Properties

Label 916.203
Modulus $916$
Conductor $916$
Order $38$
Real no
Primitive yes
Minimal yes
Parity odd

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([19,4]))
 
pari: [g,chi] = znchar(Mod(203,916))
 

Basic properties

Modulus: \(916\)
Conductor: \(916\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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 916.p

\(\chi_{916}(27,\cdot)\) \(\chi_{916}(43,\cdot)\) \(\chi_{916}(203,\cdot)\) \(\chi_{916}(271,\cdot)\) \(\chi_{916}(443,\cdot)\) \(\chi_{916}(447,\cdot)\) \(\chi_{916}(475,\cdot)\) \(\chi_{916}(511,\cdot)\) \(\chi_{916}(515,\cdot)\) \(\chi_{916}(519,\cdot)\) \(\chi_{916}(579,\cdot)\) \(\chi_{916}(619,\cdot)\) \(\chi_{916}(623,\cdot)\) \(\chi_{916}(683,\cdot)\) \(\chi_{916}(703,\cdot)\) \(\chi_{916}(731,\cdot)\) \(\chi_{916}(747,\cdot)\) \(\chi_{916}(791,\cdot)\)

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: 38.0.2472960613492762938009352687218362626942035203162587025151809700624254848124906369677396881178624.1

Values on generators

\((459,693)\) → \((-1,e\left(\frac{2}{19}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{15}{38}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{21}{38}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{27}{38}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{7}{38}\right)\)\(e\left(\frac{3}{19}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 916 }(203,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{916}(203,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(203,r) e\left(\frac{r}{458}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 916 }(203,·),\chi_{ 916 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{916}(203,\cdot),\chi_{916}(1,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(203,r) \chi_{916}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 916 }(203,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{916}(203,·)) = \sum_{r \in \Z/916\Z} \chi_{916}(203,r) e\left(\frac{1 r + 2 r^{-1}}{916}\right) = -16.5401782137+21.2508214034i \)