from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,26]))
pari: [g,chi] = znchar(Mod(67,230))
Basic properties
| Modulus: | \(230\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 230.l
\(\chi_{230}(7,\cdot)\) \(\chi_{230}(17,\cdot)\) \(\chi_{230}(33,\cdot)\) \(\chi_{230}(37,\cdot)\) \(\chi_{230}(43,\cdot)\) \(\chi_{230}(53,\cdot)\) \(\chi_{230}(57,\cdot)\) \(\chi_{230}(63,\cdot)\) \(\chi_{230}(67,\cdot)\) \(\chi_{230}(83,\cdot)\) \(\chi_{230}(97,\cdot)\) \(\chi_{230}(103,\cdot)\) \(\chi_{230}(107,\cdot)\) \(\chi_{230}(113,\cdot)\) \(\chi_{230}(143,\cdot)\) \(\chi_{230}(153,\cdot)\) \(\chi_{230}(157,\cdot)\) \(\chi_{230}(203,\cdot)\) \(\chi_{230}(217,\cdot)\) \(\chi_{230}(227,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((47,51)\) → \((i,e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 230 }(67, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)