Properties

Label 407.bc
Modulus $407$
Conductor $407$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(407, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([54,40]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(9,407))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(407\)
Conductor: \(407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
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: $\Q(\zeta_{45})$
Fixed field: 45.45.16527441490674381067348948889146522048718287580785548225593089356497792068001085725808918264262962961.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{407}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{407}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{407}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{407}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{407}(70,\cdot)\) \(1\) \(1\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{407}(71,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{407}(81,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{407}(86,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{407}(108,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{407}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{407}(181,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{407}(192,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{407}(201,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{407}(218,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{407}(229,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{407}(234,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{407}(256,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{407}(268,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{407}(312,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{407}(345,\cdot)\) \(1\) \(1\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{407}(366,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{407}(367,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{407}(377,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{407}(379,\cdot)\) \(1\) \(1\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\)