from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2835, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([37,0,18]))
pari: [g,chi] = znchar(Mod(191,2835))
Basic properties
| Modulus: | \(2835\) | |
| Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{567}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2835.ek
\(\chi_{2835}(191,\cdot)\) \(\chi_{2835}(221,\cdot)\) \(\chi_{2835}(506,\cdot)\) \(\chi_{2835}(536,\cdot)\) \(\chi_{2835}(821,\cdot)\) \(\chi_{2835}(851,\cdot)\) \(\chi_{2835}(1136,\cdot)\) \(\chi_{2835}(1166,\cdot)\) \(\chi_{2835}(1451,\cdot)\) \(\chi_{2835}(1481,\cdot)\) \(\chi_{2835}(1766,\cdot)\) \(\chi_{2835}(1796,\cdot)\) \(\chi_{2835}(2081,\cdot)\) \(\chi_{2835}(2111,\cdot)\) \(\chi_{2835}(2396,\cdot)\) \(\chi_{2835}(2426,\cdot)\) \(\chi_{2835}(2711,\cdot)\) \(\chi_{2835}(2741,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1541,1702,2026)\) → \((e\left(\frac{37}{54}\right),1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2835 }(191, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) |
sage: chi.jacobi_sum(n)