LMFDB - Dirichlet character orbit 355008.mc

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(355008, base_ring=CyclotomicField(14448)) M = H._module chi = DirichletCharacter(H, M([7224,2709,0,2336])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(67, 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("355008.67"); 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:	$118336$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$14448$	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 118336.gb	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{14448})$	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 14448 polynomial (not computed)	comment:Fixed field sage:chi.fixed_field()

First 17 of 4032 characters in Galois orbit

Character	$-1$	$1$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$25$	$29$	$31$
$\chi_{355008}(67,\cdot)$	$-1$	$1$	$e\left(\frac{5669}{14448}\right)$	$e\left(\frac{1019}{1032}\right)$	$e\left(\frac{827}{4816}\right)$	$e\left(\frac{13915}{14448}\right)$	$e\left(\frac{919}{3612}\right)$	$e\left(\frac{305}{336}\right)$	$e\left(\frac{6655}{7224}\right)$	$e\left(\frac{5669}{7224}\right)$	$e\left(\frac{9991}{14448}\right)$	$e\left(\frac{638}{903}\right)$
$\chi_{355008}(139,\cdot)$	$-1$	$1$	$e\left(\frac{6179}{14448}\right)$	$e\left(\frac{197}{1032}\right)$	$e\left(\frac{2445}{4816}\right)$	$e\left(\frac{10141}{14448}\right)$	$e\left(\frac{1117}{3612}\right)$	$e\left(\frac{71}{336}\right)$	$e\left(\frac{1081}{7224}\right)$	$e\left(\frac{6179}{7224}\right)$	$e\left(\frac{12289}{14448}\right)$	$e\left(\frac{632}{903}\right)$
$\chi_{355008}(187,\cdot)$	$-1$	$1$	$e\left(\frac{3751}{14448}\right)$	$e\left(\frac{553}{1032}\right)$	$e\left(\frac{729}{4816}\right)$	$e\left(\frac{3065}{14448}\right)$	$e\left(\frac{401}{3612}\right)$	$e\left(\frac{235}{336}\right)$	$e\left(\frac{5549}{7224}\right)$	$e\left(\frac{3751}{7224}\right)$	$e\left(\frac{13757}{14448}\right)$	$e\left(\frac{526}{903}\right)$
$\chi_{355008}(283,\cdot)$	$-1$	$1$	$e\left(\frac{3343}{14448}\right)$	$e\left(\frac{385}{1032}\right)$	$e\left(\frac{1361}{4816}\right)$	$e\left(\frac{305}{14448}\right)$	$e\left(\frac{965}{3612}\right)$	$e\left(\frac{19}{336}\right)$	$e\left(\frac{4229}{7224}\right)$	$e\left(\frac{3343}{7224}\right)$	$e\left(\frac{9029}{14448}\right)$	$e\left(\frac{892}{903}\right)$
$\chi_{355008}(427,\cdot)$	$-1$	$1$	$e\left(\frac{8747}{14448}\right)$	$e\left(\frac{101}{1032}\right)$	$e\left(\frac{4133}{4816}\right)$	$e\left(\frac{11365}{14448}\right)$	$e\left(\frac{2029}{3612}\right)$	$e\left(\frac{47}{336}\right)$	$e\left(\frac{3865}{7224}\right)$	$e\left(\frac{1523}{7224}\right)$	$e\left(\frac{2953}{14448}\right)$	$e\left(\frac{878}{903}\right)$
$\chi_{355008}(547,\cdot)$	$-1$	$1$	$e\left(\frac{1277}{14448}\right)$	$e\left(\frac{971}{1032}\right)$	$e\left(\frac{2531}{4816}\right)$	$e\left(\frac{8851}{14448}\right)$	$e\left(\frac{1891}{3612}\right)$	$e\left(\frac{233}{336}\right)$	$e\left(\frac{3919}{7224}\right)$	$e\left(\frac{1277}{7224}\right)$	$e\left(\frac{5839}{14448}\right)$	$e\left(\frac{116}{903}\right)$
