Properties

Label 147.6.e.b
Level $147$
Weight $6$
Character orbit 147.e
Analytic conductor $23.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(1\)
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.

\(f(q)\) \(=\) \( q - 10 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (68 \zeta_{6} - 68) q^{4} - 106 \zeta_{6} q^{5} - 90 q^{6} + 360 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 10 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (68 \zeta_{6} - 68) q^{4} - 106 \zeta_{6} q^{5} - 90 q^{6} + 360 q^{8} - 81 \zeta_{6} q^{9} + (1060 \zeta_{6} - 1060) q^{10} + (92 \zeta_{6} - 92) q^{11} + 612 \zeta_{6} q^{12} - 670 q^{13} - 954 q^{15} - 1424 \zeta_{6} q^{16} + (222 \zeta_{6} - 222) q^{17} + (810 \zeta_{6} - 810) q^{18} - 908 \zeta_{6} q^{19} + 7208 q^{20} + 920 q^{22} + 1176 \zeta_{6} q^{23} + ( - 3240 \zeta_{6} + 3240) q^{24} + (8111 \zeta_{6} - 8111) q^{25} + 6700 \zeta_{6} q^{26} - 729 q^{27} + 1118 q^{29} + 9540 \zeta_{6} q^{30} + ( - 3696 \zeta_{6} + 3696) q^{31} + (2720 \zeta_{6} - 2720) q^{32} + 828 \zeta_{6} q^{33} + 2220 q^{34} + 5508 q^{36} - 4182 \zeta_{6} q^{37} + (9080 \zeta_{6} - 9080) q^{38} + (6030 \zeta_{6} - 6030) q^{39} - 38160 \zeta_{6} q^{40} + 6662 q^{41} - 3700 q^{43} - 6256 \zeta_{6} q^{44} + (8586 \zeta_{6} - 8586) q^{45} + ( - 11760 \zeta_{6} + 11760) q^{46} - 7056 \zeta_{6} q^{47} - 12816 q^{48} + 81110 q^{50} + 1998 \zeta_{6} q^{51} + ( - 45560 \zeta_{6} + 45560) q^{52} + ( - 37578 \zeta_{6} + 37578) q^{53} + 7290 \zeta_{6} q^{54} + 9752 q^{55} - 8172 q^{57} - 11180 \zeta_{6} q^{58} + ( - 32700 \zeta_{6} + 32700) q^{59} + ( - 64872 \zeta_{6} + 64872) q^{60} - 10802 \zeta_{6} q^{61} - 36960 q^{62} - 18368 q^{64} + 71020 \zeta_{6} q^{65} + ( - 8280 \zeta_{6} + 8280) q^{66} + (64996 \zeta_{6} - 64996) q^{67} - 15096 \zeta_{6} q^{68} + 10584 q^{69} - 61320 q^{71} - 29160 \zeta_{6} q^{72} + ( - 38922 \zeta_{6} + 38922) q^{73} + (41820 \zeta_{6} - 41820) q^{74} + 72999 \zeta_{6} q^{75} + 61744 q^{76} + 60300 q^{78} + 88096 \zeta_{6} q^{79} + (150944 \zeta_{6} - 150944) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 66620 \zeta_{6} q^{82} - 71892 q^{83} + 23532 q^{85} + 37000 \zeta_{6} q^{86} + ( - 10062 \zeta_{6} + 10062) q^{87} + (33120 \zeta_{6} - 33120) q^{88} + 111818 \zeta_{6} q^{89} + 85860 q^{90} - 79968 q^{92} - 33264 \zeta_{6} q^{93} + (70560 \zeta_{6} - 70560) q^{94} + (96248 \zeta_{6} - 96248) q^{95} + 24480 \zeta_{6} q^{96} + 150846 q^{97} + 7452 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{2} + 9 q^{3} - 68 q^{4} - 106 q^{5} - 180 q^{6} + 720 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{2} + 9 q^{3} - 68 q^{4} - 106 q^{5} - 180 q^{6} + 720 q^{8} - 81 q^{9} - 1060 q^{10} - 92 q^{11} + 612 q^{12} - 1340 q^{13} - 1908 q^{15} - 1424 q^{16} - 222 q^{17} - 810 q^{18} - 908 q^{19} + 14416 q^{20} + 1840 q^{22} + 1176 q^{23} + 3240 q^{24} - 8111 q^{25} + 6700 q^{26} - 1458 q^{27} + 2236 q^{29} + 9540 q^{30} + 3696 q^{31} - 2720 q^{32} + 828 q^{33} + 4440 q^{34} + 11016 q^{36} - 4182 q^{37} - 9080 q^{38} - 6030 q^{39} - 38160 q^{40} + 13324 q^{41} - 7400 q^{43} - 6256 q^{44} - 8586 q^{45} + 11760 q^{46} - 7056 q^{47} - 25632 q^{48} + 162220 q^{50} + 1998 q^{51} + 45560 q^{52} + 37578 q^{53} + 7290 q^{54} + 