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Properties

Label 3600.eg
Modulus $3600$
Conductor $400$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([0,5,0,9])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(37,3600)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(3600\)
Conductor: \(400\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 400.bn
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.0.104857600000000000000000000000000000000000.2

Characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\)
\(\chi_{3600}(37,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{3600}(253,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{3600}(973,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{3600}(1477,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{3600}(2197,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{3600}(2413,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{3600}(2917,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{3600}(3133,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{10}\right)\)