# Properties

 Label 147.4.a.e 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

 $$f(q)$$ $$=$$ $$q - q^{2} + 3q^{3} - 7q^{4} - 12q^{5} - 3q^{6} + 15q^{8} + 9q^{9} + O(q^{10})$$ $$q - q^{2} + 3q^{3} - 7q^{4} - 12q^{5} - 3q^{6} + 15q^{8} + 9q^{9} + 12q^{10} + 20q^{11} - 21q^{12} + 84q^{13} - 36q^{15} + 41q^{16} + 96q^{17} - 9q^{18} - 12q^{19} + 84q^{20} - 20q^{22} - 176q^{23} + 45q^{24} + 19q^{25} - 84q^{26} + 27q^{27} + 58q^{29} + 36q^{30} + 264q^{31} - 161q^{32} + 60q^{33} - 96q^{34} - 63q^{36} + 258q^{37} + 12q^{38} + 252q^{39} - 180q^{40} + 156q^{43} - 140q^{44} - 108q^{45} + 176q^{46} + 408q^{47} + 123q^{48} - 19q^{50} + 288q^{51} - 588q^{52} - 722q^{53} - 27q^{54} - 240q^{55} - 36q^{57} - 58q^{58} - 492q^{59} + 252q^{60} + 492q^{61} - 264q^{62} - 167q^{64} - 1008q^{65} - 60q^{66} + 412q^{67} - 672q^{68} - 528q^{69} + 296q^{71} + 135q^{72} - 240q^{73} - 258q^{74} + 57q^{75} + 84q^{76} - 252q^{78} + 776q^{79} - 492q^{80} + 81q^{81} - 924q^{83} - 1152q^{85} - 156q^{86} + 174q^{87} + 300q^{88} + 744q^{89} + 108q^{90} + 1232q^{92} + 792q^{93} - 408q^{94} + 144q^{95} - 483q^{96} + 168q^{97} + 180q^{99} + O(q^{100})$$

## 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
−1.00000 3.00000 −7.00000 −12.0000 −3.00000 0 15.0000 9.00000 12.0000
 $$n$$: 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.e yes 1
3.b odd 2 1 441.4.a.h 1
4.b odd 2 1 2352.4.a.b 1
7.b odd 2 1 147.4.a.d 1
7.c even 3 2 147.4.e.e 2
7.d odd 6 2 147.4.e.f 2
21.c even 2 1 441.4.a.g 1
21.g even 6 2 441.4.e.g 2
21.h odd 6 2 441.4.e.f 2
28.d even 2 1 2352.4.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 7.b odd 2 1
147.4.a.e yes 1 1.a even 1 1 trivial
147.4.e.e 2 7.c even 3 2
147.4.e.f 2 7.d odd 6 2
441.4.a.g 1 21.c even 2 1
441.4.a.h 1 3.b odd 2 1
441.4.e.f 2 21.h odd 6 2
441.4.e.g 2 21.g even 6 2
2352.4.a.b 1 4.b odd 2 1
2352.4.a.bi 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} + 1$$ $$T_{5} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-3 + T$$
$5$ $$12 + T$$
$7$ $$T$$
$11$ $$-20 + T$$
$13$ $$-84 + T$$
$17$ $$-96 + T$$
$19$ $$12 + T$$
$23$ $$176 + T$$
$29$ $$-58 + T$$
$31$ $$-264 + T$$
$37$ $$-258 + T$$
$41$ $$T$$
$43$ $$-156 + T$$
$47$ $$-408 + T$$
$53$ $$722 + T$$
$59$ $$492 + T$$
$61$ $$-492 + T$$
$67$ $$-412 + T$$
$71$ $$-296 + T$$
$73$ $$240 + T$$
$79$ $$-776 + T$$
$83$ $$924 + T$$
$89$ $$-744 + T$$
$97$ $$-168 + T$$