from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,50,48]))
pari: [g,chi] = znchar(Mod(311,7200))
Basic properties
| Modulus: | \(7200\) | |
| Conductor: | \(3600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3600}(3011,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7200.he
\(\chi_{7200}(311,\cdot)\) \(\chi_{7200}(1031,\cdot)\) \(\chi_{7200}(1271,\cdot)\) \(\chi_{7200}(1991,\cdot)\) \(\chi_{7200}(2471,\cdot)\) \(\chi_{7200}(2711,\cdot)\) \(\chi_{7200}(3191,\cdot)\) \(\chi_{7200}(3431,\cdot)\) \(\chi_{7200}(3911,\cdot)\) \(\chi_{7200}(4631,\cdot)\) \(\chi_{7200}(4871,\cdot)\) \(\chi_{7200}(5591,\cdot)\) \(\chi_{7200}(6071,\cdot)\) \(\chi_{7200}(6311,\cdot)\) \(\chi_{7200}(6791,\cdot)\) \(\chi_{7200}(7031,\cdot)\)
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
\((6751,901,6401,577)\) → \((-1,-i,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7200 }(311, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)