from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,18]))
pari: [g,chi] = znchar(Mod(931,1840))
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 \(\chi_{368}(195,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cw
\(\chi_{1840}(11,\cdot)\) \(\chi_{1840}(51,\cdot)\) \(\chi_{1840}(171,\cdot)\) \(\chi_{1840}(251,\cdot)\) \(\chi_{1840}(291,\cdot)\) \(\chi_{1840}(411,\cdot)\) \(\chi_{1840}(451,\cdot)\) \(\chi_{1840}(571,\cdot)\) \(\chi_{1840}(651,\cdot)\) \(\chi_{1840}(891,\cdot)\) \(\chi_{1840}(931,\cdot)\) \(\chi_{1840}(971,\cdot)\) \(\chi_{1840}(1091,\cdot)\) \(\chi_{1840}(1171,\cdot)\) \(\chi_{1840}(1211,\cdot)\) \(\chi_{1840}(1331,\cdot)\) \(\chi_{1840}(1371,\cdot)\) \(\chi_{1840}(1491,\cdot)\) \(\chi_{1840}(1571,\cdot)\) \(\chi_{1840}(1811,\cdot)\)
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: | 44.44.4141890260646712580912980965306954513336276372715662057543551492310346739946349214617837764608.1 |
Values on generators
\((1151,1381,737,1201)\) → \((-1,-i,1,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(931, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) |
sage: chi.jacobi_sum(n)