from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,26]))
pari: [g,chi] = znchar(Mod(226,5625))
Basic properties
| Modulus: | \(5625\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{125}(121,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5625.t
\(\chi_{5625}(226,\cdot)\) \(\chi_{5625}(451,\cdot)\) \(\chi_{5625}(676,\cdot)\) \(\chi_{5625}(901,\cdot)\) \(\chi_{5625}(1351,\cdot)\) \(\chi_{5625}(1576,\cdot)\) \(\chi_{5625}(1801,\cdot)\) \(\chi_{5625}(2026,\cdot)\) \(\chi_{5625}(2476,\cdot)\) \(\chi_{5625}(2701,\cdot)\) \(\chi_{5625}(2926,\cdot)\) \(\chi_{5625}(3151,\cdot)\) \(\chi_{5625}(3601,\cdot)\) \(\chi_{5625}(3826,\cdot)\) \(\chi_{5625}(4051,\cdot)\) \(\chi_{5625}(4276,\cdot)\) \(\chi_{5625}(4726,\cdot)\) \(\chi_{5625}(4951,\cdot)\) \(\chi_{5625}(5176,\cdot)\) \(\chi_{5625}(5401,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 25 polynomial |
Values on generators
\((4376,1252)\) → \((1,e\left(\frac{13}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5625 }(226, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |
sage: chi.jacobi_sum(n)