from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,16]))
pari: [g,chi] = znchar(Mod(1741,1840))
Basic properties
| Modulus: | \(1840\) | |
| 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 \(\chi_{368}(269,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cv
\(\chi_{1840}(101,\cdot)\) \(\chi_{1840}(141,\cdot)\) \(\chi_{1840}(261,\cdot)\) \(\chi_{1840}(301,\cdot)\) \(\chi_{1840}(381,\cdot)\) \(\chi_{1840}(501,\cdot)\) \(\chi_{1840}(541,\cdot)\) \(\chi_{1840}(581,\cdot)\) \(\chi_{1840}(821,\cdot)\) \(\chi_{1840}(901,\cdot)\) \(\chi_{1840}(1021,\cdot)\) \(\chi_{1840}(1061,\cdot)\) \(\chi_{1840}(1181,\cdot)\) \(\chi_{1840}(1221,\cdot)\) \(\chi_{1840}(1301,\cdot)\) \(\chi_{1840}(1421,\cdot)\) \(\chi_{1840}(1461,\cdot)\) \(\chi_{1840}(1501,\cdot)\) \(\chi_{1840}(1741,\cdot)\) \(\chi_{1840}(1821,\cdot)\)
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
\((1151,1381,737,1201)\) → \((1,-i,1,e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(1741, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) |
sage: chi.jacobi_sum(n)