from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2940, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,0,0,1]))
pari: [g,chi] = znchar(Mod(811,2940))
Basic properties
Modulus: | \(2940\) | |
Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{196}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2940.cl
\(\chi_{2940}(811,\cdot)\) \(\chi_{2940}(1231,\cdot)\) \(\chi_{2940}(1651,\cdot)\) \(\chi_{2940}(2071,\cdot)\) \(\chi_{2940}(2491,\cdot)\) \(\chi_{2940}(2911,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
Fixed field: | 14.14.21972068264574400934821888.1 |
Values on generators
\((1471,1961,1177,1081)\) → \((-1,1,1,e\left(\frac{1}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2940 }(811, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) |
sage: chi.jacobi_sum(n)