1$	$e\left(\frac{13}{812}\right)$	$e\left(\frac{148}{203}\right)$	$e\left(\frac{395}{406}\right)$	$e\left(\frac{13}{406}\right)$	$e\left(\frac{79}{812}\right)$	$e\left(\frac{393}{406}\right)$	$e\left(\frac{605}{812}\right)$	$e\left(\frac{115}{116}\right)$	$e\left(\frac{337}{812}\right)$	$e\left(\frac{803}{812}\right)$
$\chi_{13688}(43,\cdot)$	$-1$	$1$	$e\left(\frac{625}{812}\right)$	$e\left(\frac{26}{203}\right)$	$e\left(\frac{127}{406}\right)$	$e\left(\frac{219}{406}\right)$	$e\left(\frac{675}{812}\right)$	$e\left(\frac{187}{406}\right)$	$e\left(\frac{729}{812}\right)$	$e\left(\frac{59}{116}\right)$	$e\left(\frac{649}{812}\right)$	$e\left(\frac{67}{812}\right)$
$\chi_{13688}(131,\cdot)$	$-1$	$1$	$e\left(\frac{499}{812}\right)$	$e\left(\frac{75}{203}\right)$	$e\left(\frac{15}{406}\right)$	$e\left(\frac{93}{406}\right)$	$e\left(\frac{409}{812}\right)$	$e\left(\frac{313}{406}\right)$	$e\left(\frac{799}{812}\right)$	$e\left(\frac{33}{116}\right)$	$e\left(\frac{131}{812}\right)$	$e\left(\frac{529}{812}\right)$
$\chi_{13688}(155,\cdot)$	$-1$	$1$	$e\left(\frac{423}{812}\right)$	$e\left(\frac{53}{203}\right)$	$e\left(\frac{173}{406}\right)$	$e\left(\frac{17}{406}\right)$	$e\left(\frac{197}{812}\right)$	$e\left(\frac{389}{406}\right)$	$e\left(\frac{635}{812}\right)$	$e\left(\frac{21}{116}\right)$	$e\left(\frac{347}{812}\right)$	$e\left(\frac{769}{812}\right)$
$\chi_{13688}(195,\cdot)$	$-1$	$1$	$e\left(\frac{85}{812}\right)$	$e\left(\frac{62}{203}\right)$	$e\left(\frac{53}{406}\right)$	$e\left(\frac{85}{406}\right)$	$e\left(\frac{579}{812}\right)$	$e\left(\frac{321}{406}\right)$	$e\left(\frac{333}{812}\right)$	$e\left(\frac{47}{116}\right)$	$e\left(\frac{517}{812}\right)$	$e\left(\frac{191}{812}\right)$
$\chi_{13688}(211,\cdot)$	$-1$	$1$	$e\left(\frac{715}{812}\right)$	$e\left(\frac{20}{203}\right)$	$e\left(\frac{207}{406}\right)$	$e\left(\frac{309}{406}\right)$	$e\left(\frac{285}{812}\right)$	$e\left(\frac{97}{406}\right)$	$e\left(\frac{795}{812}\right)$	$e\left(\frac{61}{116}\right)$	$e\left(\frac{671}{812}\right)$	$e\left(\frac{317}{812}\right)$
$\chi_{13688}(259,\cdot)$	$-1$	$1$	$e\left(\frac{495}{812}\right)$	$e\left(\frac{170}{203}\right)$	$e\left(\frac{237}{406}\right)$	$e\left(\frac{89}{406}\right)$	$e\left(\frac{697}{812}\right)$	$e\left(\frac{317}{406}\right)$	$e\left(\frac{363}{812}\right)$	$e\left(\frac{69}{116}\right)$	$e\left(\frac{527}{812}\right)$	$e\left(\frac{157}{812}\right)$
$\chi_{13688}(275,\cdot)$	$-1$	$1$	$e\left(\frac{177}{812}\right)$	$e\left(\frac{110}{203}\right)$	$e\left(\frac{225}{406}\right)$	$e\left(\frac{177}{406}\right)$	$e\left(\frac{451}{812}\right)$	$e\left(\frac{229}{406}\right)$	$e\left(\frac{617}{812}\right)$	$e\left(\frac{31}{116}\right)$	$e\left(\frac{341}{812}\right)$	$e\left(\frac{627}{812}\right)$
$\chi_{13688}(387,\cdot)$	$-1$	$1$	$e\left(\frac{619}{812}\right)$	$e\left(\frac{67}{203}\right)$	$e\left(\frac{257}{406}\right)$	$e\left(\frac{213}{406}\right)$	$e\left(\frac{701}{812}\right)$	$e\left(\frac{193}{406}\right)$	$e\left(\frac{75}{812}\right)$	$e\left(\frac{113}{116}\right)$	$e\left(\frac{431}{812}\right)$	$e\left(\frac{321}{812}\right)$
