# Properties

 Label 147.4.a.l 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 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

## 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.l 3
3.b odd 2 1 441.4.a.s 3
4.b odd 2 1 2352.4.a.ci 3
7.b odd 2 1 147.4.a.m 3
7.c even 3 2 21.4.e.b 6
7.d odd 6 2 147.4.e.n 6
21.c even 2 1 441.4.a.t 3
21.g even 6 2 441.4.e.w 6
21.h odd 6 2 63.4.e.c 6
28.d even 2 1 2352.4.a.cg 3
28.g odd 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.c even 3 2
63.4.e.c 6 21.h odd 6 2
147.4.a.l 3 1.a even 1 1 trivial
147.4.a.m 3 7.b odd 2 1
147.4.e.n 6 7.d odd 6 2
336.4.q.k 6 28.g odd 6 2
441.4.a.s 3 3.b odd 2 1
441.4.a.t 3 21.c even 2 1
441.4.e.w 6 21.g even 6 2
2352.4.a.cg 3 28.d even 2 1
2352.4.a.ci 3 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$

## 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

$p$ $F_p(T)$
$2$ $$1 - T - 10 T^{3} - 64 T^{5} + 512 T^{6}$$
$3$ $$( 1 + 3 T )^{3}$$
$5$ $$1 - 11 T + 183 T^{2} - 1514 T^{3} + 22875 T^{4} - 171875 T^{5} + 1953125 T^{6}$$
$7$ 1
$11$ $$1 - 35 T + 2625 T^{2} - 102734 T^{3} + 3493875 T^{4} - 62004635 T^{5} + 2357947691 T^{6}$$
$13$ $$1 - 62 T + 7016 T^{2} - 253976 T^{3} + 15414152 T^{4} - 299262158 T^{5} + 10604499373 T^{6}$$
$17$ $$1 - 48 T + 12339 T^{2} - 358752 T^{3} + 60621507 T^{4} - 1158603312 T^{5} + 118587876497 T^{6}$$
$19$ $$1 + 202 T + 32858 T^{2} + 3004840 T^{3} + 225373022 T^{4} + 9503267962 T^{5} + 322687697779 T^{6}$$
$23$ $$1 - 216 T + 35829 T^{2} - 3675600 T^{3} + 435931443 T^{4} - 31975752024 T^{5} + 1801152661463 T^{6}$$
$29$ $$1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 1285178355 T^{4} - 31525636013 T^{5} + 14507145975869 T^{6}$$
$31$ $$1 + 95 T + 79372 T^{2} + 5648467 T^{3} + 2364571252 T^{4} + 84312849695 T^{5} + 26439622160671 T^{6}$$
$37$ $$1 - 262 T + 166048 T^{2} - 26591324 T^{3} + 8410829344 T^{4} - 672220319158 T^{5} + 129961739795077 T^{6}$$
$41$ $$1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 12955976343 T^{4} - 1159025434804 T^{5} + 327381934393961 T^{6}$$
$43$ $$1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 13210724106 T^{4} - 2275690697640 T^{5} + 502592611936843 T^{6}$$
$47$ $$1 + 210 T + 64953 T^{2} + 48724788 T^{3} + 6743615319 T^{4} + 2263635219090 T^{5} + 1119130473102767 T^{6}$$
$53$ $$1 - 393 T + 365895 T^{2} - 83847930 T^{3} + 54473349915 T^{4} - 8710593923697 T^{5} + 3299763591802133 T^{6}$$
$59$ $$1 - 1143 T + 749241 T^{2} - 369027450 T^{3} + 153878367339 T^{4} - 48212349951663 T^{5} + 8662995818654939 T^{6}$$
$61$ $$1 + 70 T + 340043 T^{2} - 52853660 T^{3} + 77183300183 T^{4} + 3606426205270 T^{5} + 11694146092834141 T^{6}$$
$67$ $$1 + 628 T + 597326 T^{2} + 405751330 T^{3} + 179653559738 T^{4} + 56807864002132 T^{5} + 27206534396294947 T^{6}$$
$71$ $$1 - 318 T + 742929 T^{2} - 256167372 T^{3} + 265902461319 T^{4} - 40735890286878 T^{5} + 45848500718449031 T^{6}$$
$73$ $$1 - 988 T + 1162696 T^{2} - 625490474 T^{3} + 452308509832 T^{4} - 149518215573532 T^{5} + 58871586708267913 T^{6}$$
$79$ $$1 - 861 T + 1221216 T^{2} - 655056821 T^{3} + 602107115424 T^{4} - 209298299203581 T^{5} + 119851595982618319 T^{6}$$
$83$ $$1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 905564802315 T^{4} - 169682053778511 T^{5} + 186940255267540403 T^{6}$$
$89$ $$1 - 1766 T + 2392827 T^{2} - 2476945964 T^{3} + 1686868857363 T^{4} - 877668959837126 T^{5} + 350356403707485209 T^{6}$$
$97$ $$1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 1979057473987 T^{4} - 15826468093651 T^{5} + 760231058654565217 T^{6}$$