from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5733, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,10,35]))
pari: [g,chi] = znchar(Mod(1180,5733))
Basic properties
| Modulus: | \(5733\) | |
| Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{637}(543,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5733.ii
\(\chi_{5733}(1180,\cdot)\) \(\chi_{5733}(1486,\cdot)\) \(\chi_{5733}(1999,\cdot)\) \(\chi_{5733}(2305,\cdot)\) \(\chi_{5733}(2818,\cdot)\) \(\chi_{5733}(3124,\cdot)\) \(\chi_{5733}(3637,\cdot)\) \(\chi_{5733}(3943,\cdot)\) \(\chi_{5733}(4456,\cdot)\) \(\chi_{5733}(4762,\cdot)\) \(\chi_{5733}(5275,\cdot)\) \(\chi_{5733}(5581,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.16423600478713504434070778628293678810006717122913176085381268066336462525553883868883157384200587461557.2 |
Values on generators
\((2549,1522,5293)\) → \((1,e\left(\frac{5}{21}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 5733 }(1180, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(-1\) | \(e\left(\frac{19}{42}\right)\) |
sage: chi.jacobi_sum(n)