from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4002, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,30,11]))
pari: [g,chi] = znchar(Mod(157,4002))
Basic properties
| Modulus: | \(4002\) | |
| 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 \(\chi_{667}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4002.bh
\(\chi_{4002}(157,\cdot)\) \(\chi_{4002}(481,\cdot)\) \(\chi_{4002}(655,\cdot)\) \(\chi_{4002}(1003,\cdot)\) \(\chi_{4002}(1027,\cdot)\) \(\chi_{4002}(1201,\cdot)\) \(\chi_{4002}(1351,\cdot)\) \(\chi_{4002}(1525,\cdot)\) \(\chi_{4002}(1699,\cdot)\) \(\chi_{4002}(1723,\cdot)\) \(\chi_{4002}(1873,\cdot)\) \(\chi_{4002}(1897,\cdot)\) \(\chi_{4002}(2245,\cdot)\) \(\chi_{4002}(2593,\cdot)\) \(\chi_{4002}(2767,\cdot)\) \(\chi_{4002}(2917,\cdot)\) \(\chi_{4002}(2941,\cdot)\) \(\chi_{4002}(3115,\cdot)\) \(\chi_{4002}(3787,\cdot)\) \(\chi_{4002}(3961,\cdot)\)
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
\((2669,3133,553)\) → \((1,e\left(\frac{15}{22}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 4002 }(157, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) |
sage: chi.jacobi_sum(n)