Properties

Label 297.97
Modulus $297$
Conductor $297$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,54]))
 
pari: [g,chi] = znchar(Mod(97,297))
 

Basic properties

Modulus: \(297\)
Conductor: \(297\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 297.u

\(\chi_{297}(4,\cdot)\) \(\chi_{297}(16,\cdot)\) \(\chi_{297}(25,\cdot)\) \(\chi_{297}(31,\cdot)\) \(\chi_{297}(49,\cdot)\) \(\chi_{297}(58,\cdot)\) \(\chi_{297}(70,\cdot)\) \(\chi_{297}(97,\cdot)\) \(\chi_{297}(103,\cdot)\) \(\chi_{297}(115,\cdot)\) \(\chi_{297}(124,\cdot)\) \(\chi_{297}(130,\cdot)\) \(\chi_{297}(148,\cdot)\) \(\chi_{297}(157,\cdot)\) \(\chi_{297}(169,\cdot)\) \(\chi_{297}(196,\cdot)\) \(\chi_{297}(202,\cdot)\) \(\chi_{297}(214,\cdot)\) \(\chi_{297}(223,\cdot)\) \(\chi_{297}(229,\cdot)\) \(\chi_{297}(247,\cdot)\) \(\chi_{297}(256,\cdot)\) \(\chi_{297}(268,\cdot)\) \(\chi_{297}(295,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: 45.45.940750991812442660132286068374700929301371405411689329186316523736000865610069804102171689.1

Values on generators

\((56,244)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 297 }(97, a) \) \(1\)\(1\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 297 }(97,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 297 }(97,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 297 }(97,·),\chi_{ 297 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 297 }(97,·)) \;\) at \(\; a,b = \) e.g. 1,2