# Properties

 Label 147.4.a.m Level $147$ Weight $4$ Character orbit 147.a Self dual yes Analytic conductor $8.673$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.57516.1 Defining polynomial: $$x^{3} - x^{2} - 24 x + 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 3 q^{3} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + 3 \beta_{1} q^{6} + ( 10 + 9 \beta_{1} + \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 3 q^{3} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + 3 \beta_{1} q^{6} + ( 10 + 9 \beta_{1} + \beta_{2} ) q^{8} + 9 q^{9} + ( 22 - 11 \beta_{1} + \beta_{2} ) q^{10} + ( 12 - \beta_{1} - 3 \beta_{2} ) q^{11} + ( 24 + 3 \beta_{1} + 3 \beta_{2} ) q^{12} + ( -19 - 5 \beta_{1} + \beta_{2} ) q^{13} + ( -12 + 3 \beta_{1} - 3 \beta_{2} ) q^{15} + ( 74 + 19 \beta_{1} + \beta_{2} ) q^{16} + ( -16 - 4 \beta_{2} ) q^{17} + 9 \beta_{1} q^{18} + ( 65 + 7 \beta_{1} + \beta_{2} ) q^{19} + ( -150 + 11 \beta_{1} - 3 \beta_{2} ) q^{20} + ( 2 - 13 \beta_{1} - \beta_{2} ) q^{22} + ( 80 - 24 \beta_{1} - 4 \beta_{2} ) q^{23} + ( 30 + 27 \beta_{1} + 3 \beta_{2} ) q^{24} + ( 53 - 29 \beta_{1} + \beta_{2} ) q^{25} + ( -86 - 16 \beta_{1} - 5 \beta_{2} ) q^{26} + 27 q^{27} + ( 26 - 25 \beta_{1} + 5 \beta_{2} ) q^{29} + ( 66 - 33 \beta_{1} + 3 \beta_{2} ) q^{30} + ( 39 - 22 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 218 + 29 \beta_{1} + 11 \beta_{2} ) q^{32} + ( 36 - 3 \beta_{1} - 9 \beta_{2} ) q^{33} + ( 24 - 48 \beta_{1} ) q^{34} + ( 72 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 81 + 19 \beta_{1} + \beta_{2} ) q^{37} + ( 106 + 80 \beta_{1} + 7 \beta_{2} ) q^{38} + ( -57 - 15 \beta_{1} + 3 \beta_{2} ) q^{39} + ( 18 - 75 \beta_{1} + 3 \beta_{2} ) q^{40} + ( -82 + 2 \beta_{1} + 14 \beta_{2} ) q^{41} + ( 143 - 69 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -298 - 11 \beta_{1} + 11 \beta_{2} ) q^{44} + ( -36 + 9 \beta_{1} - 9 \beta_{2} ) q^{45} + ( -360 + 24 \beta_{1} - 24 \beta_{2} ) q^{46} + ( 46 + 72 \beta_{1} + 28 \beta_{2} ) q^{47} + ( 222 + 57 \beta_{1} + 3 \beta_{2} ) q^{48} + ( -470 + 32 \beta_{1} - 29 \beta_{2} ) q^{50} + ( -48 - 12 \beta_{2} ) q^{51} + ( -74 - 102 \beta_{1} - 24 \beta_{2} ) q^{52} + ( 154 - 69 \beta_{1} - 11 \beta_{2} ) q^{53} + 27 \beta_{1} q^{54} + ( 350 - 19 \beta_{1} - 25 \beta_{2} ) q^{55} + ( 195 + 21 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -430 + 41 \beta_{1} - 25 \beta_{2} ) q^{58} + ( -358 - 69 \beta_{1} + 29 \beta_{2} ) q^{59} + ( -450 + 33 \beta_{1} - 9 \beta_{2} ) q^{60} + ( -10 + 100 \beta_{1} - 20 \beta_{2} ) q^{61} + ( -364 + 33 \beta_{1} - 22 \beta_{2} ) q^{62} + ( -194 + 183 \beta_{1} + 21 \beta_{2} ) q^{64} + ( -174 + 50 \beta_{1} + 18 \beta_{2} ) q^{65} + ( 6 - 39 \beta_{1} - 3 \beta_{2} ) q^{66} + ( -215 + 17 \beta_{1} + 47 \beta_{2} ) q^{67} + ( -640 - 24 \beta_{1} - 16 \beta_{2} ) q^{68} + ( 240 - 72 \beta_{1} - 12 \beta_{2} ) q^{69} + ( 66 + 120 \beta_{1} + 12 \beta_{2} ) q^{71} + ( 90 + 81 \beta_{1} + 9 \beta_{2} ) q^{72} + ( -363 + 101 \beta_{1} + 23 \beta_{2} ) q^{73} + ( 298 + 108 \beta_{1} + 19 \beta_{2} ) q^{74} + ( 159 - 87 \beta_{1} + 3 \beta_{2} ) q^{75} + ( 718 + 186 \beta_{1} + 72 \beta_{2} ) q^{76} + ( -258 - 48 \beta_{1} - 15 \beta_{2} ) q^{78} + ( 299 - 36 \beta_{1} + 48 \beta_{2} ) q^{79} + ( -18 - 121 \beta_{1} - 51 \beta_{2} ) q^{80} + 81 q^{81} + ( -52 + 32 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -156 - 51 \beta_{1} + 27 \beta_{2} ) q^{83} + ( 624 - 72 \beta_{1} ) q^{85} + ( -1086 + 50 \beta_{1} - 69 \beta_{2} ) q^{86} + ( 78 - 75 \beta_{1} + 15 \beta_{2} ) q^{87} + ( -258 - 117 \beta_{1} - 3 \beta_{2} ) q^{88} + ( -532 - 170 \beta_{1} - 22 \beta_{2} ) q^{89} + ( 198 - 99 \beta_{1} + 9 \beta_{2} ) q^{90} + ( -112 - 336 \beta_{1} + 56 \beta_{2} ) q^{92} + ( 117 - 66 \beta_{1} + 6 \beta_{2} ) q^{93} + ( 984 + 342 \beta_{1} + 72 \beta_{2} ) q^{94} + ( -246 + 2 \beta_{1} - 54 \beta_{2} ) q^{95} + ( 654 + 87 \beta_{1} + 33 \beta_{2} ) q^{96} + ( -24 + 53 \beta_{1} - 49 \beta_{2} ) q^{97} + ( 108 - 9 \beta_{1} - 27 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + 9q^{3} + 25q^{4} - 11q^{5} + 3q^{6} + 39q^{8} + 27q^{9} + O(q^{10})$$ $$3q + q^{2} + 9q^{3} + 25q^{4} - 11q^{5} + 3q^{6} + 39q^{8} + 27q^{9} + 55q^{10} + 35q^{11} + 75q^{12} - 62q^{13} - 33q^{15} + 241q^{16} - 48q^{17} + 9q^{18} + 202q^{19} - 439q^{20} - 7q^{22} + 216q^{23} + 117q^{24} + 130q^{25} - 274q^{26} + 81q^{27} + 53q^{29} + 165q^{30} + 95q^{31} + 683q^{32} + 105q^{33} + 24q^{34} + 225q^{36} + 262q^{37} + 398q^{38} - 186q^{39} - 21q^{40} - 244q^{41} + 360q^{43} - 905q^{44} - 99q^{45} - 1056q^{46} + 210q^{47} + 723q^{48} - 1378q^{50} - 144q^{51} - 324q^{52} + 393q^{53} + 27q^{54} + 1031q^{55} + 606q^{57} - 1249q^{58} - 1143q^{59} - 1317q^{60} + 70q^{61} - 1059q^{62} - 399q^{64} - 472q^{65} - 21q^{66} - 628q^{67} - 1944q^{68} + 648q^{69} + 318q^{71} + 351q^{72} - 988q^{73} + 1002q^{74} + 390q^{75} + 2340q^{76} - 822q^{78} + 861q^{79} - 175q^{80} + 243q^{81} - 124q^{82} - 519q^{83} + 1800q^{85} - 3208q^{86} + 159q^{87} - 891q^{88} - 1766q^{89} + 495q^{90} - 672q^{92} + 285q^{93} + 3294q^{94} - 736q^{95} + 2049q^{96} - 19q^{97} + 315q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 24 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 16$$

