from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([9,10,0]))
pari: [g,chi] = znchar(Mod(1864,4025))
Basic properties
Modulus: | \(4025\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(114,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.bu
\(\chi_{4025}(254,\cdot)\) \(\chi_{4025}(1059,\cdot)\) \(\chi_{4025}(1404,\cdot)\) \(\chi_{4025}(1864,\cdot)\) \(\chi_{4025}(2209,\cdot)\) \(\chi_{4025}(2669,\cdot)\) \(\chi_{4025}(3014,\cdot)\) \(\chi_{4025}(3819,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.35434884492252294752034913472016341984272003173828125.1 |
Values on generators
\((2577,1151,3501)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 4025 }(1864, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)