# Properties

 Modulus $394$ Structure $$C_{196}$$ Order $196$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(394)

pari: g = idealstar(,394,2)

## Character group

 sage: G.order()  pari: g.no Order = 196 sage: H.invariants()  pari: g.cyc Structure = $$C_{196}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{394}(199,\cdot)$

## First 32 of 196 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{394}(1,\cdot)$$ 394.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{394}(3,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{29}{196}\right)$$ $$e\left(\frac{37}{196}\right)$$ $$e\left(\frac{81}{98}\right)$$ $$e\left(\frac{29}{98}\right)$$ $$e\left(\frac{153}{196}\right)$$ $$e\left(\frac{17}{196}\right)$$ $$e\left(\frac{33}{98}\right)$$ $$e\left(\frac{163}{196}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{191}{196}\right)$$
$$\chi_{394}(5,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{37}{196}\right)$$ $$e\left(\frac{81}{196}\right)$$ $$e\left(\frac{29}{98}\right)$$ $$e\left(\frac{37}{98}\right)$$ $$e\left(\frac{33}{196}\right)$$ $$e\left(\frac{69}{196}\right)$$ $$e\left(\frac{59}{98}\right)$$ $$e\left(\frac{39}{196}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{95}{196}\right)$$
$$\chi_{394}(7,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{81}{98}\right)$$ $$e\left(\frac{29}{98}\right)$$ $$e\left(\frac{37}{49}\right)$$ $$e\left(\frac{32}{49}\right)$$ $$e\left(\frac{59}{98}\right)$$ $$e\left(\frac{61}{98}\right)$$ $$e\left(\frac{6}{49}\right)$$ $$e\left(\frac{43}{98}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{57}{98}\right)$$
$$\chi_{394}(9,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{29}{98}\right)$$ $$e\left(\frac{37}{98}\right)$$ $$e\left(\frac{32}{49}\right)$$ $$e\left(\frac{29}{49}\right)$$ $$e\left(\frac{55}{98}\right)$$ $$e\left(\frac{17}{98}\right)$$ $$e\left(\frac{33}{49}\right)$$ $$e\left(\frac{65}{98}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{93}{98}\right)$$
$$\chi_{394}(11,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{153}{196}\right)$$ $$e\left(\frac{33}{196}\right)$$ $$e\left(\frac{59}{98}\right)$$ $$e\left(\frac{55}{98}\right)$$ $$e\left(\frac{57}{196}\right)$$ $$e\left(\frac{137}{196}\right)$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{103}{196}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{75}{196}\right)$$
$$\chi_{394}(13,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{17}{196}\right)$$ $$e\left(\frac{69}{196}\right)$$ $$e\left(\frac{61}{98}\right)$$ $$e\left(\frac{17}{98}\right)$$ $$e\left(\frac{137}{196}\right)$$ $$e\left(\frac{37}{196}\right)$$ $$e\left(\frac{43}{98}\right)$$ $$e\left(\frac{55}{196}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{139}{196}\right)$$
$$\chi_{394}(15,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{33}{98}\right)$$ $$e\left(\frac{59}{98}\right)$$ $$e\left(\frac{6}{49}\right)$$ $$e\left(\frac{33}{49}\right)$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{43}{98}\right)$$ $$e\left(\frac{46}{49}\right)$$ $$e\left(\frac{3}{98}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{45}{98}\right)$$
$$\chi_{394}(17,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{163}{196}\right)$$ $$e\left(\frac{39}{196}\right)$$ $$e\left(\frac{43}{98}\right)$$ $$e\left(\frac{65}{98}\right)$$ $$e\left(\frac{103}{196}\right)$$ $$e\left(\frac{55}{196}\right)$$ $$e\left(\frac{3}{98}\right)$$ $$e\left(\frac{193}{196}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{53}{196}\right)$$
$$\chi_{394}(19,\cdot)$$ 394.e 14 no $$1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$1$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{394}(21,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{191}{196}\right)$$ $$e\left(\frac{95}{196}\right)$$ $$e\left(\frac{57}{98}\right)$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{75}{196}\right)$$ $$e\left(\frac{139}{196}\right)$$ $$e\left(\frac{45}{98}\right)$$ $$e\left(\frac{53}{196}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{109}{196}\right)$$
