# Properties

 Label 1148.cd Modulus $1148$ Conductor $41$ Order $40$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(1148, base_ring=CyclotomicField(40))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,0,7]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(29,1148))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1148$$ Conductor: $$41$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$40$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 41.h 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$
$$\chi_{1148}(29,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{1148}(253,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{1148}(281,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{1148}(309,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-i$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{1148}(393,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{1148}(421,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$i$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{1148}(477,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-i$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{1148}(505,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$i$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{1148}(561,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$i$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{1148}(589,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-i$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{1148}(645,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$i$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{1148}(673,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{1148}(757,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-i$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{1148}(785,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{1148}(813,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{1148}(1037,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$