Properties

 Label 147.4.e.a Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} + 12 q^{6} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q -4 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} + 12 q^{6} -9 \zeta_{6} q^{9} + ( -72 + 72 \zeta_{6} ) q^{10} + ( 50 - 50 \zeta_{6} ) q^{11} -24 \zeta_{6} q^{12} -36 q^{13} + 54 q^{15} + 64 \zeta_{6} q^{16} + ( -126 + 126 \zeta_{6} ) q^{17} + ( -36 + 36 \zeta_{6} ) q^{18} + 72 \zeta_{6} q^{19} + 144 q^{20} -200 q^{22} -14 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} + 144 \zeta_{6} q^{26} + 27 q^{27} + 158 q^{29} -216 \zeta_{6} q^{30} + ( 36 - 36 \zeta_{6} ) q^{31} + ( 256 - 256 \zeta_{6} ) q^{32} + 150 \zeta_{6} q^{33} + 504 q^{34} + 72 q^{36} + 162 \zeta_{6} q^{37} + ( 288 - 288 \zeta_{6} ) q^{38} + ( 108 - 108 \zeta_{6} ) q^{39} -270 q^{41} -324 q^{43} + 400 \zeta_{6} q^{44} + ( -162 + 162 \zeta_{6} ) q^{45} + ( -56 + 56 \zeta_{6} ) q^{46} + 72 \zeta_{6} q^{47} -192 q^{48} + 796 q^{50} -378 \zeta_{6} q^{51} + ( 288 - 288 \zeta_{6} ) q^{52} + ( 22 - 22 \zeta_{6} ) q^{53} -108 \zeta_{6} q^{54} -900 q^{55} -216 q^{57} -632 \zeta_{6} q^{58} + ( -468 + 468 \zeta_{6} ) q^{59} + ( -432 + 432 \zeta_{6} ) q^{60} -792 \zeta_{6} q^{61} -144 q^{62} -512 q^{64} + 648 \zeta_{6} q^{65} + ( 600 - 600 \zeta_{6} ) q^{66} + ( -232 + 232 \zeta_{6} ) q^{67} -1008 \zeta_{6} q^{68} + 42 q^{69} -734 q^{71} + ( -180 + 180 \zeta_{6} ) q^{73} + ( 648 - 648 \zeta_{6} ) q^{74} -597 \zeta_{6} q^{75} -576 q^{76} -432 q^{78} -236 \zeta_{6} q^{79} + ( 1152 - 1152 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 1080 \zeta_{6} q^{82} + 36 q^{83} + 2268 q^{85} + 1296 \zeta_{6} q^{86} + ( -474 + 474 \zeta_{6} ) q^{87} -234 \zeta_{6} q^{89} + 648 q^{90} + 112 q^{92} + 108 \zeta_{6} q^{93} + ( 288 - 288 \zeta_{6} ) q^{94} + ( 1296 - 1296 \zeta_{6} ) q^{95} + 768 \zeta_{6} q^{96} + 468 q^{97} -450 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 3q^{3} - 8q^{4} - 18q^{5} + 24q^{6} - 9q^{9} + O(q^{10})$$ $$2q - 4q^{2} - 3q^{3} - 8q^{4} - 18q^{5} + 24q^{6} - 9q^{9} - 72q^{10} + 50q^{11} - 24q^{12} - 72q^{13} + 108q^{15} + 64q^{16} - 126q^{17} - 36q^{18} + 72q^{19} + 288q^{20} - 400q^{22} - 14q^{23} - 199q^{25} + 144q^{26} + 54q^{27} + 316q^{29} - 216q^{30} + 36q^{31} + 256q^{32} + 150q^{33} + 1008q^{34} + 144q^{36} + 162q^{37} + 288q^{38} + 108q^{39} - 540q^{41} - 648q^{43} + 400q^{44} - 162q^{45} - 56q^{46} + 72q^{47} - 384q^{48} + 1592q^{50} - 378q^{51} + 288q^{52} + 22q^{53} - 108q^{54} - 1800q^{55} - 432q^{57} - 632q^{58} - 468q^{59} - 432q^{60} - 792q^{61} - 288q^{62} - 1024q^{64} + 648q^{65} + 600q^{66} - 232q^{67} - 1008q^{68} + 84q^{69} - 1468q^{71} - 180q^{73} + 648q^{74} - 597q^{75} - 1152q^{76} - 864q^{78} - 236q^{79} + 1152q^{80} - 81q^{81} + 1080q^{82} + 72q^{83} + 4536q^{85} + 1296q^{86} - 474q^{87} - 234q^{89} + 1296q^{90} + 224q^{92} + 108q^{93} + 288q^{94} + 1296q^{95} + 768q^{96} + 936q^{97} - 900q^{99} + O(q^{100})$$

