LMFDB - Dirichlet character orbit 409600.hf

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409600, base_ring=CyclotomicField(10240)) M = H._module chi = DirichletCharacter(H, M([5120,9325,3072])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(39, 409600)) 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("409600.39"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$409600$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$204800$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$10240$	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 204800.gx	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	no
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{10240})$	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 10240 polynomial (not computed)	comment:Fixed field sage:chi.fixed_field()

First 15 of 4096 characters in Galois orbit

Character	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{409600}(39,\cdot)$	$-1$	$1$	$e\left(\frac{8039}{10240}\right)$	$e\left(\frac{717}{1024}\right)$	$e\left(\frac{2919}{5120}\right)$	$e\left(\frac{7857}{10240}\right)$	$e\left(\frac{2243}{10240}\right)$	$e\left(\frac{219}{2560}\right)$	$e\left(\frac{8011}{10240}\right)$	$e\left(\frac{4969}{10240}\right)$	$e\left(\frac{3131}{5120}\right)$	$e\left(\frac{3637}{10240}\right)$
$\chi_{409600}(119,\cdot)$	$-1$	$1$	$e\left(\frac{1117}{10240}\right)$	$e\left(\frac{287}{1024}\right)$	$e\left(\frac{1117}{5120}\right)$	$e\left(\frac{7211}{10240}\right)$	$e\left(\frac{6929}{10240}\right)$	$e\left(\frac{2457}{2560}\right)$	$e\left(\frac{5993}{10240}\right)$	$e\left(\frac{3987}{10240}\right)$	$e\left(\frac{2233}{5120}\right)$	$e\left(\frac{3351}{10240}\right)$
$\chi_{409600}(279,\cdot)$	$-1$	$1$	$e\left(\frac{5353}{10240}\right)$	$e\left(\frac{675}{1024}\right)$	$e\left(\frac{233}{5120}\right)$	$e\left(\frac{4479}{10240}\right)$	$e\left(\frac{4301}{10240}\right)$	$e\left(\frac{1333}{2560}\right)$	$e\left(\frac{9157}{10240}\right)$	$e\left(\frac{1863}{10240}\right)$	$e\left(\frac{5077}{5120}\right)$	$e\left(\frac{5819}{10240}\right)$
$\chi_{409600}(359,\cdot)$	$-1$	$1$	$e\left(\frac{6271}{10240}\right)$	$e\left(\frac{469}{1024}\right)$	$e\left(\frac{1151}{5120}\right)$	$e\left(\frac{2393}{10240}\right)$	$e\left(\frac{7227}{10240}\right)$	$e\left(\frac{531}{2560}\right)$	$e\left(\frac{4099}{10240}\right)$	$e\left(\frac{721}{10240}\right)$	$e\left(\frac{3699}{5120}\right)$	$e\left(\frac{8573}{10240}\right)$
$\chi_{409600}(439,\cdot)$	$-1$	$1$	$e\left(\frac{6389}{10240}\right)$	$e\left(\frac{679}{1024}\right)$	$e\left(\frac{1269}{5120}\right)$	$e\left(\frac{10067}{10240}\right)$	$e\left(\frac{6153}{10240}\right)$	$e\left(\frac{2129}{2560}\right)$	$e\left(\frac{1441}{10240}\right)$	$e\left(\frac{2939}{10240}\right)$	$e\left(\frac{2161}{5120}\right)$	$e\left(\frac{8927}{10240}\right)$
$\chi_{409600}(519,\cdot)$	$-1$	$1$	$e\left(\frac{587}{10240}\right)$	$e\left(\frac{281}{1024}\right)$	$e\left(\frac{587}{5120}\right)$	$e\left(\frac{1901}{10240}\right)$	$e\left(\frac{6199}{10240}\right)$	$e\left(\frac{1007}{2560}\right)$	$e\left(\frac{6303}{10240}\right)$	$e\left(\frac{3397}{10240}\right)$	$e\left(\frac{463}{5120}\right)$	$e\left(\frac{1761}{10240}\right)$
$\chi_{409600}(679,\cdot)$	$-1$	$1$	$e\left(\frac{1943}{10240}\right)$	$e\left(\frac{733}{1024}\right)$	$e\left(\frac{1943}{5120}\right)$	$e\left(\frac{9729}{10240}\right)$	$e\left(\frac{9651}{10240}\right)$	$e\left(\frac{843}{2560}\right)$	$e\left(\frac{7867}{10240}\right)$	$e\left(\frac{9273}{10240}\right)$	$e\left(\frac{1707}{5120}\right)$	$e\left(\frac{5829}{10240}\right)$
