# Properties

 Label 147.6.a.l Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 97 x^{2} + 7 x + 294$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -9 q^{3} + ( 18 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -9 + 9 \beta_{1} ) q^{6} + ( 38 - 27 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{8} + 81 q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -9 q^{3} + ( 18 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -9 + 9 \beta_{1} ) q^{6} + ( 38 - 27 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{8} + 81 q^{9} + ( -76 + 23 \beta_{1} + \beta_{2} ) q^{10} + ( 107 - 8 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{11} + ( -162 + 9 \beta_{1} - 9 \beta_{2} ) q^{12} + ( -96 - 76 \beta_{1} + \beta_{2} - 9 \beta_{3} ) q^{13} + ( 9 - 18 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{15} + ( 840 - 85 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{16} + ( -92 + 108 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 81 - 81 \beta_{1} ) q^{18} + ( -72 - 220 \beta_{1} + \beta_{2} + 15 \beta_{3} ) q^{19} + ( -1168 - 29 \beta_{1} + 9 \beta_{2} + 30 \beta_{3} ) q^{20} + ( 448 - 419 \beta_{1} - \beta_{2} - 12 \beta_{3} ) q^{22} + ( 1804 - 300 \beta_{1} + 8 \beta_{2} + 20 \beta_{3} ) q^{23} + ( -342 + 243 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{24} + ( 636 + 80 \beta_{1} - 95 \beta_{2} - 45 \beta_{3} ) q^{25} + ( 3865 - 116 \beta_{1} + 103 \beta_{2} - 20 \beta_{3} ) q^{26} -729 q^{27} + ( 147 - 254 \beta_{1} - 103 \beta_{2} + 5 \beta_{3} ) q^{29} + ( 684 - 207 \beta_{1} - 9 \beta_{2} ) q^{30} + ( 1523 + 130 \beta_{1} - 94 \beta_{2} - 24 \beta_{3} ) q^{31} + ( 3948 - 131 \beta_{1} + 71 \beta_{2} + 50 \beta_{3} ) q^{32} + ( -963 + 72 \beta_{1} - 81 \beta_{2} - 27 \beta_{3} ) q^{33} + ( -5256 - 312 \beta_{1} - 96 \beta_{2} - 24 \beta_{3} ) q^{34} + ( 1458 - 81 \beta_{1} + 81 \beta_{2} ) q^{36} + ( 3508 + 1028 \beta_{1} - 95 \beta_{2} - 9 \beta_{3} ) q^{37} + ( 10321 + 316 \beta_{1} + 175 \beta_{2} + 28 \beta_{3} ) q^{38} + ( 864 + 684 \beta_{1} - 9 \beta_{2} + 81 \beta_{3} ) q^{39} + ( 1932 + 633 \beta_{1} - 93 \beta_{2} + 42 \beta_{3} ) q^{40} + ( -1128 + 328 \beta_{1} + 62 \beta_{2} - 142 \beta_{3} ) q^{41} + ( 7142 + 816 \beta_{1} + 93 \beta_{2} - 33 \beta_{3} ) q^{43} + ( 17864 - 379 \beta_{1} + 167 \beta_{2} - 118 \beta_{3} ) q^{44} + ( -81 + 162 \beta_{1} - 81 \beta_{2} - 81 \beta_{3} ) q^{45} + ( 16008 - 1752 \beta_{1} + 240 \beta_{2} + 24 \beta_{3} ) q^{46} + ( 3818 - 324 \beta_{1} - 56 \beta_{2} + 28 \beta_{3} ) q^{47} + ( -7560 + 765 \beta_{1} - 9 \beta_{2} + 54 \beta_{3} ) q^{48} + ( -2399 + 2404 \beta_{1} + 55 \beta_{2} + 100 \beta_{3} ) q^{50} + ( 828 - 972 \beta_{1} - 72 \beta_{2} + 36 \beta_{3} ) q^{51} + ( 13450 - 6036 \beta_{1} + 144 \beta_{2} + 42 \beta_{3} ) q^{52} + ( 3095 + 1338 \beta_{1} + 13 \beta_{2} - 239 \beta_{3} ) q^{53} + ( -729 + 729 \beta_{1} ) q^{54} + ( -17987 - 506 \beta_{1} + 335 \beta_{2} + 315 \beta_{3} ) q^{55} + ( 648 + 1980 \beta_{1} - 9 \beta_{2} - 135 \beta_{3} ) q^{57} + ( 12154 + 4171 \beta_{1} + 239 \beta_{2} + 216 \beta_{3} ) q^{58} + ( -9011 + 888 \beta_{1} - 163 \beta_{2} + 71 \beta_{3} ) q^{59} + ( 10512 + 261 \beta_{1} - 81 \beta_{2} - 270 \beta_{3} ) q^{60} + ( 1566 - 796 \beta_{1} + 52 \beta_{2} - 240 \beta_{3} ) q^{61} + ( -4505 + 1875 \beta_{1} - 58 \beta_{2} + 140 \beta_{3} ) q^{62} + ( -17600 - 3189 \beta_{1} - 51 \beta_{2} + 150 \beta_{3} ) q^{64} + ( 16136 + 1660 \beta_{1} - 330 \beta_{2} + 246 \beta_{3} ) q^{65} + ( -4032 + 3771 \beta_{1} + 9 \beta_{2} + 108 \beta_{3} ) q^{66} + ( -2286 + 2764 \beta_{1} - 193 \beta_{2} + 465 \beta_{3} ) q^{67} + ( 13312 + 5280 \beta_{1} + 128 \beta_{2} + 272 \beta_{3} ) q^{68} + ( -16236 + 2700 \beta_{1} - 72 \beta_{2} - 180 \beta_{3} ) q^{69} + ( 21390 + 3660 \beta_{1} - 24 \beta_{2} - 180 \beta_{3} ) q^{71} + ( 3078 - 2187 \beta_{1} + 81 \beta_{2} - 162 \beta_{3} ) q^{72} + ( -14074 - 3056 \beta_{1} + 143 \beta_{2} + 93 \beta_{3} ) q^{73} + ( -46915 + 216 \beta_{1} - 1001 \beta_{2} + 172 \beta_{3} ) q^{74} + ( -5724 - 720 \beta_{1} + 855 \beta_{2} + 405 \beta_{3} ) q^{75} + ( -3062 - 9924 \beta_{1} - 432 \beta_{2} - 774 \beta_{3} ) q^{76} + ( -34785 + 1044 \beta_{1} - 927 \beta_{2} + 180 \beta_{3} ) q^{78} + ( -11635 + 2286 \beta_{1} - 96 \beta_{2} - 786 \beta_{3} ) q^{79} + ( 6920 + 3607 \beta_{1} - 1047 \beta_{2} - 690 \beta_{3} ) q^{80} + 6561 q^{81} + ( -13322 - 4112 \beta_{1} + 98 \beta_{2} - 408 \beta_{3} ) q^{82} + ( 50001 + 6432 \beta_{1} - 1005 \beta_{2} + 129 \beta_{3} ) q^{83} + ( 9732 - 2988 \beta_{1} - 192 \beta_{2} + 372 \beta_{3} ) q^{85} + ( -31705 - 11582 \beta_{1} - 717 \beta_{2} - 252 \beta_{3} ) q^{86} + ( -1323 + 2286 \beta_{1} + 927 \beta_{2} - 45 \beta_{3} ) q^{87} + ( 25668 - 13545 \beta_{1} + 765 \beta_{2} - 186 \beta_{3} ) q^{88} + ( 20966 - 9508 \beta_{1} - 1582 \beta_{2} - 622 \beta_{3} ) q^{89} + ( -6156 + 1863 \beta_{1} + 81 \beta_{2} ) q^{90} + ( 44224 - 15792 \beta_{1} + 1424 \beta_{2} - 1072 \beta_{3} ) q^{92} + ( -13707 - 1170 \beta_{1} + 846 \beta_{2} + 216 \beta_{3} ) q^{93} + ( 18798 - 990 \beta_{1} + 240 \beta_{2} + 168 \beta_{3} ) q^{94} + ( -55528 + 3820 \beta_{1} + 1542 \beta_{2} + 294 \beta_{3} ) q^{95} + ( -35532 + 1179 \beta_{1} - 639 \beta_{2} - 450 \beta_{3} ) q^{96} + ( 47705 - 464 \beta_{1} + 863 \beta_{2} + 669 \beta_{3} ) q^{97} + ( 8667 - 648 \beta_{1} + 729 \beta_{2} + 243 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} - 36q^{3} + 69q^{4} - 27q^{6} + 123q^{8} + 324q^{9} + O(q^{10})$$ $$4q + 3q^{2} - 36q^{3} + 69q^{4} - 27q^{6} + 123q^{8} + 324q^{9} - 283q^{10} + 402q^{11} - 621q^{12} - 462q^{13} + 3273q^{16} - 276q^{17} + 243q^{18} - 510q^{19} - 4719q^{20} + 1375q^{22} + 6900q^{23} - 1107q^{24} + 2814q^{25} + 15138q^{26} - 2916q^{27} + 540q^{29} + 2547q^{30} + 6410q^{31} + 15519q^{32} - 3618q^{33} - 21144q^{34} + 5589q^{36} + 15250q^{37} + 41250q^{38} + 4158q^{39} + 8547q^{40} - 4308q^{41} + 29198q^{43} + 70743q^{44} + 61800q^{46} + 15060q^{47} - 29457q^{48} - 7302q^{50} + 2484q^{51} + 47476q^{52} + 13692q^{53} - 2187q^{54} - 73124q^{55} + 4590q^{57} + 52309q^{58} - 34830q^{59} + 42471q^{60} + 5364q^{61} - 16029q^{62} - 73487q^{64} + 66864q^{65} - 12375q^{66} - 5994q^{67} + 58272q^{68} - 62100q^{69} + 89268q^{71} + 9963q^{72} - 59638q^{73} - 185442q^{74} - 25326q^{75} - 21308q^{76} - 136242q^{78} - 44062q^{79} + 33381q^{80} + 26244q^{81} - 57596q^{82} + 208446q^{83} + 36324q^{85} - 136968q^{86} - 4860q^{87} + 87597q^{88} + 77520q^{89} - 22923q^{90} + 158256q^{92} - 57690q^{93} + 73722q^{94} - 221376q^{95} - 139671q^{96} + 188630q^{97} + 32562q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 97 x^{2} + 7 x + 294$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 49$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} - 89 \nu + 52$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 49$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 91 \beta_{1} + 46$$

