# Properties

 Label 147.4.a.g 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: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 4.00000 12.0000 0 0 9.00000 16.0000
 $$n$$: 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.g 1
3.b odd 2 1 441.4.a.b 1
4.b odd 2 1 2352.4.a.l 1
7.b odd 2 1 21.4.a.b 1
7.c even 3 2 147.4.e.b 2
7.d odd 6 2 147.4.e.c 2
21.c even 2 1 63.4.a.a 1
21.g even 6 2 441.4.e.m 2
21.h odd 6 2 441.4.e.n 2
28.d even 2 1 336.4.a.h 1
35.c odd 2 1 525.4.a.b 1
35.f even 4 2 525.4.d.b 2
56.e even 2 1 1344.4.a.i 1
56.h odd 2 1 1344.4.a.w 1
84.h odd 2 1 1008.4.a.m 1
105.g even 2 1 1575.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 7.b odd 2 1
63.4.a.a 1 21.c even 2 1
147.4.a.g 1 1.a even 1 1 trivial
147.4.e.b 2 7.c even 3 2
147.4.e.c 2 7.d odd 6 2
336.4.a.h 1 28.d even 2 1
441.4.a.b 1 3.b odd 2 1
441.4.e.m 2 21.g even 6 2
441.4.e.n 2 21.h odd 6 2
525.4.a.b 1 35.c odd 2 1
525.4.d.b 2 35.f even 4 2
1008.4.a.m 1 84.h odd 2 1
1344.4.a.i 1 56.e even 2 1
1344.4.a.w 1 56.h odd 2 1
1575.4.a.k 1 105.g even 2 1
2352.4.a.l 1 4.b odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$-3 + T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$-62 + T$$
$13$ $$-62 + T$$
$17$ $$84 + T$$
$19$ $$100 + T$$
$23$ $$42 + T$$
$29$ $$10 + T$$
$31$ $$-48 + T$$
$37$ $$246 + T$$
$41$ $$-248 + T$$
$43$ $$-68 + T$$
$47$ $$324 + T$$
$53$ $$-258 + T$$
$59$ $$120 + T$$
$61$ $$622 + T$$
$67$ $$-904 + T$$
$71$ $$678 + T$$
$73$ $$-642 + T$$
$79$ $$-740 + T$$
$83$ $$468 + T$$
$89$ $$200 + T$$
$97$ $$-1266 + T$$