Properties

Label 147.4.e.h
Level $147$
Weight $4$
Character orbit 147.e
Analytic conductor $8.673$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + 9 q^{6} + 21 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q + 3 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + 9 q^{6} + 21 q^{8} -9 \zeta_{6} q^{9} + ( 9 - 9 \zeta_{6} ) q^{10} + ( 15 - 15 \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{12} + 64 q^{13} -9 q^{15} + 71 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} + ( 27 - 27 \zeta_{6} ) q^{18} -16 \zeta_{6} q^{19} + 3 q^{20} + 45 q^{22} + 84 \zeta_{6} q^{23} + ( 63 - 63 \zeta_{6} ) q^{24} + ( 116 - 116 \zeta_{6} ) q^{25} + 192 \zeta_{6} q^{26} -27 q^{27} -297 q^{29} -27 \zeta_{6} q^{30} + ( -253 + 253 \zeta_{6} ) q^{31} + ( -45 + 45 \zeta_{6} ) q^{32} -45 \zeta_{6} q^{33} + 252 q^{34} + 9 q^{36} + 316 \zeta_{6} q^{37} + ( 48 - 48 \zeta_{6} ) q^{38} + ( 192 - 192 \zeta_{6} ) q^{39} -63 \zeta_{6} q^{40} -360 q^{41} + 26 q^{43} + 15 \zeta_{6} q^{44} + ( -27 + 27 \zeta_{6} ) q^{45} + ( -252 + 252 \zeta_{6} ) q^{46} -30 \zeta_{6} q^{47} + 213 q^{48} + 348 q^{50} -252 \zeta_{6} q^{51} + ( -64 + 64 \zeta_{6} ) q^{52} + ( -363 + 363 \zeta_{6} ) q^{53} -81 \zeta_{6} q^{54} -45 q^{55} -48 q^{57} -891 \zeta_{6} q^{58} + ( -15 + 15 \zeta_{6} ) q^{59} + ( 9 - 9 \zeta_{6} ) q^{60} -118 \zeta_{6} q^{61} -759 q^{62} + 433 q^{64} -192 \zeta_{6} q^{65} + ( 135 - 135 \zeta_{6} ) q^{66} + ( 370 - 370 \zeta_{6} ) q^{67} + 84 \zeta_{6} q^{68} + 252 q^{69} -342 q^{71} -189 \zeta_{6} q^{72} + ( 362 - 362 \zeta_{6} ) q^{73} + ( -948 + 948 \zeta_{6} ) q^{74} -348 \zeta_{6} q^{75} + 16 q^{76} + 576 q^{78} -467 \zeta_{6} q^{79} + ( 213 - 213 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} -1080 \zeta_{6} q^{82} -477 q^{83} -252 q^{85} + 78 \zeta_{6} q^{86} + ( -891 + 891 \zeta_{6} ) q^{87} + ( 315 - 315 \zeta_{6} ) q^{88} + 906 \zeta_{6} q^{89} -81 q^{90} -84 q^{92} + 759 \zeta_{6} q^{93} + ( 90 - 90 \zeta_{6} ) q^{94} + ( -48 + 48 \zeta_{6} ) q^{95} + 135 \zeta_{6} q^{96} -503 q^{97} -135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 3q^{3} - q^{4} - 3q^{5} + 18q^{6} + 42q^{8} - 9q^{9} + O(q^{10}) \) \( 2q + 3q^{2} + 3q^{3} - q^{4} - 3q^{5} + 18q^{6} + 42q^{8} - 9q^{9} + 9q^{10} + 15q^{11} + 3q^{12} + 128q^{13} - 18q^{15} + 71q^{16} + 84q^{17} + 27q^{18} - 16q^{19} + 6q^{20} + 90q^{22} + 84q^{23} + 63q^{24} + 116q^{25} + 192q^{26} - 54q^{27} - 594q^{29} - 27q^{30} - 253q^{31} - 45q^{32} - 45q^{33} + 504q^{34} + 18q^{36} + 316q^{37} + 48q^{38} + 192q^{39} - 63q^{40} - 720q^{41} + 52q^{43} + 15q^{44} - 27q^{45} - 252q^{46} - 30q^{47} + 426q^{48} + 696q^{50} - 252q^{51} - 64q^{52} - 