$\chi_{13688}(427,\cdot)$	$-1$	$1$	$e\left(\frac{337}{812}\right)$	$e\left(\frac{167}{203}\right)$	$e\left(\frac{277}{406}\right)$	$e\left(\frac{337}{406}\right)$	$e\left(\frac{299}{812}\right)$	$e\left(\frac{69}{406}\right)$	$e\left(\frac{193}{812}\right)$	$e\left(\frac{99}{116}\right)$	$e\left(\frac{741}{812}\right)$	$e\left(\frac{79}{812}\right)$
$\chi_{13688}(443,\cdot)$	$-1$	$1$	$e\left(\frac{547}{812}\right)$	$e\left(\frac{153}{203}\right)$	$e\left(\frac{193}{406}\right)$	$e\left(\frac{141}{406}\right)$	$e\left(\frac{201}{812}\right)$	$e\left(\frac{265}{406}\right)$	$e\left(\frac{347}{812}\right)$	$e\left(\frac{65}{116}\right)$	$e\left(\frac{251}{812}\right)$	$e\left(\frac{121}{812}\right)$
$\chi_{13688}(467,\cdot)$	$-1$	$1$	$e\left(\frac{53}{812}\right)$	$e\left(\frac{10}{203}\right)$	$e\left(\frac{205}{406}\right)$	$e\left(\frac{53}{406}\right)$	$e\left(\frac{447}{812}\right)$	$e\left(\frac{353}{406}\right)$	$e\left(\frac{93}{812}\right)$	$e\left(\frac{103}{116}\right)$	$e\left(\frac{437}{812}\right)$	$e\left(\frac{463}{812}\right)$
$\chi_{13688}(483,\cdot)$	$-1$	$1$	$e\left(\frac{129}{812}\right)$	$e\left(\frac{32}{203}\right)$	$e\left(\frac{47}{406}\right)$	$e\left(\frac{129}{406}\right)$	$e\left(\frac{659}{812}\right)$	$e\left(\frac{277}{406}\right)$	$e\left(\frac{257}{812}\right)$	$e\left(\frac{115}{116}\right)$	$e\left(\frac{221}{812}\right)$	$e\left(\frac{223}{812}\right)$
$\chi_{13688}(627,\cdot)$	$-1$	$1$	$e\left(\frac{307}{812}\right)$	$e\left(\frac{169}{203}\right)$	$e\left(\frac{115}{406}\right)$	$e\left(\frac{307}{406}\right)$	$e\left(\frac{429}{812}\right)$	$e\left(\frac{99}{406}\right)$	$e\left(\frac{171}{812}\right)$	$e\left(\frac{21}{116}\right)$	$e\left(\frac{463}{812}\right)$	$e\left(\frac{537}{812}\right)$
$\chi_{13688}(659,\cdot)$	$-1$	$1$	$e\left(\frac{57}{812}\right)$	$e\left(\frac{118}{203}\right)$	$e\left(\frac{389}{406}\right)$	$e\left(\frac{57}{406}\right)$	$e\left(\frac{159}{812}\right)$	$e\left(\frac{349}{406}\right)$	$e\left(\frac{529}{812}\right)$	$e\left(\frac{67}{116}\right)$	$e\left(\frac{41}{812}\right)$	$e\left(\frac{23}{812}\right)$
$\chi_{13688}(699,\cdot)$	$-1$	$1$	$e\left(\frac{81}{812}\right)$	$e\left(\frac{157}{203}\right)$	$e\left(\frac{275}{406}\right)$	$e\left(\frac{81}{406}\right)$	$e\left(\frac{55}{812}\right)$	$e\left(\frac{325}{406}\right)$	$e\left(\frac{709}{812}\right)$	$e\left(\frac{83}{116}\right)$	$e\left(\frac{101}{812}\right)$	$e\left(\frac{631}{812}\right)$
$\chi_{13688}(739,\cdot)$	$-1$	$1$	$e\left(\frac{457}{812}\right)$	$e\left(\frac{159}{203}\right)$	$e\left(\frac{113}{406}\right)$	$e\left(\frac{51}{406}\right)$	$e\left(\frac{591}{812}\right)$	$e\left(\frac{355}{406}\right)$	$e\left(\frac{281}{812}\right)$	$e\left(\frac{63}{116}\right)$	$e\left(\frac{229}{812}\right)$	$e\left(\frac{683}{812}\right)$