## 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
 10.1812 1.79080 −1.74818 −9.22385
−9.18123 −9.00000 52.2950 −22.0716 82.6311 0 −186.333 81.0000 202.644
1.2 −0.790805 −9.00000 −31.3746 104.192 7.11724 0 50.1170 81.0000 −82.3953
1.3 2.74818 −9.00000 −24.4475 −58.3673 −24.7336 0 −155.128 81.0000 −160.404
1.4 10.2239 −9.00000 72.5272 −23.7528 −92.0147 0 414.344 81.0000 −242.845
 $$n$$: 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 147.6.a.l 4
3.b odd 2 1 441.6.a.v 4
7.b odd 2 1 147.6.a.m 4
7.c even 3 2 147.6.e.o 8
7.d odd 6 2 21.6.e.c 8
21.c even 2 1 441.6.a.w 4
21.g even 6 2 63.6.e.e 8
28.f even 6 2 336.6.q.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 7.d odd 6 2
63.6.e.e 8 21.g even 6 2
147.6.a.l 4 1.a even 1 1 trivial
147.6.a.m 4 7.b odd 2 1
147.6.e.o 8 7.c even 3 2
336.6.q.j 8 28.f even 6 2
441.6.a.v 4 3.b odd 2 1
441.6.a.w 4 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2}^{4} - 3 T_{2}^{3} - 94 T_{2}^{2} + 186 T_{2} + 204$$ $$T_{5}^{4} - 7657 T_{5}^{2} - 302700 T_{5} - 3188244$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$204 + 186 T - 94 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$( 9 + T )^{4}$$
$5$ $$-3188244 - 302700 T - 7657 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$-1682132124 + 100810572 T - 238363 T^{2} - 402 T^{3} + T^{4}$$
$13$ $$149501563456 - 515112852 T - 1148423 T^{2} + 462 T^{3} + T^{4}$$
$17$ $$-50104147968 - 894878208 T - 1594752 T^{2} + 276 T^{3} + T^{4}$$
$19$ $$7391138416576 - 797823780 T - 5896871 T^{2} + 510 T^{3} + T^{4}$$
$23$ $$3007939608576 + 17122936320 T + 7121856 T^{2} - 6900 T^{3} + T^{4}$$
$29$ $$408027025117872 + 62527747272 T - 52650397 T^{2} - 540 T^{3} + T^{4}$$
$31$ $$86716089209547 + 58618529034 T - 17713868 T^{2} - 6410 T^{3} + T^{4}$$
$37$ $$-5042926288839456 + 1497067500180 T - 46856207 T^{2} - 15250 T^{3} + T^{4}$$
$41$ $$-1856858915261952 + 1101575496480 T - 192741244 T^{2} + 4308 T^{3} + T^{4}$$
$43$ $$-991662745581932 - 199921376588 T + 199961493 T^{2} - 29198 T^{3} + T^{4}$$
$47$ $$-270685655359056 + 44937987408 T + 50649744 T^{2} - 15060 T^{3} + T^{4}$$
$53$ $$-8505482723267472 + 8038879393320 T - 497907429 T^{2} - 13692 T^{3} + T^{4}$$
$59$ $$-2578852214901936 - 461620404360 T + 188256441 T^{2} + 34830 T^{3} + T^{4}$$
$61$ $$17942190625624624 + 3379722031440 T - 532596176 T^{2} - 5364 T^{3} + T^{4}$$
$67$ $$550087288501666684 + 4941755739000 T - 2845994891 T^{2} + 5994 T^{3} + T^{4}$$
$71$ $$21932335650275568 + 16377596837712 T + 1521744768 T^{2} - 89268 T^{3} + T^{4}$$
$73$ $$-122130292613870700 - 20327373037020 T + 362940181 T^{2} + 59638 T^{3} + T^{4}$$
$79$ $$165231063841623259 + 14857064631634 T - 3781804908 T^{2} + 44062 T^{3} + T^{4}$$
$83$ $$-41533908097096407132 + 738604511000820 T + 7607249829 T^{2} - 208446 T^{3} + T^{4}$$
$89$ $$-47322044296216531968 + 1866206095720704 T - 16806213508 T^{2} - 77520 T^{3} + T^{4}$$
$97$ $$-11638556269792123644 + 99054118022220 T + 9271508101 T^{2} - 188630 T^{3} + T^{4}$$