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Properties

Label 113.h
Modulus $113$
Conductor $113$
Order $28$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(113, base_ring=CyclotomicField(28)) M = H._module chi = DirichletCharacter(H, M([3])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(2,113)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(113\)
Conductor: \(113\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(28\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{113}(2,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{113}(8,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{113}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{113}(32,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{113}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{113}(56,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{113}(57,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{113}(60,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{113}(81,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{113}(99,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{113}(105,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{113}(111,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{13}{14}\right)\)