Properties

Label 121.g
Modulus $121$
Conductor $121$
Order $55$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(121, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(4,121))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(121\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial

First 31 of 40 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{121}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{121}(5,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{121}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{121}(15,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{121}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{121}(20,\cdot)\) \(1\) \(1\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{121}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{121}(26,\cdot)\) \(1\) \(1\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{121}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{121}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{121}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{121}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{121}(42,\cdot)\) \(1\) \(1\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{121}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{121}(48,\cdot)\) \(1\) \(1\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{121}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{121}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{121}(58,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{121}(59,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{121}(60,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{121}(64,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{121}(69,\cdot)\) \(1\) \(1\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{121}(70,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{121}(71,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{121}(75,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{121}(80,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{121}(82,\cdot)\) \(1\) \(1\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{121}(86,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{121}(91,\cdot)\) \(1\) \(1\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{121}(92,\cdot)\) \(1\) \(1\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{121}(93,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\)