Properties

Label 276.197
Modulus $276$
Conductor $69$
Order $22$
Real no
Primitive no
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(276, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,14]))
 
pari: [g,chi] = znchar(Mod(197,276))
 

Basic properties

Modulus: \(276\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{69}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 276.n

\(\chi_{276}(29,\cdot)\) \(\chi_{276}(41,\cdot)\) \(\chi_{276}(77,\cdot)\) \(\chi_{276}(101,\cdot)\) \(\chi_{276}(173,\cdot)\) \(\chi_{276}(197,\cdot)\) \(\chi_{276}(209,\cdot)\) \(\chi_{276}(233,\cdot)\) \(\chi_{276}(257,\cdot)\) \(\chi_{276}(269,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.0.304011857053427966889939263171547.1

Values on generators

\((139,185,97)\) → \((1,-1,e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{5}{22}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 276 }(197,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{276}(197,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(197,r) e\left(\frac{r}{138}\right) = -15.9943073925+4.4924526746i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 276 }(197,·),\chi_{ 276 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{276}(197,\cdot),\chi_{276}(1,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(197,r) \chi_{276}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 276 }(197,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{276}(197,·)) = \sum_{r \in \Z/276\Z} \chi_{276}(197,r) e\left(\frac{1 r + 2 r^{-1}}{276}\right) = -0.0 \)