from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,7,33]))
pari: [g,chi] = znchar(Mod(209,7056))
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 \(\chi_{441}(209,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7056.gz
\(\chi_{7056}(209,\cdot)\) \(\chi_{7056}(545,\cdot)\) \(\chi_{7056}(1217,\cdot)\) \(\chi_{7056}(1553,\cdot)\) \(\chi_{7056}(2225,\cdot)\) \(\chi_{7056}(2561,\cdot)\) \(\chi_{7056}(3569,\cdot)\) \(\chi_{7056}(4241,\cdot)\) \(\chi_{7056}(4577,\cdot)\) \(\chi_{7056}(5249,\cdot)\) \(\chi_{7056}(6257,\cdot)\) \(\chi_{7056}(6593,\cdot)\)
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
\((6175,1765,785,4609)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{11}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7056 }(209, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)