from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,0,13,10]))
pari: [g,chi] = znchar(Mod(417,4400))
Basic properties
Modulus: | \(4400\) | |
Conductor: | \(275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{275}(142,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4400.hw
\(\chi_{4400}(417,\cdot)\) \(\chi_{4400}(1297,\cdot)\) \(\chi_{4400}(1473,\cdot)\) \(\chi_{4400}(2177,\cdot)\) \(\chi_{4400}(2353,\cdot)\) \(\chi_{4400}(3233,\cdot)\) \(\chi_{4400}(3937,\cdot)\) \(\chi_{4400}(4113,\cdot)\)
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: | 20.20.75487840807181783020496368408203125.1 |
Values on generators
\((2751,3301,177,1201)\) → \((1,1,e\left(\frac{13}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4400 }(417, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(-i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)