from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8020, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,20,33]))
pari: [g,chi] = znchar(Mod(369,8020))
Basic properties
Modulus: | \(8020\) | |
Conductor: | \(2005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2005}(369,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8020.cj
\(\chi_{8020}(369,\cdot)\) \(\chi_{8020}(589,\cdot)\) \(\chi_{8020}(1089,\cdot)\) \(\chi_{8020}(1329,\cdot)\) \(\chi_{8020}(1569,\cdot)\) \(\chi_{8020}(2909,\cdot)\) \(\chi_{8020}(2969,\cdot)\) \(\chi_{8020}(4249,\cdot)\) \(\chi_{8020}(4309,\cdot)\) \(\chi_{8020}(5649,\cdot)\) \(\chi_{8020}(5889,\cdot)\) \(\chi_{8020}(6129,\cdot)\) \(\chi_{8020}(6629,\cdot)\) \(\chi_{8020}(6849,\cdot)\) \(\chi_{8020}(7369,\cdot)\) \(\chi_{8020}(7869,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((4011,6417,7221)\) → \((1,-1,e\left(\frac{33}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 8020 }(369, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
sage: chi.jacobi_sum(n)