from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,51,10]))
pari: [g,chi] = znchar(Mod(597,7600))
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.hs
\(\chi_{7600}(597,\cdot)\) \(\chi_{7600}(677,\cdot)\) \(\chi_{7600}(1053,\cdot)\) \(\chi_{7600}(1133,\cdot)\) \(\chi_{7600}(2117,\cdot)\) \(\chi_{7600}(2197,\cdot)\) \(\chi_{7600}(2573,\cdot)\) \(\chi_{7600}(2653,\cdot)\) \(\chi_{7600}(3637,\cdot)\) \(\chi_{7600}(3717,\cdot)\) \(\chi_{7600}(4173,\cdot)\) \(\chi_{7600}(5237,\cdot)\) \(\chi_{7600}(5613,\cdot)\) \(\chi_{7600}(6677,\cdot)\) \(\chi_{7600}(7133,\cdot)\) \(\chi_{7600}(7213,\cdot)\)
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
\((4751,5701,5777,401)\) → \((1,i,e\left(\frac{17}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 7600 }(597, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(-i\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)