from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9036, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,44]))
pari: [g,chi] = znchar(Mod(1259,9036))
Basic properties
Modulus: | \(9036\) | |
Conductor: | \(3012\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3012}(1259,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9036.bl
\(\chi_{9036}(1259,\cdot)\) \(\chi_{9036}(1511,\cdot)\) \(\chi_{9036}(2339,\cdot)\) \(\chi_{9036}(4067,\cdot)\) \(\chi_{9036}(4139,\cdot)\) \(\chi_{9036}(4283,\cdot)\) \(\chi_{9036}(4643,\cdot)\) \(\chi_{9036}(5111,\cdot)\) \(\chi_{9036}(5723,\cdot)\) \(\chi_{9036}(5867,\cdot)\) \(\chi_{9036}(6479,\cdot)\) \(\chi_{9036}(6551,\cdot)\) \(\chi_{9036}(6767,\cdot)\) \(\chi_{9036}(7091,\cdot)\) \(\chi_{9036}(7271,\cdot)\) \(\chi_{9036}(7343,\cdot)\) \(\chi_{9036}(7379,\cdot)\) \(\chi_{9036}(8243,\cdot)\) \(\chi_{9036}(8603,\cdot)\) \(\chi_{9036}(8783,\cdot)\)
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
\((4519,2009,1261)\) → \((-1,-1,e\left(\frac{22}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 9036 }(1259, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) |
sage: chi.jacobi_sum(n)