from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,10,48]))
pari: [g,chi] = znchar(Mod(157,3850))
Basic properties
| Modulus: | \(3850\) | |
| Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{385}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3850.ge
\(\chi_{3850}(157,\cdot)\) \(\chi_{3850}(257,\cdot)\) \(\chi_{3850}(493,\cdot)\) \(\chi_{3850}(1193,\cdot)\) \(\chi_{3850}(1307,\cdot)\) \(\chi_{3850}(1543,\cdot)\) \(\chi_{3850}(1643,\cdot)\) \(\chi_{3850}(1907,\cdot)\) \(\chi_{3850}(2007,\cdot)\) \(\chi_{3850}(2357,\cdot)\) \(\chi_{3850}(2693,\cdot)\) \(\chi_{3850}(2957,\cdot)\) \(\chi_{3850}(3293,\cdot)\) \(\chi_{3850}(3393,\cdot)\) \(\chi_{3850}(3657,\cdot)\) \(\chi_{3850}(3743,\cdot)\)
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
\((2927,2201,1751)\) → \((i,e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3850 }(157, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) |
sage: chi.jacobi_sum(n)