from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6840, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,6,9,4]))
pari: [g,chi] = znchar(Mod(137,6840))
Basic properties
| Modulus: | \(6840\) | |
| Conductor: | \(855\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{855}(137,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6840.nd
\(\chi_{6840}(137,\cdot)\) \(\chi_{6840}(473,\cdot)\) \(\chi_{6840}(833,\cdot)\) \(\chi_{6840}(1073,\cdot)\) \(\chi_{6840}(2057,\cdot)\) \(\chi_{6840}(2153,\cdot)\) \(\chi_{6840}(2873,\cdot)\) \(\chi_{6840}(4577,\cdot)\) \(\chi_{6840}(4793,\cdot)\) \(\chi_{6840}(4937,\cdot)\) \(\chi_{6840}(5177,\cdot)\) \(\chi_{6840}(6257,\cdot)\)
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: | 36.36.36045670002337036813834863966937246686386512405362460211785986211962997913360595703125.1 |
Values on generators
\((1711,3421,5321,2737,6481)\) → \((1,1,e\left(\frac{1}{6}\right),i,e\left(\frac{1}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6840 }(137, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) |
sage: chi.jacobi_sum(n)