Properties

Label 3528.2993
Modulus 35283528
Conductor 441441
Order 4242
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,0,35,10]))
 
Copy content pari:[g,chi] = znchar(Mod(2993,3528))
 

Basic properties

Modulus: 35283528
Conductor: 441441
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4242
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ441(347,)\chi_{441}(347,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3528.gc

χ3528(65,)\chi_{3528}(65,\cdot) χ3528(473,)\chi_{3528}(473,\cdot) χ3528(977,)\chi_{3528}(977,\cdot) χ3528(1073,)\chi_{3528}(1073,\cdot) χ3528(1481,)\chi_{3528}(1481,\cdot) χ3528(1577,)\chi_{3528}(1577,\cdot) χ3528(1985,)\chi_{3528}(1985,\cdot) χ3528(2081,)\chi_{3528}(2081,\cdot) χ3528(2489,)\chi_{3528}(2489,\cdot) χ3528(2585,)\chi_{3528}(2585,\cdot) χ3528(2993,)\chi_{3528}(2993,\cdot) χ3528(3089,)\chi_{3528}(3089,\cdot)

Copy content sage:chi.galois_orbit()
 
Copy content pari: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: Number field defined by a degree 42 polynomial

Values on generators

(2647,1765,785,1081)(2647,1765,785,1081)(1,1,e(56),e(521))(1,1,e\left(\frac{5}{6}\right),e\left(\frac{5}{21}\right))

First values

aa 1-11155111113131717191923232525292931313737
χ3528(2993,a) \chi_{ 3528 }(2993, a) 1-111e(114)e\left(\frac{1}{14}\right)e(514)e\left(\frac{5}{14}\right)e(1121)e\left(\frac{11}{21}\right)e(1942)e\left(\frac{19}{42}\right)e(13)e\left(\frac{1}{3}\right)e(314)e\left(\frac{3}{14}\right)e(17)e\left(\frac{1}{7}\right)e(542)e\left(\frac{5}{42}\right)e(13)e\left(\frac{1}{3}\right)e(1321)e\left(\frac{13}{21}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ3528(2993,a)   \chi_{ 3528 }(2993,a) \; at   a=\;a = e.g. 2