# Properties

 Label 147.4.e.m Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} -3 \beta_{1} q^{3} + ( 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{5} + ( 3 + 3 \beta_{2} ) q^{6} + ( -41 - 5 \beta_{2} ) q^{8} + ( -9 - 9 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} -3 \beta_{1} q^{3} + ( 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{5} + ( 3 + 3 \beta_{2} ) q^{6} + ( -41 - 5 \beta_{2} ) q^{8} + ( -9 - 9 \beta_{1} ) q^{9} + ( 30 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{10} + ( -8 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{11} + ( 21 + 21 \beta_{1} + 9 \beta_{3} ) q^{12} + ( -14 + 12 \beta_{2} ) q^{13} + ( 6 + 6 \beta_{2} ) q^{15} + ( -55 - 55 \beta_{1} - 27 \beta_{3} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{18} + ( 44 + 44 \beta_{1} - 24 \beta_{3} ) q^{19} + ( -98 - 26 \beta_{2} ) q^{20} + ( -132 - 12 \beta_{2} ) q^{22} + ( -20 - 20 \beta_{1} + 34 \beta_{3} ) q^{23} + ( 123 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{24} + ( -65 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{25} + ( 154 + 154 \beta_{1} + 10 \beta_{3} ) q^{26} -27 q^{27} + ( -138 + 24 \beta_{2} ) q^{29} + ( 90 + 90 \beta_{1} + 18 \beta_{3} ) q^{30} + ( 16 \beta_{1} + 72 \beta_{2} - 72 \beta_{3} ) q^{31} + ( -105 \beta_{1} + 69 \beta_{2} - 69 \beta_{3} ) q^{32} + ( -24 - 24 \beta_{1} + 30 \beta_{3} ) q^{33} + ( -30 - 6 \beta_{2} ) q^{34} + ( 63 + 27 \beta_{2} ) q^{36} + ( 106 + 106 \beta_{1} + 36 \beta_{3} ) q^{37} + ( -292 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{38} + ( 42 \beta_{1} + 36 \beta_{2} - 36 \beta_{3} ) q^{39} + ( -222 - 222 \beta_{1} - 102 \beta_{3} ) q^{40} + ( 210 + 30 \beta_{2} ) q^{41} + ( 212 - 48 \beta_{2} ) q^{43} + ( -364 - 364 \beta_{1} - 76 \beta_{3} ) q^{44} + ( -18 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} ) q^{45} + ( 456 \beta_{1} - 48 \beta_{2} + 48 \beta_{3} ) q^{46} + ( -40 - 40 \beta_{1} + 68 \beta_{3} ) q^{47} + ( -165 - 81 \beta_{2} ) q^{48} + ( -103 + 41 \beta_{2} ) q^{50} + ( 6 + 6 \beta_{1} + 6 \beta_{3} ) q^{51} + ( 406 \beta_{1} - 78 \beta_{2} + 78 \beta_{3} ) q^{52} + ( -554 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{53} + ( -27 - 27 \beta_{1} - 27 \beta_{3} ) q^{54} + ( -264 - 24 \beta_{2} ) q^{55} + ( 132 - 72 \beta_{2} ) q^{57} + ( 198 + 198 \beta_{1} - 90 \beta_{3} ) q^{58} + ( -460 \beta_{1} - 116 \beta_{2} + 116 \beta_{3} ) q^{59} + ( 294 \beta_{1} - 78 \beta_{2} + 78 \beta_{3} ) q^{60} + ( -250 - 250 \beta_{1} + 72 \beta_{3} ) q^{61} + ( 992 + 128 \beta_{2} ) q^{62} + ( 631 + 27 \beta_{2} ) q^{64} + ( 308 + 308 \beta_{1} + 20 \beta_{3} ) q^{65} + ( 396 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{66} + ( 20 \beta_{1} - 108 \beta_{2} + 108 \beta_{3} ) q^{67} + ( -98 - 98 \beta_{1} - 26 \beta_{3} ) q^{68} + ( -60 + 102 \beta_{2} ) q^{69} + ( 492 - 30 \beta_{2} ) q^{71} + ( 369 + 369 \beta_{1} + 45 \beta_{3} ) q^{72} + ( -530 