$$\chi_{394}(23,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{40}{49}\right)$$ $$e\left(\frac{24}{49}\right)$$ $$e\left(\frac{19}{49}\right)$$ $$e\left(\frac{31}{49}\right)$$ $$e\left(\frac{37}{49}\right)$$ $$e\left(\frac{15}{49}\right)$$ $$e\left(\frac{15}{49}\right)$$ $$e\left(\frac{17}{49}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{10}{49}\right)$$
$$\chi_{394}(25,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{37}{98}\right)$$ $$e\left(\frac{81}{98}\right)$$ $$e\left(\frac{29}{49}\right)$$ $$e\left(\frac{37}{49}\right)$$ $$e\left(\frac{33}{98}\right)$$ $$e\left(\frac{69}{98}\right)$$ $$e\left(\frac{10}{49}\right)$$ $$e\left(\frac{39}{98}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{95}{98}\right)$$
$$\chi_{394}(27,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{87}{196}\right)$$ $$e\left(\frac{111}{196}\right)$$ $$e\left(\frac{47}{98}\right)$$ $$e\left(\frac{87}{98}\right)$$ $$e\left(\frac{67}{196}\right)$$ $$e\left(\frac{51}{196}\right)$$ $$e\left(\frac{1}{98}\right)$$ $$e\left(\frac{97}{196}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{181}{196}\right)$$
$$\chi_{394}(29,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{12}{49}\right)$$ $$e\left(\frac{17}{49}\right)$$ $$e\left(\frac{40}{49}\right)$$ $$e\left(\frac{24}{49}\right)$$ $$e\left(\frac{16}{49}\right)$$ $$e\left(\frac{29}{49}\right)$$ $$e\left(\frac{29}{49}\right)$$ $$e\left(\frac{10}{49}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{49}\right)$$
$$\chi_{394}(31,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{41}{196}\right)$$ $$e\left(\frac{5}{196}\right)$$ $$e\left(\frac{3}{98}\right)$$ $$e\left(\frac{41}{98}\right)$$ $$e\left(\frac{169}{196}\right)$$ $$e\left(\frac{193}{196}\right)$$ $$e\left(\frac{23}{98}\right)$$ $$e\left(\frac{75}{196}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{47}{196}\right)$$
$$\chi_{394}(33,\cdot)$$ 394.e 14 no $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$1$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{394}(35,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{3}{196}\right)$$ $$e\left(\frac{139}{196}\right)$$ $$e\left(\frac{5}{98}\right)$$ $$e\left(\frac{3}{98}\right)$$ $$e\left(\frac{151}{196}\right)$$ $$e\left(\frac{191}{196}\right)$$ $$e\left(\frac{71}{98}\right)$$ $$e\left(\frac{125}{196}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{196}\right)$$
$$\chi_{394}(37,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{15}{49}\right)$$ $$e\left(\frac{9}{49}\right)$$ $$e\left(\frac{1}{49}\right)$$ $$e\left(\frac{30}{49}\right)$$ $$e\left(\frac{20}{49}\right)$$ $$e\left(\frac{24}{49}\right)$$ $$e\left(\frac{24}{49}\right)$$ $$e\left(\frac{37}{49}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{16}{49}\right)$$
$$\chi_{394}(39,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{23}{98}\right)$$ $$e\left(\frac{53}{98}\right)$$ $$e\left(\frac{22}{49}\right)$$ $$e\left(\frac{23}{49}\right)$$ $$e\left(\frac{47}{98}\right)$$ $$e\left(\frac{27}{98}\right)$$ $$e\left(\frac{38}{49}\right)$$ $$e\left(\frac{11}{98}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{67}{98}\right)$$
$$\chi_{394}(41,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{57}{98}\right)$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{46}{49}\right)$$ $$e\left(\frac{8}{49}\right)$$ $$e\left(\frac{27}{98}\right)$$ $$e\left(\frac{3}{98}\right)$$ $$e\left(\frac{26}{49}\right)$$ $$e\left(\frac{23}{98}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{51}{98}\right)$$
$$\chi_{394}(43,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{3}{98}\right)$$ $$e\left(\frac{41}{98}\right)$$ $$e\left(\frac{5}{49}\right)$$ $$e\left(\frac{3}{49}\right)$$ $$e\left(\frac{53}{98}\right)$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{22}{49}\right)$$ $$e\left(\frac{27}{98}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{98}\right)$$
