Properties

Label 3528.461
Modulus 35283528
Conductor 35283528
Order 4242
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,7,39]))
 
pari: [g,chi] = znchar(Mod(461,3528))
 

Basic properties

Modulus: 35283528
Conductor: 35283528
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 4242
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 3528.ev

χ3528(461,)\chi_{3528}(461,\cdot) χ3528(797,)\chi_{3528}(797,\cdot) χ3528(965,)\chi_{3528}(965,\cdot) χ3528(1301,)\chi_{3528}(1301,\cdot) χ3528(1805,)\chi_{3528}(1805,\cdot) χ3528(1973,)\chi_{3528}(1973,\cdot) χ3528(2309,)\chi_{3528}(2309,\cdot) χ3528(2477,)\chi_{3528}(2477,\cdot) χ3528(2813,)\chi_{3528}(2813,\cdot) χ3528(2981,)\chi_{3528}(2981,\cdot) χ3528(3317,)\chi_{3528}(3317,\cdot) χ3528(3485,)\chi_{3528}(3485,\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(ζ21)\Q(\zeta_{21})
Fixed field: 42.42.25461370422593899025010993639968319493979540256968222705473361226986223705326760477191735540542813852487110361088.1

Values on generators

(2647,1765,785,1081)(2647,1765,785,1081)(1,1,e(16),e(1314))(1,-1,e\left(\frac{1}{6}\right),e\left(\frac{13}{14}\right))

First values

aa 1-11155111113131717191923232525292931313737
χ3528(461,a) \chi_{ 3528 }(461, a) 1111e(1142)e\left(\frac{11}{42}\right)e(1721)e\left(\frac{17}{21}\right)e(1021)e\left(\frac{10}{21}\right)e(57)e\left(\frac{5}{7}\right)11e(542)e\left(\frac{5}{42}\right)e(1121)e\left(\frac{11}{21}\right)e(821)e\left(\frac{8}{21}\right)e(56)e\left(\frac{5}{6}\right)e(314)e\left(\frac{3}{14}\right)
sage: chi.jacobi_sum(n)
 
χ3528(461,a)   \chi_{ 3528 }(461,a) \; at   a=\;a = e.g. 2