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 $2$

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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
0.500000 + 0.866025i
0.500000 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:

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

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