Properties

Label 147.6.c.a
Level $147$
Weight $6$
Character orbit 147.c
Analytic conductor $23.576$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta q^{3} + 32 q^{4} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta q^{3} + 32 q^{4} - 243 q^{9} + 288 \beta q^{12} + 659 \beta q^{13} + 1024 q^{16} + 93 \beta q^{19} - 3125 q^{25} - 2187 \beta q^{27} + 5975 \beta q^{31} - 7776 q^{36} + 6661 q^{37} - 17793 q^{39} - 22475 q^{43} + 9216 \beta q^{48} + 21088 \beta q^{52} - 2511 q^{57} + 25076 \beta q^{61} + 32768 q^{64} - 37939 q^{67} - 27009 \beta q^{73} - 28125 \beta q^{75} + 2976 \beta q^{76} + 90857 q^{79} + 59049 q^{81} - 161325 q^{93} - 73688 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{4} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{4} - 486 q^{9} + 2048 q^{16} - 6250 q^{25} - 15552 q^{36} + 13322 q^{37} - 35586 q^{39} - 44950 q^{43} - 5022 q^{57} + 65536 q^{64} - 75878 q^{67} + 181714 q^{79} + 118098 q^{81} - 322650 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
0.500000 0.866025i
0.500000 + 0.866025i
0 15.5885i 32.0000 0 0 0 0 −243.000 0
146.2 0 15.5885i 32.0000 0 0 0 0 −243.000 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 \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.c.a 2
3.b odd 2 1 CM 147.6.c.a 2
7.b odd 2 1 inner 147.6.c.a 2
7.c even 3 1 21.6.g.a 2
7.d odd 6 1 21.6.g.a 2
21.c even 2 1 inner 147.6.c.a 2
21.g even 6 1 21.6.g.a 2
21.h odd 6 1 21.6.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.g.a 2 7.c even 3 1
21.6.g.a 2 7.d odd 6 1
21.6.g.a 2 21.g even 6 1
21.6.g.a 2 21.h odd 6 1
147.6.c.a 2 1.a even 1 1 trivial
147.6.c.a 2 3.b odd 2 1 CM
147.6.c.a 2 7.b odd 2 1 inner
147.6.c.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 243 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1302843 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 25947 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 107101875 \) Copy content Toggle raw display
$37$ \( (T - 6661)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 22475)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 1886417328 \) Copy content Toggle raw display
$67$ \( (T + 37939)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2188458243 \) Copy content Toggle raw display
$79$ \( (T - 90857)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 16289764032 \) Copy content Toggle raw display
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