from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8034, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,0,31]))
pari: [g,chi] = znchar(Mod(1067,8034))
Basic properties
| Modulus: | \(8034\) | |
| Conductor: | \(309\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{309}(140,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8034.cj
\(\chi_{8034}(209,\cdot)\) \(\chi_{8034}(443,\cdot)\) \(\chi_{8034}(1067,\cdot)\) \(\chi_{8034}(1223,\cdot)\) \(\chi_{8034}(2393,\cdot)\) \(\chi_{8034}(2705,\cdot)\) \(\chi_{8034}(2861,\cdot)\) \(\chi_{8034}(3797,\cdot)\) \(\chi_{8034}(3953,\cdot)\) \(\chi_{8034}(4265,\cdot)\) \(\chi_{8034}(4421,\cdot)\) \(\chi_{8034}(4811,\cdot)\) \(\chi_{8034}(6293,\cdot)\) \(\chi_{8034}(6995,\cdot)\) \(\chi_{8034}(7073,\cdot)\) \(\chi_{8034}(7541,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | 34.34.342523005011894297428856269332610453116457630461733441736562419892654124149.1 |
Values on generators
\((5357,1237,5773)\) → \((-1,1,e\left(\frac{31}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8034 }(1067, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) |
sage: chi.jacobi_sum(n)