# Properties

 Modulus $41$ Structure $$C_{40}$$ Order $40$

Show commands for: Pari/GP / SageMath

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(41)

pari: g = idealstar(,41,2)

## Character group

 sage: G.order()  pari: g.no Order = 40 sage: H.invariants()  pari: g.cyc Structure = $$C_{40}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{41}(6,\cdot)$

## First 32 of 40 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{41}(1,\cdot)$$ 41.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{41}(2,\cdot)$$ 41.g 20 yes $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$-i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$
$$\chi_{41}(3,\cdot)$$ 41.e 8 yes $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$i$$ $$1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{41}(4,\cdot)$$ 41.f 10 yes $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{41}(5,\cdot)$$ 41.g 20 yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{41}(6,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{40}\right)$$
$$\chi_{41}(7,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{37}{40}\right)$$
$$\chi_{41}(8,\cdot)$$ 41.g 20 yes $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{41}(9,\cdot)$$ 41.c 4 yes $$1$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$i$$ $$-1$$ $$-1$$ $$1$$ $$i$$
$$\chi_{41}(10,\cdot)$$ 41.d 5 yes $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{41}(11,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{40}\right)$$
$$\chi_{41}(12,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{40}\right)$$
$$\chi_{41}(13,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{40}\right)$$
$$\chi_{41}(14,\cdot)$$ 41.e 8 yes $$-1$$ $$1$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$-i$$ $$1$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{41}(15,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{31}{40}\right)$$
$$\chi_{41}(16,\cdot)$$ 41.d 5 yes $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{41}(17,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{40}\right)$$
$$\chi_{41}(18,\cdot)$$ 41.d 5 yes $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{41}(19,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{27}{40}\right)$$
$$\chi_{41}(20,\cdot)$$ 41.g 20 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$-i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{41}(21,\cdot)$$ 41.g 20 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{41}(22,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{40}\right)$$
$$\chi_{41}(23,\cdot)$$ 41.f 10 yes $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{41}(24,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{39}{40}\right)$$
$$\chi_{41}(25,\cdot)$$ 41.f 10 yes $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{41}(26,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{40}\right)$$
$$\chi_{41}(27,\cdot)$$ 41.e 8 yes $$-1$$ $$1$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$-i$$ $$1$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{41}(28,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{33}{40}\right)$$
$$\chi_{41}(29,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$i$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{21}{40}\right)$$
$$\chi_{41}(30,\cdot)$$ 41.h 40 yes $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{29}{40}\right)$$
$$\chi_{41}(31,\cdot)$$ 41.f 10 yes $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{41}(32,\cdot)$$ 41.c 4 yes $$1$$ $$1$$ $$-1$$ $$-i$$ $$1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$-1$$ $$1$$ $$-i$$