Properties

Label 245.121
Modulus $245$
Conductor $49$
Order $21$
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(245, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,38]))
 
pari: [g,chi] = znchar(Mod(121,245))
 

Basic properties

Modulus: \(245\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 245.q

\(\chi_{245}(11,\cdot)\) \(\chi_{245}(16,\cdot)\) \(\chi_{245}(46,\cdot)\) \(\chi_{245}(51,\cdot)\) \(\chi_{245}(81,\cdot)\) \(\chi_{245}(86,\cdot)\) \(\chi_{245}(121,\cdot)\) \(\chi_{245}(151,\cdot)\) \(\chi_{245}(156,\cdot)\) \(\chi_{245}(186,\cdot)\) \(\chi_{245}(191,\cdot)\) \(\chi_{245}(221,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})^+\)

Values on generators

\((197,101)\) → \((1,e\left(\frac{19}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{21}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 245 }(121,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{245}(121,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(121,r) e\left(\frac{2r}{245}\right) = 2.9696217453+6.3388758222i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 245 }(121,·),\chi_{ 245 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{245}(121,\cdot),\chi_{245}(1,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(121,r) \chi_{245}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 245 }(121,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{245}(121,·)) = \sum_{r \in \Z/245\Z} \chi_{245}(121,r) e\left(\frac{1 r + 2 r^{-1}}{245}\right) = 2.3811863822+-31.7747240519i \)