$\chi_{13688}(859,\cdot)$	$-1$	$1$	$e\left(\frac{503}{812}\right)$	$e\left(\frac{183}{203}\right)$	$e\left(\frac{199}{406}\right)$	$e\left(\frac{97}{406}\right)$	$e\left(\frac{121}{812}\right)$	$e\left(\frac{309}{406}\right)$	$e\left(\frac{423}{812}\right)$	$e\left(\frac{113}{116}\right)$	$e\left(\frac{547}{812}\right)$	$e\left(\frac{89}{812}\right)$
$\chi_{13688}(891,\cdot)$	$-1$	$1$	$e\left(\frac{1}{812}\right)$	$e\left(\frac{27}{203}\right)$	$e\left(\frac{249}{406}\right)$	$e\left(\frac{1}{406}\right)$	$e\left(\frac{131}{812}\right)$	$e\left(\frac{405}{406}\right)$	$e\left(\frac{109}{812}\right)$	$e\left(\frac{107}{116}\right)$	$e\left(\frac{713}{812}\right)$	$e\left(\frac{499}{812}\right)$
$\chi_{13688}(939,\cdot)$	$-1$	$1$	$e\left(\frac{517}{812}\right)$	$e\left(\frac{155}{203}\right)$	$e\left(\frac{31}{406}\right)$	$e\left(\frac{111}{406}\right)$	$e\left(\frac{331}{812}\right)$	$e\left(\frac{295}{406}\right)$	$e\left(\frac{325}{812}\right)$	$e\left(\frac{103}{116}\right)$	$e\left(\frac{785}{812}\right)$	$e\left(\frac{579}{812}\right)$
$\chi_{13688}(955,\cdot)$	$-1$	$1$	$e\left(\frac{187}{812}\right)$	$e\left(\frac{177}{203}\right)$	$e\left(\frac{279}{406}\right)$	$e\left(\frac{187}{406}\right)$	$e\left(\frac{137}{812}\right)$	$e\left(\frac{219}{406}\right)$	$e\left(\frac{83}{812}\right)$	$e\left(\frac{57}{116}\right)$	$e\left(\frac{163}{812}\right)$	$e\left(\frac{745}{812}\right)$
$\chi_{13688}(1059,\cdot)$	$-1$	$1$	$e\left(\frac{751}{812}\right)$	$e\left(\frac{180}{203}\right)$	$e\left(\frac{239}{406}\right)$	$e\left(\frac{345}{406}\right)$	$e\left(\frac{129}{812}\right)$	$e\left(\frac{61}{406}\right)$	$e\left(\frac{659}{812}\right)$	$e\left(\frac{85}{116}\right)$	$e\left(\frac{355}{812}\right)$	$e\left(\frac{417}{812}\right)$
$\chi_{13688}(1075,\cdot)$	$-1$	$1$	$e\left(\frac{789}{812}\right)$	$e\left(\frac{191}{203}\right)$	$e\left(\frac{363}{406}\right)$	$e\left(\frac{383}{406}\right)$	$e\left(\frac{235}{812}\right)$	$e\left(\frac{23}{406}\right)$	$e\left(\frac{741}{812}\right)$	$e\left(\frac{91}{116}\right)$	$e\left(\frac{653}{812}\right)$	$e\left(\frac{703}{812}\right)$
$\chi_{13688}(1099,\cdot)$	$-1$	$1$	$e\left(\frac{655}{812}\right)$	$e\left(\frac{24}{203}\right)$	$e\left(\frac{289}{406}\right)$	$e\left(\frac{249}{406}\right)$	$e\left(\frac{545}{812}\right)$	$e\left(\frac{157}{406}\right)$	$e\left(\frac{751}{812}\right)$	$e\left(\frac{21}{116}\right)$	$e\left(\frac{115}{812}\right)$	$e\left(\frac{421}{812}\right)$
$\chi_{13688}(1123,\cdot)$	$-1$	$1$	$e\left(\frac{729}{812}\right)$	$e\left(\frac{195}{203}\right)$	$e\left(\frac{39}{406}\right)$	$e\left(\frac{323}{406}\right)$	$e\left(\frac{495}{812}\right)$	$e\left(\frac{83}{406}\right)$	$e\left(\frac{697}{812}\right)$	$e\left(\frac{51}{116}\right)$	$e\left(\frac{97}{812}\right)$	$e\left(\frac{807}{812}\right)$
$\chi_{13688}(1139,\cdot)$	$-1$	$1$	$e\left(\frac{491}{812}\right)$	$e\left(\frac{62}{203}\right)$	$e\left(\frac{53}{406}\right)$	$e\left(\frac{85}{406}\right)$	$e\left(\frac{173}{812}\right)$	$e\left(\frac{321}{406}\right)$	$e\left(\frac{739}{812}\right)$	$e\left(\frac{105}{116}\right)$	$e\left(\frac{111}{812}\right)$	$e\left(\frac{597}{812}\right)$
