Properties

Label 4050.1157
Modulus 40504050
Conductor 405405
Order 108108
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(108)) M = H._module chi = DirichletCharacter(H, M([22,27]))
 
Copy content pari:[g,chi] = znchar(Mod(1157,4050))
 

Basic properties

Modulus: 40504050
Conductor: 405405
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 108108
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ405(347,)\chi_{405}(347,\cdot)
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)
 

Galois orbit 4050.bm

χ4050(257,)\chi_{4050}(257,\cdot) χ4050(293,)\chi_{4050}(293,\cdot) χ4050(407,)\chi_{4050}(407,\cdot) χ4050(443,)\chi_{4050}(443,\cdot) χ4050(707,)\chi_{4050}(707,\cdot) χ4050(743,)\chi_{4050}(743,\cdot) χ4050(857,)\chi_{4050}(857,\cdot) χ4050(893,)\chi_{4050}(893,\cdot) χ4050(1157,)\chi_{4050}(1157,\cdot) χ4050(1193,)\chi_{4050}(1193,\cdot) χ4050(1307,)\chi_{4050}(1307,\cdot) χ4050(1343,)\chi_{4050}(1343,\cdot) χ4050(1607,)\chi_{4050}(1607,\cdot) χ4050(1643,)\chi_{4050}(1643,\cdot) χ4050(1757,)\chi_{4050}(1757,\cdot) χ4050(1793,)\chi_{4050}(1793,\cdot) χ4050(2057,)\chi_{4050}(2057,\cdot) χ4050(2093,)\chi_{4050}(2093,\cdot) χ4050(2207,)\chi_{4050}(2207,\cdot) χ4050(2243,)\chi_{4050}(2243,\cdot) χ4050(2507,)\chi_{4050}(2507,\cdot) χ4050(2543,)\chi_{4050}(2543,\cdot) χ4050(2657,)\chi_{4050}(2657,\cdot) χ4050(2693,)\chi_{4050}(2693,\cdot) χ4050(2957,)\chi_{4050}(2957,\cdot) χ4050(2993,)\chi_{4050}(2993,\cdot) χ4050(3107,)\chi_{4050}(3107,\cdot) χ4050(3143,)\chi_{4050}(3143,\cdot) χ4050(3407,)\chi_{4050}(3407,\cdot) χ4050(3443,)\chi_{4050}(3443,\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(ζ108)\Q(\zeta_{108})
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

(2351,3727)(2351,3727)(e(1154),i)(e\left(\frac{11}{54}\right),i)

First values

aa 1-11177111113131717191923232929313137374141
χ4050(1157,a) \chi_{ 4050 }(1157, a) 1111e(55108)e\left(\frac{55}{108}\right)e(3554)e\left(\frac{35}{54}\right)e(41108)e\left(\frac{41}{108}\right)e(3536)e\left(\frac{35}{36}\right)e(518)e\left(\frac{5}{18}\right)e(107108)e\left(\frac{107}{108}\right)e(127)e\left(\frac{1}{27}\right)e(227)e\left(\frac{2}{27}\right)e(2936)e\left(\frac{29}{36}\right)e(4354)e\left(\frac{43}{54}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ4050(1157,a)   \chi_{ 4050 }(1157,a) \; at   a=\;a = e.g. 2