# Properties

 Label 147.3.b.f Level $147$ Weight $3$ Character orbit 147.b Analytic conductor $4.005$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 147.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.00545988610$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.65856.1 Defining polynomial: $$x^{4} + 14 x^{2} + 21$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -3 + 2 \beta_{2} ) q^{4} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - 5 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{8} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -3 + 2 \beta_{2} ) q^{4} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - 5 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{8} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{9} + ( -7 - \beta_{2} ) q^{10} + 2 \beta_{1} q^{11} + ( 6 - 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( 9 - \beta_{2} ) q^{13} + ( 9 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{15} + ( 9 - 4 \beta_{2} ) q^{16} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 15 - 8 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{18} + ( -3 + 5 \beta_{2} ) q^{19} + ( 3 - 4 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{20} + ( -14 + 4 \beta_{2} ) q^{22} + ( -2 + 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{23} + ( 33 + 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{24} + ( -3 - 10 \beta_{2} ) q^{25} + ( -1 + 12 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{26} + ( -3 - 5 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{29} + ( -3 - 5 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{30} + ( -34 - 2 \beta_{2} ) q^{31} + ( 4 + \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{32} + ( -6 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{33} + 6 \beta_{2} q^{34} + ( 24 + 13 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} ) q^{36} + ( 4 + 14 \beta_{2} ) q^{37} + ( 5 - 18 \beta_{1} - 5 \beta_{2} + 10 \beta_{3} ) q^{38} + ( -3 + 11 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{39} + ( -21 - 15 \beta_{2} ) q^{40} + ( -8 + 22 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{41} + ( -40 - 6 \beta_{2} ) q^{43} + ( 4 - 18 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{44} + ( 33 - 2 \beta_{1} + 13 \beta_{2} - 4 \beta_{3} ) q^{45} + ( -42 + 18 \beta_{2} ) q^{46} + ( -4 - 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{47} + ( -12 + 17 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{48} + ( -10 + 27 \beta_{1} + 10 \beta_{2} - 20 \beta_{3} ) q^{50} + ( -24 - 6 \beta_{3} ) q^{51} + ( -41 + 21 \beta_{2} ) q^{52} + ( -16 + 10 \beta_{1} + 16 \beta_{2} - 32 \beta_{3} ) q^{53} + ( 30 + 23 \beta_{1} - \beta_{2} - 17 \beta_{3} ) q^{54} + ( -14 - 2 \beta_{2} ) q^{55} + ( 15 - 13 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{57} + ( 28 - 2 \beta_{2} ) q^{58} + ( 1 + 26 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} + ( 39 + \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{60} + ( 39 - 7 \beta_{2} ) q^{61} + ( -2 - 28 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{62} + ( 1 - 18 \beta_{2} ) q^{64} + ( 6 + 2 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{65} + ( 12 - 22 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{66} + ( -6 - 8 \beta_{2} ) q^{67} + ( -2 - 10 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -42 - 6 \beta_{1} + 12 \beta_{2} ) q^{69} + ( 6 + 18 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{71} + ( -9 + 36 \beta_{1} + 36 \beta_{2} - 18 \beta_{3} ) q^{72} + ( 8 - 26 \beta_{2} ) q^{73} + ( 14 - 38 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} ) q^{74} + ( -30 + 17 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} ) q^{75} + ( 79 - 21 \beta_{2} ) q^{76} + ( -45 - 11 \beta_{1} + 22 \beta_{2} + 8 \beta_{3} ) q^{78} + ( 32 - 36 \beta_{2} ) q^{79} + ( -3 + 8 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{80} + ( -15 + 20 \beta_{1} - 22 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -98 + 52 \beta_{2} ) q^{82} + ( 9 + 18 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{83} + ( 42 + 18 \beta_{2} ) q^{85} + ( -6 - 22 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{86} + ( -12 + 4 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} ) q^{87} + ( 42 - 24 \beta_{2} ) q^{88} + ( 16 - 42 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} ) q^{89} + ( 39 + 2 \beta_{1} - 13 \beta_{2} + 22 \beta_{3} ) q^{90} + ( 10 - 64 \beta_{1} - 10 \beta_{2} + 20 \beta_{3} ) q^{92} + ( -6 - 30 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{93} + ( 84 - 12 \beta_{2} ) q^{94} + ( 12 - 10 \beta_{1} - 12 \beta_{2} + 24 \beta_{3} ) q^{95} + ( 33 - 5 \beta_{1} + 10 \beta_{2} + 17 \beta_{3} ) q^{96} + ( 2 - 8 \beta_{2} ) q^{97} + ( 30 - 16 