# Properties

 Label 2601.y Modulus $2601$ Conductor $2601$ Order $51$ Real no Primitive yes Minimal yes Parity even

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(2601, base_ring=CyclotomicField(102))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([68,12]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(52,2601))

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Basic properties

 Modulus: $$2601$$ Conductor: $$2601$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$51$$ 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: $\Q(\zeta_{51})$ Fixed field: Number field defined by a degree 51 polynomial

## First 31 of 32 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{2601}(52,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{4}{51}\right)$$
$$\chi_{2601}(103,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$
$$\chi_{2601}(205,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$
$$\chi_{2601}(256,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$
$$\chi_{2601}(358,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$
$$\chi_{2601}(409,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$
$$\chi_{2601}(511,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$
$$\chi_{2601}(562,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$
$$\chi_{2601}(664,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$
$$\chi_{2601}(715,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{5}{51}\right)$$
$$\chi_{2601}(817,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$
$$\chi_{2601}(970,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$
$$\chi_{2601}(1021,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$
$$\chi_{2601}(1123,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{37}{51}\right)$$
$$\chi_{2601}(1174,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$
$$\chi_{2601}(1276,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$
$$\chi_{2601}(1327,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$
$$\chi_{2601}(1429,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$
$$\chi_{2601}(1480,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$
$$\chi_{2601}(1582,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$
$$\chi_{2601}(1633,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$
$$\chi_{2601}(1786,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$
$$\chi_{2601}(1888,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{5}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$
$$\chi_{2601}(1939,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$
$$\chi_{2601}(2041,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{29}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{7}{51}\right)$$
$$\chi_{2601}(2092,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{31}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{50}{51}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$
$$\chi_{2601}(2194,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{40}{51}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$
$$\chi_{2601}(2245,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{44}{51}\right)$$ $$e\left(\frac{37}{51}\right)$$ $$e\left(\frac{4}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$
$$\chi_{2601}(2347,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{41}{51}\right)$$ $$e\left(\frac{32}{51}\right)$$ $$e\left(\frac{25}{51}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{7}{51}\right)$$ $$e\left(\frac{2}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{31}{51}\right)$$
$$\chi_{2601}(2398,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{46}{51}\right)$$ $$e\left(\frac{20}{51}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{16}{51}\right)$$ $$e\left(\frac{35}{51}\right)$$
$$\chi_{2601}(2500,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{47}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{10}{51}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{13}{51}\right)$$ $$e\left(\frac{11}{51}\right)$$ $$e\left(\frac{8}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$