Properties

Label 147.4.e.e
Level $147$
Weight $4$
Character orbit 147.e
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( 7 - 7 \zeta_{6} ) q^{4} + 12 \zeta_{6} q^{5} -3 q^{6} + 15 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( 7 - 7 \zeta_{6} ) q^{4} + 12 \zeta_{6} q^{5} -3 q^{6} + 15 q^{8} -9 \zeta_{6} q^{9} + ( -12 + 12 \zeta_{6} ) q^{10} + ( -20 + 20 \zeta_{6} ) q^{11} + 21 \zeta_{6} q^{12} + 84 q^{13} -36 q^{15} -41 \zeta_{6} q^{16} + ( -96 + 96 \zeta_{6} ) q^{17} + ( 9 - 9 \zeta_{6} ) q^{18} + 12 \zeta_{6} q^{19} + 84 q^{20} -20 q^{22} + 176 \zeta_{6} q^{23} + ( -45 + 45 \zeta_{6} ) q^{24} + ( -19 + 19 \zeta_{6} ) q^{25} + 84 \zeta_{6} q^{26} + 27 q^{27} + 58 q^{29} -36 \zeta_{6} q^{30} + ( -264 + 264 \zeta_{6} ) q^{31} + ( 161 - 161 \zeta_{6} ) q^{32} -60 \zeta_{6} q^{33} -96 q^{34} -63 q^{36} -258 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{38} + ( -252 + 252 \zeta_{6} ) q^{39} + 180 \zeta_{6} q^{40} + 156 q^{43} + 140 \zeta_{6} q^{44} + ( 108 - 108 \zeta_{6} ) q^{45} + ( -176 + 176 \zeta_{6} ) q^{46} -408 \zeta_{6} q^{47} + 123 q^{48} -19 q^{50} -288 \zeta_{6} q^{51} + ( 588 - 588 \zeta_{6} ) q^{52} + ( 722 - 722 \zeta_{6} ) q^{53} + 27 \zeta_{6} q^{54} -240 q^{55} -36 q^{57} + 58 \zeta_{6} q^{58} + ( 492 - 492 \zeta_{6} ) q^{59} + ( -252 + 252 \zeta_{6} ) q^{60} -492 \zeta_{6} q^{61} -264 q^{62} -167 q^{64} + 1008 \zeta_{6} q^{65} + ( 60 - 60 \zeta_{6} ) q^{66} + ( -412 + 412 \zeta_{6} ) q^{67} + 672 \zeta_{6} q^{68} -528 q^{69} + 296 q^{71} -135 \zeta_{6} q^{72} + ( 240 - 240 \zeta_{6} ) q^{73} + ( 258 - 258 \zeta_{6} ) q^{74} -57 \zeta_{6} q^{75} + 84 q^{76} -252 q^{78} -776 \zeta_{6} q^{79} + ( 492 - 492 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} -924 q^{83} -1152 q^{85} + 156 \zeta_{6} q^{86} + ( -174 + 174 \zeta_{6} ) q^{87} + ( -300 + 300 \zeta_{6} ) q^{88} -744 \zeta_{6} q^{89} + 108 q^{90} + 1232 q^{92} -792 \zeta_{6} q^{93} + ( 408 - 408 \zeta_{6} ) q^{94} + ( -144 + 144 \zeta_{6} ) q^{95} + 483 \zeta_{6} q^{96} + 168 q^{97} + 180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 3q^{3} + 7q^{4} + 12q^{5} - 6q^{6} + 30q^{8} - 9q^{9} + O(q^{10}) \) \( 2q + q^{2} - 3q^{3} + 7q^{4} + 12q^{5} - 6q^{6} + 30q^{8} - 9q^{9} - 12q^{10} - 20q^{11} + 21q^{12} + 168q^{13} - 72q^{15} - 41q^{16} - 96q^{17} + 9q^{18} + 12q^{19} + 168q^{20} - 40q^{22} + 176q^{23} - 45q^{24} - 19q^{25} + 84q^{26} + 54q^{27} + 116q^{29} - 36q^{30} - 264q^{31} + 161q^{32} - 60q^{33} - 192q^{34} - 126q^{36} - 258q^{37} - 12q^{38} - 252q^{39} + 180q^{40} + 