Properties

Label 421.48
Modulus $421$
Conductor $421$
Order $35$
Real no
Primitive yes
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(421, base_ring=CyclotomicField(70))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([68]))
 
pari: [g,chi] = znchar(Mod(48,421))
 

Basic properties

Modulus: \(421\)
Conductor: \(421\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 421.p

\(\chi_{421}(27,\cdot)\) \(\chi_{421}(48,\cdot)\) \(\chi_{421}(49,\cdot)\) \(\chi_{421}(60,\cdot)\) \(\chi_{421}(68,\cdot)\) \(\chi_{421}(78,\cdot)\) \(\chi_{421}(85,\cdot)\) \(\chi_{421}(139,\cdot)\) \(\chi_{421}(190,\cdot)\) \(\chi_{421}(199,\cdot)\) \(\chi_{421}(232,\cdot)\) \(\chi_{421}(290,\cdot)\) \(\chi_{421}(291,\cdot)\) \(\chi_{421}(296,\cdot)\) \(\chi_{421}(307,\cdot)\) \(\chi_{421}(308,\cdot)\) \(\chi_{421}(315,\cdot)\) \(\chi_{421}(317,\cdot)\) \(\chi_{421}(321,\cdot)\) \(\chi_{421}(341,\cdot)\) \(\chi_{421}(357,\cdot)\) \(\chi_{421}(366,\cdot)\) \(\chi_{421}(376,\cdot)\) \(\chi_{421}(414,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: 35.35.168109113671617086350535469888345045006842268262463039332288137975520602419531747373726681.1

Values on generators

\(2\) → \(e\left(\frac{34}{35}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{9}{35}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 421 }(48,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{421}(48,\cdot)) = \sum_{r\in \Z/421\Z} \chi_{421}(48,r) e\left(\frac{2r}{421}\right) = 18.2208472845+9.4340195164i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 421 }(48,·),\chi_{ 421 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{421}(48,\cdot),\chi_{421}(1,\cdot)) = \sum_{r\in \Z/421\Z} \chi_{421}(48,r) \chi_{421}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 421 }(48,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{421}(48,·)) = \sum_{r \in \Z/421\Z} \chi_{421}(48,r) e\left(\frac{1 r + 2 r^{-1}}{421}\right) = -16.0034936953+1.4403405023i \)