sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(344, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,21,10]))
pari: [g,chi] = znchar(Mod(53,344))
Basic properties
| Modulus: | \(344\) | |
| Conductor: | \(344\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 344.z
\(\chi_{344}(13,\cdot)\) \(\chi_{344}(53,\cdot)\) \(\chi_{344}(101,\cdot)\) \(\chi_{344}(109,\cdot)\) \(\chi_{344}(117,\cdot)\) \(\chi_{344}(181,\cdot)\) \(\chi_{344}(189,\cdot)\) \(\chi_{344}(197,\cdot)\) \(\chi_{344}(229,\cdot)\) \(\chi_{344}(253,\cdot)\) \(\chi_{344}(325,\cdot)\) \(\chi_{344}(341,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.2011999877834826766225008958075022926316813554075780070378415668274435623250777079808.1 |
Values on generators
\((87,173,89)\) → \((1,-1,e\left(\frac{5}{21}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{344}(53,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(53,r) e\left(\frac{r}{172}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{344}(53,\cdot),\chi_{344}(1,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(53,r) \chi_{344}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{344}(53,·))
= \sum_{r \in \Z/344\Z}
\chi_{344}(53,r) e\left(\frac{1 r + 2 r^{-1}}{344}\right)
= -0.1010967375+-0.4429337477i \)