LMFDB - Dirichlet character orbit 1305.ct

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([70,42,39])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(14, 1305)) 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("1305.14"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$1305$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$1305$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$84$	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:	even	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{84})$	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 84 polynomial	comment:Fixed field sage:chi.fixed_field()

Characters in Galois orbit

Character	$-1$	$1$	$2$	$4$	$7$	$8$	$11$	$13$	$14$	$16$	$17$	$19$
$\chi_{1305}(14,\cdot)$	$1$	$1$	$e\left(\frac{67}{84}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{17}{42}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{37}{84}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{17}{84}\right)$	$e\left(\frac{4}{21}\right)$	$-i$	$e\left(\frac{5}{28}\right)$
$\chi_{1305}(119,\cdot)$	$1$	$1$	$e\left(\frac{71}{84}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{53}{84}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{13}{84}\right)$	$e\left(\frac{8}{21}\right)$	$-i$	$e\left(\frac{17}{28}\right)$
$\chi_{1305}(164,\cdot)$	$1$	$1$	$e\left(\frac{83}{84}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{17}{84}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{1}{84}\right)$	$e\left(\frac{20}{21}\right)$	$-i$	$e\left(\frac{25}{28}\right)$
$\chi_{1305}(329,\cdot)$	$1$	$1$	$e\left(\frac{13}{84}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{31}{84}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{71}{84}\right)$	$e\left(\frac{13}{21}\right)$	$i$	$e\left(\frac{11}{28}\right)$
$\chi_{1305}(374,\cdot)$	$1$	$1$	$e\left(\frac{1}{84}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{67}{84}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{83}{84}\right)$	$e\left(\frac{1}{21}\right)$	$i$	$e\left(\frac{3}{28}\right)$
$\chi_{1305}(479,\cdot)$	$1$	$1$	$e\left(\frac{53}{84}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{23}{84}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{31}{84}\right)$	$e\left(\frac{11}{21}\right)$	$i$	$e\left(\frac{19}{28}\right)$
$\chi_{1305}(524,\cdot)$	$1$	$1$	$e\left(\frac{59}{84}\right)$	$e\left(\frac{17}{42}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{5}{84}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{25}{84}\right)$	$e\left(\frac{17}{21}\right)$	$-i$	$e\left(\frac{9}{28}\right)$
$\chi_{1305}(554,\cdot)$	$1$	$1$	$e\left(\frac{43}{84}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{25}{84}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{41}{84}\right)$	$e\left(\frac{1}{21}\right)$	$-i$	$e\left(\frac{17}{28}\right)$
$\chi_{1305}(569,\cdot)$	$1$	$1$	$e\left(\frac{5}{84}\right)$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{83}{84}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{79}{84}\right)$	$e\left(\frac{5}{21}\right)$	$i$	$e\left(\frac{15}{28}\right)$
$\chi_{1305}(599,\cdot)$	$1$	$1$	$e\left(\frac{55}{84}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{73}{84}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{29}{84}\right)$	$e\left(\frac{13}{21}\right)$	$-i$	$e\left(\frac{25}{28}\right)$
$\chi_{1305}(659,\cdot)$	$1$	$1$	$e\left(\frac{23}{84}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{29}{84}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{61}{84}\right)$	$e\left(\frac{2}{21}\right)$	$-i$	$e\left(\frac{13}{28}\right)$
$\chi_{1305}(704,\cdot)$	$1$	$1$	$e\left(\frac{65}{84}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{71}{84}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{19}{84}\right)$	$e\left(\frac{2}{21}\right)$	$i$	$e\left(\frac{27}{28}\right)$
$\chi_{1305}(794,\cdot)$	$1$	$1$	$e\left(\frac{47}{84}\right)$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{41}{84}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{37}{84}\right)$	$e\left(\frac{5}{21}\right)$	$-i$	$e\left(\frac{1}{28}\right)$
$\chi_{1305}(839,\cdot)$	$1$	$1$	$e\left(\frac{17}{84}\right)$	$e\left(\frac{17}{42}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{47}{84}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{67}{84}\right)$	$e\left(\frac{17}{21}\right)$	$i$	$e\left(\frac{23}{28}\right)$
$\chi_{1305}(884,\cdot)$	$1$	$1$	$e\left(\frac{11}{84}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{65}{84}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{73}{84}\right)$	$e\left(\frac{11}{21}\right)$	$-i$	$e\left(\frac{5}{28}\right)$
$\chi_{1305}(914,\cdot)$	$1$	$1$	$e\left(\frac{25}{84}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{17}{42}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{79}{84}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{59}{84}\right)$	$e\left(\frac{4}{21}\right)$	$i$	$e\left(\frac{19}{28}\right)$
$\chi_{1305}(959,\cdot)$	$1$	$1$	$e\left(\frac{31}{84}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{61}{84}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{53}{84}\right)$	$e\left(\frac{10}{21}\right)$	$-i$	$e\left(\frac{9}{28}\right)$
$\chi_{1305}(1004,\cdot)$	$1$	$1$	$e\left(\frac{61}{84}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{55}{84}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{23}{84}\right)$	$e\left(\frac{19}{21}\right)$	$i$	$e\left(\frac{15}{28}\right)$
$\chi_{1305}(1094,\cdot)$	$1$	$1$	$e\left(\frac{79}{84}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{1}{84}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{5}{84}\right)$	$e\left(\frac{16}{21}\right)$	$-i$	$e\left(\frac{13}{28}\right)$
$\chi_{1305}(1139,\cdot)$	$1$	$1$	$e\left(\frac{37}{84}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{43}{84}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{47}{84}\right)$	$e\left(\frac{16}{21}\right)$	$i$	$e\left(\frac{27}{28}\right)$
$\chi_{1305}(1199,\cdot)$	$1$	$1$	$e\left(\frac{41}{84}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{59}{84}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{43}{84}\right)$	$e\left(\frac{20}{21}\right)$	$i$	$e\left(\frac{11}{28}\right)$
$\chi_{1305}(1229,\cdot)$	$1$	$1$	$e\left(\frac{19}{84}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{13}{84}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{65}{84}\right)$	$e\left(\frac{19}{21}\right)$	$-i$	$e\left(\frac{1}{28}\right)$
$\chi_{1305}(1244,\cdot)$	$1$	$1$	$e\left(\frac{29}{84}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{11}{84}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{55}{84}\right)$	$e\left(\frac{8}{21}\right)$	$i$	$e\left(\frac{3}{28}\right)$
$\chi_{1305}(1274,\cdot)$	$1$	$1$	$e\left(\frac{73}{84}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{19}{84}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{11}{84}\right)$	$e\left(\frac{10}{21}\right)$	$i$	$e\left(\frac{23}{28}\right)$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Character	\(-1\)	\(1\)	\(2\)	\(4\)	\(7\)	\(8\)	\(11\)	\(13\)	\(14\)	\(16\)	\(17\)	\(19\)
\(\chi_{1305}(14,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{67}{84}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{37}{84}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{17}{84}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(-i\)	\(e\left(\frac{5}{28}\right)\)
