# Properties

 Label 147.4.e.b 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: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} + 12 q^{6} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q -4 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} + 12 q^{6} -9 \zeta_{6} q^{9} + ( -16 + 16 \zeta_{6} ) q^{10} + ( -62 + 62 \zeta_{6} ) q^{11} -24 \zeta_{6} q^{12} + 62 q^{13} + 12 q^{15} + 64 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} + ( -36 + 36 \zeta_{6} ) q^{18} + 100 \zeta_{6} q^{19} + 32 q^{20} + 248 q^{22} + 42 \zeta_{6} q^{23} + ( 109 - 109 \zeta_{6} ) q^{25} -248 \zeta_{6} q^{26} + 27 q^{27} -10 q^{29} -48 \zeta_{6} q^{30} + ( -48 + 48 \zeta_{6} ) q^{31} + ( 256 - 256 \zeta_{6} ) q^{32} -186 \zeta_{6} q^{33} -336 q^{34} + 72 q^{36} + 246 \zeta_{6} q^{37} + ( 400 - 400 \zeta_{6} ) q^{38} + ( -186 + 186 \zeta_{6} ) q^{39} + 248 q^{41} + 68 q^{43} -496 \zeta_{6} q^{44} + ( -36 + 36 \zeta_{6} ) q^{45} + ( 168 - 168 \zeta_{6} ) q^{46} + 324 \zeta_{6} q^{47} -192 q^{48} -436 q^{50} + 252 \zeta_{6} q^{51} + ( -496 + 496 \zeta_{6} ) q^{52} + ( -258 + 258 \zeta_{6} ) q^{53} -108 \zeta_{6} q^{54} + 248 q^{55} -300 q^{57} + 40 \zeta_{6} q^{58} + ( 120 - 120 \zeta_{6} ) q^{59} + ( -96 + 96 \zeta_{6} ) q^{60} + 622 \zeta_{6} q^{61} + 192 q^{62} -512 q^{64} -248 \zeta_{6} q^{65} + ( -744 + 744 \zeta_{6} ) q^{66} + ( -904 + 904 \zeta_{6} ) q^{67} + 672 \zeta_{6} q^{68} -126 q^{69} -678 q^{71} + ( -642 + 642 \zeta_{6} ) q^{73} + ( 984 - 984 \zeta_{6} ) q^{74} + 327 \zeta_{6} q^{75} -800 q^{76} + 744 q^{78} -740 \zeta_{6} q^{79} + ( 256 - 256 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} -992 \zeta_{6} q^{82} -468 q^{83} -336 q^{85} -272 \zeta_{6} q^{86} + ( 30 - 30 \zeta_{6} ) q^{87} + 200 \zeta_{6} q^{89} + 144 q^{90} -336 q^{92} -144 \zeta_{6} q^{93} + ( 1296 - 1296 \zeta_{6} ) q^{94} + ( 400 - 400 \zeta_{6} ) q^{95} + 768 \zeta_{6} q^{96} + 1266 q^{97} + 558 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 3q^{3} - 8q^{4} - 4q^{5} + 24q^{6} - 9q^{9} + O(q^{10})$$ $$2q - 4q^{2} - 3q^{3} - 8q^{4} - 4q^{5} + 24q^{6} - 9q^{9} - 16q^{10} - 62q^{11} - 24q^{12} + 124q^{13} + 24q^{15} + 64q^{16} + 84q^{17} - 36q^{18} + 100q^{19} + 64q^{20} + 496q^{22} + 42q^{23} + 109q^{25} - 248q^{26} + 54q^{27} - 20q^{29} - 48q^{30} - 48q^{31} + 256q^{32} - 186q^{33} - 672q^{34} + 144q^{36} + 246q^{37} + 400q^{38} - 186q^{39} + 496q^{41} + 136q^{43} - 496q^{44} - 36q^{45} + 168q^{46} + 324q^{47} - 384q^{48} - 872q^{50} + 252q^{51} - 496q^{52} - 258q^{53} - 108q^{54} + 496q^{55} - 600q^{57} + 40q^{58} + 120q^{59} - 