19504 q^{55} - 16344 q^{57} - 11180 q^{58} + 32700 q^{59} + 64872 q^{60} - 10802 q^{61} - 73920 q^{62} - 36736 q^{64} + 71020 q^{65} + 8280 q^{66} - 64996 q^{67} - 15096 q^{68} + 21168 q^{69} - 122640 q^{71} - 29160 q^{72} + 38922 q^{73} - 41820 q^{74} + 72999 q^{75} + 123488 q^{76} + 120600 q^{78} + 88096 q^{79} - 150944 q^{80} - 6561 q^{81} - 66620 q^{82} - 143784 q^{83} + 47064 q^{85} + 37000 q^{86} + 10062 q^{87} - 33120 q^{88} + 111818 q^{89} + 171720 q^{90} - 159936 q^{92} - 33264 q^{93} - 70560 q^{94} - 96248 q^{95} + 24480 q^{96} + 301692 q^{97} + 14904 q^{99}+O(q^{100}) \) 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
−5.00000 8.66025i 4.50000 7.79423i −34.0000 + 58.8897i −53.0000 91.7987i −90.0000 0 360.000 −40.5000 70.1481i −530.000 + 917.987i
79.1 −5.00000 + 8.66025i 4.50000 + 7.79423i −34.0000 58.8897i −53.0000 + 91.7987i −90.0000 0 360.000 −40.5000 + 70.1481i −530.000 917.987i
\(n\): 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.b 2
7.b odd 2 1 147.6.e.a 2
7.c even 3 1 147.6.a.g 1
7.c even 3 1 inner 147.6.e.b 2
7.d odd 6 1 21.6.a.d 1
7.d odd 6 1 147.6.e.a 2
21.g even 6 1 63.6.a.a 1
21.h odd 6 1 441.6.a.b 1
28.f even 6 1 336.6.a.a 1
35.i odd 6 1 525.6.a.a 1
35.k even 12 2 525.6.d.a 2
84.j odd 6 1 1008.6.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 7.d odd 6 1
63.6.a.a 1 21.g even 6 1
147.6.a.g 1 7.c even 3 1
147.6.e.a 2 7.b odd 2 1
147.6.e.a 2 7.d odd 6 1
147.6.e.b 2 1.a even 1 1 trivial
147.6.e.b 2 7.c even 3 1 inner
336.6.a.a 1 28.f even 6 1
441.6.a.b 1 21.h odd 6 1
525.6.a.a 1 35.i odd 6 1
525.6.d.a 2 35.k even 12 2
1008.6.a.bc 1 84.j odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 10T_{2} + 100 \) Copy content Toggle raw display
\( T_{5}^{2} + 106T_{5} + 11236 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 106T + 11236 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 92T + 8464 \) Copy content Toggle raw display
$13$ \( (T + 670)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 222T + 49284 \) Copy content Toggle raw display
$19$ \( T^{2} + 908T + 824464 \) Copy content Toggle raw display
$23$ \( T^{2} - 1176 T + 1382976 \) Copy content Toggle raw display
$29$ \( (T - 1118)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3696 T + 13660416 \) Copy content Toggle raw display
$37$ \( T^{2} + 4182 T + 17489124 \) Copy content Toggle raw display
$41$ \( (T - 6662)^{2} \) Copy content Toggle raw display
$43$ \( (T + 3700)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7056 T + 49787136 \) Copy content Toggle raw display
$53$ \( T^{2} - 37578 T + 1412106084 \) Copy content Toggle raw display
$59$ \( T^{2} - 32700 T + 1069290000 \) Copy content Toggle raw display
$61$ \( T^{2} + 10802 T + 116683204 \) Copy content Toggle raw display
$67$ \( T^{2} + 64996 T + 4224480016 \) Copy content Toggle raw display
$71$ \( (T + 61320)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 38922 T + 1514922084 \) Copy content Toggle raw display
$79$ \( T^{2} - 88096 T + 7760905216 \) Copy content Toggle raw display
$83$ \( (T + 71892)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 111818 T + 12503265124 \) Copy content Toggle raw display
$97$ \( (T - 150846)^{2} \) Copy content Toggle raw display
show more
show less