from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(540))
M = H._module
chi = DirichletCharacter(H, M([10,27]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,2025))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2025\) | |
| Conductor: | \(2025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(540\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{540})$ |
| Fixed field: | Number field defined by a degree 540 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2025}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{540}\right)\) | \(e\left(\frac{37}{270}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{11}{270}\right)\) | \(e\left(\frac{53}{540}\right)\) | \(e\left(\frac{83}{135}\right)\) | \(e\left(\frac{37}{135}\right)\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{71}{90}\right)\) |
| \(\chi_{2025}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{407}{540}\right)\) | \(e\left(\frac{137}{270}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{121}{270}\right)\) | \(e\left(\frac{43}{540}\right)\) | \(e\left(\frac{103}{135}\right)\) | \(e\left(\frac{2}{135}\right)\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) |
| \(\chi_{2025}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{463}{540}\right)\) | \(e\left(\frac{193}{270}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{269}{270}\right)\) | \(e\left(\frac{167}{540}\right)\) | \(e\left(\frac{17}{135}\right)\) | \(e\left(\frac{58}{135}\right)\) | \(e\left(\frac{53}{180}\right)\) | \(e\left(\frac{59}{90}\right)\) |
| \(\chi_{2025}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{529}{540}\right)\) | \(e\left(\frac{259}{270}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{169}{180}\right)\) | \(e\left(\frac{77}{270}\right)\) | \(e\left(\frac{101}{540}\right)\) | \(e\left(\frac{41}{135}\right)\) | \(e\left(\frac{124}{135}\right)\) | \(e\left(\frac{59}{180}\right)\) | \(e\left(\frac{47}{90}\right)\) |
| \(\chi_{2025}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{540}\right)\) | \(e\left(\frac{47}{270}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{137}{180}\right)\) | \(e\left(\frac{211}{270}\right)\) | \(e\left(\frac{133}{540}\right)\) | \(e\left(\frac{58}{135}\right)\) | \(e\left(\frac{47}{135}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) |
| \(\chi_{2025}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{540}\right)\) | \(e\left(\frac{91}{270}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{91}{180}\right)\) | \(e\left(\frac{173}{270}\right)\) | \(e\left(\frac{539}{540}\right)\) | \(e\left(\frac{29}{135}\right)\) | \(e\left(\frac{91}{135}\right)\) | \(e\left(\frac{101}{180}\right)\) | \(e\left(\frac{53}{90}\right)\) |
| \(\chi_{2025}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{481}{540}\right)\) | \(e\left(\frac{211}{270}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{143}{270}\right)\) | \(e\left(\frac{149}{540}\right)\) | \(e\left(\frac{134}{135}\right)\) | \(e\left(\frac{76}{135}\right)\) | \(e\left(\frac{71}{180}\right)\) | \(e\left(\frac{23}{90}\right)\) |
| \(\chi_{2025}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{540}\right)\) | \(e\left(\frac{23}{270}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{23}{180}\right)\) | \(e\left(\frac{109}{270}\right)\) | \(e\left(\frac{427}{540}\right)\) | \(e\left(\frac{37}{135}\right)\) | \(e\left(\frac{23}{135}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{49}{90}\right)\) |
| \(\chi_{2025}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{540}\right)\) | \(e\left(\frac{179}{270}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{89}{180}\right)\) | \(e\left(\frac{97}{270}\right)\) | \(e\left(\frac{1}{540}\right)\) | \(e\left(\frac{106}{135}\right)\) | \(e\left(\frac{44}{135}\right)\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) |
| \(\chi_{2025}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{540}\right)\) | \(e\left(\frac{259}{270}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{77}{270}\right)\) | \(e\left(\frac{371}{540}\right)\) | \(e\left(\frac{41}{135}\right)\) | \(e\left(\frac{124}{135}\right)\) | \(e\left(\frac{149}{180}\right)\) | \(e\left(\frac{47}{90}\right)\) |
| \(\chi_{2025}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{433}{540}\right)\) | \(e\left(\frac{163}{270}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{209}{270}\right)\) | \(e\left(\frac{197}{540}\right)\) | \(e\left(\frac{92}{135}\right)\) | \(e\left(\frac{28}{135}\right)\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{89}{90}\right)\) |
| \(\chi_{2025}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{371}{540}\right)\) | \(e\left(\frac{101}{270}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{11}{180}\right)\) | \(e\left(\frac{103}{270}\right)\) | \(e\left(\frac{79}{540}\right)\) | \(e\left(\frac{4}{135}\right)\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) |
| \(\chi_{2025}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{540}\right)\) | \(e\left(\frac{41}{270}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{253}{270}\right)\) | \(e\left(\frac{409}{540}\right)\) | \(e\left(\frac{19}{135}\right)\) | \(e\left(\frac{41}{135}\right)\) | \(e\left(\frac{91}{180}\right)\) | \(e\left(\frac{13}{90}\right)\) |
| \(\chi_{2025}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{427}{540}\right)\) | \(e\left(\frac{157}{270}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{251}{270}\right)\) | \(e\left(\frac{203}{540}\right)\) | \(e\left(\frac{53}{135}\right)\) | \(e\left(\frac{22}{135}\right)\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{41}{90}\right)\) |
| \(\chi_{2025}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{540}\right)\) | \(e\left(\frac{179}{270}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{97}{270}\right)\) | \(e\left(\frac{271}{540}\right)\) | \(e\left(\frac{106}{135}\right)\) | \(e\left(\frac{44}{135}\right)\) | \(e\left(\frac{169}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) |
