from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5445, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,10]))
pari: [g,chi] = znchar(Mod(4762,5445))
Basic properties
Modulus: | \(5445\) | |
Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{605}(527,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5445.cd
\(\chi_{5445}(208,\cdot)\) \(\chi_{5445}(307,\cdot)\) \(\chi_{5445}(703,\cdot)\) \(\chi_{5445}(802,\cdot)\) \(\chi_{5445}(1198,\cdot)\) \(\chi_{5445}(1297,\cdot)\) \(\chi_{5445}(1792,\cdot)\) \(\chi_{5445}(2188,\cdot)\) \(\chi_{5445}(2287,\cdot)\) \(\chi_{5445}(2683,\cdot)\) \(\chi_{5445}(3178,\cdot)\) \(\chi_{5445}(3277,\cdot)\) \(\chi_{5445}(3673,\cdot)\) \(\chi_{5445}(3772,\cdot)\) \(\chi_{5445}(4168,\cdot)\) \(\chi_{5445}(4267,\cdot)\) \(\chi_{5445}(4663,\cdot)\) \(\chi_{5445}(4762,\cdot)\) \(\chi_{5445}(5158,\cdot)\) \(\chi_{5445}(5257,\cdot)\)
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: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((3026,4357,3511)\) → \((1,i,e\left(\frac{5}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 5445 }(4762, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) |
sage: chi.jacobi_sum(n)