LMFDB - Dirichlet character orbit 355008.lx

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(355008, base_ring=CyclotomicField(7224)) M = H._module chi = DirichletCharacter(H, M([0,903,0,1544])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(25, 355008)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("355008.25"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$355008$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$59168$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$7224$	comment:Order sage:chi.multiplicative_order() gp:charorder(g,chi) magma:Order(chi);
Real:	no	comment:Whether the character is real sage:chi.multiplicative_order() <= 2 gp:charorder(g,chi) <= 2 magma:Order(chi) le 2;
Primitive:	no, induced from 59168.ew	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	no
Parity:	even	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{7224})$	comment:Field of values of chi sage:CyclotomicField(chi.multiplicative_order()) gp:nfinit(polcyclo(charorder(g,chi))) magma:CyclotomicField(Order(chi));
Fixed field:	Number field defined by a degree 7224 polynomial (not computed)	comment:Fixed field sage:chi.fixed_field()

First 6 of 2016 characters in Galois orbit

Character	$-1$	$1$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$25$	$29$	$31$
$\chi_{355008}(25,\cdot)$	$1$	$1$	$e\left(\frac{2711}{7224}\right)$	$e\left(\frac{317}{516}\right)$	$e\left(\frac{593}{2408}\right)$	$e\left(\frac{1129}{7224}\right)$	$e\left(\frac{493}{1806}\right)$	$e\left(\frac{83}{168}\right)$	$e\left(\frac{697}{3612}\right)$	$e\left(\frac{2711}{3612}\right)$	$e\left(\frac{997}{7224}\right)$	$e\left(\frac{262}{903}\right)$
$\chi_{355008}(169,\cdot)$	$1$	$1$	$e\left(\frac{1129}{7224}\right)$	$e\left(\frac{475}{516}\right)$	$e\left(\frac{1391}{2408}\right)$	$e\left(\frac{6575}{7224}\right)$	$e\left(\frac{1529}{1806}\right)$	$e\left(\frac{13}{168}\right)$	$e\left(\frac{1103}{3612}\right)$	$e\left(\frac{1129}{3612}\right)$	$e\left(\frac{6107}{7224}\right)$	$e\left(\frac{710}{903}\right)$
$\chi_{355008}(697,\cdot)$	$1$	$1$	$e\left(\frac{4379}{7224}\right)$	$e\left(\frac{245}{516}\right)$	$e\left(\frac{2117}{2408}\right)$	$e\left(\frac{1789}{7224}\right)$	$e\left(\frac{1693}{1806}\right)$	$e\left(\frac{47}{168}\right)$	$e\left(\frac{2269}{3612}\right)$	$e\left(\frac{767}{3612}\right)$	$e\left(\frac{7153}{7224}\right)$	$e\left(\frac{244}{903}\right)$
$\chi_{355008}(745,\cdot)$	$1$	$1$	$e\left(\frac{4121}{7224}\right)$	$e\left(\frac{503}{516}\right)$	$e\left(\frac{2375}{2408}\right)$	$e\left(\frac{5143}{7224}\right)$	$e\left(\frac{403}{1806}\right)$	$e\left(\frac{29}{168}\right)$	$e\left(\frac{3559}{3612}\right)$	$e\left(\frac{509}{3612}\right)$	$e\left(\frac{2251}{7224}\right)$	$e\left(\frac{760}{903}\right)$
$\chi_{355008}(841,\cdot)$	$1$	$1$	$e\left(\frac{997}{7224}\right)$	$e\left(\frac{451}{516}\right)$	$e\left(\frac{179}{2408}\right)$	$e\left(\frac{6107}{7224}\right)$	$e\left(\frac{1499}{1806}\right)$	$e\left(\frac{121}{168}\right)$	$e\left(\frac{251}{3612}\right)$	$e\left(\frac{997}{3612}\right)$	$e\left(\frac{7127}{7224}\right)$	$e\left(\frac{575}{903}\right)$
$\chi_{355008}(1321,\cdot)$	$1$	$1$	$e\left(\frac{1153}{7224}\right)$	$e\left(\frac{151}{516}\right)$	$e\left(\frac{79}{2408}\right)$	$e\left(\frac{2063}{7224}\right)$	$e\left(\frac{221}{1806}\right)$	$e\left(\frac{85}{168}\right)$	$e\left(\frac{2243}{3612}\right)$	$e\left(\frac{1153}{3612}\right)$	$e\left(\frac{11}{7224}\right)$	$e\left(\frac{242}{903}\right)$

