from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2624, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,20,37]))
pari: [g,chi] = znchar(Mod(97,2624))
Basic properties
| Modulus: | \(2624\) | |
| Conductor: | \(328\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{328}(261,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2624.dr
\(\chi_{2624}(97,\cdot)\) \(\chi_{2624}(417,\cdot)\) \(\chi_{2624}(481,\cdot)\) \(\chi_{2624}(545,\cdot)\) \(\chi_{2624}(609,\cdot)\) \(\chi_{2624}(673,\cdot)\) \(\chi_{2624}(801,\cdot)\) \(\chi_{2624}(1249,\cdot)\) \(\chi_{2624}(1377,\cdot)\) \(\chi_{2624}(1441,\cdot)\) \(\chi_{2624}(1505,\cdot)\) \(\chi_{2624}(1569,\cdot)\) \(\chi_{2624}(1633,\cdot)\) \(\chi_{2624}(1953,\cdot)\) \(\chi_{2624}(2145,\cdot)\) \(\chi_{2624}(2529,\cdot)\)
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: | 40.0.912788483257978497757884926199917783690257306123427760963531833190283833440731136.1 |
Values on generators
\((575,1477,129)\) → \((1,-1,e\left(\frac{37}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2624 }(97, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(-i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)