\(\chi_{1305}(119,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{71}{84}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{53}{84}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{13}{84}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(-i\)	\(e\left(\frac{17}{28}\right)\)
\(\chi_{1305}(164,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{83}{84}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{17}{84}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{1}{84}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(-i\)	\(e\left(\frac{25}{28}\right)\)
\(\chi_{1305}(329,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{13}{84}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{31}{84}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{71}{84}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(i\)	\(e\left(\frac{11}{28}\right)\)
\(\chi_{1305}(374,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{1}{84}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{67}{84}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{83}{84}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(i\)	\(e\left(\frac{3}{28}\right)\)
\(\chi_{1305}(479,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{53}{84}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{23}{84}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{31}{84}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(i\)	\(e\left(\frac{19}{28}\right)\)
\(\chi_{1305}(524,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{59}{84}\right)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{5}{84}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{25}{84}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(-i\)	\(e\left(\frac{9}{28}\right)\)
\(\chi_{1305}(554,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{43}{84}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{25}{84}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{41}{84}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(-i\)	\(e\left(\frac{17}{28}\right)\)
\(\chi_{1305}(569,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{5}{84}\right)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{83}{84}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{79}{84}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(i\)	\(e\left(\frac{15}{28}\right)\)
\(\chi_{1305}(599,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{55}{84}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{73}{84}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{29}{84}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(-i\)	\(e\left(\frac{25}{28}\right)\)
\(\chi_{1305}(659,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{23}{84}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{29}{84}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{61}{84}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(-i\)	\(e\left(\frac{13}{28}\right)\)
\(\chi_{1305}(704,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{65}{84}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{71}{84}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{19}{84}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(i\)	\(e\left(\frac{27}{28}\right)\)
\(\chi_{1305}(794,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{47}{84}\right)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{41}{84}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{37}{84}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(-i\)	\(e\left(\frac{1}{28}\right)\)
\(\chi_{1305}(839,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{17}{84}\right)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{47}{84}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{67}{84}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(i\)	\(e\left(\frac{23}{28}\right)\)
\(\chi_{1305}(884,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{11}{84}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{65}{84}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{73}{84}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(-i\)	\(e\left(\frac{5}{28}\right)\)
\(\chi_{1305}(914,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{25}{84}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{79}{84}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{59}{84}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(i\)	\(e\left(\frac{19}{28}\right)\)
\(\chi_{1305}(959,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{31}{84}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{61}{84}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{53}{84}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(-i\)	\(e\left(\frac{9}{28}\right)\)
\(\chi_{1305}(1004,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{61}{84}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{55}{84}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{23}{84}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(i\)	\(e\left(\frac{15}{28}\right)\)
\(\chi_{1305}(1094,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{79}{84}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{1}{84}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{5}{84}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(-i\)	\(e\left(\frac{13}{28}\right)\)
\(\chi_{1305}(1139,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{37}{84}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{43}{84}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{47}{84}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(i\)	\(e\left(\frac{27}{28}\right)\)
\(\chi_{1305}(1199,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{41}{84}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{59}{84}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{43}{84}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(i\)	\(e\left(\frac{11}{28}\right)\)
\(\chi_{1305}(1229,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{19}{84}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{13}{84}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{65}{84}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(-i\)	\(e\left(\frac{1}{28}\right)\)
\(\chi_{1305}(1244,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{29}{84}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{11}{84}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{55}{84}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(i\)	\(e\left(\frac{3}{28}\right)\)
\(\chi_{1305}(1274,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{73}{84}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{19}{84}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{11}{84}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(i\)	\(e\left(\frac{23}{28}\right)\)

Introduction

L-functions

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Varieties

Fields

Representations

Groups

Database

Properties

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Basic properties

Related number fields

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	1305.ct
Modulus	$1305$
Conductor	$1305$
Order	$84$
Real	no
Primitive	yes
Minimal	yes
Parity	even

Modulus:	\(1305\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(1305\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(84\)	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:	even	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);