Character values

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

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

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
 0.5 + 0.866025i 0.5 − 0.866025i
−2.00000 3.46410i −1.50000 + 2.59808i −4.00000 + 6.92820i −9.00000 15.5885i 12.0000 0 0 −4.50000 7.79423i −36.0000 + 62.3538i
79.1 −2.00000 + 3.46410i −1.50000 2.59808i −4.00000 6.92820i −9.00000 + 15.5885i 12.0000 0 0 −4.50000 + 7.79423i −36.0000 62.3538i
 $$n$$: 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.a 2
3.b odd 2 1 441.4.e.o 2
7.b odd 2 1 147.4.e.d 2
7.c even 3 1 147.4.a.h yes 1
7.c even 3 1 inner 147.4.e.a 2
7.d odd 6 1 147.4.a.f 1
7.d odd 6 1 147.4.e.d 2
21.c even 2 1 441.4.e.l 2
21.g even 6 1 441.4.a.c 1
21.g even 6 1 441.4.e.l 2
21.h odd 6 1 441.4.a.a 1
21.h odd 6 1 441.4.e.o 2
28.f even 6 1 2352.4.a.t 1
28.g odd 6 1 2352.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 7.d odd 6 1
147.4.a.h yes 1 7.c even 3 1
147.4.e.a 2 1.a even 1 1 trivial
147.4.e.a 2 7.c even 3 1 inner
147.4.e.d 2 7.b odd 2 1
147.4.e.d 2 7.d odd 6 1
441.4.a.a 1 21.h odd 6 1
441.4.a.c 1 21.g even 6 1
441.4.e.l 2 21.c even 2 1
441.4.e.l 2 21.g even 6 1
441.4.e.o 2 3.b odd 2 1
441.4.e.o 2 21.h odd 6 1
2352.4.a.s 1 28.g odd 6 1
2352.4.a.t 1 28.f even 6 1

Hecke kernels

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

 $$T_{2}^{2} + 4 T_{2} + 16$$ $$T_{5}^{2} + 18 T_{5} + 324$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$324 + 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$2500 - 50 T + T^{2}$$
$13$ $$( 36 + T )^{2}$$
$17$ $$15876 + 126 T + T^{2}$$
$19$ $$5184 - 72 T + T^{2}$$
$23$ $$196 + 14 T + T^{2}$$
$29$ $$( -158 + T )^{2}$$
$31$ $$1296 - 36 T + T^{2}$$
$37$ $$26244 - 162 T + T^{2}$$
$41$ $$( 270 + T )^{2}$$
$43$ $$( 324 + T )^{2}$$
$47$ $$5184 - 72 T + T^{2}$$
$53$ $$484 - 22 T + T^{2}$$
$59$ $$219024 + 468 T + T^{2}$$
$61$ $$627264 + 792 T + T^{2}$$
$67$ $$53824 + 232 T + T^{2}$$
$71$ $$( 734 + T )^{2}$$
$73$ $$32400 + 180 T + T^{2}$$
$79$ $$55696 + 236 T + T^{2}$$
$83$ $$( -36 + T )^{2}$$
$89$ $$54756 + 234 T + T^{2}$$
$97$ $$( -468 + T )^{2}$$