from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7935, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,38]))
pari: [g,chi] = znchar(Mod(5113,7935))
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 \(\chi_{115}(53,\cdot)\) | 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
\((5291,4762,7411)\) → \((1,-i,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 7935 }(5113, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)