# Properties

 Label 78.3.l.a Level $78$ Weight $3$ Character orbit 78.l Analytic conductor $2.125$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,3,Mod(7,78)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("78.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 78.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.12534606201$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + (z^3 - z^2 - z) * q^2 + (z^3 + z) * q^3 + 2*z * q^4 + (4*z^3 + 2*z^2 - 2*z - 4) * q^5 + (-2*z^3 - z^2 + z - 1) * q^6 + (5*z^3 - 5*z^2) * q^7 + (-2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{10} + (10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 8 \zeta_{12} - 2) q^{11} + (4 \zeta_{12}^{2} - 2) q^{12} + 13 \zeta_{12} q^{13} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12} - 5) q^{14} + ( - 6 \zeta_{12} - 6) q^{15} + 4 \zeta_{12}^{2} q^{16} + (4 \zeta_{12}^{2} - 12 \zeta_{12} + 4) q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} + (12 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 19 \zeta_{12} + 19) q^{19} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12} - 8) q^{20} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 10) q^{21} + ( - 4 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 2 \zeta_{12} + 18) q^{22} + ( - 18 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 18 \zeta_{12} + 16) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + (\zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{25} + ( - 13 \zeta_{12}^{3} - 13) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10) q^{28} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{29} + (6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{30} + ( - 29 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 29) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + (18 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 12 \zeta_{12} - 12) q^{33} + (16 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 16) q^{34} + ( - 20 \zeta_{12}^{3} + 10 \zeta_{12}) q^{35} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + (23 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 13 \zeta_{12} - 36) q^{37} + (26 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{38} + (26 \zeta_{12}^{2} - 13) q^{39} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 12) q^{40} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 20 \zeta_{12} - 28) q^{41} + ( - 5 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 5 \zeta_{12}) q^{42} + (5 \zeta_{12}^{2} + 24 \zeta_{12} + 5) q^{43} + (20 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 20) q^{44} + ( - 6 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + (16 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 26 \zeta_{12} - 26) q^{46} + (18 \zeta_{12}^{3} + 30 \zeta_{12}^{2} + 30 \zeta_{12} + 