Properties

Label 297.41
Modulus 297297
Conductor 297297
Order 9090
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([85,27]))
 
pari: [g,chi] = znchar(Mod(41,297))
 

Basic properties

Modulus: 297297
Conductor: 297297
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 9090
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.x

χ297(2,)\chi_{297}(2,\cdot) χ297(29,)\chi_{297}(29,\cdot) χ297(41,)\chi_{297}(41,\cdot) χ297(50,)\chi_{297}(50,\cdot) χ297(68,)\chi_{297}(68,\cdot) χ297(74,)\chi_{297}(74,\cdot) χ297(83,)\chi_{297}(83,\cdot) χ297(95,)\chi_{297}(95,\cdot) χ297(101,)\chi_{297}(101,\cdot) χ297(128,)\chi_{297}(128,\cdot) χ297(140,)\chi_{297}(140,\cdot) χ297(149,)\chi_{297}(149,\cdot) χ297(167,)\chi_{297}(167,\cdot) χ297(173,)\chi_{297}(173,\cdot) χ297(182,)\chi_{297}(182,\cdot) χ297(194,)\chi_{297}(194,\cdot) χ297(200,)\chi_{297}(200,\cdot) χ297(227,)\chi_{297}(227,\cdot) χ297(239,)\chi_{297}(239,\cdot) χ297(248,)\chi_{297}(248,\cdot) χ297(266,)\chi_{297}(266,\cdot) χ297(272,)\chi_{297}(272,\cdot) χ297(281,)\chi_{297}(281,\cdot) χ297(293,)\chi_{297}(293,\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(ζ45)\Q(\zeta_{45})
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

(56,244)(56,244)(e(1718),e(310))(e\left(\frac{17}{18}\right),e\left(\frac{3}{10}\right))

First values

aa 1-111224455778810101313141416161717
χ297(41,a) \chi_{ 297 }(41, a) 1111e(1145)e\left(\frac{11}{45}\right)e(2245)e\left(\frac{22}{45}\right)e(8390)e\left(\frac{83}{90}\right)e(1990)e\left(\frac{19}{90}\right)e(1115)e\left(\frac{11}{15}\right)e(16)e\left(\frac{1}{6}\right)e(7790)e\left(\frac{77}{90}\right)e(4190)e\left(\frac{41}{90}\right)e(4445)e\left(\frac{44}{45}\right)e(1315)e\left(\frac{13}{15}\right)
sage: chi.jacobi_sum(n)
 
χ297(41,a)   \chi_{ 297 }(41,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ297(41,))   \tau_{ a }( \chi_{ 297 }(41,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ297(41,),χ297(n,))   J(\chi_{ 297 }(41,·),\chi_{ 297 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ297(41,))  K(a,b,\chi_{ 297 }(41,·)) \; at   a,b=\; a,b = e.g. 1,2