$\chi_{409600}(759,\cdot)$	$-1$	$1$	$e\left(\frac{9101}{10240}\right)$	$e\left(\frac{559}{1024}\right)$	$e\left(\frac{3981}{5120}\right)$	$e\left(\frac{5243}{10240}\right)$	$e\left(\frac{2817}{10240}\right)$	$e\left(\frac{1801}{2560}\right)$	$e\left(\frac{4569}{10240}\right)$	$e\left(\frac{4451}{10240}\right)$	$e\left(\frac{4649}{5120}\right)$	$e\left(\frac{6823}{10240}\right)$
$\chi_{409600}(839,\cdot)$	$-1$	$1$	$e\left(\frac{99}{10240}\right)$	$e\left(\frac{801}{1024}\right)$	$e\left(\frac{99}{5120}\right)$	$e\left(\frac{5397}{10240}\right)$	$e\left(\frac{7343}{10240}\right)$	$e\left(\frac{39}{2560}\right)$	$e\left(\frac{8791}{10240}\right)$	$e\left(\frac{8109}{10240}\right)$	$e\left(\frac{2311}{5120}\right)$	$e\left(\frac{297}{10240}\right)$
$\chi_{409600}(919,\cdot)$	$-1$	$1$	$e\left(\frac{537}{10240}\right)$	$e\left(\frac{435}{1024}\right)$	$e\left(\frac{537}{5120}\right)$	$e\left(\frac{5071}{10240}\right)$	$e\left(\frac{7869}{10240}\right)$	$e\left(\frac{677}{2560}\right)$	$e\left(\frac{5173}{10240}\right)$	$e\left(\frac{4887}{10240}\right)$	$e\left(\frac{4933}{5120}\right)$	$e\left(\frac{1611}{10240}\right)$
$\chi_{409600}(1079,\cdot)$	$-1$	$1$	$e\left(\frac{9253}{10240}\right)$	$e\left(\frac{951}{1024}\right)$	$e\left(\frac{4133}{5120}\right)$	$e\left(\frac{2979}{10240}\right)$	$e\left(\frac{7161}{10240}\right)$	$e\left(\frac{1473}{2560}\right)$	$e\left(\frac{5137}{10240}\right)$	$e\left(\frac{8523}{10240}\right)$	$e\left(\frac{4577}{5120}\right)$	$e\left(\frac{7279}{10240}\right)$
$\chi_{409600}(1159,\cdot)$	$-1$	$1$	$e\left(\frac{7291}{10240}\right)$	$e\left(\frac{809}{1024}\right)$	$e\left(\frac{2171}{5120}\right)$	$e\left(\frac{1213}{10240}\right)$	$e\left(\frac{5927}{10240}\right)$	$e\left(\frac{1631}{2560}\right)$	$e\left(\frac{8719}{10240}\right)$	$e\left(\frac{5141}{10240}\right)$	$e\left(\frac{1599}{5120}\right)$	$e\left(\frac{1393}{10240}\right)$
$\chi_{409600}(1239,\cdot)$	$-1$	$1$	$e\left(\frac{4529}{10240}\right)$	$e\left(\frac{59}{1024}\right)$	$e\left(\frac{4529}{5120}\right)$	$e\left(\frac{9207}{10240}\right)$	$e\left(\frac{693}{10240}\right)$	$e\left(\frac{1629}{2560}\right)$	$e\left(\frac{4461}{10240}\right)$	$e\left(\frac{5119}{10240}\right)$	$e\left(\frac{3581}{5120}\right)$	$e\left(\frac{3347}{10240}\right)$
$\chi_{409600}(1319,\cdot)$	$-1$	$1$	$e\left(\frac{6087}{10240}\right)$	$e\left(\frac{749}{1024}\right)$	$e\left(\frac{967}{5120}\right)$	$e\left(\frac{1361}{10240}\right)$	$e\left(\frac{6819}{10240}\right)$	$e\left(\frac{1467}{2560}\right)$	$e\left(\frac{7723}{10240}\right)$	$e\left(\frac{3337}{10240}\right)$	$e\left(\frac{283}{5120}\right)$	$e\left(\frac{8021}{10240}\right)$
$\chi_{409600}(1479,\cdot)$	$-1$	$1$	$e\left(\frac{1683}{10240}\right)$	$e\left(\frac{305}{1024}\right)$	$e\left(\frac{1683}{5120}\right)$	$e\left(\frac{9829}{10240}\right)$	$e\left(\frac{1951}{10240}\right)$	$e\left(\frac{663}{2560}\right)$	$e\left(\frac{6087}{10240}\right)$	$e\left(\frac{4733}{10240}\right)$	$e\left(\frac{3447}{5120}\right)$	$e\left(\frac{5049}{10240}\right)$

