from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,11,17]))
pari: [g,chi] = znchar(Mod(197,8712))
Basic properties
Modulus: | \(8712\) | |
Conductor: | \(2904\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2904}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8712.cb
\(\chi_{8712}(197,\cdot)\) \(\chi_{8712}(989,\cdot)\) \(\chi_{8712}(1781,\cdot)\) \(\chi_{8712}(2573,\cdot)\) \(\chi_{8712}(3365,\cdot)\) \(\chi_{8712}(4157,\cdot)\) \(\chi_{8712}(4949,\cdot)\) \(\chi_{8712}(5741,\cdot)\) \(\chi_{8712}(7325,\cdot)\) \(\chi_{8712}(8117,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | 22.22.7575717132715709690047940165082998996969028019885770276864.1 |
Values on generators
\((6535,4357,1937,5689)\) → \((1,-1,-1,e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 8712 }(197, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)