Properties

Label 145.23
Modulus $145$
Conductor $145$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(145, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,20]))
 
pari: [g,chi] = znchar(Mod(23,145))
 

Basic properties

Modulus: \(145\)
Conductor: \(145\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 145.p

\(\chi_{145}(7,\cdot)\) \(\chi_{145}(23,\cdot)\) \(\chi_{145}(52,\cdot)\) \(\chi_{145}(53,\cdot)\) \(\chi_{145}(78,\cdot)\) \(\chi_{145}(82,\cdot)\) \(\chi_{145}(83,\cdot)\) \(\chi_{145}(103,\cdot)\) \(\chi_{145}(107,\cdot)\) \(\chi_{145}(112,\cdot)\) \(\chi_{145}(123,\cdot)\) \(\chi_{145}(132,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: 28.0.59692812354437378574125162510614243030548095703125.1

Values on generators

\((117,31)\) → \((-i,e\left(\frac{5}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(-i\)\(e\left(\frac{3}{28}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 145 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{145}(23,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(23,r) e\left(\frac{2r}{145}\right) = -12.0353250182+-0.3885250399i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 145 }(23,·),\chi_{ 145 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{145}(23,\cdot),\chi_{145}(1,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(23,r) \chi_{145}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 145 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{145}(23,·)) = \sum_{r \in \Z/145\Z} \chi_{145}(23,r) e\left(\frac{1 r + 2 r^{-1}}{145}\right) = -16.4223099094+1.8503499374i \)