# Properties

 Label 85600.gp Modulus $85600$ Conductor $2675$ Order $530$ Real no Primitive no Minimal yes Parity odd

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85600, base_ring=CyclotomicField(530))

M = H._module

chi = DirichletCharacter(H, M([0,0,53,115]))

chi.galois_orbit()

[g,chi] = znchar(Mod(129,85600))

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Basic properties

 Modulus: $$85600$$ Conductor: $$2675$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$530$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 2675.v sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{265})$ Fixed field: Number field defined by a degree 530 polynomial (not computed)

## First 31 of 208 characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$
$$\chi_{85600}(129,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{471}{530}\right)$$ $$e\left(\frac{44}{53}\right)$$ $$e\left(\frac{206}{265}\right)$$ $$e\left(\frac{99}{265}\right)$$ $$e\left(\frac{497}{530}\right)$$ $$e\left(\frac{157}{265}\right)$$ $$e\left(\frac{192}{265}\right)$$ $$e\left(\frac{381}{530}\right)$$ $$e\left(\frac{293}{530}\right)$$ $$e\left(\frac{353}{530}\right)$$
$$\chi_{85600}(609,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{347}{530}\right)$$ $$e\left(\frac{17}{53}\right)$$ $$e\left(\frac{82}{265}\right)$$ $$e\left(\frac{78}{265}\right)$$ $$e\left(\frac{239}{530}\right)$$ $$e\left(\frac{204}{265}\right)$$ $$e\left(\frac{79}{265}\right)$$ $$e\left(\frac{517}{530}\right)$$ $$e\left(\frac{271}{530}\right)$$ $$e\left(\frac{511}{530}\right)$$
$$\chi_{85600}(769,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{349}{530}\right)$$ $$e\left(\frac{20}{53}\right)$$ $$e\left(\frac{84}{265}\right)$$ $$e\left(\frac{151}{265}\right)$$ $$e\left(\frac{303}{530}\right)$$ $$e\left(\frac{28}{265}\right)$$ $$e\left(\frac{68}{265}\right)$$ $$e\left(\frac{19}{530}\right)$$ $$e\left(\frac{297}{530}\right)$$ $$e\left(\frac{517}{530}\right)$$
$$\chi_{85600}(929,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{271}{530}\right)$$ $$e\left(\frac{9}{53}\right)$$ $$e\left(\frac{6}{265}\right)$$ $$e\left(\frac{219}{265}\right)$$ $$e\left(\frac{457}{530}\right)$$ $$e\left(\frac{2}{265}\right)$$ $$e\left(\frac{232}{265}\right)$$ $$e\left(\frac{361}{530}\right)$$ $$e\left(\frac{343}{530}\right)$$ $$e\left(\frac{283}{530}\right)$$
$$\chi_{85600}(1409,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{530}\right)$$ $$e\left(\frac{37}{53}\right)$$ $$e\left(\frac{7}{265}\right)$$ $$e\left(\frac{123}{265}\right)$$ $$e\left(\frac{489}{530}\right)$$ $$e\left(\frac{179}{265}\right)$$ $$e\left(\frac{94}{265}\right)$$ $$e\left(\frac{377}{530}\right)$$ $$e\left(\frac{91}{530}\right)$$ $$e\left(\frac{21}{530}\right)$$
$$\chi_{85600}(1569,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{39}{530}\right)$$ $$e\left(\frac{32}{53}\right)$$ $$e\left(\frac{39}{265}\right)$$ $$e\left(\frac{231}{265}\right)$$ $$e\left(\frac{453}{530}\right)$$ $$e\left(\frac{13}{265}\right)$$ $$e\left(\frac{183}{265}\right)$$ $$e\left(\frac{359}{530}\right)$$ $$e\left(\frac{507}{530}\right)$$ $$e\left(\frac{117}{530}\right)$$
