from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1104, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,28]))
pari: [g,chi] = znchar(Mod(13,1104))
Basic properties
| Modulus: | \(1104\) | |
| Conductor: | \(368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{368}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1104.bs
\(\chi_{1104}(13,\cdot)\) \(\chi_{1104}(85,\cdot)\) \(\chi_{1104}(133,\cdot)\) \(\chi_{1104}(301,\cdot)\) \(\chi_{1104}(325,\cdot)\) \(\chi_{1104}(349,\cdot)\) \(\chi_{1104}(397,\cdot)\) \(\chi_{1104}(445,\cdot)\) \(\chi_{1104}(469,\cdot)\) \(\chi_{1104}(541,\cdot)\) \(\chi_{1104}(565,\cdot)\) \(\chi_{1104}(637,\cdot)\) \(\chi_{1104}(685,\cdot)\) \(\chi_{1104}(853,\cdot)\) \(\chi_{1104}(877,\cdot)\) \(\chi_{1104}(901,\cdot)\) \(\chi_{1104}(949,\cdot)\) \(\chi_{1104}(997,\cdot)\) \(\chi_{1104}(1021,\cdot)\) \(\chi_{1104}(1093,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952.1 |
Values on generators
\((415,277,737,97)\) → \((1,-i,1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1104 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) |
sage: chi.jacobi_sum(n)