# Properties

 Label 147.4.e.k Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ 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] = [147,4,Mod(67,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not 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: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} - 3 \beta_{2} q^{3} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{4} + ( - 10 \beta_{2} + 7 \beta_1 - 10) q^{5} + ( - 3 \beta_{3} - 3) q^{6} + ( - 11 \beta_{3} - 9) q^{8} + ( - 9 \beta_{2} - 9) q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^2 - 3*b2 * q^3 + (-2*b3 - 5*b2 - 2*b1) * q^4 + (-10*b2 + 7*b1 - 10) * q^5 + (-3*b3 - 3) * q^6 + (-11*b3 - 9) * q^8 + (-9*b2 - 9) * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} - 3 \beta_{2} q^{3} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{4} + ( - 10 \beta_{2} + 7 \beta_1 - 10) q^{5} + ( - 3 \beta_{3} - 3) q^{6} + ( - 11 \beta_{3} - 9) q^{8} + ( - 9 \beta_{2} - 9) q^{9} + ( - 17 \beta_{3} + 24 \beta_{2} - 17 \beta_1) q^{10} + ( - 24 \beta_{3} - 10 \beta_{2} - 24 \beta_1) q^{11} + ( - 15 \beta_{2} - 6 \beta_1 - 15) q^{12} + (25 \beta_{3} + 52) q^{13} + ( - 21 \beta_{3} - 30) q^{15} + ( - 9 \beta_{2} - 36 \beta_1 - 9) q^{16} + (45 \beta_{3} + 58 \beta_{2} + 45 \beta_1) q^{17} + ( - 9 \beta_{3} + 9 \beta_{2} - 9 \beta_1) q^{18} + ( - 96 \beta_{2} - 22 \beta_1 - 96) q^{19} + ( - 15 \beta_{3} - 22) q^{20} + (14 \beta_{3} + 38) q^{22} + ( - 14 \beta_{2} - 28 \beta_1 - 14) q^{23} + ( - 33 \beta_{3} + 27 \beta_{2} - 33 \beta_1) q^{24} + ( - 140 \beta_{3} + 73 \beta_{2} - 140 \beta_1) q^{25} + ( - 102 \beta_{2} + 77 \beta_1 - 102) q^{26} - 27 q^{27} + ( - 62 \beta_{3} + 148) q^{29} + (72 \beta_{2} - 51 \beta_1 + 72) q^{30} + (50 \beta_{3} - 52 \beta_{2} + 50 \beta_1) q^{31} + ( - 61 \beta_{3} + 9 \beta_{2} - 61 \beta_1) q^{32} + ( - 30 \beta_{2} - 72 \beta_1 - 30) q^{33} + (13 \beta_{3} - 32) q^{34} + (18 \beta_{3} - 45) q^{36} + (124 \beta_{2} + 48 \beta_1 + 124) q^{37} + ( - 74 \beta_{3} + 52 \beta_{2} - 74 \beta_1) q^{38} + (75 \beta_{3} - 156 \beta_{2} + 75 \beta_1) q^{39} + (244 \beta_{2} - 173 \beta_1 + 244) q^{40} + ( - 219 \beta_{3} + 10) q^{41} + ( - 100 \beta_{3} - 360) q^{43} + ( - 146 \beta_{2} - 140 \beta_1 - 146) q^{44} + ( - 63 \beta_{3} + 90 \beta_{2} - 63 \beta_1) q^{45} + (14 \beta_{3} - 42 \beta_{2} + 14 \beta_1) q^{46} + (48 \beta_{2} + 