from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7350, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,1]))
pari: [g,chi] = znchar(Mod(101,7350))
Basic properties
| Modulus: | \(7350\) | |
| 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 7350.cj
\(\chi_{7350}(101,\cdot)\) \(\chi_{7350}(551,\cdot)\) \(\chi_{7350}(1151,\cdot)\) \(\chi_{7350}(1601,\cdot)\) \(\chi_{7350}(2201,\cdot)\) \(\chi_{7350}(2651,\cdot)\) \(\chi_{7350}(3251,\cdot)\) \(\chi_{7350}(3701,\cdot)\) \(\chi_{7350}(4301,\cdot)\) \(\chi_{7350}(4751,\cdot)\) \(\chi_{7350}(5351,\cdot)\) \(\chi_{7350}(6851,\cdot)\)
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
\((4901,1177,2551)\) → \((-1,1,e\left(\frac{1}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7350 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)