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Character	\(-1\)	\(1\)	\(5\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(25\)	\(29\)	\(31\)
\(\chi_{355008}(25,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{2711}{7224}\right)\)	\(e\left(\frac{317}{516}\right)\)	\(e\left(\frac{593}{2408}\right)\)	\(e\left(\frac{1129}{7224}\right)\)	\(e\left(\frac{493}{1806}\right)\)	\(e\left(\frac{83}{168}\right)\)	\(e\left(\frac{697}{3612}\right)\)	\(e\left(\frac{2711}{3612}\right)\)	\(e\left(\frac{997}{7224}\right)\)	\(e\left(\frac{262}{903}\right)\)
\(\chi_{355008}(169,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{1129}{7224}\right)\)	\(e\left(\frac{475}{516}\right)\)	\(e\left(\frac{1391}{2408}\right)\)	\(e\left(\frac{6575}{7224}\right)\)	\(e\left(\frac{1529}{1806}\right)\)	\(e\left(\frac{13}{168}\right)\)	\(e\left(\frac{1103}{3612}\right)\)	\(e\left(\frac{1129}{3612}\right)\)	\(e\left(\frac{6107}{7224}\right)\)	\(e\left(\frac{710}{903}\right)\)
\(\chi_{355008}(697,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{4379}{7224}\right)\)	\(e\left(\frac{245}{516}\right)\)	\(e\left(\frac{2117}{2408}\right)\)	\(e\left(\frac{1789}{7224}\right)\)	\(e\left(\frac{1693}{1806}\right)\)	\(e\left(\frac{47}{168}\right)\)	\(e\left(\frac{2269}{3612}\right)\)	\(e\left(\frac{767}{3612}\right)\)	\(e\left(\frac{7153}{7224}\right)\)	\(e\left(\frac{244}{903}\right)\)
\(\chi_{355008}(745,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{4121}{7224}\right)\)	\(e\left(\frac{503}{516}\right)\)	\(e\left(\frac{2375}{2408}\right)\)	\(e\left(\frac{5143}{7224}\right)\)	\(e\left(\frac{403}{1806}\right)\)	\(e\left(\frac{29}{168}\right)\)	\(e\left(\frac{3559}{3612}\right)\)	\(e\left(\frac{509}{3612}\right)\)	\(e\left(\frac{2251}{7224}\right)\)	\(e\left(\frac{760}{903}\right)\)
\(\chi_{355008}(841,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{997}{7224}\right)\)	\(e\left(\frac{451}{516}\right)\)	\(e\left(\frac{179}{2408}\right)\)	\(e\left(\frac{6107}{7224}\right)\)	\(e\left(\frac{1499}{1806}\right)\)	\(e\left(\frac{121}{168}\right)\)	\(e\left(\frac{251}{3612}\right)\)	\(e\left(\frac{997}{3612}\right)\)	\(e\left(\frac{7127}{7224}\right)\)	\(e\left(\frac{575}{903}\right)\)
\(\chi_{355008}(1321,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{1153}{7224}\right)\)	\(e\left(\frac{151}{516}\right)\)	\(e\left(\frac{79}{2408}\right)\)	\(e\left(\frac{2063}{7224}\right)\)	\(e\left(\frac{221}{1806}\right)\)	\(e\left(\frac{85}{168}\right)\)	\(e\left(\frac{2243}{3612}\right)\)	\(e\left(\frac{1153}{3612}\right)\)	\(e\left(\frac{11}{7224}\right)\)	\(e\left(\frac{242}{903}\right)\)

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First 6 of 2016 characters in Galois orbit

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	355008.lx
Modulus	$355008$
Conductor	$59168$
Order	$7224$
Real	no
Primitive	no
Minimal	no
Parity	even

Modulus:	\(355008\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(59168\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(7224\)	comment:Order sage:chi.multiplicative_order() gp:charorder(g,chi) magma:Order(chi);
Real:	no	comment:Whether the character is real sage:chi.multiplicative_order() <= 2 gp:charorder(g,chi) <= 2 magma:Order(chi) le 2;
Primitive:	no, induced from 59168.ew	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	no
Parity:	even	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);