from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7440, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,15,26]))
pari: [g,chi] = znchar(Mod(49,7440))
Basic properties
| Modulus: | \(7440\) | |
| Conductor: | \(155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{155}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7440.jq
\(\chi_{7440}(49,\cdot)\) \(\chi_{7440}(289,\cdot)\) \(\chi_{7440}(1249,\cdot)\) \(\chi_{7440}(3169,\cdot)\) \(\chi_{7440}(3889,\cdot)\) \(\chi_{7440}(5329,\cdot)\) \(\chi_{7440}(5569,\cdot)\) \(\chi_{7440}(6529,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.17485477327500765872904889567178150559785186767578125.1 |
Values on generators
\((6511,1861,4961,2977,5521)\) → \((1,1,1,-1,e\left(\frac{13}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7440 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) |
sage: chi.jacobi_sum(n)