Properties

Label 1800.313
Modulus $1800$
Conductor $225$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,40,57]))
 
pari: [g,chi] = znchar(Mod(313,1800))
 

Basic properties

Modulus: \(1800\)
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 \(\chi_{225}(88,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.dq

\(\chi_{1800}(97,\cdot)\) \(\chi_{1800}(313,\cdot)\) \(\chi_{1800}(337,\cdot)\) \(\chi_{1800}(553,\cdot)\) \(\chi_{1800}(673,\cdot)\) \(\chi_{1800}(697,\cdot)\) \(\chi_{1800}(817,\cdot)\) \(\chi_{1800}(913,\cdot)\) \(\chi_{1800}(1033,\cdot)\) \(\chi_{1800}(1177,\cdot)\) \(\chi_{1800}(1273,\cdot)\) \(\chi_{1800}(1417,\cdot)\) \(\chi_{1800}(1537,\cdot)\) \(\chi_{1800}(1633,\cdot)\) \(\chi_{1800}(1753,\cdot)\) \(\chi_{1800}(1777,\cdot)\)

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

\((1351,901,1001,577)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{19}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(313, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{2}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(313,a) \;\) at \(\;a = \) e.g. 2