from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,39]))
pari: [g,chi] = znchar(Mod(17,2700))
Basic properties
Modulus: | \(2700\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(167,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2700.cg
\(\chi_{2700}(17,\cdot)\) \(\chi_{2700}(197,\cdot)\) \(\chi_{2700}(233,\cdot)\) \(\chi_{2700}(413,\cdot)\) \(\chi_{2700}(737,\cdot)\) \(\chi_{2700}(773,\cdot)\) \(\chi_{2700}(953,\cdot)\) \(\chi_{2700}(1097,\cdot)\) \(\chi_{2700}(1277,\cdot)\) \(\chi_{2700}(1313,\cdot)\) \(\chi_{2700}(1637,\cdot)\) \(\chi_{2700}(1817,\cdot)\) \(\chi_{2700}(1853,\cdot)\) \(\chi_{2700}(2033,\cdot)\) \(\chi_{2700}(2177,\cdot)\) \(\chi_{2700}(2573,\cdot)\)
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
\((1351,1001,2377)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{13}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 2700 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)