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Properties

Label 3636.bt
Modulus $3636$
Conductor $404$
Order $50$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3636, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([25,0,48])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(19, 3636)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3636.19"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(3636\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(404\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(50\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from 404.o
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{3636}(19,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{7}{50}\right)\)
\(\chi_{3636}(559,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{11}{50}\right)\)
\(\chi_{3636}(631,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{41}{50}\right)\)
\(\chi_{3636}(703,\cdot)\) \(-1\) \(1\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{9}{50}\right)\)
\(\chi_{3636}(775,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{19}{50}\right)\)
\(\chi_{3636}(1135,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{49}{50}\right)\)
\(\chi_{3636}(1243,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{3}{50}\right)\)
\(\chi_{3636}(1495,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{17}{50}\right)\)
\(\chi_{3636}(1531,\cdot)\) \(-1\) \(1\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{43}{50}\right)\)
\(\chi_{3636}(1567,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{31}{50}\right)\)
\(\chi_{3636}(1603,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{47}{50}\right)\)
\(\chi_{3636}(1855,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{27}{50}\right)\)
\(\chi_{3636}(1999,\cdot)\) \(-1\) \(1\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{1}{50}\right)\)
\(\chi_{3636}(2179,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{39}{50}\right)\)
\(\chi_{3636}(2503,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{13}{50}\right)\)
\(\chi_{3636}(2899,\cdot)\) \(-1\) \(1\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{23}{50}\right)\)
\(\chi_{3636}(3007,\cdot)\) \(-1\) \(1\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{37}{50}\right)\)
\(\chi_{3636}(3187,\cdot)\) \(-1\) \(1\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{29}{50}\right)\)
\(\chi_{3636}(3223,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{21}{50}\right)\)
\(\chi_{3636}(3439,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{33}{50}\right)\)