from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,16]))
pari: [g,chi] = znchar(Mod(131,1840))
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 \(\chi_{368}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.ci
\(\chi_{1840}(131,\cdot)\) \(\chi_{1840}(211,\cdot)\) \(\chi_{1840}(331,\cdot)\) \(\chi_{1840}(371,\cdot)\) \(\chi_{1840}(491,\cdot)\) \(\chi_{1840}(531,\cdot)\) \(\chi_{1840}(611,\cdot)\) \(\chi_{1840}(731,\cdot)\) \(\chi_{1840}(771,\cdot)\) \(\chi_{1840}(811,\cdot)\) \(\chi_{1840}(1051,\cdot)\) \(\chi_{1840}(1131,\cdot)\) \(\chi_{1840}(1251,\cdot)\) \(\chi_{1840}(1291,\cdot)\) \(\chi_{1840}(1411,\cdot)\) \(\chi_{1840}(1451,\cdot)\) \(\chi_{1840}(1531,\cdot)\) \(\chi_{1840}(1651,\cdot)\) \(\chi_{1840}(1691,\cdot)\) \(\chi_{1840}(1731,\cdot)\)
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: | 44.0.7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952.1 |
Values on generators
\((1151,1381,737,1201)\) → \((-1,-i,1,e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(131, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) |
sage: chi.jacobi_sum(n)