Properties

Label 3630.7
Modulus $3630$
Conductor $605$
Order $220$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3630)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,55,14]))
 
pari: [g,chi] = znchar(Mod(7,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bv

\(\chi_{3630}(7,\cdot)\) \(\chi_{3630}(13,\cdot)\) \(\chi_{3630}(73,\cdot)\) \(\chi_{3630}(127,\cdot)\) \(\chi_{3630}(193,\cdot)\) \(\chi_{3630}(217,\cdot)\) \(\chi_{3630}(277,\cdot)\) \(\chi_{3630}(283,\cdot)\) \(\chi_{3630}(337,\cdot)\) \(\chi_{3630}(343,\cdot)\) \(\chi_{3630}(523,\cdot)\) \(\chi_{3630}(547,\cdot)\) \(\chi_{3630}(607,\cdot)\) \(\chi_{3630}(613,\cdot)\) \(\chi_{3630}(667,\cdot)\) \(\chi_{3630}(673,\cdot)\) \(\chi_{3630}(733,\cdot)\) \(\chi_{3630}(787,\cdot)\) \(\chi_{3630}(853,\cdot)\) \(\chi_{3630}(877,\cdot)\) \(\chi_{3630}(937,\cdot)\) \(\chi_{3630}(943,\cdot)\) \(\chi_{3630}(997,\cdot)\) \(\chi_{3630}(1003,\cdot)\) \(\chi_{3630}(1063,\cdot)\) \(\chi_{3630}(1117,\cdot)\) \(\chi_{3630}(1267,\cdot)\) \(\chi_{3630}(1273,\cdot)\) \(\chi_{3630}(1327,\cdot)\) \(\chi_{3630}(1333,\cdot)\) ...

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

Values on generators

\((1211,727,3511)\) → \((1,i,e\left(\frac{7}{110}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{153}{220}\right)\)\(e\left(\frac{39}{220}\right)\)\(e\left(\frac{81}{220}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{203}{220}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{15}{44}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial