Properties

Label 147.6.a.d
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 9 q^{3} - 28 q^{4} - 11 q^{5} - 18 q^{6} + 120 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 9 q^{3} - 28 q^{4} - 11 q^{5} - 18 q^{6} + 120 q^{8} + 81 q^{9} + 22 q^{10} + 269 q^{11} - 252 q^{12} + 308 q^{13} - 99 q^{15} + 656 q^{16} - 1896 q^{17} - 162 q^{18} + 164 q^{19} + 308 q^{20} - 538 q^{22} - 3264 q^{23} + 1080 q^{24} - 3004 q^{25} - 616 q^{26} + 729 q^{27} + 2417 q^{29} + 198 q^{30} - 2841 q^{31} - 5152 q^{32} + 2421 q^{33} + 3792 q^{34} - 2268 q^{36} - 11328 q^{37} - 328 q^{38} + 2772 q^{39} - 1320 q^{40} + 16856 q^{41} - 7894 q^{43} - 7532 q^{44} - 891 q^{45} + 6528 q^{46} - 21102 q^{47} + 5904 q^{48} + 6008 q^{50} - 17064 q^{51} - 8624 q^{52} - 29691 q^{53} - 1458 q^{54} - 2959 q^{55} + 1476 q^{57} - 4834 q^{58} + 8163 q^{59} + 2772 q^{60} - 15166 q^{61} + 5682 q^{62} - 10688 q^{64} - 3388 q^{65} - 4842 q^{66} - 32078 q^{67} + 53088 q^{68} - 29376 q^{69} - 38274 q^{71} + 9720 q^{72} - 34866 q^{73} + 22656 q^{74} - 27036 q^{75} - 4592 q^{76} - 5544 q^{78} + 13529 q^{79} - 7216 q^{80} + 6561 q^{81} - 33712 q^{82} + 68103 q^{83} + 20856 q^{85} + 15788 q^{86} + 21753 q^{87} + 32280 q^{88} + 114922 q^{89} + 1782 q^{90} + 91392 q^{92} - 25569 q^{93} + 42204 q^{94} - 1804 q^{95} - 46368 q^{96} - 154959 q^{97} + 21789 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 9.00000 −28.0000 −11.0000 −18.0000 0 120.000 81.0000 22.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.a.d 1
3.b odd 2 1 441.6.a.h 1
7.b odd 2 1 147.6.a.c 1
7.c even 3 2 147.6.e.g 2
7.d odd 6 2 21.6.e.a 2
21.c even 2 1 441.6.a.g 1
21.g even 6 2 63.6.e.a 2
28.f even 6 2 336.6.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 7.d odd 6 2
63.6.e.a 2 21.g even 6 2
147.6.a.c 1 7.b odd 2 1
147.6.a.d 1 1.a even 1 1 trivial
147.6.e.g 2 7.c even 3 2
336.6.q.b 2 28.f even 6 2
441.6.a.g 1 21.c even 2 1
441.6.a.h 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{5} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 11 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 269 \) Copy content Toggle raw display
$13$ \( T - 308 \) Copy content Toggle raw display
$17$ \( T + 1896 \) Copy content Toggle raw display
$19$ \( T - 164 \) Copy content Toggle raw display
$23$ \( T + 3264 \) Copy content Toggle raw display
$29$ \( T - 2417 \) Copy content Toggle raw display
$31$ \( T + 2841 \) Copy content Toggle raw display
$37$ \( T + 11328 \) Copy content Toggle raw display
$41$ \( T - 16856 \) Copy content Toggle raw display
$43$ \( T + 7894 \) Copy content Toggle raw display
$47$ \( T + 21102 \) Copy content Toggle raw display
$53$ \( T + 29691 \) Copy content Toggle raw display
$59$ \( T - 8163 \) Copy content Toggle raw display
$61$ \( T + 15166 \) Copy content Toggle raw display
$67$ \( T + 32078 \) Copy content Toggle raw display
$71$ \( T + 38274 \) Copy content Toggle raw display
$73$ \( T + 34866 \) Copy content Toggle raw display
$79$ \( T - 13529 \) Copy content Toggle raw display
$83$ \( T - 68103 \) Copy content Toggle raw display
$89$ \( T - 114922 \) Copy content Toggle raw display
$97$ \( T + 154959 \) Copy content Toggle raw display
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