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Character	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{409600}(39,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{8039}{10240}\right)\)	\(e\left(\frac{717}{1024}\right)\)	\(e\left(\frac{2919}{5120}\right)\)	\(e\left(\frac{7857}{10240}\right)\)	\(e\left(\frac{2243}{10240}\right)\)	\(e\left(\frac{219}{2560}\right)\)	\(e\left(\frac{8011}{10240}\right)\)	\(e\left(\frac{4969}{10240}\right)\)	\(e\left(\frac{3131}{5120}\right)\)	\(e\left(\frac{3637}{10240}\right)\)
\(\chi_{409600}(119,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1117}{10240}\right)\)	\(e\left(\frac{287}{1024}\right)\)	\(e\left(\frac{1117}{5120}\right)\)	\(e\left(\frac{7211}{10240}\right)\)	\(e\left(\frac{6929}{10240}\right)\)	\(e\left(\frac{2457}{2560}\right)\)	\(e\left(\frac{5993}{10240}\right)\)	\(e\left(\frac{3987}{10240}\right)\)	\(e\left(\frac{2233}{5120}\right)\)	\(e\left(\frac{3351}{10240}\right)\)
\(\chi_{409600}(279,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{5353}{10240}\right)\)	\(e\left(\frac{675}{1024}\right)\)	\(e\left(\frac{233}{5120}\right)\)	\(e\left(\frac{4479}{10240}\right)\)	\(e\left(\frac{4301}{10240}\right)\)	\(e\left(\frac{1333}{2560}\right)\)	\(e\left(\frac{9157}{10240}\right)\)	\(e\left(\frac{1863}{10240}\right)\)	\(e\left(\frac{5077}{5120}\right)\)	\(e\left(\frac{5819}{10240}\right)\)
\(\chi_{409600}(359,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{6271}{10240}\right)\)	\(e\left(\frac{469}{1024}\right)\)	\(e\left(\frac{1151}{5120}\right)\)	\(e\left(\frac{2393}{10240}\right)\)	\(e\left(\frac{7227}{10240}\right)\)	\(e\left(\frac{531}{2560}\right)\)	\(e\left(\frac{4099}{10240}\right)\)	\(e\left(\frac{721}{10240}\right)\)	\(e\left(\frac{3699}{5120}\right)\)	\(e\left(\frac{8573}{10240}\right)\)
\(\chi_{409600}(439,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{6389}{10240}\right)\)	\(e\left(\frac{679}{1024}\right)\)	\(e\left(\frac{1269}{5120}\right)\)	\(e\left(\frac{10067}{10240}\right)\)	\(e\left(\frac{6153}{10240}\right)\)	\(e\left(\frac{2129}{2560}\right)\)	\(e\left(\frac{1441}{10240}\right)\)	\(e\left(\frac{2939}{10240}\right)\)	\(e\left(\frac{2161}{5120}\right)\)	\(e\left(\frac{8927}{10240}\right)\)
\(\chi_{409600}(519,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{587}{10240}\right)\)	\(e\left(\frac{281}{1024}\right)\)	\(e\left(\frac{587}{5120}\right)\)	\(e\left(\frac{1901}{10240}\right)\)	\(e\left(\frac{6199}{10240}\right)\)	\(e\left(\frac{1007}{2560}\right)\)	\(e\left(\frac{6303}{10240}\right)\)	\(e\left(\frac{3397}{10240}\right)\)	\(e\left(\frac{463}{5120}\right)\)	\(e\left(\frac{1761}{10240}\right)\)
\(\chi_{409600}(679,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1943}{10240}\right)\)	\(e\left(\frac{733}{1024}\right)\)	\(e\left(\frac{1943}{5120}\right)\)	\(e\left(\frac{9729}{10240}\right)\)	\(e\left(\frac{9651}{10240}\right)\)	\(e\left(\frac{843}{2560}\right)\)	\(e\left(\frac{7867}{10240}\right)\)	\(e\left(\frac{9273}{10240}\right)\)	\(e\left(\frac{1707}{5120}\right)\)	\(e\left(\frac{5829}{10240}\right)\)
\(\chi_{409600}(759,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{9101}{10240}\right)\)	\(e\left(\frac{559}{1024}\right)\)	\(e\left(\frac{3981}{5120}\right)\)	\(e\left(\frac{5243}{10240}\right)\)	\(e\left(\frac{2817}{10240}\right)\)	\(e\left(\frac{1801}{2560}\right)\)	\(e\left(\frac{4569}{10240}\right)\)	\(e\left(\frac{4451}{10240}\right)\)	\(e\left(\frac{4649}{5120}\right)\)	\(e\left(\frac{6823}{10240}\right)\)