$\chi_{13688}(1163,\cdot)$	$-1$	$1$	$e\left(\frac{305}{812}\right)$	$e\left(\frac{115}{203}\right)$	$e\left(\frac{23}{406}\right)$	$e\left(\frac{305}{406}\right)$	$e\left(\frac{167}{812}\right)$	$e\left(\frac{101}{406}\right)$	$e\left(\frac{765}{812}\right)$	$e\left(\frac{39}{116}\right)$	$e\left(\frac{661}{812}\right)$	$e\left(\frac{351}{812}\right)$
$\chi_{13688}(1171,\cdot)$	$-1$	$1$	$e\left(\frac{545}{812}\right)$	$e\left(\frac{99}{203}\right)$	$e\left(\frac{101}{406}\right)$	$e\left(\frac{139}{406}\right)$	$e\left(\frac{751}{812}\right)$	$e\left(\frac{267}{406}\right)$	$e\left(\frac{129}{812}\right)$	$e\left(\frac{83}{116}\right)$	$e\left(\frac{449}{812}\right)$	$e\left(\frac{747}{812}\right)$
$\chi_{13688}(1203,\cdot)$	$-1$	$1$	$e\left(\frac{205}{812}\right)$	$e\left(\frac{54}{203}\right)$	$e\left(\frac{295}{406}\right)$	$e\left(\frac{205}{406}\right)$	$e\left(\frac{59}{812}\right)$	$e\left(\frac{201}{406}\right)$	$e\left(\frac{421}{812}\right)$	$e\left(\frac{11}{116}\right)$	$e\left(\frac{5}{812}\right)$	$e\left(\frac{795}{812}\right)$
$\chi_{13688}(1291,\cdot)$	$-1$	$1$	$e\left(\frac{275}{812}\right)$	$e\left(\frac{117}{203}\right)$	$e\left(\frac{267}{406}\right)$	$e\left(\frac{275}{406}\right)$	$e\left(\frac{297}{812}\right)$	$e\left(\frac{131}{406}\right)$	$e\left(\frac{743}{812}\right)$	$e\left(\frac{77}{116}\right)$	$e\left(\frac{383}{812}\right)$	$e\left(\frac{809}{812}\right)$
$\chi_{13688}(1331,\cdot)$	$-1$	$1$	$e\left(\frac{39}{812}\right)$	$e\left(\frac{38}{203}\right)$	$e\left(\frac{373}{406}\right)$	$e\left(\frac{39}{406}\right)$	$e\left(\frac{237}{812}\right)$	$e\left(\frac{367}{406}\right)$	$e\left(\frac{191}{812}\right)$	$e\left(\frac{113}{116}\right)$	$e\left(\frac{199}{812}\right)$	$e\left(\frac{785}{812}\right)$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Character	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(17\)	\(19\)	\(21\)
\(\chi_{13688}(11,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{13}{812}\right)\)	\(e\left(\frac{148}{203}\right)\)	\(e\left(\frac{395}{406}\right)\)	\(e\left(\frac{13}{406}\right)\)	\(e\left(\frac{79}{812}\right)\)	\(e\left(\frac{393}{406}\right)\)	\(e\left(\frac{605}{812}\right)\)	\(e\left(\frac{115}{116}\right)\)	\(e\left(\frac{337}{812}\right)\)	\(e\left(\frac{803}{812}\right)\)
\(\chi_{13688}(43,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{625}{812}\right)\)	\(e\left(\frac{26}{203}\right)\)	\(e\left(\frac{127}{406}\right)\)	\(e\left(\frac{219}{406}\right)\)	\(e\left(\frac{675}{812}\right)\)	\(e\left(\frac{187}{406}\right)\)	\(e\left(\frac{729}{812}\right)\)	\(e\left(\frac{59}{116}\right)\)	\(e\left(\frac{649}{812}\right)\)	\(e\left(\frac{67}{812}\right)\)
