from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5376, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,31,16,16]))
pari: [g,chi] = znchar(Mod(41,5376))
Basic properties
| Modulus: | \(5376\) | |
| Conductor: | \(2688\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2688}(461,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5376.db
\(\chi_{5376}(41,\cdot)\) \(\chi_{5376}(377,\cdot)\) \(\chi_{5376}(713,\cdot)\) \(\chi_{5376}(1049,\cdot)\) \(\chi_{5376}(1385,\cdot)\) \(\chi_{5376}(1721,\cdot)\) \(\chi_{5376}(2057,\cdot)\) \(\chi_{5376}(2393,\cdot)\) \(\chi_{5376}(2729,\cdot)\) \(\chi_{5376}(3065,\cdot)\) \(\chi_{5376}(3401,\cdot)\) \(\chi_{5376}(3737,\cdot)\) \(\chi_{5376}(4073,\cdot)\) \(\chi_{5376}(4409,\cdot)\) \(\chi_{5376}(4745,\cdot)\) \(\chi_{5376}(5081,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.4489912604053908534055314729400632754872833954383027744299245049859706934263808.1 |
Values on generators
\((2815,5125,1793,4609)\) → \((1,e\left(\frac{31}{32}\right),-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5376 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(i\) | \(e\left(\frac{7}{32}\right)\) |
sage: chi.jacobi_sum(n)