363q^{53} - 81q^{54} - 90q^{55} - 96q^{57} - 891q^{58} - 15q^{59} + 9q^{60} - 118q^{61} - 1518q^{62} + 866q^{64} - 192q^{65} + 135q^{66} + 370q^{67} + 84q^{68} + 504q^{69} - 684q^{71} - 189q^{72} + 362q^{73} - 948q^{74} - 348q^{75} + 32q^{76} + 1152q^{78} - 467q^{79} + 213q^{80} - 81q^{81} - 1080q^{82} - 954q^{83} - 504q^{85} + 78q^{86} - 891q^{87} + 315q^{88} + 906q^{89} - 162q^{90} - 168q^{92} + 759q^{93} + 90q^{94} - 48q^{95} + 135q^{96} - 1006q^{97} - 270q^{99} + O(q^{100}) \)

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
1.50000 + 2.59808i 1.50000 2.59808i −0.500000 + 0.866025i −1.50000 2.59808i 9.00000 0 21.0000 −4.50000 7.79423i 4.50000 7.79423i
79.1 1.50000 2.59808i 1.50000 + 2.59808i −0.500000 0.866025i −1.50000 + 2.59808i 9.00000 0 21.0000 −4.50000 + 7.79423i 4.50000 + 7.79423i
\(n\): 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.h 2
3.b odd 2 1 441.4.e.c 2
7.b odd 2 1 21.4.e.a 2
7.c even 3 1 147.4.a.a 1
7.c even 3 1 inner 147.4.e.h 2
7.d odd 6 1 21.4.e.a 2
7.d odd 6 1 147.4.a.b 1
21.c even 2 1 63.4.e.a 2
21.g even 6 1 63.4.e.a 2
21.g even 6 1 441.4.a.l 1
21.h odd 6 1 441.4.a.k 1
21.h odd 6 1 441.4.e.c 2
28.d even 2 1 336.4.q.e 2
28.f even 6 1 336.4.q.e 2
28.f even 6 1 2352.4.a.i 1
28.g odd 6 1 2352.4.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 7.b odd 2 1
21.4.e.a 2 7.d odd 6 1
63.4.e.a 2 21.c even 2 1
63.4.e.a 2 21.g even 6 1
147.4.a.a 1 7.c even 3 1
147.4.a.b 1 7.d odd 6 1
147.4.e.h 2 1.a even 1 1 trivial
147.4.e.h 2 7.c even 3 1 inner
336.4.q.e 2 28.d even 2 1
336.4.q.e 2 28.f even 6 1
441.4.a.k 1 21.h odd 6 1
441.4.a.l 1 21.g even 6 1
441.4.e.c 2 3.b odd 2 1
441.4.e.c 2 21.h odd 6 1
2352.4.a.i 1 28.f even 6 1
2352.4.a.bd 1 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 \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{2} - 3 T_{2} + 9 \)
\( T_{5}^{2} + 3 T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 3 T + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 225 - 15 T + T^{2} \)
$13$ \( ( -64 + T )^{2} \)
$17$ \( 7056 - 84 T + T^{2} \)
$19$ \( 256 + 16 T + T^{2} \)
$23$ \( 7056 - 84 T + T^{2} \)
$29$ \( ( 297 + T )^{2} \)
$31$ \( 64009 + 253 T + T^{2} \)
$37$ \( 99856 - 316 T + T^{2} \)
$41$ \( ( 360 + T )^{2} \)
$43$ \( ( -26 + T )^{2} \)
$47$ \( 900 + 30 T + T^{2} \)
$53$ \( 131769 + 363 T + T^{2} \)
$59$ \( 225 + 15 T + T^{2} \)
$61$ \( 13924 + 118 T + T^{2} \)
$67$ \( 136900 - 370 T + T^{2} \)
$71$ \( ( 342 + T )^{2} \)
$73$ \( 131044 - 362 T + T^{2} \)
$79$ \( 218089 + 467 T + T^{2} \)
$83$ \( ( 477 + T )^{2} \)
$89$ \( 820836 - 906 T + T^{2} \)
$97$ \( ( 503 + T )^{2} \)
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