\(\chi_{13688}(131,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{499}{812}\right)\)	\(e\left(\frac{75}{203}\right)\)	\(e\left(\frac{15}{406}\right)\)	\(e\left(\frac{93}{406}\right)\)	\(e\left(\frac{409}{812}\right)\)	\(e\left(\frac{313}{406}\right)\)	\(e\left(\frac{799}{812}\right)\)	\(e\left(\frac{33}{116}\right)\)	\(e\left(\frac{131}{812}\right)\)	\(e\left(\frac{529}{812}\right)\)
\(\chi_{13688}(155,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{423}{812}\right)\)	\(e\left(\frac{53}{203}\right)\)	\(e\left(\frac{173}{406}\right)\)	\(e\left(\frac{17}{406}\right)\)	\(e\left(\frac{197}{812}\right)\)	\(e\left(\frac{389}{406}\right)\)	\(e\left(\frac{635}{812}\right)\)	\(e\left(\frac{21}{116}\right)\)	\(e\left(\frac{347}{812}\right)\)	\(e\left(\frac{769}{812}\right)\)
\(\chi_{13688}(195,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{85}{812}\right)\)	\(e\left(\frac{62}{203}\right)\)	\(e\left(\frac{53}{406}\right)\)	\(e\left(\frac{85}{406}\right)\)	\(e\left(\frac{579}{812}\right)\)	\(e\left(\frac{321}{406}\right)\)	\(e\left(\frac{333}{812}\right)\)	\(e\left(\frac{47}{116}\right)\)	\(e\left(\frac{517}{812}\right)\)	\(e\left(\frac{191}{812}\right)\)
\(\chi_{13688}(211,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{715}{812}\right)\)	\(e\left(\frac{20}{203}\right)\)	\(e\left(\frac{207}{406}\right)\)	\(e\left(\frac{309}{406}\right)\)	\(e\left(\frac{285}{812}\right)\)	\(e\left(\frac{97}{406}\right)\)	\(e\left(\frac{795}{812}\right)\)	\(e\left(\frac{61}{116}\right)\)	\(e\left(\frac{671}{812}\right)\)	\(e\left(\frac{317}{812}\right)\)
\(\chi_{13688}(259,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{495}{812}\right)\)	\(e\left(\frac{170}{203}\right)\)	\(e\left(\frac{237}{406}\right)\)	\(e\left(\frac{89}{406}\right)\)	\(e\left(\frac{697}{812}\right)\)	\(e\left(\frac{317}{406}\right)\)	\(e\left(\frac{363}{812}\right)\)	\(e\left(\frac{69}{116}\right)\)	\(e\left(\frac{527}{812}\right)\)	\(e\left(\frac{157}{812}\right)\)
\(\chi_{13688}(275,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{177}{812}\right)\)	\(e\left(\frac{110}{203}\right)\)	\(e\left(\frac{225}{406}\right)\)	\(e\left(\frac{177}{406}\right)\)	\(e\left(\frac{451}{812}\right)\)	\(e\left(\frac{229}{406}\right)\)	\(e\left(\frac{617}{812}\right)\)	\(e\left(\frac{31}{116}\right)\)	\(e\left(\frac{341}{812}\right)\)	\(e\left(\frac{627}{812}\right)\)
\(\chi_{13688}(387,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{619}{812}\right)\)	\(e\left(\frac{67}{203}\right)\)	\(e\left(\frac{257}{406}\right)\)	\(e\left(\frac{213}{406}\right)\)	\(e\left(\frac{701}{812}\right)\)	\(e\left(\frac{193}{406}\right)\)	\(e\left(\frac{75}{812}\right)\)	\(e\left(\frac{113}{116}\right)\)	\(e\left(\frac{431}{812}\right)\)	\(e\left(\frac{321}{812}\right)\)
\(\chi_{13688}(427,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{337}{812}\right)\)	\(e\left(\frac{167}{203}\right)\)	\(e\left(\frac{277}{406}\right)\)	\(e\left(\frac{337}{406}\right)\)	\(e\left(\frac{299}{812}\right)\)	\(e\left(\frac{69}{406}\right)\)	\(e\left(\frac{193}{812}\right)\)	\(e\left(\frac{99}{116}\right)\)	\(e\left(\frac{741}{812}\right)\)	\(e\left(\frac{79}{812}\right)\)
