Properties

Label 900.337
Modulus $900$
Conductor $225$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(900, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,20,27]))
 
pari: [g,chi] = znchar(Mod(337,900))
 

Basic properties

Modulus: \(900\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(112,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 900.bv

\(\chi_{900}(13,\cdot)\) \(\chi_{900}(97,\cdot)\) \(\chi_{900}(133,\cdot)\) \(\chi_{900}(277,\cdot)\) \(\chi_{900}(313,\cdot)\) \(\chi_{900}(337,\cdot)\) \(\chi_{900}(373,\cdot)\) \(\chi_{900}(517,\cdot)\) \(\chi_{900}(553,\cdot)\) \(\chi_{900}(637,\cdot)\) \(\chi_{900}(673,\cdot)\) \(\chi_{900}(697,\cdot)\) \(\chi_{900}(733,\cdot)\) \(\chi_{900}(817,\cdot)\) \(\chi_{900}(853,\cdot)\) \(\chi_{900}(877,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((451,101,577)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 900 }(337,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 900 }(337,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 900 }(337,·),\chi_{ 900 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 900 }(337,·)) \;\) at \(\; a,b = \) e.g. 1,2