Properties

Label 3724.dh
Modulus $3724$
Conductor $931$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,6,7]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(141,3724))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 931.br
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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.4012683575982151476651625353295463545613564177862358851055893007057333972430504464680793872531537508239499.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(23\) \(25\) \(27\)
\(\chi_{3724}(141,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{3724}(449,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{3724}(673,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{3724}(1205,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{3724}(1513,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{3724}(1737,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{3724}(2045,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{3724}(2269,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{3724}(2577,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{3724}(2801,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{3724}(3109,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{3724}(3641,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{9}{14}\right)\)