from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,25,21,27]))
pari: [g,chi] = znchar(Mod(4109,9900))
Basic properties
Modulus: | \(9900\) | |
Conductor: | \(2475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2475}(1634,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9900.kn
\(\chi_{9900}(29,\cdot)\) \(\chi_{9900}(569,\cdot)\) \(\chi_{9900}(1289,\cdot)\) \(\chi_{9900}(4109,\cdot)\) \(\chi_{9900}(6629,\cdot)\) \(\chi_{9900}(7169,\cdot)\) \(\chi_{9900}(7409,\cdot)\) \(\chi_{9900}(7889,\cdot)\)
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: | 30.30.17200029144406711299460730183653019214710733642693885059316016850061714649200439453125.1 |
Values on generators
\((4951,5501,2377,4501)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{7}{10}\right),e\left(\frac{9}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9900 }(4109, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)