# Properties

 Label 147.4.e.d Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (8 \zeta_{6} - 8) q^{4} + 18 \zeta_{6} q^{5} - 12 q^{6} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q - 4*z * q^2 + (-3*z + 3) * q^3 + (8*z - 8) * q^4 + 18*z * q^5 - 12 * q^6 - 9*z * q^9 $$q - 4 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (8 \zeta_{6} - 8) q^{4} + 18 \zeta_{6} q^{5} - 12 q^{6} - 9 \zeta_{6} q^{9} + ( - 72 \zeta_{6} + 72) q^{10} + ( - 50 \zeta_{6} + 50) q^{11} + 24 \zeta_{6} q^{12} + 36 q^{13} + 54 q^{15} + 64 \zeta_{6} q^{16} + ( - 126 \zeta_{6} + 126) q^{17} + (36 \zeta_{6} - 36) q^{18} - 72 \zeta_{6} q^{19} - 144 q^{20} - 200 q^{22} - 14 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} - 144 \zeta_{6} q^{26} - 27 q^{27} + 158 q^{29} - 216 \zeta_{6} q^{30} + (36 \zeta_{6} - 36) q^{31} + ( - 256 \zeta_{6} + 256) q^{32} - 150 \zeta_{6} q^{33} - 504 q^{34} + 72 q^{36} + 162 \zeta_{6} q^{37} + (288 \zeta_{6} - 288) q^{38} + ( - 108 \zeta_{6} + 108) q^{39} + 270 q^{41} - 324 q^{43} + 400 \zeta_{6} q^{44} + ( - 162 \zeta_{6} + 162) q^{45} + (56 \zeta_{6} - 56) q^{46} - 72 \zeta_{6} q^{47} + 192 q^{48} + 796 q^{50} - 378 \zeta_{6} q^{51} + (288 \zeta_{6} - 288) q^{52} + ( - 22 \zeta_{6} + 22) q^{53} + 108 \zeta_{6} q^{54} + 900 q^{55} - 216 q^{57} - 632 \zeta_{6} q^{58} + ( - 468 \zeta_{6} + 468) q^{59} + (432 \zeta_{6} - 432) q^{60} + 792 \zeta_{6} q^{61} + 144 q^{62} - 512 q^{64} + 648 \zeta_{6} q^{65} + (600 \zeta_{6} - 600) q^{66} + (232 \zeta_{6} - 232) q^{67} + 1008 \zeta_{6} q^{68} - 42 q^{69} - 734 q^{71} + ( - 180 \zeta_{6} + 180) q^{73} + ( - 648 \zeta_{6} + 648) q^{74} + 597 \zeta_{6} q^{75} + 576 q^{76} - 432 q^{78} - 236 \zeta_{6} q^{79} + (1152 \zeta_{6} - 1152) q^{80} + (81 \zeta_{6} - 81) q^{81} - 1080 \zeta_{6} q^{82} - 36 q^{83} + 2268 q^{85} + 1296 \zeta_{6} q^{86} + ( - 474 \zeta_{6} + 474) q^{87} + 234 \zeta_{6} q^{89} - 648 q^{90} + 112 q^{92} + 108 \zeta_{6} q^{93} + (288 \zeta_{6} - 288) q^{94} + ( - 1296 \zeta_{6} + 1296) q^{95} - 768 \zeta_{6} q^{96} - 468 q^{97} - 450 q^{99} +O(q^{100})$$ q - 4*z * q^2 + (-3*z + 3) * q^3 + (8*z - 8) * q^4 + 18*z * q^5 - 12 * q^6 - 9*z * q^9 + (-72*z + 72) * q^10 + (-50*z + 50) * q^11 + 24*z * q^12 + 36 * q^13 + 54 * q^15 + 64*z * q^16 + (-126*z + 126) * q^17 + (36*z - 36) * q^18 - 72*z * q^19 - 144 * q^20 - 200 * q^22 - 14*z * q^23 + (199*z - 199) * q^25 - 144*z * q^26 - 27 * q^27 + 158 * q^29 - 216*z * q^30 + (36*z - 36) * q^31 + (-256*z + 256) * q^32 - 150*z * q^33 - 504 * q^34 + 72 * q^36 + 162*z * q^37 + (288*z - 288) * q^38 + (-108*z + 108) * q^39 + 270 * q^41 - 324 * q^43 + 400*z * q^44 + (-162*z + 162) * q^45 + (56*z - 56) * q^46 - 72*z * q^47 + 192 * q^48 + 796 * q^50 - 378*z * q^51 + (288*z - 288) * q^52 + (-22*z + 22) * q^53 + 108*z * q^54 + 900 * q^55 - 216 * q^57 - 632*z * q^58 + (-468*z + 468) * q^59 + (432*z - 432) * q^60 + 792*z * q^61 + 144 * q^62 - 512 * q^64 + 648*z * q^65 + (600*z - 600) * q^66 + (232*z - 232) * q^67 + 1008*z * q^68 - 42 * q^69 - 734 * q^71 + (-180*z + 180) * q^73 + (-648*z + 648) * q^74 + 597*z * q^75 + 576 * q^76 - 432 * q^78 - 236*z * q^79 + (1152*z - 1152) * q^80 + (81*z - 81) * q^81 - 1080*z * q^82 - 36 * q^83 + 2268 * q^85 + 1296*z * q^86 + (-474*z + 474) * q^87 + 234*z * q^89 - 648 * q^90 + 112 * q^92 + 108*z * q^93 + (288*z - 288) * q^94 + (-1296*z + 1296) * q^95 - 768*z * q^96 - 468 * q^97 - 450 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 3 q^{3} - 8 q^{4} + 18 q^{5} - 24 q^{6} - 9 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 + 3 * q^3 - 8 * q^4 + 18 * q^5 - 24 * q^6 - 9 * q^9 $$2 q - 4 q^{2} + 3 q^{3} - 8 q^{4} + 18 q^{5} - 24 q^{6} - 9 q^{9} + 72 q^{10} + 50 q^{11} + 24 q^{12} + 72 q^{13} + 108 q^{15} + 64 q^{16} + 126 q^{17} - 36 q^{18} - 72 q^{19} - 288 q^{20} - 400 q^{22} - 14 q^{23} - 199 q^{25} - 144 q^{26} - 54 q^{27} + 316 q^{29} - 216 q^{30} - 36 q^{31} + 256 q^{32} - 150 q^{33} - 1008 q^{34} + 144 q^{36} + 162 q^{37} - 288 q^{38} + 108 q^{39} + 540 q^{41} - 648 q^{43} + 400 q^{44} + 162 q^{45} - 56 q^{46} - 72 q^{47} + 384 q^{48} + 1592 q^{50} - 378 q^{51} - 288 q^{52} + 22 q^{53} + 108 q^{54} + 1800 q^{55} - 432 q^{57} - 632 q^{58} + 468 q^{59} - 432 q^{60} + 792 q^{61} + 288 q^{62} - 1024 q^{64} + 648 q^{65} - 600 q^{66} - 232 q^{67} + 1008 q^{68} - 84 q^{69} - 1468 q^{71} + 180 q^{73} + 648 q^{74} + 597 q^{75} + 1152 q^{76} - 864 q^{78} - 236 q^{79} - 1152 q^{80} - 81 q^{81} - 1080 q^{82} - 72 q^{83} + 4536 q^{85} + 1296 q^{86} + 474 q^{87} + 234 q^{89} - 1296 q^{90} + 224 q^{92} + 108 q^{93} - 288 q^{94} + 1296 q^{95} - 768 q^{96} - 936 q^{97} - 900 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 + 3 * q^3 - 8 * q^4 + 18 * q^5 - 24 * q^6 - 9 * q^9 + 72 * q^10 + 50 * q^11 + 24 * q^12 + 72 * q^13 + 108 * q^15 + 64 * q^16 + 126 * q^17 - 36 * q^18 - 72 * q^19 - 288 * q^20 - 400 * q^22 - 14 * q^23 - 199 * q^25 - 144 * q^26 - 54 * q^27 + 316 * q^29 - 216 * q^30 - 36 * q^31 + 256 * q^32 - 150 * q^33 - 1008 * q^34 + 144 * q^36 + 162 * q^37 - 288 * q^38 + 108 * q^39 + 540 * q^41 - 648 * q^43 + 400 * q^44 + 162 * q^45 - 56 * q^46 - 72 * q^47 + 384 * q^48 + 1592 * q^50 - 378 * q^51 - 288 * q^52 + 22 * q^53 + 108 * q^54 + 1800 * q^55 - 432 * q^57 - 632 * q^58 + 468 * q^59 - 432 * q^60 + 792 * q^61 + 288 * q^62 - 1024 * q^64 + 648 * q^65 - 600 * q^66 - 232 * q^67 + 1008 * q^68 - 84 * q^69 - 1468 * q^71 + 180 * q^73 + 648 * q^74 + 597 * q^75 + 1152 * q^76 - 864 * q^78 - 236 * q^79 - 1152 * q^80 - 81 * q^81 - 1080 * q^82 - 72 * q^83 + 4536 * q^85 + 1296 * q^86 + 474 * q^87 + 234 * q^89 - 1296 * q^90 + 224 * q^92 + 108 * q^93 - 288 * q^94 + 1296 * q^95 - 768 * q^96 - 936 * q^97 - 900 * q^99

## Character values

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

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−2.00000 3.46410i 1.50000 2.59808i −4.00000 + 6.92820i 9.00000 + 15.5885i −12.0000 0 0 −4.50000 7.79423i 36.0000 62.3538i
79.1 −2.00000 + 3.46410i 1.50000 + 2.59808i −4.00000 6.92820i 9.00000 15.5885i −12.0000 0 0 −4.50000 + 7.79423i 36.0000 + 62.3538i
 $$n$$: 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.d 2
3.b odd 2 1 441.4.e.l 2
7.b odd 2 1 147.4.e.a 2
7.c even 3 1 147.4.a.f 1
7.c even 3 1 inner 147.4.e.d 2
7.d odd 6 1 147.4.a.h yes 1
7.d odd 6 1 147.4.e.a 2
21.c even 2 1 441.4.e.o 2
21.g even 6 1 441.4.a.a 1
21.g even 6 1 441.4.e.o 2
21.h odd 6 1 441.4.a.c 1
21.h odd 6 1 441.4.e.l 2
28.f even 6 1 2352.4.a.s 1
28.g odd 6 1 2352.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 7.c even 3 1
147.4.a.h yes 1 7.d odd 6 1
147.4.e.a 2 7.b odd 2 1
147.4.e.a 2 7.d odd 6 1
147.4.e.d 2 1.a even 1 1 trivial
147.4.e.d 2 7.c even 3 1 inner
441.4.a.a 1 21.g even 6 1
441.4.a.c 1 21.h odd 6 1
441.4.e.l 2 3.b odd 2 1
441.4.e.l 2 21.h odd 6 1
441.4.e.o 2 21.c even 2 1
441.4.e.o 2 21.g even 6 1
2352.4.a.s 1 28.f even 6 1
2352.4.a.t 1 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 $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} + 4T_{2} + 16$$ T2^2 + 4*T2 + 16 $$T_{5}^{2} - 18T_{5} + 324$$ T5^2 - 18*T5 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T + 16$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2} - 18T + 324$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 50T + 2500$$
$13$ $$(T - 36)^{2}$$
$17$ $$T^{2} - 126T + 15876$$
$19$ $$T^{2} + 72T + 5184$$
$23$ $$T^{2} + 14T + 196$$
$29$ $$(T - 158)^{2}$$
$31$ $$T^{2} + 36T + 1296$$
$37$ $$T^{2} - 162T + 26244$$
$41$ $$(T - 270)^{2}$$
$43$ $$(T + 324)^{2}$$
$47$ $$T^{2} + 72T + 5184$$
$53$ $$T^{2} - 22T + 484$$
$59$ $$T^{2} - 468T + 219024$$
$61$ $$T^{2} - 792T + 627264$$
$67$ $$T^{2} + 232T + 53824$$
$71$ $$(T + 734)^{2}$$
$73$ $$T^{2} - 180T + 32400$$
$79$ $$T^{2} + 236T + 55696$$
$83$ $$(T + 36)^{2}$$
$89$ $$T^{2} - 234T + 54756$$
$97$ $$(T + 468)^{2}$$