from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,44,30]))
pari: [g,chi] = znchar(Mod(193,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.ey
\(\chi_{7728}(193,\cdot)\) \(\chi_{7728}(289,\cdot)\) \(\chi_{7728}(625,\cdot)\) \(\chi_{7728}(961,\cdot)\) \(\chi_{7728}(1297,\cdot)\) \(\chi_{7728}(2881,\cdot)\) \(\chi_{7728}(3889,\cdot)\) \(\chi_{7728}(3985,\cdot)\) \(\chi_{7728}(4225,\cdot)\) \(\chi_{7728}(4993,\cdot)\) \(\chi_{7728}(5233,\cdot)\) \(\chi_{7728}(5329,\cdot)\) \(\chi_{7728}(5569,\cdot)\) \(\chi_{7728}(6241,\cdot)\) \(\chi_{7728}(6337,\cdot)\) \(\chi_{7728}(6673,\cdot)\) \(\chi_{7728}(6913,\cdot)\) \(\chi_{7728}(7249,\cdot)\) \(\chi_{7728}(7345,\cdot)\) \(\chi_{7728}(7585,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((1,1,1,e\left(\frac{2}{3}\right),e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(193, a) \) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)