from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,20,34]))
pari: [g,chi] = znchar(Mod(197,7200))
Basic properties
| Modulus: | \(7200\) | |
| Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2400}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7200.gm
\(\chi_{7200}(197,\cdot)\) \(\chi_{7200}(413,\cdot)\) \(\chi_{7200}(917,\cdot)\) \(\chi_{7200}(1133,\cdot)\) \(\chi_{7200}(1637,\cdot)\) \(\chi_{7200}(1853,\cdot)\) \(\chi_{7200}(2573,\cdot)\) \(\chi_{7200}(3077,\cdot)\) \(\chi_{7200}(3797,\cdot)\) \(\chi_{7200}(4013,\cdot)\) \(\chi_{7200}(4517,\cdot)\) \(\chi_{7200}(4733,\cdot)\) \(\chi_{7200}(5237,\cdot)\) \(\chi_{7200}(5453,\cdot)\) \(\chi_{7200}(6173,\cdot)\) \(\chi_{7200}(6677,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.1348884380735497228084799435251384320000000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((6751,901,6401,577)\) → \((1,e\left(\frac{1}{8}\right),-1,e\left(\frac{17}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7200 }(197, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)