from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(41,1205))
Basic properties
Modulus: | \(1205\) | |
Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{241}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1205.bv
\(\chi_{1205}(41,\cdot)\) \(\chi_{1205}(61,\cdot)\) \(\chi_{1205}(116,\cdot)\) \(\chi_{1205}(236,\cdot)\) \(\chi_{1205}(246,\cdot)\) \(\chi_{1205}(366,\cdot)\) \(\chi_{1205}(421,\cdot)\) \(\chi_{1205}(441,\cdot)\) \(\chi_{1205}(561,\cdot)\) \(\chi_{1205}(676,\cdot)\) \(\chi_{1205}(696,\cdot)\) \(\chi_{1205}(771,\cdot)\) \(\chi_{1205}(916,\cdot)\) \(\chi_{1205}(991,\cdot)\) \(\chi_{1205}(1011,\cdot)\) \(\chi_{1205}(1126,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((242,971)\) → \((1,e\left(\frac{1}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 1205 }(41, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{11}{20}\right)\) | \(-1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) |
sage: chi.jacobi_sum(n)