Properties

Label 354.275
Modulus $354$
Conductor $177$
Order $58$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([29,37]))
 
pari: [g,chi] = znchar(Mod(275,354))
 

Basic properties

Modulus: \(354\)
Conductor: \(177\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{177}(98,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 354.g

\(\chi_{354}(11,\cdot)\) \(\chi_{354}(23,\cdot)\) \(\chi_{354}(47,\cdot)\) \(\chi_{354}(65,\cdot)\) \(\chi_{354}(77,\cdot)\) \(\chi_{354}(83,\cdot)\) \(\chi_{354}(89,\cdot)\) \(\chi_{354}(101,\cdot)\) \(\chi_{354}(113,\cdot)\) \(\chi_{354}(131,\cdot)\) \(\chi_{354}(149,\cdot)\) \(\chi_{354}(155,\cdot)\) \(\chi_{354}(161,\cdot)\) \(\chi_{354}(173,\cdot)\) \(\chi_{354}(179,\cdot)\) \(\chi_{354}(185,\cdot)\) \(\chi_{354}(191,\cdot)\) \(\chi_{354}(209,\cdot)\) \(\chi_{354}(215,\cdot)\) \(\chi_{354}(221,\cdot)\) \(\chi_{354}(227,\cdot)\) \(\chi_{354}(233,\cdot)\) \(\chi_{354}(269,\cdot)\) \(\chi_{354}(275,\cdot)\) \(\chi_{354}(305,\cdot)\) \(\chi_{354}(329,\cdot)\) \(\chi_{354}(335,\cdot)\) \(\chi_{354}(347,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((119,61)\) → \((-1,e\left(\frac{37}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 354 }(275, a) \) \(1\)\(1\)\(e\left(\frac{19}{58}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{41}{58}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{19}{29}\right)\)\(e\left(\frac{21}{58}\right)\)\(e\left(\frac{15}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 354 }(275,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 354 }(275,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 354 }(275,·),\chi_{ 354 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 354 }(275,·)) \;\) at \(\; a,b = \) e.g. 1,2