Properties

Label 7605.406
Modulus $7605$
Conductor $169$
Order $39$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,74]))
 
pari: [g,chi] = znchar(Mod(406,7605))
 

Basic properties

Modulus: \(7605\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(68,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7605.eg

\(\chi_{7605}(406,\cdot)\) \(\chi_{7605}(451,\cdot)\) \(\chi_{7605}(1576,\cdot)\) \(\chi_{7605}(1621,\cdot)\) \(\chi_{7605}(2161,\cdot)\) \(\chi_{7605}(2206,\cdot)\) \(\chi_{7605}(2746,\cdot)\) \(\chi_{7605}(2791,\cdot)\) \(\chi_{7605}(3331,\cdot)\) \(\chi_{7605}(3376,\cdot)\) \(\chi_{7605}(3916,\cdot)\) \(\chi_{7605}(3961,\cdot)\) \(\chi_{7605}(4501,\cdot)\) \(\chi_{7605}(4546,\cdot)\) \(\chi_{7605}(5086,\cdot)\) \(\chi_{7605}(5131,\cdot)\) \(\chi_{7605}(5671,\cdot)\) \(\chi_{7605}(5716,\cdot)\) \(\chi_{7605}(6256,\cdot)\) \(\chi_{7605}(6301,\cdot)\) \(\chi_{7605}(6841,\cdot)\) \(\chi_{7605}(6886,\cdot)\) \(\chi_{7605}(7426,\cdot)\) \(\chi_{7605}(7471,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((6761,1522,6931)\) → \((1,1,e\left(\frac{37}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 7605 }(406, a) \) \(1\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7605 }(406,a) \;\) at \(\;a = \) e.g. 2