from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8370, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,15,3]))
pari: [g,chi] = znchar(Mod(89,8370))
Basic properties
| Modulus: | \(8370\) | |
| Conductor: | \(1395\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1395}(1019,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8370.ei
\(\chi_{8370}(89,\cdot)\) \(\chi_{8370}(449,\cdot)\) \(\chi_{8370}(2069,\cdot)\) \(\chi_{8370}(2879,\cdot)\) \(\chi_{8370}(4679,\cdot)\) \(\chi_{8370}(6029,\cdot)\) \(\chi_{8370}(7469,\cdot)\) \(\chi_{8370}(7649,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.1666373156411033096512892331065440049780728838302642543803350006103515625.1 |
Values on generators
\((7751,6697,4591)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8370 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)