from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,197))
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: | \(49\) | 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 49 polynomial |
First 31 of 42 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{197}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) |
| \(\chi_{197}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{37}{49}\right)\) |
| \(\chi_{197}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) |
| \(\chi_{197}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) |
| \(\chi_{197}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{16}{49}\right)\) |
| \(\chi_{197}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) |
| \(\chi_{197}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{20}{49}\right)\) |
| \(\chi_{197}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{30}{49}\right)\) |
| \(\chi_{197}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{26}{49}\right)\) |
| \(\chi_{197}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) |
| \(\chi_{197}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) |
| \(\chi_{197}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) |
| \(\chi_{197}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) |
| \(\chi_{197}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) |
| \(\chi_{197}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) |
| \(\chi_{197}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) |
| \(\chi_{197}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) |
| \(\chi_{197}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{45}{49}\right)\) |
| \(\chi_{197}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) |
| \(\chi_{197}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{6}{49}\right)\) |
| \(\chi_{197}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{34}{49}\right)\) |
| \(\chi_{197}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{36}{49}\right)\) |
| \(\chi_{197}(90,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{34}{49}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) |
| \(\chi_{197}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) |
| \(\chi_{197}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{1}{49}\right)\) |
| \(\chi_{197}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) |
| \(\chi_{197}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) |
| \(\chi_{197}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{19}{49}\right)\) |
| \(\chi_{197}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) |
| \(\chi_{197}(142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{47}{49}\right)\) |
| \(\chi_{197}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) |