$\chi_{355008}(619,\cdot)$	$-1$	$1$	$e\left(\frac{1531}{14448}\right)$	$e\left(\frac{853}{1032}\right)$	$e\left(\frac{485}{4816}\right)$	$e\left(\frac{5045}{14448}\right)$	$e\left(\frac{389}{3612}\right)$	$e\left(\frac{127}{336}\right)$	$e\left(\frac{3041}{7224}\right)$	$e\left(\frac{1531}{7224}\right)$	$e\left(\frac{10553}{14448}\right)$	$e\left(\frac{499}{903}\right)$
$\chi_{355008}(787,\cdot)$	$-1$	$1$	$e\left(\frac{7681}{14448}\right)$	$e\left(\frac{775}{1032}\right)$	$e\left(\frac{4415}{4816}\right)$	$e\left(\frac{2879}{14448}\right)$	$e\left(\frac{227}{3612}\right)$	$e\left(\frac{13}{336}\right)$	$e\left(\frac{1691}{7224}\right)$	$e\left(\frac{457}{7224}\right)$	$e\left(\frac{3419}{14448}\right)$	$e\left(\frac{586}{903}\right)$
$\chi_{355008}(883,\cdot)$	$-1$	$1$	$e\left(\frac{5993}{14448}\right)$	$e\left(\frac{1031}{1032}\right)$	$e\left(\frac{1175}{4816}\right)$	$e\left(\frac{14407}{14448}\right)$	$e\left(\frac{3127}{3612}\right)$	$e\left(\frac{101}{336}\right)$	$e\left(\frac{2179}{7224}\right)$	$e\left(\frac{5993}{7224}\right)$	$e\left(\frac{10771}{14448}\right)$	$e\left(\frac{188}{903}\right)$
$\chi_{355008}(955,\cdot)$	$-1$	$1$	$e\left(\frac{2983}{14448}\right)$	$e\left(\frac{601}{1032}\right)$	$e\left(\frac{4185}{4816}\right)$	$e\left(\frac{2969}{14448}\right)$	$e\left(\frac{2525}{3612}\right)$	$e\left(\frac{283}{336}\right)$	$e\left(\frac{5189}{7224}\right)$	$e\left(\frac{2983}{7224}\right)$	$e\left(\frac{6557}{14448}\right)$	$e\left(\frac{790}{903}\right)$
$\chi_{355008}(1003,\cdot)$	$-1$	$1$	$e\left(\frac{1627}{14448}\right)$	$e\left(\frac{589}{1032}\right)$	$e\left(\frac{53}{4816}\right)$	$e\left(\frac{1445}{14448}\right)$	$e\left(\frac{2381}{3612}\right)$	$e\left(\frac{79}{336}\right)$	$e\left(\frac{377}{7224}\right)$	$e\left(\frac{1627}{7224}\right)$	$e\left(\frac{617}{14448}\right)$	$e\left(\frac{466}{903}\right)$
$\chi_{355008}(1027,\cdot)$	$-1$	$1$	$e\left(\frac{13589}{14448}\right)$	$e\left(\frac{395}{1032}\right)$	$e\left(\frac{1307}{4816}\right)$	$e\left(\frac{13099}{14448}\right)$	$e\left(\frac{2719}{3612}\right)$	$e\left(\frac{209}{336}\right)$	$e\left(\frac{7207}{7224}\right)$	$e\left(\frac{6365}{7224}\right)$	$e\left(\frac{6583}{14448}\right)$	$e\left(\frac{173}{903}\right)$
$\chi_{355008}(1099,\cdot)$	$-1$	$1$	$e\left(\frac{3779}{14448}\right)$	$e\left(\frac{605}{1032}\right)$	$e\left(\frac{3613}{4816}\right)$	$e\left(\frac{13453}{14448}\right)$	$e\left(\frac{1885}{3612}\right)$	$e\left(\frac{263}{336}\right)$	$e\left(\frac{2665}{7224}\right)$	$e\left(\frac{3779}{7224}\right)$	$e\left(\frac{625}{14448}\right)$	$e\left(\frac{554}{903}\right)$
$\chi_{355008}(1171,\cdot)$	$-1$	$1$	$e\left(\frac{10673}{14448}\right)$	$e\left(\frac{287}{1032}\right)$	$e\left(\frac{2991}{4816}\right)$	$e\left(\frac{8671}{14448}\right)$	$e\left(\frac{907}{3612}\right)$	$e\left(\frac{29}{336}\right)$	$e\left(\frac{4147}{7224}\right)$	$e\left(\frac{3449}{7224}\right)$	$e\left(\frac{14011}{14448}\right)$	$e\left(\frac{611}{903}\right)$
