from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3672, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,26,27]))
pari: [g,chi] = znchar(Mod(497,3672))
Basic properties
| Modulus: | \(3672\) | |
| Conductor: | \(459\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{459}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3672.dr
\(\chi_{3672}(353,\cdot)\) \(\chi_{3672}(497,\cdot)\) \(\chi_{3672}(761,\cdot)\) \(\chi_{3672}(905,\cdot)\) \(\chi_{3672}(1577,\cdot)\) \(\chi_{3672}(1721,\cdot)\) \(\chi_{3672}(1985,\cdot)\) \(\chi_{3672}(2129,\cdot)\) \(\chi_{3672}(2801,\cdot)\) \(\chi_{3672}(2945,\cdot)\) \(\chi_{3672}(3209,\cdot)\) \(\chi_{3672}(3353,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.14555723661975701129520414713591505717264293810334661879947459060497463940377.1 |
Values on generators
\((919,1837,137,649)\) → \((1,1,e\left(\frac{13}{18}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3672 }(497, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)