from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9196, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,18,22]))
pari: [g,chi] = znchar(Mod(45,9196))
Basic properties
Modulus: | \(9196\) | |
Conductor: | \(2299\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2299}(45,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9196.bw
\(\chi_{9196}(45,\cdot)\) \(\chi_{9196}(353,\cdot)\) \(\chi_{9196}(881,\cdot)\) \(\chi_{9196}(1189,\cdot)\) \(\chi_{9196}(1717,\cdot)\) \(\chi_{9196}(2025,\cdot)\) \(\chi_{9196}(2553,\cdot)\) \(\chi_{9196}(2861,\cdot)\) \(\chi_{9196}(3697,\cdot)\) \(\chi_{9196}(4225,\cdot)\) \(\chi_{9196}(4533,\cdot)\) \(\chi_{9196}(5061,\cdot)\) \(\chi_{9196}(5369,\cdot)\) \(\chi_{9196}(5897,\cdot)\) \(\chi_{9196}(6205,\cdot)\) \(\chi_{9196}(6733,\cdot)\) \(\chi_{9196}(7041,\cdot)\) \(\chi_{9196}(7569,\cdot)\) \(\chi_{9196}(7877,\cdot)\) \(\chi_{9196}(8405,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 33 polynomial |
Values on generators
\((4599,3269,8229)\) → \((1,e\left(\frac{3}{11}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 9196 }(45, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)