Properties

Label 3630.113
Modulus $3630$
Conductor $1815$
Order $220$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3630, base_ring=CyclotomicField(220))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,165,116]))
 
pari: [g,chi] = znchar(Mod(113,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(1815\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1815}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bt

\(\chi_{3630}(47,\cdot)\) \(\chi_{3630}(53,\cdot)\) \(\chi_{3630}(113,\cdot)\) \(\chi_{3630}(137,\cdot)\) \(\chi_{3630}(203,\cdot)\) \(\chi_{3630}(257,\cdot)\) \(\chi_{3630}(317,\cdot)\) \(\chi_{3630}(377,\cdot)\) \(\chi_{3630}(383,\cdot)\) \(\chi_{3630}(443,\cdot)\) \(\chi_{3630}(467,\cdot)\) \(\chi_{3630}(533,\cdot)\) \(\chi_{3630}(587,\cdot)\) \(\chi_{3630}(647,\cdot)\) \(\chi_{3630}(653,\cdot)\) \(\chi_{3630}(707,\cdot)\) \(\chi_{3630}(713,\cdot)\) \(\chi_{3630}(773,\cdot)\) \(\chi_{3630}(797,\cdot)\) \(\chi_{3630}(863,\cdot)\) \(\chi_{3630}(917,\cdot)\) \(\chi_{3630}(983,\cdot)\) \(\chi_{3630}(1037,\cdot)\) \(\chi_{3630}(1043,\cdot)\) \(\chi_{3630}(1103,\cdot)\) \(\chi_{3630}(1127,\cdot)\) \(\chi_{3630}(1193,\cdot)\) \(\chi_{3630}(1247,\cdot)\) \(\chi_{3630}(1307,\cdot)\) \(\chi_{3630}(1313,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((1211,727,3511)\) → \((-1,-i,e\left(\frac{29}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3630 }(113, a) \) \(1\)\(1\)\(e\left(\frac{97}{220}\right)\)\(e\left(\frac{111}{220}\right)\)\(e\left(\frac{19}{220}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{197}{220}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{19}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3630 }(113,a) \;\) at \(\;a = \) e.g. 2