$$\chi_{85600}(1729,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{451}{530}\right)$$ $$e\left(\frac{14}{53}\right)$$ $$e\left(\frac{186}{265}\right)$$ $$e\left(\frac{164}{265}\right)$$ $$e\left(\frac{387}{530}\right)$$ $$e\left(\frac{62}{265}\right)$$ $$e\left(\frac{37}{265}\right)$$ $$e\left(\frac{61}{530}\right)$$ $$e\left(\frac{33}{530}\right)$$ $$e\left(\frac{293}{530}\right)$$
$$\chi_{85600}(1889,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{103}{530}\right)$$ $$e\left(\frac{22}{53}\right)$$ $$e\left(\frac{103}{265}\right)$$ $$e\left(\frac{182}{265}\right)$$ $$e\left(\frac{381}{530}\right)$$ $$e\left(\frac{211}{265}\right)$$ $$e\left(\frac{96}{265}\right)$$ $$e\left(\frac{323}{530}\right)$$ $$e\left(\frac{279}{530}\right)$$ $$e\left(\frac{309}{530}\right)$$
$$\chi_{85600}(2369,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{299}{530}\right)$$ $$e\left(\frac{51}{53}\right)$$ $$e\left(\frac{34}{265}\right)$$ $$e\left(\frac{181}{265}\right)$$ $$e\left(\frac{293}{530}\right)$$ $$e\left(\frac{188}{265}\right)$$ $$e\left(\frac{78}{265}\right)$$ $$e\left(\frac{279}{530}\right)$$ $$e\left(\frac{177}{530}\right)$$ $$e\left(\frac{367}{530}\right)$$
$$\chi_{85600}(2529,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{91}{530}\right)$$ $$e\left(\frac{4}{53}\right)$$ $$e\left(\frac{91}{265}\right)$$ $$e\left(\frac{9}{265}\right)$$ $$e\left(\frac{527}{530}\right)$$ $$e\left(\frac{207}{265}\right)$$ $$e\left(\frac{162}{265}\right)$$ $$e\left(\frac{131}{530}\right)$$ $$e\left(\frac{123}{530}\right)$$ $$e\left(\frac{273}{530}\right)$$
$$\chi_{85600}(3169,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{379}{530}\right)$$ $$e\left(\frac{12}{53}\right)$$ $$e\left(\frac{114}{265}\right)$$ $$e\left(\frac{186}{265}\right)$$ $$e\left(\frac{203}{530}\right)$$ $$e\left(\frac{38}{265}\right)$$ $$e\left(\frac{168}{265}\right)$$ $$e\left(\frac{499}{530}\right)$$ $$e\left(\frac{157}{530}\right)$$ $$e\left(\frac{77}{530}\right)$$
$$\chi_{85600}(3489,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{203}{530}\right)$$ $$e\left(\frac{13}{53}\right)$$ $$e\left(\frac{203}{265}\right)$$ $$e\left(\frac{122}{265}\right)$$ $$e\left(\frac{401}{530}\right)$$ $$e\left(\frac{156}{265}\right)$$ $$e\left(\frac{76}{265}\right)$$ $$e\left(\frac{333}{530}\right)$$ $$e\left(\frac{519}{530}\right)$$ $$e\left(\frac{79}{530}\right)$$
$$\chi_{85600}(4129,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{291}{530}\right)$$ $$e\left(\frac{39}{53}\right)$$ $$e\left(\frac{26}{265}\right)$$ $$e\left(\frac{154}{265}\right)$$ $$e\left(\frac{37}{530}\right)$$ $$e\left(\frac{97}{265}\right)$$ $$e\left(\frac{122}{265}\right)$$ $$e\left(\frac{151}{530}\right)$$ $$e\left(\frac{73}{530}\right)$$ $$e\left(\frac{343}{530}\right)$$
$$\chi_{85600}(4609,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{467}{530}\right)$$ $$e\left(\frac{38}{53}\right)$$ $$e\left(\frac{202}{265}\right)$$ $$e\left(\frac{218}{265}\right)$$ $$e\left(\frac{369}{530}\right)$$ $$e\left(\frac{244}{265}\right)$$ $$e\left(\frac{214}{265}\right)$$ $$e\left(\frac{317}{530}\right)$$ $$e\left(\frac{241}{530}\right)$$ $$e\left(\frac{341}{530}\right)$$
$$\chi_{85600}(4929,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{51}{530}\right)$$ $$e\left(\frac{50}{53}\right)$$ $$e\left(\frac{51}{265}\right)$$ $$e\left(\frac{139}{265}\right)$$ $$e\left(\frac{307}{530}\right)$$ $$e\left(\frac{17}{265}\right)$$ $$e\left(\frac{117}{265}\right)$$ $$e\left(\frac{21}{530}\right)$$ $$e\left(\frac{133}{530}\right)$$ $$e\left(\frac{153}{530}\right)$$
