from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69696, base_ring=CyclotomicField(2640))
M = H._module
chi = DirichletCharacter(H, M([1320,165,2200,768]))
pari: [g,chi] = znchar(Mod(59,69696))
Basic properties
| Modulus: | \(69696\) | |
| Conductor: | \(69696\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2640\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 69696.ma
\(\chi_{69696}(59,\cdot)\) \(\chi_{69696}(203,\cdot)\) \(\chi_{69696}(443,\cdot)\) \(\chi_{69696}(515,\cdot)\) \(\chi_{69696}(587,\cdot)\) \(\chi_{69696}(707,\cdot)\) \(\chi_{69696}(731,\cdot)\) \(\chi_{69696}(779,\cdot)\) \(\chi_{69696}(851,\cdot)\) \(\chi_{69696}(1235,\cdot)\) \(\chi_{69696}(1307,\cdot)\) \(\chi_{69696}(1379,\cdot)\) \(\chi_{69696}(1499,\cdot)\) \(\chi_{69696}(1523,\cdot)\)
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: | $\Q(\zeta_{2640})$ |
| Fixed field: | Number field defined by a degree 2640 polynomial (not computed) |
Values on generators
\((67519,4357,54209,14401)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{5}{6}\right),e\left(\frac{16}{55}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 69696 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{1997}{2640}\right)\) | \(e\left(\frac{653}{1320}\right)\) | \(e\left(\frac{2603}{2640}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{73}{880}\right)\) | \(e\left(\frac{239}{264}\right)\) | \(e\left(\frac{677}{1320}\right)\) | \(e\left(\frac{1231}{2640}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{221}{880}\right)\) |
sage: chi.jacobi_sum(n)