Properties

Label 147.6.e.k
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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.

\(f(q)\) \(=\) \( q + 6 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} + 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 6 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} + 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} + (36 \zeta_{6} - 36) q^{10} + ( - 564 \zeta_{6} + 564) q^{11} + 36 \zeta_{6} q^{12} - 638 q^{13} + 54 q^{15} + 1136 \zeta_{6} q^{16} + ( - 882 \zeta_{6} + 882) q^{17} + ( - 486 \zeta_{6} + 486) q^{18} - 556 \zeta_{6} q^{19} - 24 q^{20} + 3384 q^{22} + 840 \zeta_{6} q^{23} + ( - 1512 \zeta_{6} + 1512) q^{24} + ( - 3089 \zeta_{6} + 3089) q^{25} - 3828 \zeta_{6} q^{26} - 729 q^{27} + 4638 q^{29} + 324 \zeta_{6} q^{30} + ( - 4400 \zeta_{6} + 4400) q^{31} + (1440 \zeta_{6} - 1440) q^{32} - 5076 \zeta_{6} q^{33} + 5292 q^{34} + 324 q^{36} + 2410 \zeta_{6} q^{37} + ( - 3336 \zeta_{6} + 3336) q^{38} + (5742 \zeta_{6} - 5742) q^{39} + 1008 \zeta_{6} q^{40} + 6870 q^{41} + 9644 q^{43} + 2256 \zeta_{6} q^{44} + ( - 486 \zeta_{6} + 486) q^{45} + (5040 \zeta_{6} - 5040) q^{46} - 18672 \zeta_{6} q^{47} + 10224 q^{48} + 18534 q^{50} - 7938 \zeta_{6} q^{51} + ( - 2552 \zeta_{6} + 2552) q^{52} + (33750 \zeta_{6} - 33750) q^{53} - 4374 \zeta_{6} q^{54} + 3384 q^{55} - 5004 q^{57} + 27828 \zeta_{6} q^{58} + (18084 \zeta_{6} - 18084) q^{59} + (216 \zeta_{6} - 216) q^{60} + 39758 \zeta_{6} q^{61} + 26400 q^{62} + 27712 q^{64} - 3828 \zeta_{6} q^{65} + ( - 30456 \zeta_{6} + 30456) q^{66} + ( - 23068 \zeta_{6} + 23068) q^{67} + 3528 \zeta_{6} q^{68} + 7560 q^{69} - 4248 q^{71} - 13608 \zeta_{6} q^{72} + (41110 \zeta_{6} - 41110) q^{73} + (14460 \zeta_{6} - 14460) q^{74} - 27801 \zeta_{6} q^{75} + 2224 q^{76} - 34452 q^{78} - 21920 \zeta_{6} q^{79} + (6816 \zeta_{6} - 6816) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 41220 \zeta_{6} q^{82} - 82452 q^{83} + 5292 q^{85} + 57864 \zeta_{6} q^{86} + ( - 41742 \zeta_{6} + 41742) q^{87} + ( - 94752 \zeta_{6} + 94752) q^{88} - 94086 \zeta_{6} q^{89} + 2916 q^{90} - 3360 q^{92} - 39600 \zeta_{6} q^{93} + ( - 112032 \zeta_{6} + 112032) q^{94} + ( - 3336 \zeta_{6} + 3336) q^{95} + 12960 \zeta_{6} q^{96} - 49442 q^{97} - 45684 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 9 q^{3} - 4 q^{4} + 6 q^{5} + 108 q^{6} + 336 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 9 q^{3} - 4 q^{4} + 6 q^{5} + 108 q^{6} + 336 q^{8} - 81 q^{9} - 36 q^{10} + 564 q^{11} + 36 q^{12} - 1276 q^{13} + 108 q^{15} + 1136 q^{16} + 882 q^{17} + 486 q^{18} - 556 q^{19} - 48 q^{20} + 6768 q^{22} + 840 q^{23} + 1512 q^{24} + 3089 q^{25} - 3828 q^{26} - 1458 