$$\chi_{85600}(5089,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{363}{530}\right)$$ $$e\left(\frac{41}{53}\right)$$ $$e\left(\frac{98}{265}\right)$$ $$e\left(\frac{132}{265}\right)$$ $$e\left(\frac{221}{530}\right)$$ $$e\left(\frac{121}{265}\right)$$ $$e\left(\frac{256}{265}\right)$$ $$e\left(\frac{243}{530}\right)$$ $$e\left(\frac{479}{530}\right)$$ $$e\left(\frac{29}{530}\right)$$
$$\chi_{85600}(5409,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{407}{530}\right)$$ $$e\left(\frac{1}{53}\right)$$ $$e\left(\frac{142}{265}\right)$$ $$e\left(\frac{148}{265}\right)$$ $$e\left(\frac{39}{530}\right)$$ $$e\left(\frac{224}{265}\right)$$ $$e\left(\frac{14}{265}\right)$$ $$e\left(\frac{417}{530}\right)$$ $$e\left(\frac{521}{530}\right)$$ $$e\left(\frac{161}{530}\right)$$
$$\chi_{85600}(5569,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{179}{530}\right)$$ $$e\left(\frac{30}{53}\right)$$ $$e\left(\frac{179}{265}\right)$$ $$e\left(\frac{41}{265}\right)$$ $$e\left(\frac{163}{530}\right)$$ $$e\left(\frac{148}{265}\right)$$ $$e\left(\frac{208}{265}\right)$$ $$e\left(\frac{479}{530}\right)$$ $$e\left(\frac{207}{530}\right)$$ $$e\left(\frac{7}{530}\right)$$
$$\chi_{85600}(5729,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{261}{530}\right)$$ $$e\left(\frac{47}{53}\right)$$ $$e\left(\frac{261}{265}\right)$$ $$e\left(\frac{119}{265}\right)$$ $$e\left(\frac{137}{530}\right)$$ $$e\left(\frac{87}{265}\right)$$ $$e\left(\frac{22}{265}\right)$$ $$e\left(\frac{201}{530}\right)$$ $$e\left(\frac{213}{530}\right)$$ $$e\left(\frac{253}{530}\right)$$
$$\chi_{85600}(6529,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{191}{530}\right)$$ $$e\left(\frac{48}{53}\right)$$ $$e\left(\frac{191}{265}\right)$$ $$e\left(\frac{214}{265}\right)$$ $$e\left(\frac{17}{530}\right)$$ $$e\left(\frac{152}{265}\right)$$ $$e\left(\frac{142}{265}\right)$$ $$e\left(\frac{141}{530}\right)$$ $$e\left(\frac{363}{530}\right)$$ $$e\left(\frac{43}{530}\right)$$
$$\chi_{85600}(6689,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{353}{530}\right)$$ $$e\left(\frac{26}{53}\right)$$ $$e\left(\frac{88}{265}\right)$$ $$e\left(\frac{32}{265}\right)$$ $$e\left(\frac{431}{530}\right)$$ $$e\left(\frac{206}{265}\right)$$ $$e\left(\frac{46}{265}\right)$$ $$e\left(\frac{83}{530}\right)$$ $$e\left(\frac{349}{530}\right)$$ $$e\left(\frac{529}{530}\right)$$
$$\chi_{85600}(7009,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{127}{530}\right)$$ $$e\left(\frac{5}{53}\right)$$ $$e\left(\frac{127}{265}\right)$$ $$e\left(\frac{263}{265}\right)$$ $$e\left(\frac{89}{530}\right)$$ $$e\left(\frac{219}{265}\right)$$ $$e\left(\frac{229}{265}\right)$$ $$e\left(\frac{177}{530}\right)$$ $$e\left(\frac{61}{530}\right)$$ $$e\left(\frac{381}{530}\right)$$
$$\chi_{85600}(7969,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{359}{530}\right)$$ $$e\left(\frac{35}{53}\right)$$ $$e\left(\frac{94}{265}\right)$$ $$e\left(\frac{251}{265}\right)$$ $$e\left(\frac{93}{530}\right)$$ $$e\left(\frac{208}{265}\right)$$ $$e\left(\frac{13}{265}\right)$$ $$e\left(\frac{179}{530}\right)$$ $$e\left(\frac{427}{530}\right)$$ $$e\left(\frac{17}{530}\right)$$
