from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2940, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,1]))
pari: [g,chi] = znchar(Mod(101,2940))
Basic properties
| Modulus: | \(2940\) | |
| Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{147}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2940.dc
\(\chi_{2940}(101,\cdot)\) \(\chi_{2940}(341,\cdot)\) \(\chi_{2940}(761,\cdot)\) \(\chi_{2940}(941,\cdot)\) \(\chi_{2940}(1181,\cdot)\) \(\chi_{2940}(1361,\cdot)\) \(\chi_{2940}(1601,\cdot)\) \(\chi_{2940}(1781,\cdot)\) \(\chi_{2940}(2021,\cdot)\) \(\chi_{2940}(2201,\cdot)\) \(\chi_{2940}(2441,\cdot)\) \(\chi_{2940}(2621,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((1471,1961,1177,1081)\) → \((1,-1,1,e\left(\frac{1}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2940 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)