\(\chi_{13688}(443,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{547}{812}\right)\)	\(e\left(\frac{153}{203}\right)\)	\(e\left(\frac{193}{406}\right)\)	\(e\left(\frac{141}{406}\right)\)	\(e\left(\frac{201}{812}\right)\)	\(e\left(\frac{265}{406}\right)\)	\(e\left(\frac{347}{812}\right)\)	\(e\left(\frac{65}{116}\right)\)	\(e\left(\frac{251}{812}\right)\)	\(e\left(\frac{121}{812}\right)\)
\(\chi_{13688}(467,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{53}{812}\right)\)	\(e\left(\frac{10}{203}\right)\)	\(e\left(\frac{205}{406}\right)\)	\(e\left(\frac{53}{406}\right)\)	\(e\left(\frac{447}{812}\right)\)	\(e\left(\frac{353}{406}\right)\)	\(e\left(\frac{93}{812}\right)\)	\(e\left(\frac{103}{116}\right)\)	\(e\left(\frac{437}{812}\right)\)	\(e\left(\frac{463}{812}\right)\)
\(\chi_{13688}(483,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{129}{812}\right)\)	\(e\left(\frac{32}{203}\right)\)	\(e\left(\frac{47}{406}\right)\)	\(e\left(\frac{129}{406}\right)\)	\(e\left(\frac{659}{812}\right)\)	\(e\left(\frac{277}{406}\right)\)	\(e\left(\frac{257}{812}\right)\)	\(e\left(\frac{115}{116}\right)\)	\(e\left(\frac{221}{812}\right)\)	\(e\left(\frac{223}{812}\right)\)
\(\chi_{13688}(627,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{307}{812}\right)\)	\(e\left(\frac{169}{203}\right)\)	\(e\left(\frac{115}{406}\right)\)	\(e\left(\frac{307}{406}\right)\)	\(e\left(\frac{429}{812}\right)\)	\(e\left(\frac{99}{406}\right)\)	\(e\left(\frac{171}{812}\right)\)	\(e\left(\frac{21}{116}\right)\)	\(e\left(\frac{463}{812}\right)\)	\(e\left(\frac{537}{812}\right)\)
\(\chi_{13688}(659,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{57}{812}\right)\)	\(e\left(\frac{118}{203}\right)\)	\(e\left(\frac{389}{406}\right)\)	\(e\left(\frac{57}{406}\right)\)	\(e\left(\frac{159}{812}\right)\)	\(e\left(\frac{349}{406}\right)\)	\(e\left(\frac{529}{812}\right)\)	\(e\left(\frac{67}{116}\right)\)	\(e\left(\frac{41}{812}\right)\)	\(e\left(\frac{23}{812}\right)\)
\(\chi_{13688}(699,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{81}{812}\right)\)	\(e\left(\frac{157}{203}\right)\)	\(e\left(\frac{275}{406}\right)\)	\(e\left(\frac{81}{406}\right)\)	\(e\left(\frac{55}{812}\right)\)	\(e\left(\frac{325}{406}\right)\)	\(e\left(\frac{709}{812}\right)\)	\(e\left(\frac{83}{116}\right)\)	\(e\left(\frac{101}{812}\right)\)	\(e\left(\frac{631}{812}\right)\)
\(\chi_{13688}(739,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{457}{812}\right)\)	\(e\left(\frac{159}{203}\right)\)	\(e\left(\frac{113}{406}\right)\)	\(e\left(\frac{51}{406}\right)\)	\(e\left(\frac{591}{812}\right)\)	\(e\left(\frac{355}{406}\right)\)	\(e\left(\frac{281}{812}\right)\)	\(e\left(\frac{63}{116}\right)\)	\(e\left(\frac{229}{812}\right)\)	\(e\left(\frac{683}{812}\right)\)
\(\chi_{13688}(859,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{503}{812}\right)\)	\(e\left(\frac{183}{203}\right)\)	\(e\left(\frac{199}{406}\right)\)	\(e\left(\frac{97}{406}\right)\)	\(e\left(\frac{121}{812}\right)\)	\(e\left(\frac{309}{406}\right)\)	\(e\left(\frac{423}{812}\right)\)	\(e\left(\frac{113}{116}\right)\)	\(e\left(\frac{547}{812}\right)\)	\(e\left(\frac{89}{812}\right)\)
