Properties

Label 147.6.a.f
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5 q^{2} - 9 q^{3} - 7 q^{4} - 94 q^{5} - 45 q^{6} - 195 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{2} - 9 q^{3} - 7 q^{4} - 94 q^{5} - 45 q^{6} - 195 q^{8} + 81 q^{9} - 470 q^{10} + 52 q^{11} + 63 q^{12} + 770 q^{13} + 846 q^{15} - 751 q^{16} + 2022 q^{17} + 405 q^{18} - 1732 q^{19} + 658 q^{20} + 260 q^{22} - 576 q^{23} + 1755 q^{24} + 5711 q^{25} + 3850 q^{26} - 729 q^{27} + 5518 q^{29} + 4230 q^{30} - 6336 q^{31} + 2485 q^{32} - 468 q^{33} + 10110 q^{34} - 567 q^{36} - 7338 q^{37} - 8660 q^{38} - 6930 q^{39} + 18330 q^{40} + 3262 q^{41} + 5420 q^{43} - 364 q^{44} - 7614 q^{45} - 2880 q^{46} - 864 q^{47} + 6759 q^{48} + 28555 q^{50} - 18198 q^{51} - 5390 q^{52} + 4182 q^{53} - 3645 q^{54} - 4888 q^{55} + 15588 q^{57} + 27590 q^{58} + 11220 q^{59} - 5922 q^{60} + 45602 q^{61} - 31680 q^{62} + 36457 q^{64} - 72380 q^{65} - 2340 q^{66} + 1396 q^{67} - 14154 q^{68} + 5184 q^{69} + 18720 q^{71} - 15795 q^{72} - 46362 q^{73} - 36690 q^{74} - 51399 q^{75} + 12124 q^{76} - 34650 q^{78} + 97424 q^{79} + 70594 q^{80} + 6561 q^{81} + 16310 q^{82} + 81228 q^{83} - 190068 q^{85} + 27100 q^{86} - 49662 q^{87} - 10140 q^{88} + 3182 q^{89} - 38070 q^{90} + 4032 q^{92} + 57024 q^{93} - 4320 q^{94} + 162808 q^{95} - 22365 q^{96} - 4914 q^{97} + 4212 q^{99}+O(q^{100}) \) 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
5.00000 −9.00000 −7.00000 −94.0000 −45.0000 0 −195.000 81.0000 −470.000
\(n\): 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.f 1
3.b odd 2 1 441.6.a.c 1
7.b odd 2 1 21.6.a.c 1
7.c even 3 2 147.6.e.d 2
7.d odd 6 2 147.6.e.c 2
21.c even 2 1 63.6.a.b 1
28.d even 2 1 336.6.a.i 1
35.c odd 2 1 525.6.a.b 1
35.f even 4 2 525.6.d.c 2
84.h odd 2 1 1008.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 7.b odd 2 1
63.6.a.b 1 21.c even 2 1
147.6.a.f 1 1.a even 1 1 trivial
147.6.e.c 2 7.d odd 6 2
147.6.e.d 2 7.c even 3 2
336.6.a.i 1 28.d even 2 1
441.6.a.c 1 3.b odd 2 1
525.6.a.b 1 35.c odd 2 1
525.6.d.c 2 35.f even 4 2
1008.6.a.a 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} - 5 \) Copy content Toggle raw display
\( T_{5} + 94 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 5 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T + 94 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 52 \) Copy content Toggle raw display
$13$ \( T - 770 \) Copy content Toggle raw display
$17$ \( T - 2022 \) Copy content Toggle raw display
$19$ \( T + 1732 \) Copy content Toggle raw display
$23$ \( T + 576 \) Copy content Toggle raw display
$29$ \( T - 5518 \) Copy content Toggle raw display
$31$ \( T + 6336 \) Copy content Toggle raw display
$37$ \( T + 7338 \) Copy content Toggle raw display
$41$ \( T - 3262 \) Copy content Toggle raw display
$43$ \( T - 5420 \) Copy content Toggle raw display
$47$ \( T + 864 \) Copy content Toggle raw display
$53$ \( T - 4182 \) Copy content Toggle raw display
$59$ \( T - 11220 \) Copy content Toggle raw display
$61$ \( T - 45602 \) Copy content Toggle raw display
$67$ \( T - 1396 \) Copy content Toggle raw display
$71$ \( T - 18720 \) Copy content Toggle raw display
$73$ \( T + 46362 \) Copy content Toggle raw display
$79$ \( T - 97424 \) Copy content Toggle raw display
$83$ \( T - 81228 \) Copy content Toggle raw display
$89$ \( T - 3182 \) Copy content Toggle raw display
$97$ \( T + 4914 \) Copy content Toggle raw display
show more
show less