from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,9,10]))
pari: [g,chi] = znchar(Mod(237,1400))
Basic properties
Modulus: | \(1400\) | |
Conductor: | \(1400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1400.ct
\(\chi_{1400}(13,\cdot)\) \(\chi_{1400}(237,\cdot)\) \(\chi_{1400}(517,\cdot)\) \(\chi_{1400}(573,\cdot)\) \(\chi_{1400}(797,\cdot)\) \(\chi_{1400}(853,\cdot)\) \(\chi_{1400}(1077,\cdot)\) \(\chi_{1400}(1133,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.882735153125000000000000000000000000000000.1 |
Values on generators
\((351,701,1177,801)\) → \((1,-1,e\left(\frac{9}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 1400 }(237, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)