from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,0,39,40]))
pari: [g,chi] = znchar(Mod(67,8400))
Basic properties
Modulus: | \(8400\) | |
Conductor: | \(2800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2800}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8400.kx
\(\chi_{8400}(67,\cdot)\) \(\chi_{8400}(1003,\cdot)\) \(\chi_{8400}(1747,\cdot)\) \(\chi_{8400}(2683,\cdot)\) \(\chi_{8400}(2923,\cdot)\) \(\chi_{8400}(3187,\cdot)\) \(\chi_{8400}(3427,\cdot)\) \(\chi_{8400}(4363,\cdot)\) \(\chi_{8400}(4603,\cdot)\) \(\chi_{8400}(4867,\cdot)\) \(\chi_{8400}(6283,\cdot)\) \(\chi_{8400}(6547,\cdot)\) \(\chi_{8400}(6787,\cdot)\) \(\chi_{8400}(7723,\cdot)\) \(\chi_{8400}(7963,\cdot)\) \(\chi_{8400}(8227,\cdot)\)
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
\((3151,2101,2801,5377,3601)\) → \((-1,-i,1,e\left(\frac{13}{20}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 8400 }(67, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)