# Properties

 Label 147.4.a.k Level $147$ Weight $4$ Character orbit 147.a Self dual yes Analytic conductor $8.673$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + 3 q^{3} + ( -5 + 2 \beta ) q^{4} + ( -10 - 7 \beta ) q^{5} + ( 3 + 3 \beta ) q^{6} + ( -9 - 11 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + 3 q^{3} + ( -5 + 2 \beta ) q^{4} + ( -10 - 7 \beta ) q^{5} + ( 3 + 3 \beta ) q^{6} + ( -9 - 11 \beta ) q^{8} + 9 q^{9} + ( -24 - 17 \beta ) q^{10} + ( -10 + 24 \beta ) q^{11} + ( -15 + 6 \beta ) q^{12} + ( -52 - 25 \beta ) q^{13} + ( -30 - 21 \beta ) q^{15} + ( 9 - 36 \beta ) q^{16} + ( -58 + 45 \beta ) q^{17} + ( 9 + 9 \beta ) q^{18} + ( -96 + 22 \beta ) q^{19} + ( 22 + 15 \beta ) q^{20} + ( 38 + 14 \beta ) q^{22} + ( 14 - 28 \beta ) q^{23} + ( -27 - 33 \beta ) q^{24} + ( 73 + 140 \beta ) q^{25} + ( -102 - 77 \beta ) q^{26} + 27 q^{27} + ( 148 - 62 \beta ) q^{29} + ( -72 - 51 \beta ) q^{30} + ( 52 + 50 \beta ) q^{31} + ( 9 + 61 \beta ) q^{32} + ( -30 + 72 \beta ) q^{33} + ( 32 - 13 \beta ) q^{34} + ( -45 + 18 \beta ) q^{36} + ( -124 + 48 \beta ) q^{37} + ( -52 - 74 \beta ) q^{38} + ( -156 - 75 \beta ) q^{39} + ( 244 + 173 \beta ) q^{40} + ( -10 + 219 \beta ) q^{41} + ( -360 - 100 \beta ) q^{43} + ( 146 - 140 \beta ) q^{44} + ( -90 - 63 \beta ) q^{45} + ( -42 - 14 \beta ) q^{46} + ( 48 - 250 \beta ) q^{47} + ( 27 - 108 \beta ) q^{48} + ( 353 + 213 \beta ) q^{50} + ( -174 + 135 \beta ) q^{51} + ( 160 + 21 \beta ) q^{52} + ( 134 - 360 \beta ) q^{53} + ( 27 + 27 \beta ) q^{54} + ( -236 - 170 \beta ) q^{55} + ( -288 + 66 \beta ) q^{57} + ( 24 + 86 \beta ) q^{58} + ( 308 + 226 \beta ) q^{59} + ( 66 + 45 \beta ) q^{60} + ( -8 + 3 \beta ) q^{61} + ( 152 + 102 \beta ) q^{62} + ( 59 + 358 \beta ) q^{64} + ( 870 + 614 \beta ) q^{65} + ( 114 + 42 \beta ) q^{66} + ( -72 - 524 \beta ) q^{67} + ( 470 - 341 \beta ) q^{68} + ( 42 - 84 \beta ) q^{69} + ( 494 - 232 \beta ) q^{71} + ( -81 - 99 \beta ) q^{72} + ( -52 - 401 \beta ) q^{73} + ( -28 - 76 \beta ) q^{74} + ( 219 + 420 \beta ) q^{75} + ( 568 - 302 \beta ) q^{76} + ( -306 - 231 \beta ) q^{78} + ( -472 + 236 \beta ) q^{79} + ( 414 + 297 \beta ) q^{80} + 81 q^{81} + ( 428 + 209 \beta ) q^{82} + ( -508 - 80 \beta ) q^{83} + ( -50 - 44 \beta ) q^{85} + ( -560 - 460 \beta ) q^{86} + ( 444 - 186 \beta ) q^{87} + ( -438 - 106 \beta ) q^{88} + ( 194 - 339 \beta ) q^{89} + ( -216 - 153 \beta ) q^{90} + ( -182 + 168 \beta ) q^{92} + ( 156 + 150 \beta ) q^{93} + ( -452 - 202 \beta ) q^{94} + ( 652 + 452 \beta ) q^{95} + ( 27 + 183 \beta ) q^{96} + ( -244 + 599 \beta ) q^{97} + ( -90 + 216 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 6q^{3} - 10q^{4} - 20q^{5} + 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 6q^{3} - 10q^{4} - 20q^{5} + 6q^{6} - 18q^{8} + 18q^{9} - 48q^{10} - 20q^{11} - 30q^{12} - 104q^{13} - 60q^{15} + 18q^{16} - 116q^{17} + 18q^{18} - 192q^{19} + 44q^{20} + 76q^{22} + 28q^{23} - 54q^{24} + 146q^{25} - 204q^{26} + 54q^{27} + 296q^{29} - 144q^{30} + 104q^{31} + 18q^{32} - 60q^{33} + 64q^{34} - 90q^{36} - 248q^{37} - 104q^{38} - 312q^{39} + 488q^{40} - 20q^{41} - 720q^{43} + 292q^{44} - 180q^{45} - 84q^{46} + 96q^{47} + 54q^{48} + 706q^{50} - 348q^{51} + 320q^{52} + 268q^{53} + 54q^{54} - 472q^{55} - 576q^{57} + 48q^{58} + 616q^{59} + 132q^{60} - 16q^{61} + 304q^{62} + 118q^{64} + 1740q^{65} + 228q^{66} - 144q^{67} + 940q^{68} + 84q^{69} + 988q^{71} - 162q^{72} - 104q^{73} - 56q^{74} + 438q^{75} + 1136q^{76} - 612q^{78} - 944q^{79} + 828q^{80} + 162q^{81} + 856q^{82} - 1016q^{83} - 100q^{85} - 1120q^{86} + 888q^{87} - 876q^{88} + 388q^{89} - 432q^{90} - 364q^{92} + 312q^{93} - 904q^{94} + 1304q^{95} + 54q^{96} - 488q^{97} - 180q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−0.414214 3.00000 −7.82843 −0.100505 −1.24264 0 6.55635 9.00000 0.0416306
1.2 2.41421 3.00000 −2.17157 −19.8995 7.24264 0 −24.5563 9.00000 −48.0416
 $$n$$: 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.k yes 2
3.b odd 2 1 441.4.a.o 2
4.b odd 2 1 2352.4.a.bl 2
7.b odd 2 1 147.4.a.j 2
7.c even 3 2 147.4.e.j 4
7.d odd 6 2 147.4.e.k 4
21.c even 2 1 441.4.a.n 2
21.g even 6 2 441.4.e.v 4
21.h odd 6 2 441.4.e.u 4
28.d even 2 1 2352.4.a.cf 2

