from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,197))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(197\) | |
| Conductor: | \(197\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(98\) | 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: | $\Q(\zeta_{49})$ |
| Fixed field: | Number field defined by a degree 98 polynomial |
First 31 of 42 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{197}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{29}{98}\right)\) |
| \(\chi_{197}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{59}{98}\right)\) |
| \(\chi_{197}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{55}{98}\right)\) |
| \(\chi_{197}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{31}{98}\right)\) |
| \(\chi_{197}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{93}{98}\right)\) |
| \(\chi_{197}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{43}{98}\right)\) |
| \(\chi_{197}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{33}{98}\right)\) |
| \(\chi_{197}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{83}{98}\right)\) |
| \(\chi_{197}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) |
| \(\chi_{197}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{27}{98}\right)\) |
| \(\chi_{197}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{53}{98}\right)\) |
| \(\chi_{197}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{75}{98}\right)\) |
| \(\chi_{197}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{45}{98}\right)\) |
| \(\chi_{197}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{1}{98}\right)\) |
| \(\chi_{197}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{87}{98}\right)\) |
| \(\chi_{197}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{85}{98}\right)\) |
| \(\chi_{197}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{5}{98}\right)\) |
| \(\chi_{197}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{51}{98}\right)\) |
| \(\chi_{197}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{25}{98}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{13}{98}\right)\) |
| \(\chi_{197}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{37}{98}\right)\) |
| \(\chi_{197}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{97}{98}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{23}{98}\right)\) |
| \(\chi_{197}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{19}{98}\right)\) |
| \(\chi_{197}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{25}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{61}{98}\right)\) |
| \(\chi_{197}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{57}{98}\right)\) |
| \(\chi_{197}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{41}{98}\right)\) |
| \(\chi_{197}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{65}{98}\right)\) |
| \(\chi_{197}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{95}{98}\right)\) |
| \(\chi_{197}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{73}{98}\right)\) |
| \(\chi_{197}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{67}{98}\right)\) |
| \(\chi_{197}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{97}{98}\right)\) |
| \(\chi_{197}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{97}{98}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{15}{98}\right)\) |