250 \beta_1 + 48) q^{47} + (108 \beta_{3} - 27) q^{48} + (213 \beta_{3} + 353) q^{50} + (174 \beta_{2} + 135 \beta_1 + 174) q^{51} + (21 \beta_{3} - 160 \beta_{2} + 21 \beta_1) q^{52} + (360 \beta_{3} + 134 \beta_{2} + 360 \beta_1) q^{53} + (27 \beta_{2} - 27 \beta_1 + 27) q^{54} + (170 \beta_{3} + 236) q^{55} + (66 \beta_{3} - 288) q^{57} + ( - 24 \beta_{2} + 86 \beta_1 - 24) q^{58} + (226 \beta_{3} - 308 \beta_{2} + 226 \beta_1) q^{59} + ( - 45 \beta_{3} + 66 \beta_{2} - 45 \beta_1) q^{60} + ( - 8 \beta_{2} - 3 \beta_1 - 8) q^{61} + ( - 102 \beta_{3} - 152) q^{62} + (358 \beta_{3} + 59) q^{64} + ( - 870 \beta_{2} + 614 \beta_1 - 870) q^{65} + (42 \beta_{3} - 114 \beta_{2} + 42 \beta_1) q^{66} + (524 \beta_{3} - 72 \beta_{2} + 524 \beta_1) q^{67} + (470 \beta_{2} + 341 \beta_1 + 470) q^{68} + (84 \beta_{3} - 42) q^{69} + ( - 232 \beta_{3} + 494) q^{71} + (81 \beta_{2} - 99 \beta_1 + 81) q^{72} + ( - 401 \beta_{3} + 52 \beta_{2} - 401 \beta_1) q^{73} + (76 \beta_{3} - 28 \beta_{2} + 76 \beta_1) q^{74} + (219 \beta_{2} - 420 \beta_1 + 219) q^{75} + (302 \beta_{3} - 568) q^{76} + ( - 231 \beta_{3} - 306) q^{78} + (472 \beta_{2} + 236 \beta_1 + 472) q^{79} + (297 \beta_{3} - 414 \beta_{2} + 297 \beta_1) q^{80} + 81 \beta_{2} q^{81} + (428 \beta_{2} - 209 \beta_1 + 428) q^{82} + (80 \beta_{3} + 508) q^{83} + ( - 44 \beta_{3} - 50) q^{85} + (560 \beta_{2} - 460 \beta_1 + 560) q^{86} + ( - 186 \beta_{3} - 444 \beta_{2} - 186 \beta_1) q^{87} + (106 \beta_{3} - 438 \beta_{2} + 106 \beta_1) q^{88} + (194 \beta_{2} + 339 \beta_1 + 194) q^{89} + (153 \beta_{3} + 216) q^{90} + (168 \beta_{3} - 182) q^{92} + ( - 156 \beta_{2} + 150 \beta_1 - 156) q^{93} + ( - 202 \beta_{3} + 452 \beta_{2} - 202 \beta_1) q^{94} + ( - 452 \beta_{3} + 652 \beta_{2} - 452 \beta_1) q^{95} + (27 \beta_{2} - 183 \beta_1 + 27) q^{96} + ( - 599 \beta_{3} + 244) q^{97} + (216 \beta_{3} - 90) q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^2 - 3*b2 * q^3 + (-2*b3 - 5*b2 - 2*b1) * q^4 + (-10*b2 + 7*b1 - 10) * q^5 + (-3*b3 - 3) * q^6 + (-11*b3 - 9) * q^8 + (-9*b2 - 9) * q^9 + (-17*b3 + 24*b2 - 17*b1) * q^10 + (-24*b3 - 10*b2 - 24*b1) * q^11 + (-15*b2 - 6*b1 - 15) * q^12 + (25*b3 + 52) * q^13 + (-21*b3 - 30) * q^15 + (-9*b2 - 36*b1 - 9) * q^16 + (45*b3 + 58*b2 + 45*b1) * q^17 + (-9*b3 + 9*b2 - 9*b1) * q^18 + (-96*b2 - 22*b1 - 96) * q^19 + (-15*b3 - 22) * q^20 + (14*b3 + 38) * q^22 + (-14*b2 - 28*b1 - 14) * q^23 + (-33*b3 + 27*b2 - 33*b1) * q^24 + (-140*b3 + 73*b2 - 140*b1) * q^25 + (-102*b2 + 77*b1 - 102) * q^26 - 27 * q^27 + (-62*b3 + 148) * q^29 + (72*b2 - 51*b1 + 72) * q^30 + (50*b3 - 52*b2 + 50*b1) * q^31 + (-61*b3 + 9*b2 - 61*b1) * q^32 + (-30*b2 - 72*b1 - 30) * q^33 + (13*b3 - 32) * q^34 + (18*b3 - 45) * q^36 + (124*b2 + 48*b1 + 124) * q^37 + (-74*b3 + 52*b2 - 74*b1) * q^38 + (75*b3 - 156*b2 + 75*b1) * q^39 + (244*b2 - 173*b1 + 244) * q^40 + (-219*b3 + 10) * q^41 + (-100*b3 - 360) * q^43 + (-146*b2 - 140*b1 - 146) * q^44 + (-63*b3 + 90*b2 - 63*b1) * q^45 + (14*b3 - 42*b2 + 14*b1) * q^46 + (48*b2 + 250*b1 + 48) * q^47 + (108*b3 - 27) * q^48 + (213*b3 + 353) * q^50 + (174*b2 + 135*b1 + 174) * q^51 + (21*b3 - 160*b2 + 21*b1) * q^52 + (360*b3 + 134*b2 + 360*b1) * q^53 + (27*b2 - 27*b1 + 27) * q^54 + (170*b3 + 236) * q^55 + (66*b3 - 288) * q^57 + (-24*b2 + 86*b1 - 24) * q^58 + (226*b3 - 308*b2 + 226*b1) * q^59 + (-45*b3 + 66*b2 - 45*b1) * q^60 + (-8*b2 - 3*b1 - 8) * q^61 + (-102*b3 - 152) * q^62 + (358*b3 + 59) * q^64 + (-870*b2 + 614*b1 - 870) * q^65 + (42*b3 - 114*b2 + 42*b1) * q^66 + (524*b3 - 72*b2 + 524*b1) * q^67 + (470*b2 + 341*b1 + 470) * q^68 + (84*b3 - 42) * q^69 + (-232*b3 + 494) * q^71 + (81*b2 - 99*b1 + 81) * q^72 + (-401*b3 + 52*b2 - 401*b1) * q^73 + (76*b3 - 28*b2 + 76*b1) * q^74 + (219*b2 - 420*b1 + 219) * q^75 + (302*b3 - 568) * q^76 + (-231*b3 - 306) * q^78 + (472*b2 + 236*b1 + 472) * q^79 + (297*b3 - 414*b2 + 297*b1) * q^80 + 81*b2 * q^81 + (428*b2 - 209*b1 + 428) * q^82 + (80*b3 + 508) * q^83 + (-44*b3 - 50) * q^85 + (560*b2 - 460*b1 + 560) * q^86 + (-186*b3 - 444*b2 - 186*b1) * q^87 + (106*b3 - 438*b2 + 106*b1) * q^88 + (194*b2 + 339*b1 + 194) * q^89 + (153*b3 + 216) * q^90 + (168*b3 - 182) * q^92 + (-156*b2 + 150*b1 - 156) * q^93 + (-202*b3 + 452*b2 - 202*b1) * q^94 + (-452*b3 + 652*b2 - 452*b1) * q^95 + (27*b2 - 183*b1 + 27) * q^96 + (-599*b3 + 244) * q^97 + (216*b3 - 90) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 6 q^{3} + 10 q^{4} - 20 q^{5} - 12 q^{6} - 36 q^{8} - 18 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 6 * q^3 + 10 * q^4 - 20 * q^5 - 12 * q^6 - 36 * q^8 - 18 * q^9 $$4 q - 2 q^{2} + 6 q^{3} + 10 q^{4} - 20 q^{5} - 12 q^{6} - 36 q^{8} - 18 q^{9} - 48 q^{10} + 20 q^{11} - 30 q^{12} + 208 q^{13} - 120 q^{15} - 18 q^{16} - 116 q^{17} - 