from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,32]))
pari: [g,chi] = znchar(Mod(309,3344))
Basic properties
Modulus: | \(3344\) | |
Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{304}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3344.ef
\(\chi_{3344}(309,\cdot)\) \(\chi_{3344}(397,\cdot)\) \(\chi_{3344}(1013,\cdot)\) \(\chi_{3344}(1277,\cdot)\) \(\chi_{3344}(1365,\cdot)\) \(\chi_{3344}(1453,\cdot)\) \(\chi_{3344}(1981,\cdot)\) \(\chi_{3344}(2069,\cdot)\) \(\chi_{3344}(2685,\cdot)\) \(\chi_{3344}(2949,\cdot)\) \(\chi_{3344}(3037,\cdot)\) \(\chi_{3344}(3125,\cdot)\)
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: | 36.36.52733281945045886724167383478270850720626086921526306402773390818541568.1 |
Values on generators
\((2927,837,2433,705)\) → \((1,i,1,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 3344 }(309, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)