from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,27,4]))
pari: [g,chi] = znchar(Mod(97,5184))
Basic properties
Modulus: | \(5184\) | |
Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{648}(421,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5184.cf
\(\chi_{5184}(97,\cdot)\) \(\chi_{5184}(481,\cdot)\) \(\chi_{5184}(673,\cdot)\) \(\chi_{5184}(1057,\cdot)\) \(\chi_{5184}(1249,\cdot)\) \(\chi_{5184}(1633,\cdot)\) \(\chi_{5184}(1825,\cdot)\) \(\chi_{5184}(2209,\cdot)\) \(\chi_{5184}(2401,\cdot)\) \(\chi_{5184}(2785,\cdot)\) \(\chi_{5184}(2977,\cdot)\) \(\chi_{5184}(3361,\cdot)\) \(\chi_{5184}(3553,\cdot)\) \(\chi_{5184}(3937,\cdot)\) \(\chi_{5184}(4129,\cdot)\) \(\chi_{5184}(4513,\cdot)\) \(\chi_{5184}(4705,\cdot)\) \(\chi_{5184}(5089,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((2431,325,1217)\) → \((1,-1,e\left(\frac{2}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 5184 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) |
sage: chi.jacobi_sum(n)