312q^{43} + 140q^{44} + 108q^{45} - 176q^{46} - 408q^{47} + 246q^{48} - 38q^{50} - 288q^{51} + 588q^{52} + 722q^{53} + 27q^{54} - 480q^{55} - 72q^{57} + 58q^{58} + 492q^{59} - 252q^{60} - 492q^{61} - 528q^{62} - 334q^{64} + 1008q^{65} + 60q^{66} - 412q^{67} + 672q^{68} - 1056q^{69} + 592q^{71} - 135q^{72} + 240q^{73} + 258q^{74} - 57q^{75} + 168q^{76} - 504q^{78} - 776q^{79} + 492q^{80} - 81q^{81} - 1848q^{83} - 2304q^{85} + 156q^{86} - 174q^{87} - 300q^{88} - 744q^{89} + 216q^{90} + 2464q^{92} - 792q^{93} + 408q^{94} - 144q^{95} + 483q^{96} + 336q^{97} + 360q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i −1.50000 + 2.59808i 3.50000 6.06218i 6.00000 + 10.3923i −3.00000 0 15.0000 −4.50000 7.79423i −6.00000 + 10.3923i
79.1 0.500000 0.866025i −1.50000 2.59808i 3.50000 + 6.06218i 6.00000 10.3923i −3.00000 0 15.0000 −4.50000 + 7.79423i −6.00000 10.3923i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.e.e 2
3.b odd 2 1 441.4.e.f 2
7.b odd 2 1 147.4.e.f 2
7.c even 3 1 147.4.a.e yes 1
7.c even 3 1 inner 147.4.e.e 2
7.d odd 6 1 147.4.a.d 1
7.d odd 6 1 147.4.e.f 2
21.c even 2 1 441.4.e.g 2
21.g even 6 1 441.4.a.g 1
21.g even 6 1 441.4.e.g 2
21.h odd 6 1 441.4.a.h 1
21.h odd 6 1 441.4.e.f 2
28.f even 6 1 2352.4.a.bi 1
28.g odd 6 1 2352.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 7.d odd 6 1
147.4.a.e yes 1 7.c even 3 1
147.4.e.e 2 1.a even 1 1 trivial
147.4.e.e 2 7.c even 3 1 inner
147.4.e.f 2 7.b odd 2 1
147.4.e.f 2 7.d odd 6 1
441.4.a.g 1 21.g even 6 1
441.4.a.h 1 21.h odd 6 1
441.4.e.f 2 3.b odd 2 1
441.4.e.f 2 21.h odd 6 1
441.4.e.g 2 21.c even 2 1
441.4.e.g 2 21.g even 6 1
2352.4.a.b 1 28.g odd 6 1
2352.4.a.bi 1 28.f even 6 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \)
\( T_{5}^{2} - 12 T_{5} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 144 - 12 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 400 + 20 T + T^{2} \)
$13$ \( ( -84 + T )^{2} \)
$17$ \( 9216 + 96 T + T^{2} \)
$19$ \( 144 - 12 T + T^{2} \)
$23$ \( 30976 - 176 T + T^{2} \)
$29$ \( ( -58 + T )^{2} \)
$31$ \( 69696 + 264 T + T^{2} \)
$37$ \( 66564 + 258 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -156 + T )^{2} \)
$47$ \( 166464 + 408 T + T^{2} \)
$53$ \( 521284 - 722 T + T^{2} \)
$59$ \( 242064 - 492 T + T^{2} \)
$61$ \( 242064 + 492 T + T^{2} \)
$67$ \( 169744 + 412 T + T^{2} \)
$71$ \( ( -296 + T )^{2} \)
$73$ \( 57600 - 240 T + T^{2} \)
$79$ \( 602176 + 776 T + T^{2} \)
$83$ \( ( 924 + T )^{2} \)
$89$ \( 553536 + 744 T + T^{2} \)
$97$ \( ( -168 + T )^{2} \)
show more
show less