from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,9,10]))
pari: [g,chi] = znchar(Mod(317,760))
Basic properties
Modulus: | \(760\) | |
Conductor: | \(760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 760.cs
\(\chi_{760}(13,\cdot)\) \(\chi_{760}(53,\cdot)\) \(\chi_{760}(117,\cdot)\) \(\chi_{760}(173,\cdot)\) \(\chi_{760}(317,\cdot)\) \(\chi_{760}(333,\cdot)\) \(\chi_{760}(357,\cdot)\) \(\chi_{760}(413,\cdot)\) \(\chi_{760}(477,\cdot)\) \(\chi_{760}(573,\cdot)\) \(\chi_{760}(637,\cdot)\) \(\chi_{760}(717,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | 36.36.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1 |
Values on generators
\((191,381,457,401)\) → \((1,-1,i,e\left(\frac{5}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 760 }(317, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)