from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7440, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,45,26]))
pari: [g,chi] = znchar(Mod(613,7440))
Basic properties
| Modulus: | \(7440\) | |
| Conductor: | \(2480\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2480}(613,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7440.lf
\(\chi_{7440}(613,\cdot)\) \(\chi_{7440}(637,\cdot)\) \(\chi_{7440}(1357,\cdot)\) \(\chi_{7440}(2533,\cdot)\) \(\chi_{7440}(3277,\cdot)\) \(\chi_{7440}(3493,\cdot)\) \(\chi_{7440}(3733,\cdot)\) \(\chi_{7440}(4237,\cdot)\) \(\chi_{7440}(4477,\cdot)\) \(\chi_{7440}(4693,\cdot)\) \(\chi_{7440}(5437,\cdot)\) \(\chi_{7440}(5653,\cdot)\) \(\chi_{7440}(5893,\cdot)\) \(\chi_{7440}(6397,\cdot)\) \(\chi_{7440}(6637,\cdot)\) \(\chi_{7440}(7333,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((6511,1861,4961,2977,5521)\) → \((1,i,1,-i,e\left(\frac{13}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7440 }(613, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)