from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2880, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,40,36]))
pari: [g,chi] = znchar(Mod(1373,2880))
Basic properties
Modulus: | \(2880\) | |
Conductor: | \(2880\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 2880.fb
\(\chi_{2880}(173,\cdot)\) \(\chi_{2880}(437,\cdot)\) \(\chi_{2880}(653,\cdot)\) \(\chi_{2880}(677,\cdot)\) \(\chi_{2880}(893,\cdot)\) \(\chi_{2880}(1157,\cdot)\) \(\chi_{2880}(1373,\cdot)\) \(\chi_{2880}(1397,\cdot)\) \(\chi_{2880}(1613,\cdot)\) \(\chi_{2880}(1877,\cdot)\) \(\chi_{2880}(2093,\cdot)\) \(\chi_{2880}(2117,\cdot)\) \(\chi_{2880}(2333,\cdot)\) \(\chi_{2880}(2597,\cdot)\) \(\chi_{2880}(2813,\cdot)\) \(\chi_{2880}(2837,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2431,901,641,577)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{5}{6}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 2880 }(1373, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(-1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) |
sage: chi.jacobi_sum(n)