from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,33,50]))
pari: [g,chi] = znchar(Mod(373,7600))
Basic properties
Modulus: | \(7600\) | |
Conductor: | \(7600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 7600.gw
\(\chi_{7600}(373,\cdot)\) \(\chi_{7600}(1437,\cdot)\) \(\chi_{7600}(1813,\cdot)\) \(\chi_{7600}(2877,\cdot)\) \(\chi_{7600}(3333,\cdot)\) \(\chi_{7600}(3413,\cdot)\) \(\chi_{7600}(4397,\cdot)\) \(\chi_{7600}(4477,\cdot)\) \(\chi_{7600}(4853,\cdot)\) \(\chi_{7600}(4933,\cdot)\) \(\chi_{7600}(5917,\cdot)\) \(\chi_{7600}(5997,\cdot)\) \(\chi_{7600}(6373,\cdot)\) \(\chi_{7600}(6453,\cdot)\) \(\chi_{7600}(7437,\cdot)\) \(\chi_{7600}(7517,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((4751,5701,5777,401)\) → \((1,i,e\left(\frac{11}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 7600 }(373, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(i\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{60}\right)\) |
sage: chi.jacobi_sum(n)