# Properties

 Label 41.h Modulus $41$ Conductor $41$ Order $40$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(41, base_ring=CyclotomicField(40))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(6,41))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$41$$ Conductor: $$41$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$40$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $$\Q(\zeta_{40})$$ Fixed field: $$\Q(\zeta_{41})$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{41}(6,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{40}\right)$$
$$\chi_{41}(7,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{37}{40}\right)$$
$$\chi_{41}(11,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{40}\right)$$
$$\chi_{41}(12,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{40}\right)$$
$$\chi_{41}(13,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{40}\right)$$
$$\chi_{41}(15,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{31}{40}\right)$$
$$\chi_{41}(17,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{40}\right)$$
$$\chi_{41}(19,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{27}{40}\right)$$
$$\chi_{41}(22,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{40}\right)$$
$$\chi_{41}(24,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{39}{40}\right)$$
$$\chi_{41}(26,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{40}\right)$$
$$\chi_{41}(28,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{33}{40}\right)$$
$$\chi_{41}(29,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{21}{40}\right)$$
$$\chi_{41}(30,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{29}{40}\right)$$
$$\chi_{41}(34,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{17}{40}\right)$$
$$\chi_{41}(35,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{23}{40}\right)$$