Properties

Label 147.7.b.a
Level 147
Weight 7
Character orbit 147.b
Self dual yes
Analytic conductor 33.818
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8179502921\)
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)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 27q^{3} + 64q^{4} + 729q^{9} + O(q^{10}) \) \( q + 27q^{3} + 64q^{4} + 729q^{9} + 1728q^{12} - 506q^{13} + 4096q^{16} + 10582q^{19} + 15625q^{25} + 19683q^{27} - 35282q^{31} + 46656q^{36} - 89206q^{37} - 13662q^{39} + 111386q^{43} + 110592q^{48} - 32384q^{52} + 285714q^{57} + 420838q^{61} + 262144q^{64} + 172874q^{67} - 638066q^{73} + 421875q^{75} + 677248q^{76} - 204622q^{79} + 531441q^{81} - 952614q^{93} + 56446q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
0
0 27.0000 64.0000 0 0 0 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.b.a 1
3.b odd 2 1 CM 147.7.b.a 1
7.b odd 2 1 3.7.b.a 1
21.c even 2 1 3.7.b.a 1
28.d even 2 1 48.7.e.a 1
35.c odd 2 1 75.7.c.a 1
35.f even 4 2 75.7.d.a 2
56.e even 2 1 192.7.e.a 1
56.h odd 2 1 192.7.e.b 1
63.l odd 6 2 81.7.d.a 2
63.o even 6 2 81.7.d.a 2
84.h odd 2 1 48.7.e.a 1
105.g even 2 1 75.7.c.a 1
105.k odd 4 2 75.7.d.a 2
168.e odd 2 1 192.7.e.a 1
168.i even 2 1 192.7.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.7.b.a 1 7.b odd 2 1
3.7.b.a 1 21.c even 2 1
48.7.e.a 1 28.d even 2 1
48.7.e.a 1 84.h odd 2 1
75.7.c.a 1 35.c odd 2 1
75.7.c.a 1 105.g even 2 1
75.7.d.a 2 35.f even 4 2
75.7.d.a 2 105.k odd 4 2
81.7.d.a 2 63.l odd 6 2
81.7.d.a 2 63.o even 6 2
147.7.b.a 1 1.a even 1 1 trivial
147.7.b.a 1 3.b odd 2 1 CM
192.7.e.a 1 56.e even 2 1
192.7.e.a 1 168.e odd 2 1
192.7.e.b 1 56.h odd 2 1
192.7.e.b 1 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\):

\( T_{2} \)
\( T_{13} + 506 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T )( 1 + 8 T ) \)
$3$ \( 1 - 27 T \)
$5$ \( ( 1 - 125 T )( 1 + 125 T ) \)
$7$ 1
$11$ \( ( 1 - 1331 T )( 1 + 1331 T ) \)
$13$ \( 1 + 506 T + 4826809 T^{2} \)
$17$ \( ( 1 - 4913 T )( 1 + 4913 T ) \)
$19$ \( 1 - 10582 T + 47045881 T^{2} \)
$23$ \( ( 1 - 12167 T )( 1 + 12167 T ) \)
$29$ \( ( 1 - 24389 T )( 1 + 24389 T ) \)
$31$ \( 1 + 35282 T + 887503681 T^{2} \)
$37$ \( 1 + 89206 T + 2565726409 T^{2} \)
$41$ \( ( 1 - 68921 T )( 1 + 68921 T ) \)
$43$ \( 1 - 111386 T + 6321363049 T^{2} \)
$47$ \( ( 1 - 103823 T )( 1 + 103823 T ) \)
$53$ \( ( 1 - 148877 T )( 1 + 148877 T ) \)
$59$ \( ( 1 - 205379 T )( 1 + 205379 T ) \)
$61$ \( 1 - 420838 T + 51520374361 T^{2} \)
$67$ \( 1 - 172874 T + 90458382169 T^{2} \)
$71$ \( ( 1 - 357911 T )( 1 + 357911 T ) \)
$73$ \( 1 + 638066 T + 151334226289 T^{2} \)
$79$ \( 1 + 204622 T + 243087455521 T^{2} \)
$83$ \( ( 1 - 571787 T )( 1 + 571787 T ) \)
$89$ \( ( 1 - 704969 T )( 1 + 704969 T ) \)
$97$ \( 1 - 56446 T + 832972004929 T^{2} \)
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