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{73} + ( 610 \beta_{1} - 178 \beta_{2} + 178 \beta_{3} ) q^{74} + ( -195 - 195 \beta_{1} + 36 \beta_{3} ) q^{75} + ( 700 + 108 \beta_{2} ) q^{76} + ( 462 + 30 \beta_{2} ) q^{78} + ( 232 + 232 \beta_{1} + 108 \beta_{3} ) q^{79} + ( -866 \beta_{1} + 218 \beta_{2} - 218 \beta_{3} ) q^{80} + 81 \beta_{1} q^{81} + ( 630 + 630 \beta_{1} + 270 \beta_{3} ) q^{82} + ( -924 - 96 \beta_{2} ) q^{83} + ( -60 - 12 \beta_{2} ) q^{85} + ( -460 - 460 \beta_{1} + 116 \beta_{3} ) q^{86} + ( 414 \beta_{1} + 72 \beta_{2} - 72 \beta_{3} ) q^{87} + ( -372 \beta_{1} + 420 \beta_{2} - 420 \beta_{3} ) q^{88} + ( 254 + 254 \beta_{1} - 142 \beta_{3} ) q^{89} + ( 270 + 54 \beta_{2} ) q^{90} + ( -1288 - 280 \beta_{2} ) q^{92} + ( 48 + 48 \beta_{1} - 216 \beta_{3} ) q^{93} + ( 912 \beta_{1} - 96 \beta_{2} + 96 \beta_{3} ) q^{94} + ( -584 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{95} + ( -315 - 315 \beta_{1} - 207 \beta_{3} ) q^{96} + ( -266 - 276 \beta_{2} ) q^{97} + ( -72 + 90 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} + 6q^{3} - 17q^{4} + 6q^{5} + 18q^{6} - 174q^{8} - 18q^{9} + O(q^{10})$$ $$4q + 3q^{2} + 6q^{3} - 17q^{4} + 6q^{5} + 18q^{6} - 174q^{8} - 18q^{9} - 66q^{10} + 6q^{11} + 51q^{12} - 32q^{13} + 36q^{15} - 137q^{16} - 6q^{17} + 27q^{18} + 64q^{19} - 444q^{20} - 552q^{22} - 6q^{23} - 261q^{24} + 118q^{25} + 318q^{26} - 108q^{27} - 504q^{29} + 198q^{30} + 40q^{31} + 279q^{32} - 18q^{33} - 132q^{34} + 306q^{36} + 248q^{37} + 588q^{38} - 48q^{39} - 546q^{40} + 900q^{41} + 752q^{43} - 804q^{44} + 54q^{45} - 960q^{46} - 12q^{47} - 822q^{48} - 330q^{50} + 18q^{51} - 890q^{52} + 1104q^{53} - 81q^{54} - 1104q^{55} + 384q^{57} + 306q^{58} + 804q^{59} - 666q^{60} - 428q^{61} + 4224q^{62} + 2578q^{64} + 636q^{65} - 828q^{66} - 148q^{67} - 222q^{68} - 36q^{69} + 1908q^{71} + 783q^{72} + 1072q^{73} - 1398q^{74} - 354q^{75} + 3016q^{76} + 1908q^{78} + 572q^{79} + 1950q^{80} - 162q^{81} + 1530q^{82} - 3888q^{83} - 264q^{85} - 804q^{86} - 756q^{87} + 1164q^{88} + 366q^{89} + 1188q^{90} - 5712q^{92} - 120q^{93} - 1920q^{94} + 1176q^{95} - 837q^{96} - 1616q^{97} - 108q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 25$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 9 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{3} + 2 \nu^{2} + 8 \nu - 25$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} - 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 14 \beta_{1} + 13$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 19$$$$)/3$$

## Character values

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

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
−1.13746 1.97014i 1.50000 2.59808i 1.41238 2.44631i −2.27492 3.94027i −6.82475 0 −24.6254 −4.50000 7.79423i −5.17525 + 8.96379i
67.2 2.63746 + 4.56821i 1.50000 2.59808i −9.91238 + 17.1687i 5.27492 + 9.13642i 15.8248 0 −62.3746 −4.50000 7.79423i −27.8248 + 48.1939i
79.1 −1.13746 + 1.97014i 1.50000 + 2.59808i 1.41238 + 2.44631i −2.27492 + 3.94027i −6.82475 0 −24.6254 −4.50000 + 7.79423i −5.17525 8.96379i
79.2 2.63746 4.56821i 1.50000 + 2.59808i −9.91238 17.1687i 5.27492 9.13642i 15.8248 0 −62.3746 −4.50000 + 7.79423i −27.8248 48.1939i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.e.m 4
3.b odd 2 1 441.4.e.p 4
7.b odd 2 1 147.4.e.l 4
7.c even 3 1 147.4.a.i 2
7.c even 3 1 inner 147.4.e.m 4
7.d odd 6 1 21.4.a.c 2
7.d odd 6 1 147.4.e.l 4
21.c even 2 1 441.4.e.q 4
21.g even 6 1 63.4.a.e 2
21.g even 6 1 441.4.e.q 4
21.h odd 6 1 441.4.a.r 2
21.h odd 6 1 441.4.e.p 4
28.f even 6 1 336.4.a.m 2
28.g odd 6 1 2352.4.a.bz 2
35.i odd 6 1 525.4.a.n 2
35.k even 12 2 525.4.d.g 4
56.j odd 6 1 1344.4.a.bg 2
56.m even 6 1 1344.4.a.bo 2
84.j odd 6 1 1008.4.a.ba 2
105.p even 6 1 1575.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 7.d odd 6 1
63.4.a.e 2 21.g even 6 1
147.4.a.i 2 7.c even 3 1
147.4.e.l 4 7.b odd 2 1
147.4.e.l 4 7.d odd 6 1
147.4.e.m 4 1.a even 1 1 trivial
147.4.e.m 4 7.c even 3 1 inner
336.4.a.m 2 28.f even 6 1
441.4.a.r 2 21.h odd 6 1
441.4.e.p 4 3.b odd 2 1
441.4.e.p 4 21.h odd 6 1
441.4.e.q 4 21.c even 2 1
441.4.e.q 4 21.g even 6 1
525.4.a.n 2 35.i odd 6 1
525.4.d.g 4 35.k even 12 2
1008.4.a.ba 2 84.j odd 6 1
1344.4.a.bg 2 56.j odd 6 1
1344.4.a.bo 2 56.m even 6 1
1575.4.a.p 2 105.p even 6 1
2352.4.a.bz 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{4} - 3 T_{2}^{3} + 21 T_{2}^{2} + 36 T_{2} + 144$$ $$T_{5}^{4} - 6 T_{5}^{3} + 84 T_{5}^{2} + 288 T_{5} + 2304$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$2304 + 288 T + 84 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$2005056 + 8496 T + 1452 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$( -1988 + 16 T + T^{2} )^{2}$$
$17$ $$2304 - 288 T + 84 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$51609856 + 459776 T + 11280 T^{2} - 64 T^{3} + T^{4}$$
$23$ $$271063296 - 98784 T + 16500 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$( 7668 + 252 T + T^{2} )^{2}$$
$31$ $$5398134784 + 2938880 T + 75072 T^{2} - 40 T^{3} + T^{4}$$
$37$ $$9560464 + 766816 T + 64596 T^{2} - 248 T^{3} + T^{4}$$
$41$ $$( 37800 - 450 T + T^{2} )^{2}$$
$43$ $$( 2512 - 376 T + T^{2} )^{2}$$
$47$ $$4337012736 - 790272 T + 66000 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$92705634576 - 336141504 T + 914340 T^{2} - 1104 T^{3} + T^{4}$$
$59$ $$908660736 + 24235776 T + 676560 T^{2} - 804 T^{3} + T^{4}$$
$61$ $$788261776 - 12016528 T + 211260 T^{2} + 428 T^{3} + T^{4}$$
$67$ $$25836061696 - 23788928 T + 182640 T^{2} + 148 T^{3} + T^{4}$$
$71$ $$( 214704 - 954 T + T^{2} )^{2}$$
$73$ $$81364139536 - 305781568 T + 863940 T^{2} - 1072 T^{3} + T^{4}$$
$79$ $$7126061056 + 48285952 T + 411600 T^{2} - 572 T^{3} + T^{4}$$
$83$ $$( 813456 + 1944 T + T^{2} )^{2}$$
$89$ $$64438807104 + 92908368 T + 387804 T^{2} - 366 T^{3} + T^{4}$$
$97$ $$( -922292 + 808 T + T^{2} )^{2}$$