$\chi_{355008}(1219,\cdot)$	$-1$	$1$	$e\left(\frac{2533}{14448}\right)$	$e\left(\frac{355}{1032}\right)$	$e\left(\frac{491}{4816}\right)$	$e\left(\frac{6299}{14448}\right)$	$e\left(\frac{863}{3612}\right)$	$e\left(\frac{193}{336}\right)$	$e\left(\frac{4583}{7224}\right)$	$e\left(\frac{2533}{7224}\right)$	$e\left(\frac{7079}{14448}\right)$	$e\left(\frac{211}{903}\right)$
$\chi_{355008}(1315,\cdot)$	$-1$	$1$	$e\left(\frac{445}{14448}\right)$	$e\left(\frac{163}{1032}\right)$	$e\left(\frac{1459}{4816}\right)$	$e\left(\frac{1523}{14448}\right)$	$e\left(\frac{2687}{3612}\right)$	$e\left(\frac{313}{336}\right)$	$e\left(\frac{2927}{7224}\right)$	$e\left(\frac{445}{7224}\right)$	$e\left(\frac{10079}{14448}\right)$	$e\left(\frac{703}{903}\right)$
$\chi_{355008}(1459,\cdot)$	$-1$	$1$	$e\left(\frac{11225}{14448}\right)$	$e\left(\frac{575}{1032}\right)$	$e\left(\frac{4119}{4816}\right)$	$e\left(\frac{13255}{14448}\right)$	$e\left(\frac{3331}{3612}\right)$	$e\left(\frac{5}{336}\right)$	$e\left(\frac{5083}{7224}\right)$	$e\left(\frac{4001}{7224}\right)$	$e\left(\frac{11059}{14448}\right)$	$e\left(\frac{647}{903}\right)$

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Higher genus families
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Character	\(-1\)	\(1\)	\(5\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(25\)	\(29\)	\(31\)
\(\chi_{355008}(67,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{5669}{14448}\right)\)	\(e\left(\frac{1019}{1032}\right)\)	\(e\left(\frac{827}{4816}\right)\)	\(e\left(\frac{13915}{14448}\right)\)	\(e\left(\frac{919}{3612}\right)\)	\(e\left(\frac{305}{336}\right)\)	\(e\left(\frac{6655}{7224}\right)\)	\(e\left(\frac{5669}{7224}\right)\)	\(e\left(\frac{9991}{14448}\right)\)	\(e\left(\frac{638}{903}\right)\)
\(\chi_{355008}(139,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{6179}{14448}\right)\)	\(e\left(\frac{197}{1032}\right)\)	\(e\left(\frac{2445}{4816}\right)\)	\(e\left(\frac{10141}{14448}\right)\)	\(e\left(\frac{1117}{3612}\right)\)	\(e\left(\frac{71}{336}\right)\)	\(e\left(\frac{1081}{7224}\right)\)	\(e\left(\frac{6179}{7224}\right)\)	\(e\left(\frac{12289}{14448}\right)\)	\(e\left(\frac{632}{903}\right)\)
\(\chi_{355008}(187,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{3751}{14448}\right)\)	\(e\left(\frac{553}{1032}\right)\)	\(e\left(\frac{729}{4816}\right)\)	\(e\left(\frac{3065}{14448}\right)\)	\(e\left(\frac{401}{3612}\right)\)	\(e\left(\frac{235}{336}\right)\)	\(e\left(\frac{5549}{7224}\right)\)	\(e\left(\frac{3751}{7224}\right)\)	\(e\left(\frac{13757}{14448}\right)\)	\(e\left(\frac{526}{903}\right)\)
\(\chi_{355008}(283,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{3343}{14448}\right)\)	\(e\left(\frac{385}{1032}\right)\)	\(e\left(\frac{1361}{4816}\right)\)	\(e\left(\frac{305}{14448}\right)\)	\(e\left(\frac{965}{3612}\right)\)	\(e\left(\frac{19}{336}\right)\)	\(e\left(\frac{4229}{7224}\right)\)	\(e\left(\frac{3343}{7224}\right)\)	\(e\left(\frac{9029}{14448}\right)\)	\(e\left(\frac{892}{903}\right)\)
