from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,14]))
pari: [g,chi] = znchar(Mod(253,4020))
Basic properties
| Modulus: | \(4020\) | |
| Conductor: | \(335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{335}(253,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.ct
\(\chi_{4020}(253,\cdot)\) \(\chi_{4020}(313,\cdot)\) \(\chi_{4020}(673,\cdot)\) \(\chi_{4020}(697,\cdot)\) \(\chi_{4020}(913,\cdot)\) \(\chi_{4020}(1057,\cdot)\) \(\chi_{4020}(1117,\cdot)\) \(\chi_{4020}(1393,\cdot)\) \(\chi_{4020}(1477,\cdot)\) \(\chi_{4020}(1717,\cdot)\) \(\chi_{4020}(2053,\cdot)\) \(\chi_{4020}(2197,\cdot)\) \(\chi_{4020}(2353,\cdot)\) \(\chi_{4020}(2857,\cdot)\) \(\chi_{4020}(2953,\cdot)\) \(\chi_{4020}(3073,\cdot)\) \(\chi_{4020}(3157,\cdot)\) \(\chi_{4020}(3757,\cdot)\) \(\chi_{4020}(3877,\cdot)\) \(\chi_{4020}(3913,\cdot)\)
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: | 44.44.5769681722973112639196821168549004411214468274227153473067543998508362785597448237240314483642578125.1 |
Values on generators
\((2011,2681,3217,1141)\) → \((1,1,-i,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(253, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(-1\) | \(e\left(\frac{21}{22}\right)\) | \(-i\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)