| \(\chi_{2025}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{540}\right)\) | \(e\left(\frac{173}{270}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{173}{180}\right)\) | \(e\left(\frac{139}{270}\right)\) | \(e\left(\frac{277}{540}\right)\) | \(e\left(\frac{67}{135}\right)\) | \(e\left(\frac{38}{135}\right)\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{79}{90}\right)\) |
| \(\chi_{2025}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{337}{540}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{71}{270}\right)\) | \(e\left(\frac{293}{540}\right)\) | \(e\left(\frac{8}{135}\right)\) | \(e\left(\frac{67}{135}\right)\) | \(e\left(\frac{107}{180}\right)\) | \(e\left(\frac{41}{90}\right)\) |
| \(\chi_{2025}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{527}{540}\right)\) | \(e\left(\frac{257}{270}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{167}{180}\right)\) | \(e\left(\frac{91}{270}\right)\) | \(e\left(\frac{463}{540}\right)\) | \(e\left(\frac{73}{135}\right)\) | \(e\left(\frac{122}{135}\right)\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) |
| \(\chi_{2025}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{540}\right)\) | \(e\left(\frac{223}{270}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{43}{180}\right)\) | \(e\left(\frac{59}{270}\right)\) | \(e\left(\frac{407}{540}\right)\) | \(e\left(\frac{77}{135}\right)\) | \(e\left(\frac{88}{135}\right)\) | \(e\left(\frac{113}{180}\right)\) | \(e\left(\frac{29}{90}\right)\) |
| \(\chi_{2025}(272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{540}\right)\) | \(e\left(\frac{19}{270}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{137}{270}\right)\) | \(e\left(\frac{341}{540}\right)\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{19}{135}\right)\) | \(e\left(\frac{119}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) |
| \(\chi_{2025}(302,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{437}{540}\right)\) | \(e\left(\frac{167}{270}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{77}{180}\right)\) | \(e\left(\frac{181}{270}\right)\) | \(e\left(\frac{13}{540}\right)\) | \(e\left(\frac{28}{135}\right)\) | \(e\left(\frac{32}{135}\right)\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) |
| \(\chi_{2025}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{540}\right)\) | \(e\left(\frac{121}{270}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{31}{180}\right)\) | \(e\left(\frac{233}{270}\right)\) | \(e\left(\frac{239}{540}\right)\) | \(e\left(\frac{89}{135}\right)\) | \(e\left(\frac{121}{135}\right)\) | \(e\left(\frac{161}{180}\right)\) | \(e\left(\frac{23}{90}\right)\) |
| \(\chi_{2025}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{540}\right)\) | \(e\left(\frac{241}{270}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{61}{180}\right)\) | \(e\left(\frac{203}{270}\right)\) | \(e\left(\frac{389}{540}\right)\) | \(e\left(\frac{59}{135}\right)\) | \(e\left(\frac{106}{135}\right)\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{83}{90}\right)\) |
| \(\chi_{2025}(338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{540}\right)\) | \(e\left(\frac{143}{270}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{79}{270}\right)\) | \(e\left(\frac{307}{540}\right)\) | \(e\left(\frac{7}{135}\right)\) | \(e\left(\frac{8}{135}\right)\) | \(e\left(\frac{133}{180}\right)\) | \(e\left(\frac{19}{90}\right)\) |
| \(\chi_{2025}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{540}\right)\) | \(e\left(\frac{29}{270}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{421}{540}\right)\) | \(e\left(\frac{76}{135}\right)\) | \(e\left(\frac{29}{135}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) |
| \(\chi_{2025}(353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{540}\right)\) | \(e\left(\frac{19}{270}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{19}{180}\right)\) | \(e\left(\frac{137}{270}\right)\) | \(e\left(\frac{71}{540}\right)\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{19}{135}\right)\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) |
| \(\chi_{2025}(362,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{540}\right)\) | \(e\left(\frac{193}{270}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{13}{180}\right)\) | \(e\left(\frac{269}{270}\right)\) | \(e\left(\frac{437}{540}\right)\) | \(e\left(\frac{17}{135}\right)\) | \(e\left(\frac{58}{135}\right)\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{59}{90}\right)\) |
| \(\chi_{2025}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{491}{540}\right)\) | \(e\left(\frac{221}{270}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{73}{270}\right)\) | \(e\left(\frac{499}{540}\right)\) | \(e\left(\frac{109}{135}\right)\) | \(e\left(\frac{86}{135}\right)\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{13}{90}\right)\) |
| \(\chi_{2025}(392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{540}\right)\) | \(e\left(\frac{161}{270}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{161}{180}\right)\) | \(e\left(\frac{223}{270}\right)\) | \(e\left(\frac{289}{540}\right)\) | \(e\left(\frac{124}{135}\right)\) | \(e\left(\frac{26}{135}\right)\) | \(e\left(\frac{151}{180}\right)\) | \(e\left(\frac{73}{90}\right)\) |
| \(\chi_{2025}(398,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{540}\right)\) | \(e\left(\frac{187}{270}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{7}{180}\right)\) | \(e\left(\frac{41}{270}\right)\) | \(e\left(\frac{443}{540}\right)\) | \(e\left(\frac{113}{135}\right)\) | \(e\left(\frac{52}{135}\right)\) | \(e\left(\frac{77}{180}\right)\) | \(e\left(\frac{11}{90}\right)\) |
| \(\chi_{2025}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{540}\right)\) | \(e\left(\frac{29}{270}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{119}{180}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{151}{540}\right)\) | \(e\left(\frac{76}{135}\right)\) | \(e\left(\frac{29}{135}\right)\) | \(e\left(\frac{49}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) |