from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8064, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,40,24]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,8064))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(8064\) | |
Conductor: | \(4032\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 4032.gw | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{8064}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{16}\right)\) |
\(\chi_{8064}(713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{16}\right)\) |
\(\chi_{8064}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{9}{16}\right)\) |
\(\chi_{8064}(1721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{16}\right)\) |
\(\chi_{8064}(2057,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{16}\right)\) |
\(\chi_{8064}(2729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{16}\right)\) |
\(\chi_{8064}(3065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{16}\right)\) |
\(\chi_{8064}(3737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{16}\right)\) |
\(\chi_{8064}(4073,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{15}{16}\right)\) |
\(\chi_{8064}(4745,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{16}\right)\) |
\(\chi_{8064}(5081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{16}\right)\) |
\(\chi_{8064}(5753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{16}\right)\) |
\(\chi_{8064}(6089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{16}\right)\) |
\(\chi_{8064}(6761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{15}{16}\right)\) |
\(\chi_{8064}(7097,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{16}\right)\) |
\(\chi_{8064}(7769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{16}\right)\) |