Properties

Label 147.6.e.g
Level $147$
Weight $6$
Character orbit 147.e
Analytic conductor $23.576$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + ( - 28 \zeta_{6} + 28) q^{4} + 11 \zeta_{6} q^{5} - 18 q^{6} + 120 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + ( - 28 \zeta_{6} + 28) q^{4} + 11 \zeta_{6} q^{5} - 18 q^{6} + 120 q^{8} - 81 \zeta_{6} q^{9} + (22 \zeta_{6} - 22) q^{10} + (269 \zeta_{6} - 269) q^{11} + 252 \zeta_{6} q^{12} + 308 q^{13} - 99 q^{15} - 656 \zeta_{6} q^{16} + ( - 1896 \zeta_{6} + 1896) q^{17} + ( - 162 \zeta_{6} + 162) q^{18} - 164 \zeta_{6} q^{19} + 308 q^{20} - 538 q^{22} + 3264 \zeta_{6} q^{23} + (1080 \zeta_{6} - 1080) q^{24} + ( - 3004 \zeta_{6} + 3004) q^{25} + 616 \zeta_{6} q^{26} + 729 q^{27} + 2417 q^{29} - 198 \zeta_{6} q^{30} + ( - 2841 \zeta_{6} + 2841) q^{31} + ( - 5152 \zeta_{6} + 5152) q^{32} - 2421 \zeta_{6} q^{33} + 3792 q^{34} - 2268 q^{36} + 11328 \zeta_{6} q^{37} + ( - 328 \zeta_{6} + 328) q^{38} + (2772 \zeta_{6} - 2772) q^{39} + 1320 \zeta_{6} q^{40} + 16856 q^{41} - 7894 q^{43} + 7532 \zeta_{6} q^{44} + ( - 891 \zeta_{6} + 891) q^{45} + (6528 \zeta_{6} - 6528) q^{46} + 21102 \zeta_{6} q^{47} + 5904 q^{48} + 6008 q^{50} + 17064 \zeta_{6} q^{51} + ( - 8624 \zeta_{6} + 8624) q^{52} + ( - 29691 \zeta_{6} + 29691) q^{53} + 1458 \zeta_{6} q^{54} - 2959 q^{55} + 1476 q^{57} + 4834 \zeta_{6} q^{58} + (8163 \zeta_{6} - 8163) q^{59} + (2772 \zeta_{6} - 2772) q^{60} + 15166 \zeta_{6} q^{61} + 5682 q^{62} - 10688 q^{64} + 3388 \zeta_{6} q^{65} + ( - 4842 \zeta_{6} + 4842) q^{66} + ( - 32078 \zeta_{6} + 32078) q^{67} - 53088 \zeta_{6} q^{68} - 29376 q^{69} - 38274 q^{71} - 9720 \zeta_{6} q^{72} + ( - 34866 \zeta_{6} + 34866) q^{73} + (22656 \zeta_{6} - 22656) q^{74} + 27036 \zeta_{6} q^{75} - 4592 q^{76} - 5544 q^{78} - 13529 \zeta_{6} q^{79} + ( - 7216 \zeta_{6} + 7216) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 33712 \zeta_{6} q^{82} + 68103 q^{83} + 20856 q^{85} - 15788 \zeta_{6} q^{86} + (21753 \zeta_{6} - 21753) q^{87} + (32280 \zeta_{6} - 32280) q^{88} - 114922 \zeta_{6} q^{89} + 1782 q^{90} + 91392 q^{92} + 25569 \zeta_{6} q^{93} + (42204 \zeta_{6} - 42204) q^{94} + ( - 1804 \zeta_{6} + 1804) q^{95} + 46368 \zeta_{6} q^{96} - 154959 q^{97} + 21789 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 9 q^{3} + 28 q^{4} + 11 q^{5} - 36 q^{6} + 240 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 9 q^{3} + 28 q^{4} + 11 q^{5} - 36 q^{6} + 240 q^{8} - 81 q^{9} - 22 q^{10} - 269 q^{11} + 252 q^{12} + 616 q^{13} - 198 q^{15} - 656 q^{16} + 1896 q^{17} + 162 q^{18} - 164 q^{19} + 616 q^{20} - 1076 q^{22} + 3264 q^{23} - 1080 q^{24} + 3004 q^{25} + 616 q^{26} + 1458 q^{27} + 4834 q^{29} - 198 q^{30} + 2841 q^{31} + 5152 q^{32} - 2421 q^{33} + 7584 q^{34} - 4536 q^{36} + 11328 q^{37} + 328 q^{38} - 2772 q^{39} + 1320 q^{40} + 33712 q^{41} - 15788 q^{43} + 7532 q^{44} + 891 q^{45} - 6528 q^{46} + 21102 q^{47} + 11808 q^{48} + 12016 q^{50} + 17064 q^{51} + 8624 q^{52} + 29691 q^{53} + 1458 q^{54} - 5918 q^{55} + 2952 q^{57} + 4834 