Properties

Label 147.6.a.b
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

\(f(q)\) \(=\) \( q - 6 q^{2} + 9 q^{3} + 4 q^{4} - 78 q^{5} - 54 q^{6} + 168 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{2} + 9 q^{3} + 4 q^{4} - 78 q^{5} - 54 q^{6} + 168 q^{8} + 81 q^{9} + 468 q^{10} + 444 q^{11} + 36 q^{12} + 442 q^{13} - 702 q^{15} - 1136 q^{16} + 126 q^{17} - 486 q^{18} - 2684 q^{19} - 312 q^{20} - 2664 q^{22} + 4200 q^{23} + 1512 q^{24} + 2959 q^{25} - 2652 q^{26} + 729 q^{27} - 5442 q^{29} + 4212 q^{30} - 80 q^{31} + 1440 q^{32} + 3996 q^{33} - 756 q^{34} + 324 q^{36} - 5434 q^{37} + 16104 q^{38} + 3978 q^{39} - 13104 q^{40} - 7962 q^{41} - 11524 q^{43} + 1776 q^{44} - 6318 q^{45} - 25200 q^{46} + 13920 q^{47} - 10224 q^{48} - 17754 q^{50} + 1134 q^{51} + 1768 q^{52} - 9594 q^{53} - 4374 q^{54} - 34632 q^{55} - 24156 q^{57} + 32652 q^{58} - 27492 q^{59} - 2808 q^{60} - 49478 q^{61} + 480 q^{62} + 27712 q^{64} - 34476 q^{65} - 23976 q^{66} - 59356 q^{67} + 504 q^{68} + 37800 q^{69} + 32040 q^{71} + 13608 q^{72} + 61846 q^{73} + 32604 q^{74} + 26631 q^{75} - 10736 q^{76} - 23868 q^{78} - 65776 q^{79} + 88608 q^{80} + 6561 q^{81} + 47772 q^{82} - 40188 q^{83} - 9828 q^{85} + 69144 q^{86} - 48978 q^{87} + 74592 q^{88} + 7974 q^{89} + 37908 q^{90} + 16800 q^{92} - 720 q^{93} - 83520 q^{94} + 209352 q^{95} + 12960 q^{96} + 143662 q^{97} + 35964 q^{99}+O(q^{100}) \) 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
−6.00000 9.00000 4.00000 −78.0000 −54.0000 0 168.000 81.0000 468.000
\(n\): 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.b 1
3.b odd 2 1 441.6.a.j 1
7.b odd 2 1 21.6.a.a 1
7.c even 3 2 147.6.e.i 2
7.d odd 6 2 147.6.e.j 2
21.c even 2 1 63.6.a.d 1
28.d even 2 1 336.6.a.r 1
35.c odd 2 1 525.6.a.d 1
35.f even 4 2 525.6.d.b 2
84.h odd 2 1 1008.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 7.b odd 2 1
63.6.a.d 1 21.c even 2 1
147.6.a.b 1 1.a even 1 1 trivial
147.6.e.i 2 7.c even 3 2
147.6.e.j 2 7.d odd 6 2
336.6.a.r 1 28.d even 2 1
441.6.a.j 1 3.b odd 2 1
525.6.a.d 1 35.c odd 2 1
525.6.d.b 2 35.f even 4 2
1008.6.a.c 1 84.h odd 2 1

Hecke kernels

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

\( T_{2} + 6 \) Copy content Toggle raw display
\( T_{5} + 78 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 6 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 78 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 444 \) Copy content Toggle raw display
$13$ \( T - 442 \) Copy content Toggle raw display
$17$ \( T - 126 \) Copy content Toggle raw display
$19$ \( T + 2684 \) Copy content Toggle raw display
$23$ \( T - 4200 \) Copy content Toggle raw display
$29$ \( T + 5442 \) Copy content Toggle raw display
$31$ \( T + 80 \) Copy content Toggle raw display
$37$ \( T + 5434 \) Copy content Toggle raw display
$41$ \( T + 7962 \) Copy content Toggle raw display
$43$ \( T + 11524 \) Copy content Toggle raw display
$47$ \( T - 13920 \) Copy content Toggle raw display
$53$ \( T + 9594 \) Copy content Toggle raw display
$59$ \( T + 27492 \) Copy content Toggle raw display
$61$ \( T + 49478 \) Copy content Toggle raw display
$67$ \( T + 59356 \) Copy content Toggle raw display
$71$ \( T - 32040 \) Copy content Toggle raw display
$73$ \( T - 61846 \) Copy content Toggle raw display
$79$ \( T + 65776 \) Copy content Toggle raw display
$83$ \( T + 40188 \) Copy content Toggle raw display
$89$ \( T - 7974 \) Copy content Toggle raw display
$97$ \( T - 143662 \) Copy content Toggle raw display
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