LMFDB - Dirichlet character orbit 343.i

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(98)) M = H._module chi = DirichletCharacter(H, M([96])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(8, 343)) 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("343.8"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$343$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$343$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$49$	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:	$\Q(\zeta_{49})$	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 49 polynomial	comment:Fixed field sage:chi.fixed_field()

First 31 of 42 characters in Galois orbit

Character	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$8$	$9$	$10$	$11$	$12$
$\chi_{343}(8,\cdot)$	$1$	$1$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{6}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{3}{49}\right)$
$\chi_{343}(15,\cdot)$	$1$	$1$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{34}{49}\right)$
$\chi_{343}(22,\cdot)$	$1$	$1$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{31}{49}\right)$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{44}{49}\right)$
$\chi_{343}(29,\cdot)$	$1$	$1$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{38}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{46}{49}\right)$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{33}{49}\right)$
$\chi_{343}(36,\cdot)$	$1$	$1$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{23}{49}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{1}{49}\right)$
$\chi_{343}(43,\cdot)$	$1$	$1$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{46}{49}\right)$
$\chi_{343}(57,\cdot)$	$1$	$1$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{8}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{24}{49}\right)$
$\chi_{343}(64,\cdot)$	$1$	$1$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{8}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{6}{49}\right)$
$\chi_{343}(71,\cdot)$	$1$	$1$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{38}{49}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{16}{49}\right)$
$\chi_{343}(78,\cdot)$	$1$	$1$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{31}{49}\right)$	$e\left(\frac{23}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{5}{49}\right)$
$\chi_{343}(85,\cdot)$	$1$	$1$	$e\left(\frac{31}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{22}{49}\right)$
$\chi_{343}(92,\cdot)$	$1$	$1$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{6}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{18}{49}\right)$
$\chi_{343}(106,\cdot)$	$1$	$1$	$e\left(\frac{30}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{6}{49}\right)$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{45}{49}\right)$
$\chi_{343}(113,\cdot)$	$1$	$1$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{31}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{27}{49}\right)$
$\chi_{343}(120,\cdot)$	$1$	$1$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{8}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{37}{49}\right)$
$\chi_{343}(127,\cdot)$	$1$	$1$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{26}{49}\right)$
$\chi_{343}(134,\cdot)$	$1$	$1$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{43}{49}\right)$
$\chi_{343}(141,\cdot)$	$1$	$1$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{23}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{39}{49}\right)$
$\chi_{343}(155,\cdot)$	$1$	$1$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{30}{49}\right)$	$e\left(\frac{17}{49}\right)$
$\chi_{343}(162,\cdot)$	$1$	$1$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{48}{49}\right)$
$\chi_{343}(169,\cdot)$	$1$	$1$	$e\left(\frac{6}{49}\right)$	$e\left(\frac{46}{49}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{9}{49}\right)$
$\chi_{343}(176,\cdot)$	$1$	$1$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{30}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{8}{49}\right)$	$e\left(\frac{47}{49}\right)$
$\chi_{343}(183,\cdot)$	$1$	$1$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{30}{49}\right)$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{38}{49}\right)$	$e\left(\frac{15}{49}\right)$
$\chi_{343}(190,\cdot)$	$1$	$1$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{31}{49}\right)$	$e\left(\frac{8}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{11}{49}\right)$
$\chi_{343}(204,\cdot)$	$1$	$1$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{38}{49}\right)$
$\chi_{343}(211,\cdot)$	$1$	$1$	$e\left(\frac{46}{49}\right)$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{23}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{20}{49}\right)$
$\chi_{343}(218,\cdot)$	$1$	$1$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{4}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{30}{49}\right)$
$\chi_{343}(225,\cdot)$	$1$	$1$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{39}{49}\right)$	$e\left(\frac{38}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{19}{49}\right)$
$\chi_{343}(232,\cdot)$	$1$	$1$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{23}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{36}{49}\right)$
$\chi_{343}(239,\cdot)$	$1$	$1$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{22}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{44}{49}\right)$	$e\left(\frac{6}{49}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{32}{49}\right)$
$\chi_{343}(253,\cdot)$	$1$	$1$	$e\left(\frac{23}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{46}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{8}{49}\right)$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{10}{49}\right)$

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	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(8\)	\(9\)	\(10\)	\(11\)	\(12\)
\(\chi_{343}(8,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{6}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)
\(\chi_{343}(15,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)
\(\chi_{343}(22,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{31}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)
\(\chi_{343}(29,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{38}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{46}{49}\right)\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)
\(\chi_{343}(36,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{23}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{1}{49}\right)\)
\(\chi_{343}(43,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{46}{49}\right)\)
\(\chi_{343}(57,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{8}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)
\(\chi_{343}(64,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{8}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{6}{49}\right)\)
\(\chi_{343}(71,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{38}{49}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)
\(\chi_{343}(78,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{31}{49}\right)\)	\(e\left(\frac{23}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)
\(\chi_{343}(85,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{31}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)
\(\chi_{343}(92,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{6}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)
\(\chi_{343}(106,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{30}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{6}{49}\right)\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)
\(\chi_{343}(113,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{31}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)
\(\chi_{343}(120,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{8}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)
\(\chi_{343}(127,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)
\(\chi_{343}(134,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)
\(\chi_{343}(141,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{23}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)
\(\chi_{343}(155,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{30}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)
\(\chi_{343}(162,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)
\(\chi_{343}(169,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{6}{49}\right)\)	\(e\left(\frac{46}{49}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)
\(\chi_{343}(176,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{30}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{8}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)
\(\chi_{343}(183,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{30}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{38}{49}\right)\)	\(e\left(\frac{15}{49}\right)\)
\(\chi_{343}(190,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{31}{49}\right)\)	\(e\left(\frac{8}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)
\(\chi_{343}(204,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{38}{49}\right)\)
\(\chi_{343}(211,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{46}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{23}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)
\(\chi_{343}(218,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{4}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{30}{49}\right)\)
\(\chi_{343}(225,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{39}{49}\right)\)	\(e\left(\frac{38}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)
\(\chi_{343}(232,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{23}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)
\(\chi_{343}(239,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{22}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)	\(e\left(\frac{6}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{32}{49}\right)\)
\(\chi_{343}(253,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{23}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{46}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{8}{49}\right)\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)

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First 31 of 42 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	343.i
Modulus	$343$
Conductor	$343$
Order	$49$
Real	no
Primitive	yes
Minimal	yes
Parity	even

Modulus:	\(343\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(343\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(49\)	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);