96q^{60} + 622q^{61} + 384q^{62} - 1024q^{64} - 248q^{65} - 744q^{66} - 904q^{67} + 672q^{68} - 252q^{69} - 1356q^{71} - 642q^{73} + 984q^{74} + 327q^{75} - 1600q^{76} + 1488q^{78} - 740q^{79} + 256q^{80} - 81q^{81} - 992q^{82} - 936q^{83} - 672q^{85} - 272q^{86} + 30q^{87} + 200q^{89} + 288q^{90} - 672q^{92} - 144q^{93} + 1296q^{94} + 400q^{95} + 768q^{96} + 2532q^{97} + 1116q^{99} + O(q^{100})$$

## 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 −2.00000 3.46410i 12.0000 0 0 −4.50000 7.79423i −8.00000 + 13.8564i
79.1 −2.00000 + 3.46410i −1.50000 2.59808i −4.00000 6.92820i −2.00000 + 3.46410i 12.0000 0 0 −4.50000 + 7.79423i −8.00000 13.8564i
 $$n$$: 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.b 2
3.b odd 2 1 441.4.e.n 2
7.b odd 2 1 147.4.e.c 2
7.c even 3 1 147.4.a.g 1
7.c even 3 1 inner 147.4.e.b 2
7.d odd 6 1 21.4.a.b 1
7.d odd 6 1 147.4.e.c 2
21.c even 2 1 441.4.e.m 2
21.g even 6 1 63.4.a.a 1
21.g even 6 1 441.4.e.m 2
21.h odd 6 1 441.4.a.b 1
21.h odd 6 1 441.4.e.n 2
28.f even 6 1 336.4.a.h 1
28.g odd 6 1 2352.4.a.l 1
35.i odd 6 1 525.4.a.b 1
35.k even 12 2 525.4.d.b 2
56.j odd 6 1 1344.4.a.w 1
56.m even 6 1 1344.4.a.i 1
84.j odd 6 1 1008.4.a.m 1
105.p even 6 1 1575.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 7.d odd 6 1
63.4.a.a 1 21.g even 6 1
147.4.a.g 1 7.c even 3 1
147.4.e.b 2 1.a even 1 1 trivial
147.4.e.b 2 7.c even 3 1 inner
147.4.e.c 2 7.b odd 2 1
147.4.e.c 2 7.d odd 6 1
336.4.a.h 1 28.f even 6 1
441.4.a.b 1 21.h odd 6 1
441.4.e.m 2 21.c even 2 1
441.4.e.m 2 21.g even 6 1
441.4.e.n 2 3.b odd 2 1
441.4.e.n 2 21.h odd 6 1
525.4.a.b 1 35.i odd 6 1
525.4.d.b 2 35.k even 12 2
1008.4.a.m 1 84.j odd 6 1
1344.4.a.i 1 56.m even 6 1
1344.4.a.w 1 56.j odd 6 1
1575.4.a.k 1 105.p even 6 1
2352.4.a.l 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 $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} + 4 T_{2} + 16$$ $$T_{5}^{2} + 4 T_{5} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3844 + 62 T + T^{2}$$
$13$ $$( -62 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$10000 - 100 T + T^{2}$$
$23$ $$1764 - 42 T + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$2304 + 48 T + T^{2}$$
$37$ $$60516 - 246 T + T^{2}$$
$41$ $$( -248 + T )^{2}$$
$43$ $$( -68 + T )^{2}$$
$47$ $$104976 - 324 T + T^{2}$$
$53$ $$66564 + 258 T + T^{2}$$
$59$ $$14400 - 120 T + T^{2}$$
$61$ $$386884 - 622 T + T^{2}$$
$67$ $$817216 + 904 T + T^{2}$$
$71$ $$( 678 + T )^{2}$$
$73$ $$412164 + 642 T + T^{2}$$
$79$ $$547600 + 740 T + T^{2}$$
$83$ $$( 468 + T )^{2}$$
$89$ $$40000 - 200 T + T^{2}$$
$97$ $$( -1266 + T )^{2}$$