Properties

Label 3600.337
Modulus $3600$
Conductor $225$
Order $60$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,20,27]))
 
pari: [g,chi] = znchar(Mod(337,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(112,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3600.fy

\(\chi_{3600}(97,\cdot)\) \(\chi_{3600}(337,\cdot)\) \(\chi_{3600}(673,\cdot)\) \(\chi_{3600}(817,\cdot)\) \(\chi_{3600}(913,\cdot)\) \(\chi_{3600}(1537,\cdot)\) \(\chi_{3600}(1633,\cdot)\) \(\chi_{3600}(1777,\cdot)\) \(\chi_{3600}(2113,\cdot)\) \(\chi_{3600}(2353,\cdot)\) \(\chi_{3600}(2497,\cdot)\) \(\chi_{3600}(2833,\cdot)\) \(\chi_{3600}(2977,\cdot)\) \(\chi_{3600}(3073,\cdot)\) \(\chi_{3600}(3217,\cdot)\) \(\chi_{3600}(3553,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3151,901,2801,577)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(337, a) \) \(-1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(337,a) \;\) at \(\;a = \) e.g. 2