Properties

Label 147.4.g.b
Level $147$
Weight $4$
Character orbit 147.g
Analytic conductor $8.673$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 + 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} + 27 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 + 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} + 27 \zeta_{6} q^{9} + ( -48 + 24 \zeta_{6} ) q^{12} + ( -36 + 72 \zeta_{6} ) q^{13} -64 \zeta_{6} q^{16} + ( -180 + 90 \zeta_{6} ) q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + ( 90 + 90 \zeta_{6} ) q^{31} -216 q^{36} + 110 \zeta_{6} q^{37} + ( -324 + 324 \zeta_{6} ) q^{39} + 520 q^{43} + ( 192 - 384 \zeta_{6} ) q^{48} + ( -288 - 288 \zeta_{6} ) q^{52} -810 q^{57} + ( 1080 - 540 \zeta_{6} ) q^{61} + 512 q^{64} + ( 880 - 880 \zeta_{6} ) q^{67} + ( 216 + 216 \zeta_{6} ) q^{73} + ( 750 - 375 \zeta_{6} ) q^{75} + ( 720 - 1440 \zeta_{6} ) q^{76} -884 \zeta_{6} q^{79} + ( -729 + 729 \zeta_{6} ) q^{81} + 810 \zeta_{6} q^{93} + ( -792 + 1584 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} - 8 q^{4} + 27 q^{9} + O(q^{10}) \) \( 2 q + 9 q^{3} - 8 q^{4} + 27 q^{9} - 72 q^{12} - 64 q^{16} - 270 q^{19} + 125 q^{25} + 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} - 864 q^{52} - 1620 q^{57} + 1620 q^{61} + 1024 q^{64} + 880 q^{67} + 648 q^{73} + 1125 q^{75} - 884 q^{79} - 729 q^{81} + 810 q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.500000 0.866025i
0.500000 + 0.866025i
0 4.50000 2.59808i −4.00000 6.92820i 0 0 0 0 13.5000 23.3827i 0
80.1 0 4.50000 + 2.59808i −4.00000 + 6.92820i 0 0 0 0 13.5000 + 23.3827i 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.g.b 2
3.b odd 2 1 CM 147.4.g.b 2
7.b odd 2 1 147.4.g.a 2
7.c even 3 1 21.4.c.a 2
7.c even 3 1 147.4.g.a 2
7.d odd 6 1 21.4.c.a 2
7.d odd 6 1 inner 147.4.g.b 2
21.c even 2 1 147.4.g.a 2
21.g even 6 1 21.4.c.a 2
21.g even 6 1 inner 147.4.g.b 2
21.h odd 6 1 21.4.c.a 2
21.h odd 6 1 147.4.g.a 2
28.f even 6 1 336.4.k.a 2
28.g odd 6 1 336.4.k.a 2
84.j odd 6 1 336.4.k.a 2
84.n even 6 1 336.4.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 7.c even 3 1
21.4.c.a 2 7.d odd 6 1
21.4.c.a 2 21.g even 6 1
21.4.c.a 2 21.h odd 6 1
147.4.g.a 2 7.b odd 2 1
147.4.g.a 2 7.c even 3 1
147.4.g.a 2 21.c even 2 1
147.4.g.a 2 21.h odd 6 1
147.4.g.b 2 1.a even 1 1 trivial
147.4.g.b 2 3.b odd 2 1 CM
147.4.g.b 2 7.d odd 6 1 inner
147.4.g.b 2 21.g even 6 1 inner
336.4.k.a 2 28.f even 6 1
336.4.k.a 2 28.g odd 6 1
336.4.k.a 2 84.j odd 6 1
336.4.k.a 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2} \)
\( T_{19}^{2} + 270 T_{19} + 24300 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 27 - 9 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 3888 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 24300 + 270 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 24300 - 270 T + T^{2} \)
$37$ \( 12100 - 110 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -520 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 874800 - 1620 T + T^{2} \)
$67$ \( 774400 - 880 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 139968 - 648 T + T^{2} \)
$79$ \( 781456 + 884 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1881792 + T^{2} \)
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