from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7935, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,2]))
pari: [g,chi] = znchar(Mod(28,7935))
Basic properties
| Modulus: | \(7935\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7935.x
\(\chi_{7935}(28,\cdot)\) \(\chi_{7935}(352,\cdot)\) \(\chi_{7935}(592,\cdot)\) \(\chi_{7935}(1417,\cdot)\) \(\chi_{7935}(1717,\cdot)\) \(\chi_{7935}(2158,\cdot)\) \(\chi_{7935}(2527,\cdot)\) \(\chi_{7935}(2908,\cdot)\) \(\chi_{7935}(3202,\cdot)\) \(\chi_{7935}(3448,\cdot)\) \(\chi_{7935}(3898,\cdot)\) \(\chi_{7935}(5113,\cdot)\) \(\chi_{7935}(5332,\cdot)\) \(\chi_{7935}(5353,\cdot)\) \(\chi_{7935}(6082,\cdot)\) \(\chi_{7935}(6178,\cdot)\) \(\chi_{7935}(6478,\cdot)\) \(\chi_{7935}(6622,\cdot)\) \(\chi_{7935}(7072,\cdot)\) \(\chi_{7935}(7288,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((5291,4762,7411)\) → \((1,-i,e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 7935 }(28, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) |
sage: chi.jacobi_sum(n)