from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,22,12]))
pari: [g,chi] = znchar(Mod(307,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(2576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2576}(307,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.fi
\(\chi_{7728}(307,\cdot)\) \(\chi_{7728}(811,\cdot)\) \(\chi_{7728}(979,\cdot)\) \(\chi_{7728}(1315,\cdot)\) \(\chi_{7728}(1651,\cdot)\) \(\chi_{7728}(1819,\cdot)\) \(\chi_{7728}(1987,\cdot)\) \(\chi_{7728}(2155,\cdot)\) \(\chi_{7728}(3163,\cdot)\) \(\chi_{7728}(3499,\cdot)\) \(\chi_{7728}(4171,\cdot)\) \(\chi_{7728}(4675,\cdot)\) \(\chi_{7728}(4843,\cdot)\) \(\chi_{7728}(5179,\cdot)\) \(\chi_{7728}(5515,\cdot)\) \(\chi_{7728}(5683,\cdot)\) \(\chi_{7728}(5851,\cdot)\) \(\chi_{7728}(6019,\cdot)\) \(\chi_{7728}(7027,\cdot)\) \(\chi_{7728}(7363,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((-1,-i,1,-1,e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(307, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)