18 q^{18} - 192 q^{19} - 88 q^{20} + 152 q^{22} - 28 q^{23} - 54 q^{24} - 146 q^{25} - 204 q^{26} - 108 q^{27} + 592 q^{29} + 144 q^{30} + 104 q^{31} - 18 q^{32} - 60 q^{33} - 128 q^{34} - 180 q^{36} + 248 q^{37} - 104 q^{38} + 312 q^{39} + 488 q^{40} + 40 q^{41} - 1440 q^{43} - 292 q^{44} - 180 q^{45} + 84 q^{46} + 96 q^{47} - 108 q^{48} + 1412 q^{50} + 348 q^{51} + 320 q^{52} - 268 q^{53} + 54 q^{54} + 944 q^{55} - 1152 q^{57} - 48 q^{58} + 616 q^{59} - 132 q^{60} - 16 q^{61} - 608 q^{62} + 236 q^{64} - 1740 q^{65} + 228 q^{66} + 144 q^{67} + 940 q^{68} - 168 q^{69} + 1976 q^{71} + 162 q^{72} - 104 q^{73} + 56 q^{74} + 438 q^{75} - 2272 q^{76} - 1224 q^{78} + 944 q^{79} + 828 q^{80} - 162 q^{81} + 856 q^{82} + 2032 q^{83} - 200 q^{85} + 1120 q^{86} + 888 q^{87} + 876 q^{88} + 388 q^{89} + 864 q^{90} - 728 q^{92} - 312 q^{93} - 904 q^{94} - 1304 q^{95} + 54 q^{96} + 976 q^{97} - 360 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 6 * q^3 + 10 * q^4 - 20 * q^5 - 12 * q^6 - 36 * q^8 - 18 * q^9 - 48 * q^10 + 20 * q^11 - 30 * q^12 + 208 * q^13 - 120 * q^15 - 18 * q^16 - 116 * q^17 - 18 * q^18 - 192 * q^19 - 88 * q^20 + 152 * q^22 - 28 * q^23 - 54 * q^24 - 146 * q^25 - 204 * q^26 - 108 * q^27 + 592 * q^29 + 144 * q^30 + 104 * q^31 - 18 * q^32 - 60 * q^33 - 128 * q^34 - 180 * q^36 + 248 * q^37 - 104 * q^38 + 312 * q^39 + 488 * q^40 + 40 * q^41 - 1440 * q^43 - 292 * q^44 - 180 * q^45 + 84 * q^46 + 96 * q^47 - 108 * q^48 + 1412 * q^50 + 348 * q^51 + 320 * q^52 - 268 * q^53 + 54 * q^54 + 944 * q^55 - 1152 * q^57 - 48 * q^58 + 616 * q^59 - 132 * q^60 - 16 * q^61 - 608 * q^62 + 236 * q^64 - 1740 * q^65 + 228 * q^66 + 144 * q^67 + 940 * q^68 - 168 * q^69 + 1976 * q^71 + 162 * q^72 - 104 * q^73 + 56 * q^74 + 438 * q^75 - 2272 * q^76 - 1224 * q^78 + 944 * q^79 + 828 * q^80 - 162 * q^81 + 856 * q^82 + 2032 * q^83 - 200 * q^85 + 1120 * q^86 + 888 * q^87 + 876 * q^88 + 388 * q^89 + 864 * q^90 - 728 * q^92 - 312 * q^93 - 904 * q^94 - 1304 * q^95 + 54 * q^96 + 976 * q^97 - 360 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.20711 2.09077i 1.50000 2.59808i 1.08579 1.88064i −9.94975 17.2335i −7.24264 0 −24.5563 −4.50000 7.79423i −24.0208 + 41.6053i
67.2 0.207107 + 0.358719i 1.50000 2.59808i 3.91421 6.77962i −0.0502525 0.0870399i 1.24264 0 6.55635 −4.50000 7.79423i 0.0208153 0.0360531i