## 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
 −4.55637 0.248072 5.30829
−4.55637 3.00000 12.7605 −17.8732 −13.6691 0 −21.6905 9.00000 81.4369
1.2 0.248072 3.00000 −7.93846 12.4346 0.744216 0 −3.95388 9.00000 3.08468
1.3 5.30829 3.00000 20.1780 −5.56140 15.9249 0 64.6443 9.00000 −29.5215
 $$n$$: 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.m 3
3.b odd 2 1 441.4.a.t 3
4.b odd 2 1 2352.4.a.cg 3
7.b odd 2 1 147.4.a.l 3
7.c even 3 2 147.4.e.n 6
7.d odd 6 2 21.4.e.b 6
21.c even 2 1 441.4.a.s 3
21.g even 6 2 63.4.e.c 6
21.h odd 6 2 441.4.e.w 6
28.d even 2 1 2352.4.a.ci 3
28.f even 6 2 336.4.q.k 6

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

 $$T_{2}^{3} - T_{2}^{2} - 24 T_{2} + 6$$ $$T_{5}^{3} + 11 T_{5}^{2} - 192 T_{5} - 1236$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6 - 24 T - T^{2} + T^{3}$$
$3$ $$( -3 + T )^{3}$$
$5$ $$-1236 - 192 T + 11 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-9564 - 1368 T - 35 T^{2} + T^{3}$$
$13$ $$-18452 + 425 T + 62 T^{2} + T^{3}$$
$17$ $$-112896 - 2400 T + 48 T^{2} + T^{3}$$
$19$ $$-233804 + 12281 T - 202 T^{2} + T^{3}$$
$23$ $$1580544 - 672 T - 216 T^{2} + T^{3}$$
$29$ $$-824976 - 20472 T - 53 T^{2} + T^{3}$$
$31$ $$11823 - 10001 T - 95 T^{2} + T^{3}$$
$37$ $$-49152 + 14089 T - 262 T^{2} + T^{3}$$
$41$ $$300384 - 18780 T + 244 T^{2} + T^{3}$$
$43$ $$18269746 - 72363 T - 360 T^{2} + T^{3}$$
$47$ $$-5119128 - 246516 T - 210 T^{2} + T^{3}$$
$53$ $$33169392 - 80736 T - 393 T^{2} + T^{3}$$
$59$ $$-100468944 + 133104 T + 1143 T^{2} + T^{3}$$
$61$ $$84631000 - 340900 T - 70 T^{2} + T^{3}$$
$67$ $$27993002 - 304963 T + 628 T^{2} + T^{3}$$
$71$ $$-28535976 - 330804 T - 318 T^{2} + T^{3}$$
$73$ $$-143207118 - 4355 T + 988 T^{2} + T^{3}$$
$79$ $$193956337 - 257901 T - 861 T^{2} + T^{3}$$
$83$ $$-47916036 - 131616 T + 519 T^{2} + T^{3}$$
$89$ $$-13004544 + 277920 T + 1766 T^{2} + T^{3}$$
$97$ $$-44776452 - 569600 T + 19 T^{2} + T^{3}$$