from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8030, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,28,15]))
pari: [g,chi] = znchar(Mod(51,8030))
Basic properties
| Modulus: | \(8030\) | |
| Conductor: | \(803\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{803}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8030.ec
\(\chi_{8030}(51,\cdot)\) \(\chi_{8030}(501,\cdot)\) \(\chi_{8030}(1251,\cdot)\) \(\chi_{8030}(1701,\cdot)\) \(\chi_{8030}(2241,\cdot)\) \(\chi_{8030}(2691,\cdot)\) \(\chi_{8030}(3891,\cdot)\) \(\chi_{8030}(4171,\cdot)\) \(\chi_{8030}(4881,\cdot)\) \(\chi_{8030}(4901,\cdot)\) \(\chi_{8030}(5161,\cdot)\) \(\chi_{8030}(5891,\cdot)\) \(\chi_{8030}(6811,\cdot)\) \(\chi_{8030}(7091,\cdot)\) \(\chi_{8030}(7541,\cdot)\) \(\chi_{8030}(7801,\cdot)\)
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: | 40.40.5086707555259206569909391789171251844572469837694708123558699476396815066424043130627934440371174648777.1 |
Values on generators
\((1607,2191,881)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{3}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8030 }(51, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) |
sage: chi.jacobi_sum(n)