# Properties

 Label 147.4.a.b Level $147$ Weight $4$ Character orbit 147.a Self dual yes Analytic conductor $8.673$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + 3 q^{3} + q^{4} - 3 q^{5} - 9 q^{6} + 21 q^{8} + 9 q^{9} + O(q^{10})$$ $$q - 3 q^{2} + 3 q^{3} + q^{4} - 3 q^{5} - 9 q^{6} + 21 q^{8} + 9 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} - 64 q^{13} - 9 q^{15} - 71 q^{16} + 84 q^{17} - 27 q^{18} - 16 q^{19} - 3 q^{20} + 45 q^{22} - 84 q^{23} + 63 q^{24} - 116 q^{25} + 192 q^{26} + 27 q^{27} - 297 q^{29} + 27 q^{30} - 253 q^{31} + 45 q^{32} - 45 q^{33} - 252 q^{34} + 9 q^{36} - 316 q^{37} + 48 q^{38} - 192 q^{39} - 63 q^{40} + 360 q^{41} + 26 q^{43} - 15 q^{44} - 27 q^{45} + 252 q^{46} - 30 q^{47} - 213 q^{48} + 348 q^{50} + 252 q^{51} - 64 q^{52} + 363 q^{53} - 81 q^{54} + 45 q^{55} - 48 q^{57} + 891 q^{58} - 15 q^{59} - 9 q^{60} - 118 q^{61} + 759 q^{62} + 433 q^{64} + 192 q^{65} + 135 q^{66} - 370 q^{67} + 84 q^{68} - 252 q^{69} - 342 q^{71} + 189 q^{72} + 362 q^{73} + 948 q^{74} - 348 q^{75} - 16 q^{76} + 576 q^{78} + 467 q^{79} + 213 q^{80} + 81 q^{81} - 1080 q^{82} + 477 q^{83} - 252 q^{85} - 78 q^{86} - 891 q^{87} - 315 q^{88} + 906 q^{89} + 81 q^{90} - 84 q^{92} - 759 q^{93} + 90 q^{94} + 48 q^{95} + 135 q^{96} + 503 q^{97} - 135 q^{99} + O(q^{100})$$

## 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
−3.00000 3.00000 1.00000 −3.00000 −9.00000 0 21.0000 9.00000 9.00000
 $$n$$: 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.b 1
3.b odd 2 1 441.4.a.l 1
4.b odd 2 1 2352.4.a.i 1
7.b odd 2 1 147.4.a.a 1
7.c even 3 2 21.4.e.a 2
7.d odd 6 2 147.4.e.h 2
21.c even 2 1 441.4.a.k 1
21.g even 6 2 441.4.e.c 2
21.h odd 6 2 63.4.e.a 2
28.d even 2 1 2352.4.a.bd 1
28.g odd 6 2 336.4.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 7.c even 3 2
63.4.e.a 2 21.h odd 6 2
147.4.a.a 1 7.b odd 2 1
147.4.a.b 1 1.a even 1 1 trivial
147.4.e.h 2 7.d odd 6 2
336.4.q.e 2 28.g odd 6 2
441.4.a.k 1 21.c even 2 1
441.4.a.l 1 3.b odd 2 1
441.4.e.c 2 21.g even 6 2
2352.4.a.i 1 4.b odd 2 1
2352.4.a.bd 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} + 3$$ $$T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T$$
$3$ $$-3 + T$$
$5$ $$3 + T$$
$7$ $$T$$
$11$ $$15 + T$$
$13$ $$64 + T$$
$17$ $$-84 + T$$
$19$ $$16 + T$$
$23$ $$84 + T$$
$29$ $$297 + T$$
$31$ $$253 + T$$
$37$ $$316 + T$$
$41$ $$-360 + T$$
$43$ $$-26 + T$$
$47$ $$30 + T$$
$53$ $$-363 + T$$
$59$ $$15 + T$$
$61$ $$118 + T$$
$67$ $$370 + T$$
$71$ $$342 + T$$
$73$ $$-362 + T$$
$79$ $$-467 + T$$
$83$ $$-477 + T$$
$89$ $$-906 + T$$
$97$ $$-503 + T$$