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 12q^{4} - 14q^{6} - 20q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 12q^{4} - 14q^{6} - 20q^{9} - 28q^{10} + 22q^{12} + 36q^{13} + 28q^{15} + 36q^{16} + 56q^{18} - 12q^{19} - 56q^{22} + 126q^{24} - 12q^{25} - 10q^{27} - 28q^{30} - 136q^{31} - 28q^{33} + 116q^{36} + 16q^{37} + 4q^{39} - 84q^{40} - 160q^{43} + 140q^{45} - 168q^{46} - 38q^{48} - 84q^{51} - 164q^{52} + 154q^{54} - 56q^{55} + 64q^{57} + 112q^{58} + 140q^{60} + 156q^{61} + 4q^{64} + 28q^{66} - 24q^{67} - 168q^{69} + 32q^{73} - 146q^{75} + 316q^{76} - 196q^{78} + 128q^{79} - 68q^{81} - 392q^{82} + 168q^{85} - 28q^{87} + 168q^{88} + 112q^{90} - 96q^{93} + 336q^{94} + 98q^{96} + 8q^{97} + 112q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 14 x^{2} + 21$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + 7$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 13 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} - 7$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} - 2 \beta_{2} - 13 \beta_{1} + 2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
50.1
 − 3.50592i − 1.30710i 1.30710i 3.50592i
3.50592i −0.822876 2.88494i −8.29150 1.24197i −10.1144 + 2.88494i 0 15.0457i −7.64575 + 4.74789i −4.35425
50.2 1.30710i 1.82288 + 2.38267i 2.29150 7.37953i 3.11438 2.38267i 0 8.22359i −2.35425 + 8.68663i −9.64575
50.3 1.30710i 1.82288 2.38267i 2.29150 7.37953i 3.11438 + 2.38267i 0 8.22359i −2.35425 8.68663i −9.64575
50.4 3.50592i −0.822876 + 2.88494i −8.29150 1.24197i −10.1144 2.88494i 0 15.0457i −7.64575 4.74789i −4.35425
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.b.f 4
3.b odd 2 1 inner 147.3.b.f 4
7.b odd 2 1 21.3.b.a 4
7.c even 3 2 147.3.h.c 8
7.d odd 6 2 147.3.h.e 8
21.c even 2 1 21.3.b.a 4
21.g even 6 2 147.3.h.e 8
21.h odd 6 2 147.3.h.c 8
28.d even 2 1 336.3.d.c 4
35.c odd 2 1 525.3.c.a 4
35.f even 4 2 525.3.f.a 8
56.e even 2 1 1344.3.d.b 4
56.h odd 2 1 1344.3.d.f 4
63.l odd 6 2 567.3.r.c 8
63.o even 6 2 567.3.r.c 8
84.h odd 2 1 336.3.d.c 4
105.g even 2 1 525.3.c.a 4
105.k odd 4 2 525.3.f.a 8
168.e odd 2 1 1344.3.d.b 4
168.i even 2 1 1344.3.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a 4 7.b odd 2 1
21.3.b.a 4 21.c even 2 1
147.3.b.f 4 1.a even 1 1 trivial
147.3.b.f 4 3.b odd 2 1 inner
147.3.h.c 8 7.c even 3 2
147.3.h.c 8 21.h odd 6 2
147.3.h.e 8 7.d odd 6 2
147.3.h.e 8 21.g even 6 2
336.3.d.c 4 28.d even 2 1
336.3.d.c 4 84.h odd 2 1
525.3.c.a 4 35.c odd 2 1
525.3.c.a 4 105.g even 2 1
525.3.f.a 8 35.f even 4 2
525.3.f.a 8 105.k odd 4 2
567.3.r.c 8 63.l odd 6 2
567.3.r.c 8 63.o even 6 2
1344.3.d.b 4 56.e even 2 1
1344.3.d.b 4 168.e odd 2 1
1344.3.d.f 4 56.h odd 2 1
1344.3.d.f 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{4} + 14 T_{2}^{2} + 21$$ $$T_{5}^{4} + 56 T_{5}^{2} + 84$$ $$T_{13}^{2} - 18 T_{13} + 74$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 5 T^{4} - 32 T^{6} + 256 T^{8}$$
$3$ $$1 - 2 T + 12 T^{2} - 18 T^{3} + 81 T^{4}$$
$5$ $$1 - 44 T^{2} + 1034 T^{4} - 27500 T^{6} + 390625 T^{8}$$
$7$ 1
$11$ $$1 - 428 T^{2} + 74630 T^{4} - 6266348 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 - 18 T + 412 T^{2} - 3042 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$1 - 988 T^{2} + 407046 T^{4} - 82518748 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + 6 T + 556 T^{2} + 2166 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 - 1444 T^{2} + 980166 T^{4} - 404090404 T^{6} + 78310985281 T^{8}$$
$29$ $$1 - 2972 T^{2} + 3611558 T^{4} - 2102039132 T^{6} + 500246412961 T^{8}$$
$31$ $$( 1 + 68 T + 3050 T^{2} + 65348 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - 8 T + 1382 T^{2} - 10952 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 1292 T^{2} + 2832038 T^{4} - 3650883212 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 + 80 T + 5046 T^{2} + 147920 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 6148 T^{2} + 19144326 T^{4} - 30000278788 T^{6} + 23811286661761 T^{8}$$
$53$ $$1 + 20 T^{2} - 13350138 T^{4} + 157809620 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 - 3676 T^{2} + 15964266 T^{4} - 44543419036 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 - 78 T + 8620 T^{2} - 290238 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 + 12 T + 8566 T^{2} + 53868 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 - 16 T + 5990 T^{2} - 85264 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 - 64 T + 4434 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 13948 T^{2} + 141899946 T^{4} - 661948661308 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 - 11468 T^{2} + 120945830 T^{4} - 719528019788 T^{6} + 3936588805702081 T^{8}$$
$97$ $$( 1 - 4 T + 18374 T^{2} - 37636 T^{3} + 88529281 T^{4} )^{2}$$
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