from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,55,57]))
pari: [g,chi] = znchar(Mod(145,7728))
Basic properties
Modulus: | \(7728\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(145,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.gm
\(\chi_{7728}(145,\cdot)\) \(\chi_{7728}(241,\cdot)\) \(\chi_{7728}(481,\cdot)\) \(\chi_{7728}(1249,\cdot)\) \(\chi_{7728}(1489,\cdot)\) \(\chi_{7728}(1585,\cdot)\) \(\chi_{7728}(2593,\cdot)\) \(\chi_{7728}(4177,\cdot)\) \(\chi_{7728}(4513,\cdot)\) \(\chi_{7728}(4849,\cdot)\) \(\chi_{7728}(5185,\cdot)\) \(\chi_{7728}(5281,\cdot)\) \(\chi_{7728}(5617,\cdot)\) \(\chi_{7728}(5857,\cdot)\) \(\chi_{7728}(5953,\cdot)\) \(\chi_{7728}(6289,\cdot)\) \(\chi_{7728}(6529,\cdot)\) \(\chi_{7728}(6865,\cdot)\) \(\chi_{7728}(6961,\cdot)\) \(\chi_{7728}(7633,\cdot)\)
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: | Number field defined by a degree 66 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((1,1,1,e\left(\frac{5}{6}\right),e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 7728 }(145, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)