# Properties

 Label 147.4.e.e Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ Analytic rank $0$ Dimension $2$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 12 \zeta_{6} q^{5} - 3 q^{6} + 15 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (3*z - 3) * q^3 + (-7*z + 7) * q^4 + 12*z * q^5 - 3 * q^6 + 15 * q^8 - 9*z * q^9 $$q + \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 12 \zeta_{6} q^{5} - 3 q^{6} + 15 q^{8} - 9 \zeta_{6} q^{9} + (12 \zeta_{6} - 12) q^{10} + (20 \zeta_{6} - 20) q^{11} + 21 \zeta_{6} q^{12} + 84 q^{13} - 36 q^{15} - 41 \zeta_{6} q^{16} + (96 \zeta_{6} - 96) q^{17} + ( - 9 \zeta_{6} + 9) q^{18} + 12 \zeta_{6} q^{19} + 84 q^{20} - 20 q^{22} + 176 \zeta_{6} q^{23} + (45 \zeta_{6} - 45) q^{24} + (19 \zeta_{6} - 19) q^{25} + 84 \zeta_{6} q^{26} + 27 q^{27} + 58 q^{29} - 36 \zeta_{6} q^{30} + (264 \zeta_{6} - 264) q^{31} + ( - 161 \zeta_{6} + 161) q^{32} - 60 \zeta_{6} q^{33} - 96 q^{34} - 63 q^{36} - 258 \zeta_{6} q^{37} + (12 \zeta_{6} - 12) q^{38} + (252 \zeta_{6} - 252) q^{39} + 180 \zeta_{6} q^{40} + 156 q^{43} + 140 \zeta_{6} q^{44} + ( - 108 \zeta_{6} + 108) q^{45} + (176 \zeta_{6} - 176) q^{46} - 408 \zeta_{6} q^{47} + 123 q^{48} - 19 q^{50} - 288 \zeta_{6} q^{51} + ( - 588 \zeta_{6} + 588) q^{52} + ( - 722 \zeta_{6} + 722) q^{53} + 27 \zeta_{6} q^{54} - 240 q^{55} - 36 q^{57} + 58 \zeta_{6} q^{58} + ( - 492 \zeta_{6} + 492) q^{59} + (252 \zeta_{6} - 252) q^{60} - 492 \zeta_{6} q^{61} - 264 q^{62} - 167 q^{64} + 1008 \zeta_{6} q^{65} + ( - 60 \zeta_{6} + 60) q^{66} + (412 \zeta_{6} - 412) q^{67} + 672 \zeta_{6} q^{68} - 528 q^{69} + 296 q^{71} - 135 \zeta_{6} q^{72} + ( - 240 \zeta_{6} + 240) q^{73} + ( - 258 \zeta_{6} + 258) q^{74} - 57 \zeta_{6} q^{75} + 84 q^{76} - 252 q^{78} - 776 \zeta_{6} q^{79} + ( - 492 \zeta_{6} + 492) q^{80} + (81 \zeta_{6} - 81) q^{81} - 924 q^{83} - 1152 q^{85} + 156 \zeta_{6} q^{86} + (174 \zeta_{6} - 174) q^{87} + (300 \zeta_{6} - 300) q^{88} - 744 \zeta_{6} q^{89} + 108 q^{90} + 1232 q^{92} - 792 \zeta_{6} q^{93} + ( - 408 \zeta_{6} + 408) q^{94} + (144 \zeta_{6} - 144) q^{95} + 483 \zeta_{6} q^{96} + 168 q^{97} + 180 q^{99} +O(q^{100})$$ q + z * q^2 + (3*z - 3) * q^3 + (-7*z + 7) * q^4 + 12*z * q^5 - 3 * q^6 + 15 * q^8 - 9*z * q^9 + (12*z - 12) * q^10 + (20*z - 20) * q^11 + 21*z * q^12 + 84 * q^13 - 36 * q^15 - 41*z * q^16 + (96*z - 96) * q^17 + (-9*z + 9) * q^18 + 12*z * q^19 + 84 * q^20 - 20 * q^22 + 176*z * q^23 + (45*z - 45) * q^24 + (19*z - 19) * q^25 + 84*z * q^26 + 27 * q^27 + 58 * q^29 - 36*z * q^30 + (264*z - 264) * q^31 + (-161*z + 161) * q^32 - 60*z * q^33 - 96 * q^34 - 63 * q^36 - 258*z * q^37 + (12*z - 12) * q^38 + (252*z - 252) * q^39 + 180*z * q^40 + 156 * q^43 + 140*z * q^44 + (-108*z + 108) * q^45 + (176*z - 176) * q^46 - 408*z * q^47 + 123 * q^48 - 19 * q^50 - 288*z * q^51 + (-588*z + 588) * q^52 + (-722*z + 722) * q^53 + 27*z * q^54 - 240 * q^55 - 36 * q^57 + 58*z * q^58 + (-492*z + 492) * q^59 + (252*z - 252) * q^60 - 492*z * q^61 - 264 * q^62 - 167 * q^64 + 1008*z * q^65 + (-60*z + 60) * q^66 + (412*z - 412) * q^67 + 672*z * q^68 - 528 * q^69 + 296 * q^71 - 135*z * q^72 + (-240*z + 240) * q^73 + (-258*z + 258) * q^74 - 57*z * q^75 + 84 * q^76 - 252 * q^78 - 776*z * q^79 + (-492*z + 492) * q^80 + (81*z - 81) * q^81 - 924 * q^83 - 1152 * q^85 + 156*z * q^86 + (174*z - 174) * q^87 + (300*z - 300) * q^88 - 744*z * q^89 + 108 * q^90 + 1232 * q^92 - 792*z * q^93 + (-408*z + 408) * q^94 + (144*z - 144) * q^95 + 483*z * q^96 + 168 * q^97 + 180 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 3 q^{3} + 7 q^{4} + 12 q^{5} - 6 q^{6} + 30 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + q^2 - 3 * q^3 + 7 * q^4 + 12 * q^5 - 6 * q^6 + 30 * q^8 - 9 * q^9 $$2 q + q^{2} - 3 q^{3} + 7 q^{4} + 12 q^{5} - 6 q^{6} + 30 q^{8} - 9 q^{9} - 12 q^{10} - 20 q^{11} + 21 q^{12} + 168 q^{13} - 72 q^{15} - 41 q^{16} - 96 q^{17} + 9 q^{18} + 12 q^{19} + 168 q^{20} - 40 q^{22} + 176 q^{23} - 45 q^{24} - 19 q^{25} + 84 q^{26} + 54 q^{27} + 116 q^{29} - 36 q^{30} - 264 q^{31} + 161 q^{32} - 60 q^{33} - 192 q^{34} - 126 q^{36} - 258 q^{37} - 12 q^{38} - 252 q^{39} + 180 q^{40} + 312 q^{43} + 140 q^{44} + 108 q^{45} - 176 q^{46} - 408 q^{47} + 246 q^{48} - 38 q^{50} - 288 q^{51} + 588 q^{52} + 722 q^{53} + 27 q^{54} - 480 q^{55} - 72 q^{57} + 58 q^{58} + 492 q^{59} - 252 q^{60} - 492 q^{61} - 528 q^{62} - 334 q^{64} + 1008 q^{65} + 60 q^{66} - 412 q^{67} + 672 q^{68} - 1056 q^{69} + 592 q^{71} - 135 q^{72} + 240 q^{73} + 258 q^{74} - 57 q^{75} + 168 q^{76} - 504 q^{78} - 776 q^{79} + 492 q^{80} - 81 q^{81} - 1848 q^{83} - 2304 q^{85} + 156 q^{86} - 174 q^{87} - 300 q^{88} - 744 q^{89} + 216 q^{90} + 2464 q^{92} - 792 q^{93} + 408 q^{94} - 144 q^{95} + 483 q^{96} + 336 q^{97} + 360 q^{99}+O(q^{100})$$ 2 * q + q^2 - 3 * q^3 + 7 * q^4 + 12 * q^5 - 6 * q^6 + 30 * q^8 - 9 * q^9 - 12 * q^10 - 20 * q^11 + 21 * q^12 + 168 * q^13 - 72 * q^15 - 41 * q^16 - 96 * q^17 + 9 * q^18 + 12 * q^19 + 168 * q^20 - 40 * q^22 + 176 * q^23 - 45 * q^24 - 19 * q^25 + 84 * q^26 + 54 * q^27 + 116 * q^29 - 36 * q^30 - 264 * q^31 + 161 * q^32 - 60 * q^33 - 192 * q^34 - 126 * q^36 - 258 * q^37 - 12 * q^38 - 252 * q^39 + 180 * q^40 + 312 * q^43 + 140 * q^44 + 108 * q^45 - 176 * q^46 - 408 * q^47 + 246 * q^48 - 38 * q^50 - 288 * q^51 + 588 * q^52 + 722 * q^53 + 27 * q^54 - 480 * q^55 - 72 * q^57 + 58 * q^58 + 492 * q^59 - 252 * q^60 - 492 * q^61 - 528 * q^62 - 334 * q^64 + 1008 * q^65 + 60 * q^66 - 412 * q^67 + 672 * q^68 - 1056 * q^69 + 592 * q^71 - 135 * q^72 + 240 * q^73 + 258 * q^74 - 57 * q^75 + 168 * q^76 - 504 * q^78 - 776 * q^79 + 492 * q^80 - 81 * q^81 - 1848 * q^83 - 2304 * q^85 + 156 * q^86 - 174 * q^87 - 300 * q^88 - 744 * q^89 + 216 * q^90 + 2464 * q^92 - 792 * q^93 + 408 * q^94 - 144 * q^95 + 483 * q^96 + 336 * q^97 + 360 * 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
0.500000 + 0.866025i −1.50000 + 2.59808i 3.50000 6.06218i 6.00000 + 10.3923i −3.00000 0 15.0000 −4.50000 7.79423i −6.00000 + 10.3923i
79.1 0.500000 0.866025i −1.50000 2.59808i 3.50000 + 6.06218i 6.00000 10.3923i −3.00000 0 15.0000 −4.50000 + 7.79423i −6.00000 10.3923i
 $$n$$: 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.e 2
3.b odd 2 1 441.4.e.f 2
7.b odd 2 1 147.4.e.f 2
7.c even 3 1 147.4.a.e yes 1
7.c even 3 1 inner 147.4.e.e 2
7.d odd 6 1 147.4.a.d 1
7.d odd 6 1 147.4.e.f 2
21.c even 2 1 441.4.e.g 2
21.g even 6 1 441.4.a.g 1
21.g even 6 1 441.4.e.g 2
21.h odd 6 1 441.4.a.h 1
21.h odd 6 1 441.4.e.f 2
28.f even 6 1 2352.4.a.bi 1
28.g odd 6 1 2352.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 7.d odd 6 1
147.4.a.e yes 1 7.c even 3 1
147.4.e.e 2 1.a even 1 1 trivial
147.4.e.e 2 7.c even 3 1 inner
147.4.e.f 2 7.b odd 2 1
147.4.e.f 2 7.d odd 6 1
441.4.a.g 1 21.g even 6 1
441.4.a.h 1 21.h odd 6 1
441.4.e.f 2 3.b odd 2 1
441.4.e.f 2 21.h odd 6 1
441.4.e.g 2 21.c even 2 1
441.4.e.g 2 21.g even 6 1
2352.4.a.b 1 28.g odd 6 1
2352.4.a.bi 1 28.f even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{5}^{2} - 12T_{5} + 144$$ T5^2 - 12*T5 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - 12T + 144$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 20T + 400$$
$13$ $$(T - 84)^{2}$$
$17$ $$T^{2} + 96T + 9216$$
$19$ $$T^{2} - 12T + 144$$
$23$ $$T^{2} - 176T + 30976$$
$29$ $$(T - 58)^{2}$$
$31$ $$T^{2} + 264T + 69696$$
$37$ $$T^{2} + 258T + 66564$$
$41$ $$T^{2}$$
$43$ $$(T - 156)^{2}$$
$47$ $$T^{2} + 408T + 166464$$
$53$ $$T^{2} - 722T + 521284$$
$59$ $$T^{2} - 492T + 242064$$
$61$ $$T^{2} + 492T + 242064$$
$67$ $$T^{2} + 412T + 169744$$
$71$ $$(T - 296)^{2}$$
$73$ $$T^{2} - 240T + 57600$$
$79$ $$T^{2} + 776T + 602176$$
$83$ $$(T + 924)^{2}$$
$89$ $$T^{2} + 744T + 553536$$
$97$ $$(T - 168)^{2}$$