from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38025, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,33,50]))
pari: [g,chi] = znchar(Mod(23,38025))
Basic properties
Modulus: | \(38025\) | |
Conductor: | \(2925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2925}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 38025.hy
\(\chi_{38025}(23,\cdot)\) \(\chi_{38025}(992,\cdot)\) \(\chi_{38025}(2513,\cdot)\) \(\chi_{38025}(7628,\cdot)\) \(\chi_{38025}(8597,\cdot)\) \(\chi_{38025}(13712,\cdot)\) \(\chi_{38025}(15233,\cdot)\) \(\chi_{38025}(16202,\cdot)\) \(\chi_{38025}(17723,\cdot)\) \(\chi_{38025}(21317,\cdot)\) \(\chi_{38025}(22838,\cdot)\) \(\chi_{38025}(25328,\cdot)\) \(\chi_{38025}(28922,\cdot)\) \(\chi_{38025}(31412,\cdot)\) \(\chi_{38025}(32933,\cdot)\) \(\chi_{38025}(36527,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((29576,9127,37351)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 38025 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(i\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) |
sage: chi.jacobi_sum(n)