from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6840, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,18,9,20]))
pari: [g,chi] = znchar(Mod(17,6840))
Basic properties
| Modulus: | \(6840\) | |
| Conductor: | \(285\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{285}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6840.oa
\(\chi_{6840}(17,\cdot)\) \(\chi_{6840}(233,\cdot)\) \(\chi_{6840}(377,\cdot)\) \(\chi_{6840}(593,\cdot)\) \(\chi_{6840}(2753,\cdot)\) \(\chi_{6840}(2897,\cdot)\) \(\chi_{6840}(3113,\cdot)\) \(\chi_{6840}(3977,\cdot)\) \(\chi_{6840}(4337,\cdot)\) \(\chi_{6840}(4697,\cdot)\) \(\chi_{6840}(5633,\cdot)\) \(\chi_{6840}(6713,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1711,3421,5321,2737,6481)\) → \((1,1,-1,i,e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6840 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) |
sage: chi.jacobi_sum(n)