18) q^{47} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{48} + ( - \zeta_{12}^{3} + 25 \zeta_{12}^{2} + \zeta_{12} - 50) q^{49} + (11 \zeta_{12}^{3} + 11 \zeta_{12}^{2} + 13 \zeta_{12} - 24) q^{50} + (12 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{51} + 26 \zeta_{12}^{2} q^{52} + ( - 46 \zeta_{12}^{3} + 92 \zeta_{12} + 18) q^{53} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{54} + ( - 24 \zeta_{12}^{3} - 60 \zeta_{12}^{2} - 24 \zeta_{12}) q^{55} + (10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{56} + (5 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26 \zeta_{12} - 5) q^{57} + (20 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{58} + (2 \zeta_{12}^{3} - 46 \zeta_{12}^{2} + 44 \zeta_{12} + 44) q^{59} + ( - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{60} + (50 \zeta_{12}^{3} + 36 \zeta_{12}^{2} - 25 \zeta_{12} - 36) q^{61} + (53 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 53 \zeta_{12} - 10) q^{62} + ( - 15 \zeta_{12} + 15) q^{63} + 8 \zeta_{12}^{3} q^{64} + (26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} - 52 \zeta_{12} - 52) q^{65} + ( - 18 \zeta_{12}^{3} + 36 \zeta_{12} + 6) q^{66} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 31 \zeta_{12} - 43) q^{67} + (8 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 8 \zeta_{12}) q^{68} + (18 \zeta_{12}^{2} + 24 \zeta_{12} + 18) q^{69} + (10 \zeta_{12}^{3} + 20 \zeta_{12}^{2} - 20 \zeta_{12} - 10) q^{70} + (22 \zeta_{12}^{3} - 46 \zeta_{12}^{2} - 68 \zeta_{12} + 68) q^{71} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + ( - 59 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 23 \zeta_{12} - 59) q^{73} + ( - 72 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 36 \zeta_{12} + 10) q^{74} + ( - 36 \zeta_{12}^{3} + \zeta_{12}^{2} + 36 \zeta_{12} - 2) q^{75} + ( - 14 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 38 \zeta_{12} - 24) q^{76} + (30 \zeta_{12}^{3} - 80 \zeta_{12}^{2} + 40) q^{77} + ( - 13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 13 \zeta_{12} + 26) q^{78} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} + 24) q^{79} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 16 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + (28 \zeta_{12}^{2} + 12 \zeta_{12} + 28) q^{82} + ( - 40 \zeta_{12}^{3} + 52 \zeta_{12}^{2} - 52 \zeta_{12} + 40) q^{83} + (10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 20 \zeta_{12} + 20) q^{84} + ( - 24 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{85} + ( - 19 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12} - 19) q^{86} + ( - 30 \zeta_{12}^{2} + 30) q^{87} + ( - 36 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 36 \zeta_{12} + 8) q^{88} + (64 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 104 \zeta_{12} + 