\(\chi_{13688}(891,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{812}\right)\)	\(e\left(\frac{27}{203}\right)\)	\(e\left(\frac{249}{406}\right)\)	\(e\left(\frac{1}{406}\right)\)	\(e\left(\frac{131}{812}\right)\)	\(e\left(\frac{405}{406}\right)\)	\(e\left(\frac{109}{812}\right)\)	\(e\left(\frac{107}{116}\right)\)	\(e\left(\frac{713}{812}\right)\)	\(e\left(\frac{499}{812}\right)\)
\(\chi_{13688}(939,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{517}{812}\right)\)	\(e\left(\frac{155}{203}\right)\)	\(e\left(\frac{31}{406}\right)\)	\(e\left(\frac{111}{406}\right)\)	\(e\left(\frac{331}{812}\right)\)	\(e\left(\frac{295}{406}\right)\)	\(e\left(\frac{325}{812}\right)\)	\(e\left(\frac{103}{116}\right)\)	\(e\left(\frac{785}{812}\right)\)	\(e\left(\frac{579}{812}\right)\)
\(\chi_{13688}(955,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{187}{812}\right)\)	\(e\left(\frac{177}{203}\right)\)	\(e\left(\frac{279}{406}\right)\)	\(e\left(\frac{187}{406}\right)\)	\(e\left(\frac{137}{812}\right)\)	\(e\left(\frac{219}{406}\right)\)	\(e\left(\frac{83}{812}\right)\)	\(e\left(\frac{57}{116}\right)\)	\(e\left(\frac{163}{812}\right)\)	\(e\left(\frac{745}{812}\right)\)
\(\chi_{13688}(1059,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{751}{812}\right)\)	\(e\left(\frac{180}{203}\right)\)	\(e\left(\frac{239}{406}\right)\)	\(e\left(\frac{345}{406}\right)\)	\(e\left(\frac{129}{812}\right)\)	\(e\left(\frac{61}{406}\right)\)	\(e\left(\frac{659}{812}\right)\)	\(e\left(\frac{85}{116}\right)\)	\(e\left(\frac{355}{812}\right)\)	\(e\left(\frac{417}{812}\right)\)
\(\chi_{13688}(1075,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{789}{812}\right)\)	\(e\left(\frac{191}{203}\right)\)	\(e\left(\frac{363}{406}\right)\)	\(e\left(\frac{383}{406}\right)\)	\(e\left(\frac{235}{812}\right)\)	\(e\left(\frac{23}{406}\right)\)	\(e\left(\frac{741}{812}\right)\)	\(e\left(\frac{91}{116}\right)\)	\(e\left(\frac{653}{812}\right)\)	\(e\left(\frac{703}{812}\right)\)
\(\chi_{13688}(1099,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{655}{812}\right)\)	\(e\left(\frac{24}{203}\right)\)	\(e\left(\frac{289}{406}\right)\)	\(e\left(\frac{249}{406}\right)\)	\(e\left(\frac{545}{812}\right)\)	\(e\left(\frac{157}{406}\right)\)	\(e\left(\frac{751}{812}\right)\)	\(e\left(\frac{21}{116}\right)\)	\(e\left(\frac{115}{812}\right)\)	\(e\left(\frac{421}{812}\right)\)
\(\chi_{13688}(1123,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{729}{812}\right)\)	\(e\left(\frac{195}{203}\right)\)	\(e\left(\frac{39}{406}\right)\)	\(e\left(\frac{323}{406}\right)\)	\(e\left(\frac{495}{812}\right)\)	\(e\left(\frac{83}{406}\right)\)	\(e\left(\frac{697}{812}\right)\)	\(e\left(\frac{51}{116}\right)\)	\(e\left(\frac{97}{812}\right)\)	\(e\left(\frac{807}{812}\right)\)
