from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,10]))
pari: [g,chi] = znchar(Mod(43,9075))
Basic properties
| Modulus: | \(9075\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9075.dq
\(\chi_{9075}(43,\cdot)\) \(\chi_{9075}(307,\cdot)\) \(\chi_{9075}(868,\cdot)\) \(\chi_{9075}(1132,\cdot)\) \(\chi_{9075}(1957,\cdot)\) \(\chi_{9075}(2518,\cdot)\) \(\chi_{9075}(3343,\cdot)\) \(\chi_{9075}(3607,\cdot)\) \(\chi_{9075}(4168,\cdot)\) \(\chi_{9075}(4432,\cdot)\) \(\chi_{9075}(4993,\cdot)\) \(\chi_{9075}(5257,\cdot)\) \(\chi_{9075}(5818,\cdot)\) \(\chi_{9075}(6082,\cdot)\) \(\chi_{9075}(6643,\cdot)\) \(\chi_{9075}(6907,\cdot)\) \(\chi_{9075}(7468,\cdot)\) \(\chi_{9075}(7732,\cdot)\) \(\chi_{9075}(8293,\cdot)\) \(\chi_{9075}(8557,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((3026,727,5326)\) → \((1,-i,e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) |
sage: chi.jacobi_sum(n)