from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1334, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([14,33]))
pari: [g,chi] = znchar(Mod(17,1334))
Basic properties
| Modulus: | \(1334\) | |
| Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{667}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1334.q
\(\chi_{1334}(17,\cdot)\) \(\chi_{1334}(99,\cdot)\) \(\chi_{1334}(157,\cdot)\) \(\chi_{1334}(191,\cdot)\) \(\chi_{1334}(249,\cdot)\) \(\chi_{1334}(273,\cdot)\) \(\chi_{1334}(365,\cdot)\) \(\chi_{1334}(389,\cdot)\) \(\chi_{1334}(447,\cdot)\) \(\chi_{1334}(481,\cdot)\) \(\chi_{1334}(539,\cdot)\) \(\chi_{1334}(563,\cdot)\) \(\chi_{1334}(655,\cdot)\) \(\chi_{1334}(911,\cdot)\) \(\chi_{1334}(1003,\cdot)\) \(\chi_{1334}(1027,\cdot)\) \(\chi_{1334}(1119,\cdot)\) \(\chi_{1334}(1201,\cdot)\) \(\chi_{1334}(1259,\cdot)\) \(\chi_{1334}(1293,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1 |
Values on generators
\((465,553)\) → \((e\left(\frac{7}{22}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1334 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) |
sage: chi.jacobi_sum(n)