# Properties

 Label 5520.fn Modulus $5520$ Conductor $1840$ 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([22,33,0,33,4]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(163,5520))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$5520$$ Conductor: $$1840$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 1840.ch 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 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$
$$\chi_{5520}(163,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{5520}(187,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{5520}(403,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{5520}(427,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{5520}(883,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{5520}(1363,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{5520}(1603,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{5520}(1843,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{5520}(1867,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{5520}(2083,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{5520}(3067,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{5520}(3307,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{5520}(3523,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{5520}(4027,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{5520}(4267,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{5520}(4723,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{5520}(4747,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{5520}(4963,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{5520}(5227,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{5520}(5467,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$