from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,25,0,22]))
pari: [g,chi] = znchar(Mod(523,7200))
Basic properties
Modulus: | \(7200\) | |
Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{800}(523,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7200.gp
\(\chi_{7200}(523,\cdot)\) \(\chi_{7200}(1027,\cdot)\) \(\chi_{7200}(1747,\cdot)\) \(\chi_{7200}(1963,\cdot)\) \(\chi_{7200}(2467,\cdot)\) \(\chi_{7200}(2683,\cdot)\) \(\chi_{7200}(3187,\cdot)\) \(\chi_{7200}(3403,\cdot)\) \(\chi_{7200}(4123,\cdot)\) \(\chi_{7200}(4627,\cdot)\) \(\chi_{7200}(5347,\cdot)\) \(\chi_{7200}(5563,\cdot)\) \(\chi_{7200}(6067,\cdot)\) \(\chi_{7200}(6283,\cdot)\) \(\chi_{7200}(6787,\cdot)\) \(\chi_{7200}(7003,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.386856262276681335905976320000000000000000000000000000000000000000000000000000000000000000000000.2 |
Values on generators
\((6751,901,6401,577)\) → \((-1,e\left(\frac{5}{8}\right),1,e\left(\frac{11}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 7200 }(523, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)