Properties

Label 2601.20
Modulus $2601$
Conductor $2601$
Order $816$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(816))
 
M = H._module
 
chi = DirichletCharacter(H, M([136,195]))
 
pari: [g,chi] = znchar(Mod(20,2601))
 

Basic properties

Modulus: \(2601\)
Conductor: \(2601\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(816\)
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 2601.bn

\(\chi_{2601}(5,\cdot)\) \(\chi_{2601}(11,\cdot)\) \(\chi_{2601}(14,\cdot)\) \(\chi_{2601}(20,\cdot)\) \(\chi_{2601}(23,\cdot)\) \(\chi_{2601}(29,\cdot)\) \(\chi_{2601}(41,\cdot)\) \(\chi_{2601}(56,\cdot)\) \(\chi_{2601}(74,\cdot)\) \(\chi_{2601}(92,\cdot)\) \(\chi_{2601}(95,\cdot)\) \(\chi_{2601}(113,\cdot)\) \(\chi_{2601}(122,\cdot)\) \(\chi_{2601}(146,\cdot)\) \(\chi_{2601}(164,\cdot)\) \(\chi_{2601}(167,\cdot)\) \(\chi_{2601}(173,\cdot)\) \(\chi_{2601}(176,\cdot)\) \(\chi_{2601}(182,\cdot)\) \(\chi_{2601}(194,\cdot)\) \(\chi_{2601}(209,\cdot)\) \(\chi_{2601}(218,\cdot)\) \(\chi_{2601}(227,\cdot)\) \(\chi_{2601}(245,\cdot)\) \(\chi_{2601}(248,\cdot)\) \(\chi_{2601}(266,\cdot)\) \(\chi_{2601}(275,\cdot)\) \(\chi_{2601}(284,\cdot)\) \(\chi_{2601}(299,\cdot)\) \(\chi_{2601}(311,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{816})$
Fixed field: Number field defined by a degree 816 polynomial (not computed)

Values on generators

\((290,2026)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{65}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2601 }(20, a) \) \(1\)\(1\)\(e\left(\frac{233}{408}\right)\)\(e\left(\frac{29}{204}\right)\)\(e\left(\frac{455}{816}\right)\)\(e\left(\frac{577}{816}\right)\)\(e\left(\frac{97}{136}\right)\)\(e\left(\frac{35}{272}\right)\)\(e\left(\frac{541}{816}\right)\)\(e\left(\frac{35}{204}\right)\)\(e\left(\frac{227}{816}\right)\)\(e\left(\frac{29}{102}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2601 }(20,a) \;\) at \(\;a = \) e.g. 2