Properties

Label 916.225
Modulus $916$
Conductor $229$
Order $19$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(916, base_ring=CyclotomicField(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,26]))
 
pari: [g,chi] = znchar(Mod(225,916))
 

Basic properties

Modulus: \(916\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{229}(225,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 916.m

\(\chi_{916}(17,\cdot)\) \(\chi_{916}(53,\cdot)\) \(\chi_{916}(57,\cdot)\) \(\chi_{916}(61,\cdot)\) \(\chi_{916}(121,\cdot)\) \(\chi_{916}(161,\cdot)\) \(\chi_{916}(165,\cdot)\) \(\chi_{916}(225,\cdot)\) \(\chi_{916}(245,\cdot)\) \(\chi_{916}(273,\cdot)\) \(\chi_{916}(289,\cdot)\) \(\chi_{916}(333,\cdot)\) \(\chi_{916}(485,\cdot)\) \(\chi_{916}(501,\cdot)\) \(\chi_{916}(661,\cdot)\) \(\chi_{916}(729,\cdot)\) \(\chi_{916}(901,\cdot)\) \(\chi_{916}(905,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 19.19.2999429662895796650415561622892044448017561.1

Values on generators

\((459,693)\) → \((1,e\left(\frac{13}{19}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{10}{19}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 916 }(225,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{916}(225,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(225,r) e\left(\frac{r}{458}\right) = 11.8204419029+27.8617507207i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 916 }(225,·),\chi_{ 916 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{916}(225,\cdot),\chi_{916}(1,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(225,r) \chi_{916}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 916 }(225,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{916}(225,·)) = \sum_{r \in \Z/916\Z} \chi_{916}(225,r) e\left(\frac{1 r + 2 r^{-1}}{916}\right) = -0.0 \)