Properties

Label 147.6.a.a
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 6 q^{2} - 9 q^{3} + 4 q^{4} - 6 q^{5} + 54 q^{6} + 168 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{2} - 9 q^{3} + 4 q^{4} - 6 q^{5} + 54 q^{6} + 168 q^{8} + 81 q^{9} + 36 q^{10} - 564 q^{11} - 36 q^{12} - 638 q^{13} + 54 q^{15} - 1136 q^{16} - 882 q^{17} - 486 q^{18} + 556 q^{19} - 24 q^{20} + 3384 q^{22} - 840 q^{23} - 1512 q^{24} - 3089 q^{25} + 3828 q^{26} - 729 q^{27} + 4638 q^{29} - 324 q^{30} - 4400 q^{31} + 1440 q^{32} + 5076 q^{33} + 5292 q^{34} + 324 q^{36} - 2410 q^{37} - 3336 q^{38} + 5742 q^{39} - 1008 q^{40} + 6870 q^{41} + 9644 q^{43} - 2256 q^{44} - 486 q^{45} + 5040 q^{46} + 18672 q^{47} + 10224 q^{48} + 18534 q^{50} + 7938 q^{51} - 2552 q^{52} + 33750 q^{53} + 4374 q^{54} + 3384 q^{55} - 5004 q^{57} - 27828 q^{58} + 18084 q^{59} + 216 q^{60} - 39758 q^{61} + 26400 q^{62} + 27712 q^{64} + 3828 q^{65} - 30456 q^{66} - 23068 q^{67} - 3528 q^{68} + 7560 q^{69} - 4248 q^{71} + 13608 q^{72} + 41110 q^{73} + 14460 q^{74} + 27801 q^{75} + 2224 q^{76} - 34452 q^{78} + 21920 q^{79} + 6816 q^{80} + 6561 q^{81} - 41220 q^{82} - 82452 q^{83} + 5292 q^{85} - 57864 q^{86} - 41742 q^{87} - 94752 q^{88} + 94086 q^{89} + 2916 q^{90} - 3360 q^{92} + 39600 q^{93} - 112032 q^{94} - 3336 q^{95} - 12960 q^{96} - 49442 q^{97} - 45684 q^{99}+O(q^{100}) \) 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 −6.00000 54.0000 0 168.000 81.0000 36.0000
\(n\): 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.a 1
3.b odd 2 1 441.6.a.i 1
7.b odd 2 1 3.6.a.a 1
7.c even 3 2 147.6.e.k 2
7.d odd 6 2 147.6.e.h 2
21.c even 2 1 9.6.a.a 1
28.d even 2 1 48.6.a.a 1
35.c odd 2 1 75.6.a.e 1
35.f even 4 2 75.6.b.b 2
56.e even 2 1 192.6.a.l 1
56.h odd 2 1 192.6.a.d 1
63.l odd 6 2 81.6.c.c 2
63.o even 6 2 81.6.c.a 2
77.b even 2 1 363.6.a.d 1
84.h odd 2 1 144.6.a.f 1
91.b odd 2 1 507.6.a.b 1
105.g even 2 1 225.6.a.a 1
105.k odd 4 2 225.6.b.b 2
112.j even 4 2 768.6.d.h 2
112.l odd 4 2 768.6.d.k 2
119.d odd 2 1 867.6.a.a 1
133.c even 2 1 1083.6.a.c 1
168.e odd 2 1 576.6.a.t 1
168.i even 2 1 576.6.a.s 1
231.h odd 2 1 1089.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 7.b odd 2 1
9.6.a.a 1 21.c even 2 1
48.6.a.a 1 28.d even 2 1
75.6.a.e 1 35.c odd 2 1
75.6.b.b 2 35.f even 4 2
81.6.c.a 2 63.o even 6 2
81.6.c.c 2 63.l odd 6 2
144.6.a.f 1 84.h odd 2 1
147.6.a.a 1 1.a even 1 1 trivial
147.6.e.h 2 7.d odd 6 2
147.6.e.k 2 7.c even 3 2
192.6.a.d 1 56.h odd 2 1
192.6.a.l 1 56.e even 2 1
225.6.a.a 1 105.g even 2 1
225.6.b.b 2 105.k odd 4 2
363.6.a.d 1 77.b even 2 1
441.6.a.i 1 3.b odd 2 1
507.6.a.b 1 91.b odd 2 1
576.6.a.s 1 168.i even 2 1
576.6.a.t 1 168.e odd 2 1
768.6.d.h 2 112.j even 4 2
768.6.d.k 2 112.l odd 4 2
867.6.a.a 1 119.d odd 2 1
1083.6.a.c 1 133.c even 2 1
1089.6.a.b 1 231.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
\( T_{5} + 6 \) 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
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 564 \) Copy content Toggle raw display
$13$ \( T + 638 \) Copy content Toggle raw display
$17$ \( T + 882 \) Copy content Toggle raw display
$19$ \( T - 556 \) Copy content Toggle raw display
$23$ \( T + 840 \) Copy content Toggle raw display
$29$ \( T - 4638 \) Copy content Toggle raw display
$31$ \( T + 4400 \) Copy content Toggle raw display
$37$ \( T + 2410 \) Copy content Toggle raw display
$41$ \( T - 6870 \) Copy content Toggle raw display
$43$ \( T - 9644 \) Copy content Toggle raw display
$47$ \( T - 18672 \) Copy content Toggle raw display
$53$ \( T - 33750 \) Copy content Toggle raw display
$59$ \( T - 18084 \) Copy content Toggle raw display
$61$ \( T + 39758 \) Copy content Toggle raw display
$67$ \( T + 23068 \) Copy content Toggle raw display
$71$ \( T + 4248 \) Copy content Toggle raw display
$73$ \( T - 41110 \) Copy content Toggle raw display
$79$ \( T - 21920 \) Copy content Toggle raw display
$83$ \( T + 82452 \) Copy content Toggle raw display
$89$ \( T - 94086 \) Copy content Toggle raw display
$97$ \( T + 49442 \) Copy content Toggle raw display
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