Properties

Label 8100.77
Modulus $8100$
Conductor $2025$
Order $540$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(540))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,290,27]))
 
pari: [g,chi] = znchar(Mod(77,8100))
 

Basic properties

Modulus: \(8100\)
Conductor: \(2025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(540\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2025}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8100.dr

\(\chi_{8100}(77,\cdot)\) \(\chi_{8100}(113,\cdot)\) \(\chi_{8100}(137,\cdot)\) \(\chi_{8100}(173,\cdot)\) \(\chi_{8100}(317,\cdot)\) \(\chi_{8100}(353,\cdot)\) \(\chi_{8100}(437,\cdot)\) \(\chi_{8100}(473,\cdot)\) \(\chi_{8100}(497,\cdot)\) \(\chi_{8100}(533,\cdot)\) \(\chi_{8100}(617,\cdot)\) \(\chi_{8100}(653,\cdot)\) \(\chi_{8100}(677,\cdot)\) \(\chi_{8100}(713,\cdot)\) \(\chi_{8100}(797,\cdot)\) \(\chi_{8100}(833,\cdot)\) \(\chi_{8100}(977,\cdot)\) \(\chi_{8100}(1013,\cdot)\) \(\chi_{8100}(1037,\cdot)\) \(\chi_{8100}(1073,\cdot)\) \(\chi_{8100}(1217,\cdot)\) \(\chi_{8100}(1253,\cdot)\) \(\chi_{8100}(1337,\cdot)\) \(\chi_{8100}(1373,\cdot)\) \(\chi_{8100}(1397,\cdot)\) \(\chi_{8100}(1433,\cdot)\) \(\chi_{8100}(1517,\cdot)\)

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: $\Q(\zeta_{540})$
Fixed field: Number field defined by a degree 540 polynomial (not computed)

Values on generators

\((4051,6401,7777)\) → \((1,e\left(\frac{29}{54}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8100 }(77, a) \) \(1\)\(1\)\(e\left(\frac{91}{108}\right)\)\(e\left(\frac{211}{270}\right)\)\(e\left(\frac{133}{540}\right)\)\(e\left(\frac{67}{180}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{247}{540}\right)\)\(e\left(\frac{131}{135}\right)\)\(e\left(\frac{19}{135}\right)\)\(e\left(\frac{1}{180}\right)\)\(e\left(\frac{179}{270}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8100 }(77,a) \;\) at \(\;a = \) e.g. 2