from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,26,11]))
pari: [g,chi] = znchar(Mod(1723,2001))
Basic properties
Modulus: | \(2001\) | |
Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{667}(389,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2001.bh
\(\chi_{2001}(157,\cdot)\) \(\chi_{2001}(244,\cdot)\) \(\chi_{2001}(481,\cdot)\) \(\chi_{2001}(592,\cdot)\) \(\chi_{2001}(655,\cdot)\) \(\chi_{2001}(766,\cdot)\) \(\chi_{2001}(916,\cdot)\) \(\chi_{2001}(940,\cdot)\) \(\chi_{2001}(1003,\cdot)\) \(\chi_{2001}(1027,\cdot)\) \(\chi_{2001}(1114,\cdot)\) \(\chi_{2001}(1201,\cdot)\) \(\chi_{2001}(1351,\cdot)\) \(\chi_{2001}(1525,\cdot)\) \(\chi_{2001}(1699,\cdot)\) \(\chi_{2001}(1723,\cdot)\) \(\chi_{2001}(1786,\cdot)\) \(\chi_{2001}(1873,\cdot)\) \(\chi_{2001}(1897,\cdot)\) \(\chi_{2001}(1960,\cdot)\)
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: | 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1 |
Values on generators
\((668,1132,553)\) → \((1,e\left(\frac{13}{22}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2001 }(1723, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)