40) q^{89} + (18 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{90} + ( - 65 \zeta_{12}^{3} + 65 \zeta_{12}^{2} - 65) q^{91} + ( - 16 \zeta_{12}^{3} + 32 \zeta_{12} + 36) q^{92} + (19 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 34 \zeta_{12} + 53) q^{93} + ( - 30 \zeta_{12}^{3} - 66 \zeta_{12}^{2} - 30 \zeta_{12}) q^{94} + ( - 10 \zeta_{12}^{2} + 42 \zeta_{12} - 10) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (71 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 84 \zeta_{12} + 84) q^{97} + ( - 50 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100})$$ q + (z^3 - z^2 - z) * q^2 + (z^3 + z) * q^3 + 2*z * q^4 + (4*z^3 + 2*z^2 - 2*z - 4) * q^5 + (-2*z^3 - z^2 + z - 1) * q^6 + (5*z^3 - 5*z^2) * q^7 + (-2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 + (-6*z^3 - 2*z^2 + 6*z + 4) * q^10 + (10*z^3 + 10*z^2 - 8*z - 2) * q^11 + (4*z^2 - 2) * q^12 + 13*z * q^13 + (-5*z^3 + 10*z - 5) * q^14 + (-6*z - 6) * q^15 + 4*z^2 * q^16 + (4*z^2 - 12*z + 4) * q^17 + (-3*z^3 + 3) * q^18 + (12*z^3 - 7*z^2 - 19*z + 19) * q^19 + (4*z^3 + 4*z^2 - 8*z - 8) * q^20 + (-10*z^3 + 5*z^2 + 5*z - 10) * q^21 + (-4*z^3 - 18*z^2 + 2*z + 18) * q^22 + (-18*z^3 - 8*z^2 + 18*z + 16) * q^23 + (-2*z^3 - 2*z^2 - 2*z + 4) * q^24 + (z^3 - 24*z^2 + 12) * q^25 + (-13*z^3 - 13) * q^26 + (3*z^3 - 6*z) * q^27 + (-10*z^3 + 10*z^2 - 10) * q^28 + (-10*z^3 - 10*z) * q^29 + (6*z^2 + 6*z + 6) * q^30 + (-29*z^3 - 5*z^2 + 5*z + 29) * q^31 + (-4*z^2 - 4*z + 4) * q^32 + (18*z^3 - 6*z^2 - 12*z - 12) * q^33 + (16*z^3 - 8*z^2 - 8*z + 16) * q^34 + (-20*z^3 + 10*z) * q^35 + (6*z^3 - 6*z) * q^36 + (23*z^3 + 23*z^2 + 13*z - 36) * q^37 + (26*z^3 - 24*z^2 + 12) * q^38 + (26*z^2 - 13) * q^39 + (-4*z^3 + 8*z + 12) * q^40 + (-8*z^3 + 8*z^2 - 20*z - 28) * q^41 + (-5*z^3 + 15*z^2 - 5*z) * q^42 + (5*z^2 + 24*z + 5) * q^43 + (20*z^3 + 4*z^2 - 4*z - 20) * q^44 + (-6*z^3 - 12*z^2 - 6*z + 6) * q^45 + (16*z^3 + 10*z^2 - 26*z - 26) * q^46 + (18*z^3 + 30*z^2 + 30*z + 18) * q^47 + (8*z^3 - 4*z) * q^48 + (-z^3 + 25*z^2 + z - 50) * q^49 + (11*z^3 + 11*z^2 + 13*z - 24) * q^50 + (12*z^3 - 24*z^2 + 12) * q^51 + 26*z^2 * q^52 + (-46*z^3 + 92*z + 18) * q^53 + (3*z^3 - 3*z^2 + 3*z + 6) * q^54 + (-24*z^3 - 60*z^2 - 24*z) * q^55 + (10*z^2 - 10*z + 10) * q^56 + (5*z^3 - 26*z^2 + 26*z - 5) * q^57 + (20*z^3 + 10*z^2 - 10*z + 10) * q^58 + (2*z^3 - 46*z^2 + 44*z + 44) * q^59 + (-12*z^2 - 12*z) * q^60 + (50*z^3 + 36*z^2 - 25*z - 36) * q^61 + (53*z^3 + 5*z^2 - 53*z - 10) * q^62 + (-15*z + 15) * q^63 + 8*z^3 * q^64 + (26*z^3 + 26*z^2 - 52*z - 52) * q^65 + (-18*z^3 + 36*z + 6) * q^66 + (-12*z^3 + 12*z^2 - 31*z - 43) * q^67 + (8*z^3 - 24*z^2 + 8*z) * q^68 + (18*z^2 + 24*z + 18) * q^69 + (10*z^3 + 20*z^2 - 20*z - 10) * q^70 + (22*z^3 - 46*z^2 - 68*z + 68) * q^71 + (-6*z^2 + 6*z + 6) * q^72 + (-59*z^3 + 23*z^2 + 23*z - 59) * q^73 + (-72*z^3 - 10*z^2 + 36*z + 10) * q^74 + (-36*z^3 + z^2 + 36*z - 2) * q^75 + (-14*z^3 - 14*z^2 + 38*z - 24) * q^76 + (30*z^3 - 80*z^2 + 40) * q^77 + (-13*z^3 - 13*z^2 - 13*z + 26) * q^78 + (-7*z^3 + 14*z + 24) * q^79 + (8*z^3 - 8*z^2 - 16*z - 8) * q^80 - 9*z^2 * q^81 + (28*z^2 + 12*z + 28) * q^82 + (-40*z^3 + 52*z^2 - 52*z + 40) * q^83 + (10*z^3 - 10*z^2 - 20*z + 20) * q^84 + (-24*z^2 + 24*z + 24) * q^85 + (-19*z^3 - 10*z^2 - 10*z - 19) * q^86 + (-30*z^2 + 30) * q^87 + (-36*z^3 - 4*z^2 + 36*z + 8) * q^88 + (64*z^3 + 64*z^2 - 104*z + 40) * q^89 + (18*z^3 + 12*z^2 - 6) * q^90 + (-65*z^3 + 65*z^2 - 65) * q^91 + (-16*z^3 + 32*z + 36) * q^92 + (19*z^3 - 19*z^2 + 34*z + 53) * q^93 + (-30*z^3 - 66*z^2 - 30*z) * q^94 + (-10*z^2 + 42*z - 10) * q^95 + (-4*z^3 - 8*z^2 + 8*z + 4) * q^96 + (71*z^3 - 13*z^2 - 84*z + 84) * q^97 + (-50*z^3 + 26*z^2 + 24*z + 24) * q^98 + (-24*z^3 - 6*z^2 - 6*z - 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 12 q^{5} - 6 q^{6} - 10 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 12 * q^5 - 6 * q^6 - 10 * q^7 - 8 * q^8 - 6 * q^9 $$4 q - 2 q^{2} - 12 q^{5} - 6 q^{6} - 10 q^{7} - 8 q^{8} - 6 q^{9} + 12 q^{10} + 12 q^{11} - 20 q^{14} - 24 q^{15} + 8 q^{16} + 24 q^{17} + 12 q^{18} + 62 q^{19} - 24 q^{20} - 30 q^{21} + 36 q^{22} + 48 q^{23} + 12 q^{24} - 52 q^{26} - 20 q^{28} + 36 q^{30} + 106 q^{31} + 8 q^{32} - 60 q^{33} + 48 q^{34} - 98 q^{37} + 48 q^{40} - 96 q^{41} + 30 q^{42} + 30 q^{43} - 72 q^{44} - 84 q^{46} + 132 q^{47} - 150 q^{49} - 74 q^{50} + 52 q^{52} + 72 q^{53} + 18 q^{54} - 120 q^{55} + 60 q^{56} - 72 q^{57} + 60 q^{58} + 84 q^{59} - 24 q^{60} - 72 q^{61} - 30 q^{62} + 60 q^{63} - 156 q^{65} + 24 q^{66} - 148 q^{67} - 48 q^{68} + 108 q^{69} + 180 q^{71} + 12 q^{72} - 190 q^{73} + 20 q^{74} - 6 q^{75} - 124 q^{76} + 78 q^{78} + 96 q^{79} - 48 q^{80} - 18 q^{81} + 168 q^{82} + 264 q^{83} + 60 q^{84} + 48 q^{85} - 96 q^{86} + 60 q^{87} + 24 q^{88} + 288 q^{89} - 130 q^{91} + 144 q^{92} + 174 q^{93} - 132 q^{94} - 60 q^{95} + 310 q^{97} + 148 q^{98} - 108 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 12 * q^5 - 6 * q^6 - 10 * q^7 - 8 * q^8 - 6 * q^9 + 12 * q^10 + 12 * q^11 - 20 * q^14 - 24 * q^15 + 8 * q^16 + 24 * q^17 + 12 * q^18 + 62 * q^19 - 24 * q^20 - 30 * q^21 + 36 * q^22 + 48 * q^23 + 12 * q^24 - 52 * q^26 - 20 * q^28 + 36 * q^30 + 106 * q^31 + 8 * q^32 - 60 * q^33 + 48 * q^34 - 98 * q^37 + 48 * q^40 - 96 * q^41 + 30 * q^42 + 30 * q^43 - 72 * q^44 - 84 * q^46 + 132 * q^47 - 150 * q^49 - 74 * q^50 + 52 * q^52 + 72 * q^53 + 18 * q^54 - 120 * q^55 + 60 * q^56 - 72 * q^57 + 60 * q^58 + 84 * q^59 - 24 * q^60 - 72 * q^61 - 30 * q^62 + 60 * q^63 - 156 * q^65 + 24 * q^66 - 148 * q^67 - 48 * q^68 + 108 * q^69 + 180 * q^71 + 12 * q^72 - 190 * q^73 + 20 * q^74 - 6 * q^75 - 124 * q^76 + 78 * q^78 + 96 * q^79 - 48 * q^80 - 18 * q^81 + 168 * q^82 + 264 * q^83 + 60 * q^84 + 48 * q^85 - 96 * q^86 + 60 * q^87 + 24 * q^88 + 288 * q^89 - 130 * q^91 + 144 * q^92 + 174 * q^93 - 132 * q^94 - 60 * q^95 + 310 * q^97 + 148 * q^98 - 108 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.36603 + 0.366025i 0.866025 1.50000i 1.73205 1.00000i −4.73205 4.73205i −0.633975 + 2.36603i −2.50000 0.669873i −2.00000 + 2.00000i −1.50000 2.59808i 8.19615 + 4.73205i
19.1 0.366025 + 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i −1.26795 + 1.26795i −2.36603 0.633975i −2.50000 + 9.33013i −2.00000 2.00000i −1.50000 2.59808i −2.19615 1.26795i
37.1 0.366025 1.36603i −0.866025 1.50000i −1.73205 1.00000i −1.26795 1.26795i −2.36603 + 0.633975i −2.50000 9.33013i −2.00000 + 2.00000i −1.50000 + 2.59808i −2.19615 + 1.26795i
67.1 −1.36603 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −4.73205 + 4.73205i −0.633975 2.36603i −2.50000 + 0.669873i −2.00000 2.00000i −1.50000 + 2.59808i 8.19615 4.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.3.l.a 4
3.b odd 2 1 234.3.bb.c 4
13.c even 3 1 1014.3.f.e 4
13.e even 6 1 1014.3.f.d 4
13.f odd 12 1 inner 78.3.l.a 4
13.f odd 12 1 1014.3.f.d 4
13.f odd 12 1 1014.3.f.e 4
39.k even 12 1 234.3.bb.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.a 4 1.a even 1 1 trivial
78.3.l.a 4 13.f odd 12 1 inner
234.3.bb.c 4 3.b odd 2 1
234.3.bb.c 4 39.k even 12 1
1014.3.f.d 4 13.e even 6 1
1014.3.f.d 4 13.f odd 12 1
1014.3.f.e 4 13.c even 3 1
1014.3.f.e 4 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12T_{5}^{3} + 72T_{5}^{2} + 144T_{5} + 144$$ acting on $$S_{3}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 144$$
$7$ $$T^{4} + 10 T^{3} + 125 T^{2} + \cdots + 625$$
$11$ $$T^{4} - 12 T^{3} + 180 T^{2} + \cdots + 24336$$
$13$ $$T^{4} - 169 T^{2} + 28561$$
$17$ $$T^{4} - 24 T^{3} + 96 T^{2} + \cdots + 9216$$
$19$ $$T^{4} - 62 T^{3} + 986 T^{2} + \cdots + 14884$$
$23$ $$T^{4} - 48 T^{3} + 636 T^{2} + \cdots + 17424$$
$29$ $$T^{4} + 300 T^{2} + 90000$$
$31$ $$T^{4} - 106 T^{3} + 5618 T^{2} + \cdots + 1868689$$
$37$ $$T^{4} + 98 T^{3} + 5882 T^{2} + \cdots + 3587236$$
$41$ $$T^{4} + 96 T^{3} + 3600 T^{2} + \cdots + 1218816$$
$43$ $$T^{4} - 30 T^{3} - 201 T^{2} + \cdots + 251001$$
$47$ $$T^{4} - 132 T^{3} + 8712 T^{2} + \cdots + 685584$$
$53$ $$(T^{2} - 36 T - 6024)^{2}$$
$59$ $$T^{4} - 84 T^{3} + 4068 T^{2} + \cdots + 16353936$$
$61$ $$T^{4} + 72 T^{3} + 5763 T^{2} + \cdots + 335241$$
$67$ $$T^{4} + 148 T^{3} + 8501 T^{2} + \cdots + 6723649$$
$71$ $$T^{4} - 180 T^{3} + \cdots + 33315984$$
$73$ $$T^{4} + 190 T^{3} + \cdots + 13830961$$
$79$ $$(T^{2} - 48 T + 429)^{2}$$
$83$ $$T^{4} - 264 T^{3} + \cdots + 21678336$$
$89$ $$T^{4} - 288 T^{3} + \cdots + 137170944$$
$97$ $$T^{4} - 310 T^{3} + 27389 T^{2} + \cdots + 8162449$$