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

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{5}^{2} + 20 T_{5} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$2 + 20 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1052 + 20 T + T^{2}$$
$13$ $$1454 + 104 T + T^{2}$$
$17$ $$-686 + 116 T + T^{2}$$
$19$ $$8248 + 192 T + T^{2}$$
$23$ $$-1372 - 28 T + T^{2}$$
$29$ $$14216 - 296 T + T^{2}$$
$31$ $$-2296 - 104 T + T^{2}$$
$37$ $$10768 + 248 T + T^{2}$$
$41$ $$-95822 + 20 T + T^{2}$$
$43$ $$109600 + 720 T + T^{2}$$
$47$ $$-122696 - 96 T + T^{2}$$
$53$ $$-241244 - 268 T + T^{2}$$
$59$ $$-7288 - 616 T + T^{2}$$
$61$ $$46 + 16 T + T^{2}$$
$67$ $$-543968 + 144 T + T^{2}$$
$71$ $$136388 - 988 T + T^{2}$$
$73$ $$-318898 + 104 T + T^{2}$$
$79$ $$111392 + 944 T + T^{2}$$
$83$ $$245264 + 1016 T + T^{2}$$
$89$ $$-192206 - 388 T + T^{2}$$
$97$ $$-658066 + 488 T + T^{2}$$