$$\chi_{394}(45,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{95}{196}\right)$$ $$e\left(\frac{155}{196}\right)$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{95}{98}\right)$$ $$e\left(\frac{143}{196}\right)$$ $$e\left(\frac{103}{196}\right)$$ $$e\left(\frac{27}{98}\right)$$ $$e\left(\frac{169}{196}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{85}{196}\right)$$
$$\chi_{394}(47,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{93}{98}\right)$$ $$e\left(\frac{95}{98}\right)$$ $$e\left(\frac{8}{49}\right)$$ $$e\left(\frac{44}{49}\right)$$ $$e\left(\frac{75}{98}\right)$$ $$e\left(\frac{41}{98}\right)$$ $$e\left(\frac{45}{49}\right)$$ $$e\left(\frac{53}{98}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{98}\right)$$
$$\chi_{394}(49,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{32}{49}\right)$$ $$e\left(\frac{29}{49}\right)$$ $$e\left(\frac{25}{49}\right)$$ $$e\left(\frac{15}{49}\right)$$ $$e\left(\frac{10}{49}\right)$$ $$e\left(\frac{12}{49}\right)$$ $$e\left(\frac{12}{49}\right)$$ $$e\left(\frac{43}{49}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{8}{49}\right)$$
$$\chi_{394}(51,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{48}{49}\right)$$ $$e\left(\frac{19}{49}\right)$$ $$e\left(\frac{13}{49}\right)$$ $$e\left(\frac{47}{49}\right)$$ $$e\left(\frac{15}{49}\right)$$ $$e\left(\frac{18}{49}\right)$$ $$e\left(\frac{18}{49}\right)$$ $$e\left(\frac{40}{49}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{12}{49}\right)$$
$$\chi_{394}(53,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{24}{49}\right)$$ $$e\left(\frac{34}{49}\right)$$ $$e\left(\frac{31}{49}\right)$$ $$e\left(\frac{48}{49}\right)$$ $$e\left(\frac{32}{49}\right)$$ $$e\left(\frac{9}{49}\right)$$ $$e\left(\frac{9}{49}\right)$$ $$e\left(\frac{20}{49}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{49}\right)$$
$$\chi_{394}(55,\cdot)$$ 394.h 98 no $$1$$ $$1$$ $$e\left(\frac{95}{98}\right)$$ $$e\left(\frac{57}{98}\right)$$ $$e\left(\frac{44}{49}\right)$$ $$e\left(\frac{46}{49}\right)$$ $$e\left(\frac{45}{98}\right)$$ $$e\left(\frac{5}{98}\right)$$ $$e\left(\frac{27}{49}\right)$$ $$e\left(\frac{71}{98}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{85}{98}\right)$$
$$\chi_{394}(57,\cdot)$$ 394.i 196 no $$-1$$ $$1$$ $$e\left(\frac{71}{196}\right)$$ $$e\left(\frac{23}{196}\right)$$ $$e\left(\frac{53}{98}\right)$$ $$e\left(\frac{71}{98}\right)$$ $$e\left(\frac{111}{196}\right)$$ $$e\left(\frac{143}{196}\right)$$ $$e\left(\frac{47}{98}\right)$$ $$e\left(\frac{149}{196}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{177}{196}\right)$$
$$\chi_{394}(59,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{19}{49}\right)$$ $$e\left(\frac{31}{49}\right)$$ $$e\left(\frac{47}{49}\right)$$ $$e\left(\frac{38}{49}\right)$$ $$e\left(\frac{9}{49}\right)$$ $$e\left(\frac{1}{49}\right)$$ $$e\left(\frac{1}{49}\right)$$ $$e\left(\frac{24}{49}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{17}{49}\right)$$
$$\chi_{394}(61,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{5}{49}\right)$$ $$e\left(\frac{3}{49}\right)$$ $$e\left(\frac{33}{49}\right)$$ $$e\left(\frac{10}{49}\right)$$ $$e\left(\frac{23}{49}\right)$$ $$e\left(\frac{8}{49}\right)$$ $$e\left(\frac{8}{49}\right)$$ $$e\left(\frac{45}{49}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{38}{49}\right)$$
$$\chi_{394}(63,\cdot)$$ 394.g 49 no $$1$$ $$1$$ $$e\left(\frac{6}{49}\right)$$ $$e\left(\frac{33}{49}\right)$$ $$e\left(\frac{20}{49}\right)$$ $$e\left(\frac{12}{49}\right)$$ $$e\left(\frac{8}{49}\right)$$ $$e\left(\frac{39}{49}\right)$$ $$e\left(\frac{39}{49}\right)$$ $$e\left(\frac{5}{49}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{26}{49}\right)$$