\(\chi_{409600}(839,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{99}{10240}\right)\)	\(e\left(\frac{801}{1024}\right)\)	\(e\left(\frac{99}{5120}\right)\)	\(e\left(\frac{5397}{10240}\right)\)	\(e\left(\frac{7343}{10240}\right)\)	\(e\left(\frac{39}{2560}\right)\)	\(e\left(\frac{8791}{10240}\right)\)	\(e\left(\frac{8109}{10240}\right)\)	\(e\left(\frac{2311}{5120}\right)\)	\(e\left(\frac{297}{10240}\right)\)
\(\chi_{409600}(919,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{537}{10240}\right)\)	\(e\left(\frac{435}{1024}\right)\)	\(e\left(\frac{537}{5120}\right)\)	\(e\left(\frac{5071}{10240}\right)\)	\(e\left(\frac{7869}{10240}\right)\)	\(e\left(\frac{677}{2560}\right)\)	\(e\left(\frac{5173}{10240}\right)\)	\(e\left(\frac{4887}{10240}\right)\)	\(e\left(\frac{4933}{5120}\right)\)	\(e\left(\frac{1611}{10240}\right)\)
\(\chi_{409600}(1079,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{9253}{10240}\right)\)	\(e\left(\frac{951}{1024}\right)\)	\(e\left(\frac{4133}{5120}\right)\)	\(e\left(\frac{2979}{10240}\right)\)	\(e\left(\frac{7161}{10240}\right)\)	\(e\left(\frac{1473}{2560}\right)\)	\(e\left(\frac{5137}{10240}\right)\)	\(e\left(\frac{8523}{10240}\right)\)	\(e\left(\frac{4577}{5120}\right)\)	\(e\left(\frac{7279}{10240}\right)\)
\(\chi_{409600}(1159,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{7291}{10240}\right)\)	\(e\left(\frac{809}{1024}\right)\)	\(e\left(\frac{2171}{5120}\right)\)	\(e\left(\frac{1213}{10240}\right)\)	\(e\left(\frac{5927}{10240}\right)\)	\(e\left(\frac{1631}{2560}\right)\)	\(e\left(\frac{8719}{10240}\right)\)	\(e\left(\frac{5141}{10240}\right)\)	\(e\left(\frac{1599}{5120}\right)\)	\(e\left(\frac{1393}{10240}\right)\)
\(\chi_{409600}(1239,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{4529}{10240}\right)\)	\(e\left(\frac{59}{1024}\right)\)	\(e\left(\frac{4529}{5120}\right)\)	\(e\left(\frac{9207}{10240}\right)\)	\(e\left(\frac{693}{10240}\right)\)	\(e\left(\frac{1629}{2560}\right)\)	\(e\left(\frac{4461}{10240}\right)\)	\(e\left(\frac{5119}{10240}\right)\)	\(e\left(\frac{3581}{5120}\right)\)	\(e\left(\frac{3347}{10240}\right)\)
\(\chi_{409600}(1319,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{6087}{10240}\right)\)	\(e\left(\frac{749}{1024}\right)\)	\(e\left(\frac{967}{5120}\right)\)	\(e\left(\frac{1361}{10240}\right)\)	\(e\left(\frac{6819}{10240}\right)\)	\(e\left(\frac{1467}{2560}\right)\)	\(e\left(\frac{7723}{10240}\right)\)	\(e\left(\frac{3337}{10240}\right)\)	\(e\left(\frac{283}{5120}\right)\)	\(e\left(\frac{8021}{10240}\right)\)
\(\chi_{409600}(1479,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1683}{10240}\right)\)	\(e\left(\frac{305}{1024}\right)\)	\(e\left(\frac{1683}{5120}\right)\)	\(e\left(\frac{9829}{10240}\right)\)	\(e\left(\frac{1951}{10240}\right)\)	\(e\left(\frac{663}{2560}\right)\)	\(e\left(\frac{6087}{10240}\right)\)	\(e\left(\frac{4733}{10240}\right)\)	\(e\left(\frac{3447}{5120}\right)\)	\(e\left(\frac{5049}{10240}\right)\)

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First 15 of 4096 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	409600.hf
Modulus	$409600$
Conductor	$204800$
Order	$10240$
Real	no
Primitive	no
Minimal	no
Parity	odd

Modulus:	\(409600\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(204800\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(10240\)	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 204800.gx	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	no
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);