Properties

Label 3311.113
Modulus $3311$
Conductor $473$
Order $70$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3311, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,56,5]))
 
pari: [g,chi] = znchar(Mod(113,3311))
 

Basic properties

Modulus: \(3311\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3311.er

\(\chi_{3311}(113,\cdot)\) \(\chi_{3311}(323,\cdot)\) \(\chi_{3311}(610,\cdot)\) \(\chi_{3311}(806,\cdot)\) \(\chi_{3311}(862,\cdot)\) \(\chi_{3311}(911,\cdot)\) \(\chi_{3311}(1016,\cdot)\) \(\chi_{3311}(1114,\cdot)\) \(\chi_{3311}(1226,\cdot)\) \(\chi_{3311}(1527,\cdot)\) \(\chi_{3311}(1709,\cdot)\) \(\chi_{3311}(1765,\cdot)\) \(\chi_{3311}(1919,\cdot)\) \(\chi_{3311}(2017,\cdot)\) \(\chi_{3311}(2066,\cdot)\) \(\chi_{3311}(2115,\cdot)\) \(\chi_{3311}(2220,\cdot)\) \(\chi_{3311}(2612,\cdot)\) \(\chi_{3311}(2731,\cdot)\) \(\chi_{3311}(2913,\cdot)\) \(\chi_{3311}(2920,\cdot)\) \(\chi_{3311}(3018,\cdot)\) \(\chi_{3311}(3221,\cdot)\) \(\chi_{3311}(3270,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1893,904,2927)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 3311 }(113, a) \) \(-1\)\(1\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{69}{70}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{3}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3311 }(113,a) \;\) at \(\;a = \) e.g. 2