# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

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

## 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.

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 $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

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

## Hecke characteristic polynomials

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