# Properties

 Label 147.4.a.h Level $147$ Weight $4$ Character orbit 147.a Self dual yes Analytic conductor $8.673$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
4.00000 3.00000 8.00000 18.0000 12.0000 0 0 9.00000 72.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.a.h yes 1
3.b odd 2 1 441.4.a.a 1
4.b odd 2 1 2352.4.a.s 1
7.b odd 2 1 147.4.a.f 1
7.c even 3 2 147.4.e.a 2
7.d odd 6 2 147.4.e.d 2
21.c even 2 1 441.4.a.c 1
21.g even 6 2 441.4.e.l 2
21.h odd 6 2 441.4.e.o 2
28.d even 2 1 2352.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 7.b odd 2 1
147.4.a.h yes 1 1.a even 1 1 trivial
147.4.e.a 2 7.c even 3 2
147.4.e.d 2 7.d odd 6 2
441.4.a.a 1 3.b odd 2 1
441.4.a.c 1 21.c even 2 1
441.4.e.l 2 21.g even 6 2
441.4.e.o 2 21.h odd 6 2
2352.4.a.s 1 4.b odd 2 1
2352.4.a.t 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} - 4$$ $$T_{5} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$-3 + T$$
$5$ $$-18 + T$$
$7$ $$T$$
$11$ $$50 + T$$
$13$ $$36 + T$$
$17$ $$-126 + T$$
$19$ $$72 + T$$
$23$ $$-14 + T$$
$29$ $$-158 + T$$
$31$ $$36 + T$$
$37$ $$162 + T$$
$41$ $$270 + T$$
$43$ $$324 + T$$
$47$ $$72 + T$$
$53$ $$22 + T$$
$59$ $$-468 + T$$
$61$ $$-792 + T$$
$67$ $$-232 + T$$
$71$ $$734 + T$$
$73$ $$-180 + T$$
$79$ $$-236 + T$$
$83$ $$-36 + T$$
$89$ $$-234 + T$$
$97$ $$-468 + T$$