\(\chi_{355008}(427,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{8747}{14448}\right)\)	\(e\left(\frac{101}{1032}\right)\)	\(e\left(\frac{4133}{4816}\right)\)	\(e\left(\frac{11365}{14448}\right)\)	\(e\left(\frac{2029}{3612}\right)\)	\(e\left(\frac{47}{336}\right)\)	\(e\left(\frac{3865}{7224}\right)\)	\(e\left(\frac{1523}{7224}\right)\)	\(e\left(\frac{2953}{14448}\right)\)	\(e\left(\frac{878}{903}\right)\)
\(\chi_{355008}(547,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1277}{14448}\right)\)	\(e\left(\frac{971}{1032}\right)\)	\(e\left(\frac{2531}{4816}\right)\)	\(e\left(\frac{8851}{14448}\right)\)	\(e\left(\frac{1891}{3612}\right)\)	\(e\left(\frac{233}{336}\right)\)	\(e\left(\frac{3919}{7224}\right)\)	\(e\left(\frac{1277}{7224}\right)\)	\(e\left(\frac{5839}{14448}\right)\)	\(e\left(\frac{116}{903}\right)\)
\(\chi_{355008}(619,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1531}{14448}\right)\)	\(e\left(\frac{853}{1032}\right)\)	\(e\left(\frac{485}{4816}\right)\)	\(e\left(\frac{5045}{14448}\right)\)	\(e\left(\frac{389}{3612}\right)\)	\(e\left(\frac{127}{336}\right)\)	\(e\left(\frac{3041}{7224}\right)\)	\(e\left(\frac{1531}{7224}\right)\)	\(e\left(\frac{10553}{14448}\right)\)	\(e\left(\frac{499}{903}\right)\)
\(\chi_{355008}(787,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{7681}{14448}\right)\)	\(e\left(\frac{775}{1032}\right)\)	\(e\left(\frac{4415}{4816}\right)\)	\(e\left(\frac{2879}{14448}\right)\)	\(e\left(\frac{227}{3612}\right)\)	\(e\left(\frac{13}{336}\right)\)	\(e\left(\frac{1691}{7224}\right)\)	\(e\left(\frac{457}{7224}\right)\)	\(e\left(\frac{3419}{14448}\right)\)	\(e\left(\frac{586}{903}\right)\)
\(\chi_{355008}(883,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{5993}{14448}\right)\)	\(e\left(\frac{1031}{1032}\right)\)	\(e\left(\frac{1175}{4816}\right)\)	\(e\left(\frac{14407}{14448}\right)\)	\(e\left(\frac{3127}{3612}\right)\)	\(e\left(\frac{101}{336}\right)\)	\(e\left(\frac{2179}{7224}\right)\)	\(e\left(\frac{5993}{7224}\right)\)	\(e\left(\frac{10771}{14448}\right)\)	\(e\left(\frac{188}{903}\right)\)
\(\chi_{355008}(955,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{2983}{14448}\right)\)	\(e\left(\frac{601}{1032}\right)\)	\(e\left(\frac{4185}{4816}\right)\)	\(e\left(\frac{2969}{14448}\right)\)	\(e\left(\frac{2525}{3612}\right)\)	\(e\left(\frac{283}{336}\right)\)	\(e\left(\frac{5189}{7224}\right)\)	\(e\left(\frac{2983}{7224}\right)\)	\(e\left(\frac{6557}{14448}\right)\)	\(e\left(\frac{790}{903}\right)\)
\(\chi_{355008}(1003,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1627}{14448}\right)\)	\(e\left(\frac{589}{1032}\right)\)	\(e\left(\frac{53}{4816}\right)\)	\(e\left(\frac{1445}{14448}\right)\)	\(e\left(\frac{2381}{3612}\right)\)	\(e\left(\frac{79}{336}\right)\)	\(e\left(\frac{377}{7224}\right)\)	\(e\left(\frac{1627}{7224}\right)\)	\(e\left(\frac{617}{14448}\right)\)	\(e\left(\frac{466}{903}\right)\)
