from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2028, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,3]))
pari: [g,chi] = znchar(Mod(25,2028))
Basic properties
| Modulus: | \(2028\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2028.bf
\(\chi_{2028}(25,\cdot)\) \(\chi_{2028}(181,\cdot)\) \(\chi_{2028}(493,\cdot)\) \(\chi_{2028}(649,\cdot)\) \(\chi_{2028}(805,\cdot)\) \(\chi_{2028}(961,\cdot)\) \(\chi_{2028}(1117,\cdot)\) \(\chi_{2028}(1273,\cdot)\) \(\chi_{2028}(1429,\cdot)\) \(\chi_{2028}(1585,\cdot)\) \(\chi_{2028}(1741,\cdot)\) \(\chi_{2028}(1897,\cdot)\)
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: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((1015,677,1861)\) → \((1,1,e\left(\frac{3}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2028 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) |
sage: chi.jacobi_sum(n)