from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3630, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,6]))
pari: [g,chi] = znchar(Mod(3607,3630))
Basic properties
| Modulus: | \(3630\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(582,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3630.bg
\(\chi_{3630}(43,\cdot)\) \(\chi_{3630}(307,\cdot)\) \(\chi_{3630}(373,\cdot)\) \(\chi_{3630}(637,\cdot)\) \(\chi_{3630}(703,\cdot)\) \(\chi_{3630}(1033,\cdot)\) \(\chi_{3630}(1297,\cdot)\) \(\chi_{3630}(1363,\cdot)\) \(\chi_{3630}(1627,\cdot)\) \(\chi_{3630}(1957,\cdot)\) \(\chi_{3630}(2023,\cdot)\) \(\chi_{3630}(2287,\cdot)\) \(\chi_{3630}(2353,\cdot)\) \(\chi_{3630}(2617,\cdot)\) \(\chi_{3630}(2683,\cdot)\) \(\chi_{3630}(2947,\cdot)\) \(\chi_{3630}(3013,\cdot)\) \(\chi_{3630}(3277,\cdot)\) \(\chi_{3630}(3343,\cdot)\) \(\chi_{3630}(3607,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((1211,727,3511)\) → \((1,i,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3630 }(3607, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) |
sage: chi.jacobi_sum(n)