from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,40]))
pari: [g,chi] = znchar(Mod(5401,7056))
Basic properties
| Modulus: | \(7056\) | |
| Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{392}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7056.gk
\(\chi_{7056}(793,\cdot)\) \(\chi_{7056}(1369,\cdot)\) \(\chi_{7056}(1801,\cdot)\) \(\chi_{7056}(2377,\cdot)\) \(\chi_{7056}(2809,\cdot)\) \(\chi_{7056}(3385,\cdot)\) \(\chi_{7056}(3817,\cdot)\) \(\chi_{7056}(4393,\cdot)\) \(\chi_{7056}(4825,\cdot)\) \(\chi_{7056}(5401,\cdot)\) \(\chi_{7056}(5833,\cdot)\) \(\chi_{7056}(6409,\cdot)\)
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: | 42.42.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1 |
Values on generators
\((6175,1765,785,4609)\) → \((1,-1,1,e\left(\frac{20}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7056 }(5401, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{41}{42}\right)\) |
sage: chi.jacobi_sum(n)