79.1 −1.20711 + 2.09077i 1.50000 + 2.59808i 1.08579 + 1.88064i −9.94975 + 17.2335i −7.24264 0 −24.5563 −4.50000 + 7.79423i −24.0208 41.6053i
79.2 0.207107 0.358719i 1.50000 + 2.59808i 3.91421 + 6.77962i −0.0502525 + 0.0870399i 1.24264 0 6.55635 −4.50000 + 7.79423i 0.0208153 + 0.0360531i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.e.k 4
3.b odd 2 1 441.4.e.v 4
7.b odd 2 1 147.4.e.j 4
7.c even 3 1 147.4.a.j 2
7.c even 3 1 inner 147.4.e.k 4
7.d odd 6 1 147.4.a.k yes 2
7.d odd 6 1 147.4.e.j 4
21.c even 2 1 441.4.e.u 4
21.g even 6 1 441.4.a.o 2
21.g even 6 1 441.4.e.u 4
21.h odd 6 1 441.4.a.n 2
21.h odd 6 1 441.4.e.v 4
28.f even 6 1 2352.4.a.bl 2
28.g odd 6 1 2352.4.a.cf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j 2 7.c even 3 1
147.4.a.k yes 2 7.d odd 6 1
147.4.e.j 4 7.b odd 2 1
147.4.e.j 4 7.d odd 6 1
147.4.e.k 4 1.a even 1 1 trivial
147.4.e.k 4 7.c even 3 1 inner
441.4.a.n 2 21.h odd 6 1
441.4.a.o 2 21.g even 6 1
441.4.e.u 4 21.c even 2 1
441.4.e.u 4 21.g even 6 1
441.4.e.v 4 3.b odd 2 1
441.4.e.v 4 21.h odd 6 1
2352.4.a.bl 2 28.f even 6 1
2352.4.a.cf 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1$$ T2^4 + 2*T2^3 + 5*T2^2 - 2*T2 + 1 $$T_{5}^{4} + 20T_{5}^{3} + 398T_{5}^{2} + 40T_{5} + 4$$ T5^4 + 20*T5^3 + 398*T5^2 + 40*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$3$ $$(T^{2} - 3 T + 9)^{2}$$
$5$ $$T^{4} + 20 T^{3} + 398 T^{2} + 40 T + 4$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 20 T^{3} + 1452 T^{2} + \cdots + 1106704$$
$13$ $$(T^{2} - 104 T + 1454)^{2}$$
$17$ $$T^{4} + 116 T^{3} + 14142 T^{2} + \cdots + 470596$$
$19$ $$T^{4} + 192 T^{3} + \cdots + 68029504$$
$23$ $$T^{4} + 28 T^{3} + 2156 T^{2} + \cdots + 1882384$$
$29$ $$(T^{2} - 296 T + 14216)^{2}$$
$31$ $$T^{4} - 104 T^{3} + 13112 T^{2} + \cdots + 5271616$$
$37$ $$T^{4} - 248 T^{3} + \cdots + 115949824$$
$41$ $$(T^{2} - 20 T - 95822)^{2}$$
$43$ $$(T^{2} + 720 T + 109600)^{2}$$
$47$ $$T^{4} - 96 T^{3} + \cdots + 15054308416$$
$53$ $$T^{4} + 268 T^{3} + \cdots + 58198667536$$
$59$ $$T^{4} - 616 T^{3} + \cdots + 53114944$$
$61$ $$T^{4} + 16 T^{3} + 210 T^{2} + \cdots + 2116$$
$67$ $$T^{4} - 144 T^{3} + \cdots + 295901185024$$
$71$ $$(T^{2} - 988 T + 136388)^{2}$$
$73$ $$T^{4} + 104 T^{3} + \cdots + 101695934404$$
$79$ $$T^{4} - 944 T^{3} + \cdots + 12408177664$$
$83$ $$(T^{2} - 1016 T + 245264)^{2}$$
$89$ $$T^{4} - 388 T^{3} + \cdots + 36943146436$$
$97$ $$(T^{2} - 488 T - 658066)^{2}$$