# Properties

 Label 7098.eb Modulus $7098$ Conductor $3549$ Order $156$ Real no Primitive no Minimal yes Parity even

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(7098, base_ring=CyclotomicField(156))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([78,104,83]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(137,7098))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$7098$$ Conductor: $$3549$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$156$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 3549.em 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: $\Q(\zeta_{156})$ Fixed field: Number field defined by a degree 156 polynomial (not computed)

## First 31 of 48 characters in Galois orbit

Character $$-1$$ $$1$$ $$5$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$37$$ $$41$$
$$\chi_{7098}(137,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{97}{156}\right)$$ $$e\left(\frac{151}{156}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{131}{156}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{113}{156}\right)$$
$$\chi_{7098}(275,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{156}\right)$$ $$e\left(\frac{113}{156}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{97}{156}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{67}{156}\right)$$
$$\chi_{7098}(305,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{67}{156}\right)$$ $$e\left(\frac{85}{156}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{113}{156}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{107}{156}\right)$$
$$\chi_{7098}(401,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{101}{156}\right)$$ $$e\left(\frac{35}{156}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{19}{156}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{145}{156}\right)$$
$$\chi_{7098}(683,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{156}\right)$$ $$e\left(\frac{127}{156}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{11}{156}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{125}{156}\right)$$
$$\chi_{7098}(821,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{155}{156}\right)$$ $$e\left(\frac{29}{156}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{145}{156}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{31}{156}\right)$$
$$\chi_{7098}(851,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{156}\right)$$ $$e\left(\frac{109}{156}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{77}{156}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{95}{156}\right)$$
$$\chi_{7098}(947,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{125}{156}\right)$$ $$e\left(\frac{119}{156}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{127}{156}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{25}{156}\right)$$
$$\chi_{7098}(1229,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{61}{156}\right)$$ $$e\left(\frac{103}{156}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{47}{156}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{137}{156}\right)$$
$$\chi_{7098}(1367,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{131}{156}\right)$$ $$e\left(\frac{101}{156}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{37}{156}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{151}{156}\right)$$
$$\chi_{7098}(1397,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{103}{156}\right)$$ $$e\left(\frac{133}{156}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{41}{156}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{83}{156}\right)$$
$$\chi_{7098}(1493,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{149}{156}\right)$$ $$e\left(\frac{47}{156}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{79}{156}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{61}{156}\right)$$
$$\chi_{7098}(1775,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{121}{156}\right)$$ $$e\left(\frac{79}{156}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{83}{156}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{149}{156}\right)$$
$$\chi_{7098}(1913,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{107}{156}\right)$$ $$e\left(\frac{17}{156}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{85}{156}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{115}{156}\right)$$
$$\chi_{7098}(1943,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{156}\right)$$ $$e\left(\frac{1}{156}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{5}{156}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{71}{156}\right)$$
$$\chi_{7098}(2039,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{156}\right)$$ $$e\left(\frac{131}{156}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{31}{156}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{97}{156}\right)$$
$$\chi_{7098}(2321,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{55}{156}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{119}{156}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{5}{156}\right)$$
$$\chi_{7098}(2459,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{156}\right)$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{133}{156}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{79}{156}\right)$$
$$\chi_{7098}(2489,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{139}{156}\right)$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{125}{156}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{59}{156}\right)$$
$$\chi_{7098}(2585,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{156}\right)$$ $$e\left(\frac{59}{156}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{139}{156}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{133}{156}\right)$$
$$\chi_{7098}(2867,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{85}{156}\right)$$ $$e\left(\frac{31}{156}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{155}{156}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{17}{156}\right)$$
$$\chi_{7098}(3005,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{59}{156}\right)$$ $$e\left(\frac{5}{156}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{43}{156}\right)$$
$$\chi_{7098}(3035,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{79}{156}\right)$$ $$e\left(\frac{49}{156}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{47}{156}\right)$$
$$\chi_{7098}(3413,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{145}{156}\right)$$ $$e\left(\frac{7}{156}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{35}{156}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{29}{156}\right)$$
$$\chi_{7098}(3551,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{156}\right)$$ $$e\left(\frac{77}{156}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{73}{156}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{7}{156}\right)$$
$$\chi_{7098}(3581,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{156}\right)$$ $$e\left(\frac{73}{156}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{53}{156}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{35}{156}\right)$$
$$\chi_{7098}(3677,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{71}{156}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{43}{156}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{49}{156}\right)$$
$$\chi_{7098}(3959,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{156}\right)$$ $$e\left(\frac{139}{156}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{71}{156}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{41}{156}\right)$$
$$\chi_{7098}(4097,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{156}\right)$$ $$e\left(\frac{149}{156}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{121}{156}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{127}{156}\right)$$
$$\chi_{7098}(4127,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{115}{156}\right)$$ $$e\left(\frac{97}{156}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{17}{156}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{23}{156}\right)$$
$$\chi_{7098}(4223,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{113}{156}\right)$$ $$e\left(\frac{155}{156}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{151}{156}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{85}{156}\right)$$