from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,27,20]))
pari: [g,chi] = znchar(Mod(37,700))
Basic properties
| Modulus: | \(700\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{175}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 700.bu
\(\chi_{700}(37,\cdot)\) \(\chi_{700}(53,\cdot)\) \(\chi_{700}(137,\cdot)\) \(\chi_{700}(177,\cdot)\) \(\chi_{700}(233,\cdot)\) \(\chi_{700}(277,\cdot)\) \(\chi_{700}(317,\cdot)\) \(\chi_{700}(333,\cdot)\) \(\chi_{700}(373,\cdot)\) \(\chi_{700}(417,\cdot)\) \(\chi_{700}(473,\cdot)\) \(\chi_{700}(513,\cdot)\) \(\chi_{700}(597,\cdot)\) \(\chi_{700}(613,\cdot)\) \(\chi_{700}(653,\cdot)\) \(\chi_{700}(697,\cdot)\)
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
\((351,477,101)\) → \((1,e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 700 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)