Properties

Label 2450.93
Modulus $2450$
Conductor $245$
Order $84$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,16]))
 
pari: [g,chi] = znchar(Mod(93,2450))
 

Basic properties

Modulus: \(2450\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(93,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2450.bn

\(\chi_{2450}(93,\cdot)\) \(\chi_{2450}(107,\cdot)\) \(\chi_{2450}(193,\cdot)\) \(\chi_{2450}(207,\cdot)\) \(\chi_{2450}(443,\cdot)\) \(\chi_{2450}(457,\cdot)\) \(\chi_{2450}(543,\cdot)\) \(\chi_{2450}(793,\cdot)\) \(\chi_{2450}(807,\cdot)\) \(\chi_{2450}(893,\cdot)\) \(\chi_{2450}(907,\cdot)\) \(\chi_{2450}(1143,\cdot)\) \(\chi_{2450}(1257,\cdot)\) \(\chi_{2450}(1493,\cdot)\) \(\chi_{2450}(1507,\cdot)\) \(\chi_{2450}(1593,\cdot)\) \(\chi_{2450}(1607,\cdot)\) \(\chi_{2450}(1857,\cdot)\) \(\chi_{2450}(1943,\cdot)\) \(\chi_{2450}(1957,\cdot)\) \(\chi_{2450}(2193,\cdot)\) \(\chi_{2450}(2207,\cdot)\) \(\chi_{2450}(2293,\cdot)\) \(\chi_{2450}(2307,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1177,101)\) → \((-i,e\left(\frac{4}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2450 }(93, a) \) \(-1\)\(1\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2450 }(93,a) \;\) at \(\;a = \) e.g. 2