from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5070, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,17]))
pari: [g,chi] = znchar(Mod(259,5070))
Basic properties
| Modulus: | \(5070\) | |
| Conductor: | \(845\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{845}(259,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5070.bq
\(\chi_{5070}(259,\cdot)\) \(\chi_{5070}(649,\cdot)\) \(\chi_{5070}(1039,\cdot)\) \(\chi_{5070}(1429,\cdot)\) \(\chi_{5070}(1819,\cdot)\) \(\chi_{5070}(2209,\cdot)\) \(\chi_{5070}(2599,\cdot)\) \(\chi_{5070}(2989,\cdot)\) \(\chi_{5070}(3769,\cdot)\) \(\chi_{5070}(4159,\cdot)\) \(\chi_{5070}(4549,\cdot)\) \(\chi_{5070}(4939,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\((1691,4057,1861)\) → \((1,-1,e\left(\frac{17}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5070 }(259, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
sage: chi.jacobi_sum(n)