↑

Properties

Label 1034.7
Modulus $1034$
Conductor $517$
Order $230$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1034, base_ring=CyclotomicField(230)) M = H._module chi = DirichletCharacter(H, M([161,160]))
 
Copy content gp:[g,chi] = znchar(Mod(7, 1034))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1034.7");
 

Basic properties

Modulus: \(1034\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(517\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(230\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{517}(7,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1034.p

\(\chi_{1034}(7,\cdot)\) \(\chi_{1034}(17,\cdot)\) \(\chi_{1034}(51,\cdot)\) \(\chi_{1034}(61,\cdot)\) \(\chi_{1034}(63,\cdot)\) \(\chi_{1034}(79,\cdot)\) \(\chi_{1034}(83,\cdot)\) \(\chi_{1034}(101,\cdot)\) \(\chi_{1034}(145,\cdot)\) \(\chi_{1034}(149,\cdot)\) \(\chi_{1034}(173,\cdot)\) \(\chi_{1034}(183,\cdot)\) \(\chi_{1034}(195,\cdot)\) \(\chi_{1034}(205,\cdot)\) \(\chi_{1034}(215,\cdot)\) \(\chi_{1034}(237,\cdot)\) \(\chi_{1034}(239,\cdot)\) \(\chi_{1034}(249,\cdot)\) \(\chi_{1034}(259,\cdot)\) \(\chi_{1034}(271,\cdot)\) \(\chi_{1034}(277,\cdot)\) \(\chi_{1034}(299,\cdot)\) \(\chi_{1034}(303,\cdot)\) \(\chi_{1034}(337,\cdot)\) \(\chi_{1034}(343,\cdot)\) \(\chi_{1034}(347,\cdot)\) \(\chi_{1034}(365,\cdot)\) \(\chi_{1034}(371,\cdot)\) \(\chi_{1034}(393,\cdot)\) \(\chi_{1034}(403,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

Values on generators

\((189,287)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{16}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 1034 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{59}{115}\right)\)\(e\left(\frac{57}{115}\right)\)\(e\left(\frac{37}{230}\right)\)\(e\left(\frac{3}{115}\right)\)\(e\left(\frac{81}{230}\right)\)\(e\left(\frac{1}{115}\right)\)\(e\left(\frac{99}{230}\right)\)\(e\left(\frac{93}{230}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{11}{23}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 1034 }(7,a) \;\) at \(\;a = \) e.g. 2