Properties

Label 4012.29
Modulus $4012$
Conductor $1003$
Order $464$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(464))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,377,224]))
 
pari: [g,chi] = znchar(Mod(29,4012))
 

Basic properties

Modulus: \(4012\)
Conductor: \(1003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(464\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1003}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4012.bm

\(\chi_{4012}(5,\cdot)\) \(\chi_{4012}(29,\cdot)\) \(\chi_{4012}(41,\cdot)\) \(\chi_{4012}(45,\cdot)\) \(\chi_{4012}(57,\cdot)\) \(\chi_{4012}(105,\cdot)\) \(\chi_{4012}(125,\cdot)\) \(\chi_{4012}(133,\cdot)\) \(\chi_{4012}(181,\cdot)\) \(\chi_{4012}(193,\cdot)\) \(\chi_{4012}(197,\cdot)\) \(\chi_{4012}(241,\cdot)\) \(\chi_{4012}(245,\cdot)\) \(\chi_{4012}(261,\cdot)\) \(\chi_{4012}(265,\cdot)\) \(\chi_{4012}(277,\cdot)\) \(\chi_{4012}(317,\cdot)\) \(\chi_{4012}(369,\cdot)\) \(\chi_{4012}(381,\cdot)\) \(\chi_{4012}(405,\cdot)\) \(\chi_{4012}(449,\cdot)\) \(\chi_{4012}(481,\cdot)\) \(\chi_{4012}(513,\cdot)\) \(\chi_{4012}(517,\cdot)\) \(\chi_{4012}(521,\cdot)\) \(\chi_{4012}(605,\cdot)\) \(\chi_{4012}(609,\cdot)\) \(\chi_{4012}(617,\cdot)\) \(\chi_{4012}(641,\cdot)\) \(\chi_{4012}(653,\cdot)\) ...

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: $\Q(\zeta_{464})$
Fixed field: Number field defined by a degree 464 polynomial (not computed)

Values on generators

\((2007,3777,3129)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{14}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 4012 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{441}{464}\right)\)\(e\left(\frac{445}{464}\right)\)\(e\left(\frac{291}{464}\right)\)\(e\left(\frac{209}{232}\right)\)\(e\left(\frac{351}{464}\right)\)\(e\left(\frac{113}{116}\right)\)\(e\left(\frac{211}{232}\right)\)\(e\left(\frac{167}{232}\right)\)\(e\left(\frac{67}{116}\right)\)\(e\left(\frac{199}{464}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4012 }(29,a) \;\) at \(\;a = \) e.g. 2