\(\chi_{355008}(1027,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{13589}{14448}\right)\)	\(e\left(\frac{395}{1032}\right)\)	\(e\left(\frac{1307}{4816}\right)\)	\(e\left(\frac{13099}{14448}\right)\)	\(e\left(\frac{2719}{3612}\right)\)	\(e\left(\frac{209}{336}\right)\)	\(e\left(\frac{7207}{7224}\right)\)	\(e\left(\frac{6365}{7224}\right)\)	\(e\left(\frac{6583}{14448}\right)\)	\(e\left(\frac{173}{903}\right)\)
\(\chi_{355008}(1099,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{3779}{14448}\right)\)	\(e\left(\frac{605}{1032}\right)\)	\(e\left(\frac{3613}{4816}\right)\)	\(e\left(\frac{13453}{14448}\right)\)	\(e\left(\frac{1885}{3612}\right)\)	\(e\left(\frac{263}{336}\right)\)	\(e\left(\frac{2665}{7224}\right)\)	\(e\left(\frac{3779}{7224}\right)\)	\(e\left(\frac{625}{14448}\right)\)	\(e\left(\frac{554}{903}\right)\)
\(\chi_{355008}(1171,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{10673}{14448}\right)\)	\(e\left(\frac{287}{1032}\right)\)	\(e\left(\frac{2991}{4816}\right)\)	\(e\left(\frac{8671}{14448}\right)\)	\(e\left(\frac{907}{3612}\right)\)	\(e\left(\frac{29}{336}\right)\)	\(e\left(\frac{4147}{7224}\right)\)	\(e\left(\frac{3449}{7224}\right)\)	\(e\left(\frac{14011}{14448}\right)\)	\(e\left(\frac{611}{903}\right)\)
\(\chi_{355008}(1219,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{2533}{14448}\right)\)	\(e\left(\frac{355}{1032}\right)\)	\(e\left(\frac{491}{4816}\right)\)	\(e\left(\frac{6299}{14448}\right)\)	\(e\left(\frac{863}{3612}\right)\)	\(e\left(\frac{193}{336}\right)\)	\(e\left(\frac{4583}{7224}\right)\)	\(e\left(\frac{2533}{7224}\right)\)	\(e\left(\frac{7079}{14448}\right)\)	\(e\left(\frac{211}{903}\right)\)
\(\chi_{355008}(1315,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{445}{14448}\right)\)	\(e\left(\frac{163}{1032}\right)\)	\(e\left(\frac{1459}{4816}\right)\)	\(e\left(\frac{1523}{14448}\right)\)	\(e\left(\frac{2687}{3612}\right)\)	\(e\left(\frac{313}{336}\right)\)	\(e\left(\frac{2927}{7224}\right)\)	\(e\left(\frac{445}{7224}\right)\)	\(e\left(\frac{10079}{14448}\right)\)	\(e\left(\frac{703}{903}\right)\)
\(\chi_{355008}(1459,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{11225}{14448}\right)\)	\(e\left(\frac{575}{1032}\right)\)	\(e\left(\frac{4119}{4816}\right)\)	\(e\left(\frac{13255}{14448}\right)\)	\(e\left(\frac{3331}{3612}\right)\)	\(e\left(\frac{5}{336}\right)\)	\(e\left(\frac{5083}{7224}\right)\)	\(e\left(\frac{4001}{7224}\right)\)	\(e\left(\frac{11059}{14448}\right)\)	\(e\left(\frac{647}{903}\right)\)

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First 17 of 4032 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.mc
Modulus	$355008$
Conductor	$118336$
Order	$14448$
Real	no
Primitive	no
Minimal	yes
Parity	odd

Modulus:	\(355008\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(118336\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(14448\)	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 118336.gb	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);