\(\chi_{13688}(1139,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{491}{812}\right)\)	\(e\left(\frac{62}{203}\right)\)	\(e\left(\frac{53}{406}\right)\)	\(e\left(\frac{85}{406}\right)\)	\(e\left(\frac{173}{812}\right)\)	\(e\left(\frac{321}{406}\right)\)	\(e\left(\frac{739}{812}\right)\)	\(e\left(\frac{105}{116}\right)\)	\(e\left(\frac{111}{812}\right)\)	\(e\left(\frac{597}{812}\right)\)
\(\chi_{13688}(1163,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{305}{812}\right)\)	\(e\left(\frac{115}{203}\right)\)	\(e\left(\frac{23}{406}\right)\)	\(e\left(\frac{305}{406}\right)\)	\(e\left(\frac{167}{812}\right)\)	\(e\left(\frac{101}{406}\right)\)	\(e\left(\frac{765}{812}\right)\)	\(e\left(\frac{39}{116}\right)\)	\(e\left(\frac{661}{812}\right)\)	\(e\left(\frac{351}{812}\right)\)
\(\chi_{13688}(1171,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{545}{812}\right)\)	\(e\left(\frac{99}{203}\right)\)	\(e\left(\frac{101}{406}\right)\)	\(e\left(\frac{139}{406}\right)\)	\(e\left(\frac{751}{812}\right)\)	\(e\left(\frac{267}{406}\right)\)	\(e\left(\frac{129}{812}\right)\)	\(e\left(\frac{83}{116}\right)\)	\(e\left(\frac{449}{812}\right)\)	\(e\left(\frac{747}{812}\right)\)
\(\chi_{13688}(1203,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{205}{812}\right)\)	\(e\left(\frac{54}{203}\right)\)	\(e\left(\frac{295}{406}\right)\)	\(e\left(\frac{205}{406}\right)\)	\(e\left(\frac{59}{812}\right)\)	\(e\left(\frac{201}{406}\right)\)	\(e\left(\frac{421}{812}\right)\)	\(e\left(\frac{11}{116}\right)\)	\(e\left(\frac{5}{812}\right)\)	\(e\left(\frac{795}{812}\right)\)
\(\chi_{13688}(1291,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{275}{812}\right)\)	\(e\left(\frac{117}{203}\right)\)	\(e\left(\frac{267}{406}\right)\)	\(e\left(\frac{275}{406}\right)\)	\(e\left(\frac{297}{812}\right)\)	\(e\left(\frac{131}{406}\right)\)	\(e\left(\frac{743}{812}\right)\)	\(e\left(\frac{77}{116}\right)\)	\(e\left(\frac{383}{812}\right)\)	\(e\left(\frac{809}{812}\right)\)
\(\chi_{13688}(1331,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{39}{812}\right)\)	\(e\left(\frac{38}{203}\right)\)	\(e\left(\frac{373}{406}\right)\)	\(e\left(\frac{39}{406}\right)\)	\(e\left(\frac{237}{812}\right)\)	\(e\left(\frac{367}{406}\right)\)	\(e\left(\frac{191}{812}\right)\)	\(e\left(\frac{113}{116}\right)\)	\(e\left(\frac{199}{812}\right)\)	\(e\left(\frac{785}{812}\right)\)

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First 31 of 336 characters in Galois orbit

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	13688.do
Modulus	$13688$
Conductor	$13688$
Order	$812$
Real	no
Primitive	yes
Minimal	yes
Parity	odd

Modulus:	\(13688\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(13688\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(812\)	comment:Order sage:chi.multiplicative_order() gp:charorder(g,chi) magma:Order(chi);
Real:	no	comment:Whether the character is real sage:chi.multiplicative_order() <= 2 gp:charorder(g,chi) <= 2 magma:Order(chi) le 2;
Primitive:	yes	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);