from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(332928, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,1,0,0]))
pari: [g,chi] = znchar(Mod(260101,332928))
Basic properties
| Modulus: | \(332928\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{128}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 332928.ln
\(\chi_{332928}(10405,\cdot)\) \(\chi_{332928}(31213,\cdot)\) \(\chi_{332928}(52021,\cdot)\) \(\chi_{332928}(72829,\cdot)\) \(\chi_{332928}(93637,\cdot)\) \(\chi_{332928}(114445,\cdot)\) \(\chi_{332928}(135253,\cdot)\) \(\chi_{332928}(156061,\cdot)\) \(\chi_{332928}(176869,\cdot)\) \(\chi_{332928}(197677,\cdot)\) \(\chi_{332928}(218485,\cdot)\) \(\chi_{332928}(239293,\cdot)\) \(\chi_{332928}(260101,\cdot)\) \(\chi_{332928}(280909,\cdot)\) \(\chi_{332928}(301717,\cdot)\) \(\chi_{332928}(322525,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((202879,260101,184961,165889)\) → \((1,e\left(\frac{1}{32}\right),1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 332928 }(260101, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(i\) | \(e\left(\frac{11}{32}\right)\) |
sage: chi.jacobi_sum(n)