from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,35,23]))
pari: [g,chi] = znchar(Mod(1265,7056))
Basic properties
Modulus: | \(7056\) | |
Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{441}(383,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7056.hb
\(\chi_{7056}(257,\cdot)\) \(\chi_{7056}(353,\cdot)\) \(\chi_{7056}(1265,\cdot)\) \(\chi_{7056}(1361,\cdot)\) \(\chi_{7056}(2369,\cdot)\) \(\chi_{7056}(3281,\cdot)\) \(\chi_{7056}(3377,\cdot)\) \(\chi_{7056}(4289,\cdot)\) \(\chi_{7056}(4385,\cdot)\) \(\chi_{7056}(5297,\cdot)\) \(\chi_{7056}(5393,\cdot)\) \(\chi_{7056}(6305,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((6175,1765,785,4609)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{23}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 7056 }(1265, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(-1\) | \(e\left(\frac{11}{21}\right)\) |
sage: chi.jacobi_sum(n)