sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(690, base_ring=CyclotomicField(44))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,11,38]))
pari: [g,chi] = znchar(Mod(7,690))
Basic properties
Modulus: | \(690\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 690.w
\(\chi_{690}(7,\cdot)\) \(\chi_{690}(37,\cdot)\) \(\chi_{690}(43,\cdot)\) \(\chi_{690}(67,\cdot)\) \(\chi_{690}(97,\cdot)\) \(\chi_{690}(103,\cdot)\) \(\chi_{690}(157,\cdot)\) \(\chi_{690}(217,\cdot)\) \(\chi_{690}(247,\cdot)\) \(\chi_{690}(283,\cdot)\) \(\chi_{690}(313,\cdot)\) \(\chi_{690}(337,\cdot)\) \(\chi_{690}(343,\cdot)\) \(\chi_{690}(373,\cdot)\) \(\chi_{690}(433,\cdot)\) \(\chi_{690}(457,\cdot)\) \(\chi_{690}(493,\cdot)\) \(\chi_{690}(517,\cdot)\) \(\chi_{690}(523,\cdot)\) \(\chi_{690}(613,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\((461,277,511)\) → \((1,i,e\left(\frac{19}{22}\right))\)
Values
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) |
Related number fields
Field of values: | \(\Q(\zeta_{44})\) |
Fixed field: | \(\Q(\zeta_{115})^+\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{690}(7,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(7,r) e\left(\frac{r}{345}\right) = 7.7979078568+-7.3615645794i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{690}(7,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(7,r) \chi_{690}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{690}(7,·))
= \sum_{r \in \Z/690\Z}
\chi_{690}(7,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right)
= 41.5825976865+-2.9740460828i \)