from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,0,23]))
pari: [g,chi] = znchar(Mod(71,4592))
Basic properties
| Modulus: | \(4592\) | |
| Conductor: | \(328\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{328}(235,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4592.gr
\(\chi_{4592}(71,\cdot)\) \(\chi_{4592}(183,\cdot)\) \(\chi_{4592}(855,\cdot)\) \(\chi_{4592}(967,\cdot)\) \(\chi_{4592}(1079,\cdot)\) \(\chi_{4592}(1751,\cdot)\) \(\chi_{4592}(1975,\cdot)\) \(\chi_{4592}(2199,\cdot)\) \(\chi_{4592}(2311,\cdot)\) \(\chi_{4592}(2535,\cdot)\) \(\chi_{4592}(2759,\cdot)\) \(\chi_{4592}(3431,\cdot)\) \(\chi_{4592}(3543,\cdot)\) \(\chi_{4592}(3655,\cdot)\) \(\chi_{4592}(4327,\cdot)\) \(\chi_{4592}(4439,\cdot)\)
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: | 40.40.912788483257978497757884926199917783690257306123427760963531833190283833440731136.1 |
Values on generators
\((575,3445,3937,785)\) → \((-1,-1,1,e\left(\frac{23}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 4592 }(71, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)