q^{58} - 8163 q^{59} - 2772 q^{60} + 15166 q^{61} + 11364 q^{62} - 21376 q^{64} + 3388 q^{65} + 4842 q^{66} + 32078 q^{67} - 53088 q^{68} - 58752 q^{69} - 76548 q^{71} - 9720 q^{72} + 34866 q^{73} - 22656 q^{74} + 27036 q^{75} - 9184 q^{76} - 11088 q^{78} - 13529 q^{79} + 7216 q^{80} - 6561 q^{81} + 33712 q^{82} + 136206 q^{83} + 41712 q^{85} - 15788 q^{86} - 21753 q^{87} - 32280 q^{88} - 114922 q^{89} + 3564 q^{90} + 182784 q^{92} + 25569 q^{93} - 42204 q^{94} + 1804 q^{95} + 46368 q^{96} - 309918 q^{97} + 43578 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
1.00000 + 1.73205i −4.50000 + 7.79423i 14.0000 24.2487i 5.50000 + 9.52628i −18.0000 0 120.000 −40.5000 70.1481i −11.0000 + 19.0526i
79.1 1.00000 1.73205i −4.50000 7.79423i 14.0000 + 24.2487i 5.50000 9.52628i −18.0000 0 120.000 −40.5000 + 70.1481i −11.0000 19.0526i
\(n\): 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 147.6.e.g 2
7.b odd 2 1 21.6.e.a 2
7.c even 3 1 147.6.a.d 1
7.c even 3 1 inner 147.6.e.g 2
7.d odd 6 1 21.6.e.a 2
7.d odd 6 1 147.6.a.c 1
21.c even 2 1 63.6.e.a 2
21.g even 6 1 63.6.e.a 2
21.g even 6 1 441.6.a.g 1
21.h odd 6 1 441.6.a.h 1
28.d even 2 1 336.6.q.b 2
28.f even 6 1 336.6.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 7.b odd 2 1
21.6.e.a 2 7.d odd 6 1
63.6.e.a 2 21.c even 2 1
63.6.e.a 2 21.g even 6 1
147.6.a.c 1 7.d odd 6 1
147.6.a.d 1 7.c even 3 1
147.6.e.g 2 1.a even 1 1 trivial
147.6.e.g 2 7.c even 3 1 inner
336.6.q.b 2 28.d even 2 1
336.6.q.b 2 28.f even 6 1
441.6.a.g 1 21.g even 6 1
441.6.a.h 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 11T_{5} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 269T + 72361 \) Copy content Toggle raw display
$13$ \( (T - 308)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1896 T + 3594816 \) Copy content Toggle raw display
$19$ \( T^{2} + 164T + 26896 \) Copy content Toggle raw display
$23$ \( T^{2} - 3264 T + 10653696 \) Copy content Toggle raw display
$29$ \( (T - 2417)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2841 T + 8071281 \) Copy content Toggle raw display
$37$ \( T^{2} - 11328 T + 128323584 \) Copy content Toggle raw display
$41$ \( (T - 16856)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7894)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 21102 T + 445294404 \) Copy content Toggle raw display
$53$ \( T^{2} - 29691 T + 881555481 \) Copy content Toggle raw display
$59$ \( T^{2} + 8163 T + 66634569 \) Copy content Toggle raw display
$61$ \( T^{2} - 15166 T + 230007556 \) Copy content Toggle raw display
$67$ \( T^{2} - 32078 T + 1028998084 \) Copy content Toggle raw display
$71$ \( (T + 38274)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 34866 T + 1215637956 \) Copy content Toggle raw display
$79$ \( T^{2} + 13529 T + 183033841 \) Copy content Toggle raw display
$83$ \( (T - 68103)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 114922 T + 13207066084 \) Copy content Toggle raw display
$97$ \( (T + 154959)^{2} \) Copy content Toggle raw display
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