from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9036, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,13]))
pari: [g,chi] = znchar(Mod(235,9036))
Basic properties
| Modulus: | \(9036\) | |
| Conductor: | \(1004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1004}(235,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9036.bh
\(\chi_{9036}(235,\cdot)\) \(\chi_{9036}(379,\cdot)\) \(\chi_{9036}(451,\cdot)\) \(\chi_{9036}(2179,\cdot)\) \(\chi_{9036}(3007,\cdot)\) \(\chi_{9036}(3259,\cdot)\) \(\chi_{9036}(4771,\cdot)\) \(\chi_{9036}(4951,\cdot)\) \(\chi_{9036}(5311,\cdot)\) \(\chi_{9036}(6175,\cdot)\) \(\chi_{9036}(6211,\cdot)\) \(\chi_{9036}(6283,\cdot)\) \(\chi_{9036}(6463,\cdot)\) \(\chi_{9036}(6787,\cdot)\) \(\chi_{9036}(7003,\cdot)\) \(\chi_{9036}(7075,\cdot)\) \(\chi_{9036}(7687,\cdot)\) \(\chi_{9036}(7831,\cdot)\) \(\chi_{9036}(8443,\cdot)\) \(\chi_{9036}(8911,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((4519,2009,1261)\) → \((-1,1,e\left(\frac{13}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 9036 }(235, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) |
sage: chi.jacobi_sum(n)