from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,6]))
pari: [g,chi] = znchar(Mod(333,430))
Basic properties
Modulus: | \(430\) | |
Conductor: | \(215\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{215}(118,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 430.r
\(\chi_{430}(27,\cdot)\) \(\chi_{430}(113,\cdot)\) \(\chi_{430}(137,\cdot)\) \(\chi_{430}(217,\cdot)\) \(\chi_{430}(223,\cdot)\) \(\chi_{430}(237,\cdot)\) \(\chi_{430}(247,\cdot)\) \(\chi_{430}(297,\cdot)\) \(\chi_{430}(303,\cdot)\) \(\chi_{430}(323,\cdot)\) \(\chi_{430}(333,\cdot)\) \(\chi_{430}(383,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.28.1407829094312471215113334241722636006626629352569580078125.1 |
Values on generators
\((87,261)\) → \((-i,e\left(\frac{3}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 430 }(333, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(i\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
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)