from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3645, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([4,9]))
pari: [g,chi] = znchar(Mod(1297,3645))
Basic properties
| Modulus: | \(3645\) | |
| Conductor: | \(135\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{135}(112,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3645.s
\(\chi_{3645}(82,\cdot)\) \(\chi_{3645}(163,\cdot)\) \(\chi_{3645}(568,\cdot)\) \(\chi_{3645}(892,\cdot)\) \(\chi_{3645}(1297,\cdot)\) \(\chi_{3645}(1378,\cdot)\) \(\chi_{3645}(1783,\cdot)\) \(\chi_{3645}(2107,\cdot)\) \(\chi_{3645}(2512,\cdot)\) \(\chi_{3645}(2593,\cdot)\) \(\chi_{3645}(2998,\cdot)\) \(\chi_{3645}(3322,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.7225377334561374804949923918873673793376691639423370361328125.1 |
Values on generators
\((731,2917)\) → \((e\left(\frac{1}{9}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3645 }(1297, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)