# Properties

 Label 5520.ft Modulus $5520$ Conductor $1380$ Order $44$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(5520, base_ring=CyclotomicField(44))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([22,0,22,33,2]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(143,5520))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$5520$$ Conductor: $$1380$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 1380.bv 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: $$\Q(\zeta_{44})$$ Fixed field: Number field defined by a degree 44 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$
$$\chi_{5520}(143,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{43}{44}\right)$$
$$\chi_{5520}(287,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{44}\right)$$
$$\chi_{5520}(383,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$
$$\chi_{5520}(527,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$
$$\chi_{5520}(1247,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$
$$\chi_{5520}(1487,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{5520}(1583,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{5520}(2687,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{29}{44}\right)$$
$$\chi_{5520}(3023,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{44}\right)$$
$$\chi_{5520}(3263,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{5520}(3503,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{44}\right)$$
$$\chi_{5520}(3743,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$
$$\chi_{5520}(4127,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{41}{44}\right)$$
$$\chi_{5520}(4223,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$
$$\chi_{5520}(4367,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{17}{44}\right)$$
$$\chi_{5520}(4607,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{25}{44}\right)$$
$$\chi_{5520}(4703,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{35}{44}\right)$$
$$\chi_{5520}(4847,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$
$$\chi_{5520}(4943,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$
$$\chi_{5520}(5327,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{44}\right)$$