$$\chi_{85600}(8129,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{491}{530}\right)$$ $$e\left(\frac{21}{53}\right)$$ $$e\left(\frac{226}{265}\right)$$ $$e\left(\frac{34}{265}\right)$$ $$e\left(\frac{77}{530}\right)$$ $$e\left(\frac{252}{265}\right)$$ $$e\left(\frac{82}{265}\right)$$ $$e\left(\frac{171}{530}\right)$$ $$e\left(\frac{23}{530}\right)$$ $$e\left(\frac{413}{530}\right)$$
$$\chi_{85600}(8289,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{443}{530}\right)$$ $$e\left(\frac{2}{53}\right)$$ $$e\left(\frac{178}{265}\right)$$ $$e\left(\frac{137}{265}\right)$$ $$e\left(\frac{131}{530}\right)$$ $$e\left(\frac{236}{265}\right)$$ $$e\left(\frac{81}{265}\right)$$ $$e\left(\frac{463}{530}\right)$$ $$e\left(\frac{459}{530}\right)$$ $$e\left(\frac{269}{530}\right)$$
$$\chi_{85600}(9569,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{479}{530}\right)$$ $$e\left(\frac{3}{53}\right)$$ $$e\left(\frac{214}{265}\right)$$ $$e\left(\frac{126}{265}\right)$$ $$e\left(\frac{223}{530}\right)$$ $$e\left(\frac{248}{265}\right)$$ $$e\left(\frac{148}{265}\right)$$ $$e\left(\frac{509}{530}\right)$$ $$e\left(\frac{397}{530}\right)$$ $$e\left(\frac{377}{530}\right)$$
$$\chi_{85600}(9889,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{313}{530}\right)$$ $$e\left(\frac{19}{53}\right)$$ $$e\left(\frac{48}{265}\right)$$ $$e\left(\frac{162}{265}\right)$$ $$e\left(\frac{211}{530}\right)$$ $$e\left(\frac{16}{265}\right)$$ $$e\left(\frac{1}{265}\right)$$ $$e\left(\frac{503}{530}\right)$$ $$e\left(\frac{359}{530}\right)$$ $$e\left(\frac{409}{530}\right)$$
$$\chi_{85600}(10369,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{529}{530}\right)$$ $$e\left(\frac{25}{53}\right)$$ $$e\left(\frac{264}{265}\right)$$ $$e\left(\frac{96}{265}\right)$$ $$e\left(\frac{233}{530}\right)$$ $$e\left(\frac{88}{265}\right)$$ $$e\left(\frac{138}{265}\right)$$ $$e\left(\frac{249}{530}\right)$$ $$e\left(\frac{517}{530}\right)$$ $$e\left(\frac{527}{530}\right)$$
$$\chi_{85600}(10529,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{351}{530}\right)$$ $$e\left(\frac{23}{53}\right)$$ $$e\left(\frac{86}{265}\right)$$ $$e\left(\frac{224}{265}\right)$$ $$e\left(\frac{367}{530}\right)$$ $$e\left(\frac{117}{265}\right)$$ $$e\left(\frac{57}{265}\right)$$ $$e\left(\frac{51}{530}\right)$$ $$e\left(\frac{323}{530}\right)$$ $$e\left(\frac{523}{530}\right)$$
$$\chi_{85600}(10689,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{333}{530}\right)$$ $$e\left(\frac{49}{53}\right)$$ $$e\left(\frac{68}{265}\right)$$ $$e\left(\frac{97}{265}\right)$$ $$e\left(\frac{321}{530}\right)$$ $$e\left(\frac{111}{265}\right)$$ $$e\left(\frac{156}{265}\right)$$ $$e\left(\frac{293}{530}\right)$$ $$e\left(\frac{89}{530}\right)$$ $$e\left(\frac{469}{530}\right)$$
$$\chi_{85600}(11009,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{237}{530}\right)$$ $$e\left(\frac{11}{53}\right)$$ $$e\left(\frac{237}{265}\right)$$ $$e\left(\frac{38}{265}\right)$$ $$e\left(\frac{429}{530}\right)$$ $$e\left(\frac{79}{265}\right)$$ $$e\left(\frac{154}{265}\right)$$ $$e\left(\frac{347}{530}\right)$$ $$e\left(\frac{431}{530}\right)$$ $$e\left(\frac{181}{530}\right)$$