from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,51,20]))
pari: [g,chi] = znchar(Mod(197,5700))
Basic properties
Modulus: | \(5700\) | |
Conductor: | \(1425\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1425}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5700.eq
\(\chi_{5700}(197,\cdot)\) \(\chi_{5700}(353,\cdot)\) \(\chi_{5700}(653,\cdot)\) \(\chi_{5700}(1037,\cdot)\) \(\chi_{5700}(1337,\cdot)\) \(\chi_{5700}(2177,\cdot)\) \(\chi_{5700}(2477,\cdot)\) \(\chi_{5700}(2633,\cdot)\) \(\chi_{5700}(2933,\cdot)\) \(\chi_{5700}(3317,\cdot)\) \(\chi_{5700}(3617,\cdot)\) \(\chi_{5700}(3773,\cdot)\) \(\chi_{5700}(4073,\cdot)\) \(\chi_{5700}(4913,\cdot)\) \(\chi_{5700}(5213,\cdot)\) \(\chi_{5700}(5597,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2851,1901,3877,4201)\) → \((1,-1,e\left(\frac{17}{20}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5700 }(197, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)