from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7350, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,2]))
pari: [g,chi] = znchar(Mod(499,7350))
Basic properties
Modulus: | \(7350\) | |
Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{245}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7350.cl
\(\chi_{7350}(499,\cdot)\) \(\chi_{7350}(1999,\cdot)\) \(\chi_{7350}(2599,\cdot)\) \(\chi_{7350}(3049,\cdot)\) \(\chi_{7350}(3649,\cdot)\) \(\chi_{7350}(4099,\cdot)\) \(\chi_{7350}(4699,\cdot)\) \(\chi_{7350}(5149,\cdot)\) \(\chi_{7350}(5749,\cdot)\) \(\chi_{7350}(6199,\cdot)\) \(\chi_{7350}(6799,\cdot)\) \(\chi_{7350}(7249,\cdot)\)
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: | 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1 |
Values on generators
\((4901,1177,2551)\) → \((1,-1,e\left(\frac{1}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7350 }(499, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)