q^{27} + 9276 q^{29} + 324 q^{30} + 4400 q^{31} - 1440 q^{32} - 5076 q^{33} + 10584 q^{34} + 648 q^{36} + 2410 q^{37} + 3336 q^{38} - 5742 q^{39} + 1008 q^{40} + 13740 q^{41} + 19288 q^{43} + 2256 q^{44} + 486 q^{45} - 5040 q^{46} - 18672 q^{47} + 20448 q^{48} + 37068 q^{50} - 7938 q^{51} + 2552 q^{52} - 33750 q^{53} - 4374 q^{54} + 6768 q^{55} - 10008 q^{57} + 27828 q^{58} - 18084 q^{59} - 216 q^{60} + 39758 q^{61} + 52800 q^{62} + 55424 q^{64} - 3828 q^{65} + 30456 q^{66} + 23068 q^{67} + 3528 q^{68} + 15120 q^{69} - 8496 q^{71} - 13608 q^{72} - 41110 q^{73} - 14460 q^{74} - 27801 q^{75} + 4448 q^{76} - 68904 q^{78} - 21920 q^{79} - 6816 q^{80} - 6561 q^{81} + 41220 q^{82} - 164904 q^{83} + 10584 q^{85} + 57864 q^{86} + 41742 q^{87} + 94752 q^{88} - 94086 q^{89} + 5832 q^{90} - 6720 q^{92} - 39600 q^{93} + 112032 q^{94} + 3336 q^{95} + 12960 q^{96} - 98884 q^{97} - 91368 q^{99}+O(q^{100}) \) 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
3.00000 + 5.19615i 4.50000 7.79423i −2.00000 + 3.46410i 3.00000 + 5.19615i 54.0000 0 168.000 −40.5000 70.1481i −18.0000 + 31.1769i
79.1 3.00000 5.19615i 4.50000 + 7.79423i −2.00000 3.46410i 3.00000 5.19615i 54.0000 0 168.000 −40.5000 + 70.1481i −18.0000 31.1769i
\(n\): 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.k 2
7.b odd 2 1 147.6.e.h 2
7.c even 3 1 147.6.a.a 1
7.c even 3 1 inner 147.6.e.k 2
7.d odd 6 1 3.6.a.a 1
7.d odd 6 1 147.6.e.h 2
21.g even 6 1 9.6.a.a 1
21.h odd 6 1 441.6.a.i 1
28.f even 6 1 48.6.a.a 1
35.i odd 6 1 75.6.a.e 1
35.k even 12 2 75.6.b.b 2
56.j odd 6 1 192.6.a.d 1
56.m even 6 1 192.6.a.l 1
63.i even 6 1 81.6.c.a 2
63.k odd 6 1 81.6.c.c 2
63.s even 6 1 81.6.c.a 2
63.t odd 6 1 81.6.c.c 2
77.i even 6 1 363.6.a.d 1
84.j odd 6 1 144.6.a.f 1
91.s odd 6 1 507.6.a.b 1
105.p even 6 1 225.6.a.a 1
105.w odd 12 2 225.6.b.b 2
112.v even 12 2 768.6.d.h 2
112.x odd 12 2 768.6.d.k 2
119.h odd 6 1 867.6.a.a 1
133.o even 6 1 1083.6.a.c 1
168.ba even 6 1 576.6.a.s 1
168.be odd 6 1 576.6.a.t 1
231.k odd 6 1 1089.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 7.d odd 6 1
9.6.a.a 1 21.g even 6 1
48.6.a.a 1 28.f even 6 1
75.6.a.e 1 35.i odd 6 1
75.6.b.b 2 35.k even 12 2
81.6.c.a 2 63.i even 6 1
81.6.c.a 2 63.s even 6 1
81.6.c.c 2 63.k odd 6 1
81.6.c.c 2 63.t odd 6 1
144.6.a.f 1 84.j odd 6 1
147.6.a.a 1 7.c even 3 1
147.6.e.h 2 7.b odd 2 1
147.6.e.h 2 7.d odd 6 1
147.6.e.k 2 1.a even 1 1 trivial
147.6.e.k 2 7.c even 3 1 inner
192.6.a.d 1 56.j odd 6 1
192.6.a.l 1 56.m even 6 1
225.6.a.a 1 105.p even 6 1
225.6.b.b 2 105.w odd 12 2
363.6.a.d 1 77.i even 6 1
441.6.a.i 1 21.h odd 6 1
507.6.a.b 1 91.s odd 6 1
576.6.a.s 1 168.ba even 6 1
576.6.a.t 1 168.be odd 6 1
768.6.d.h 2 112.v even 12 2
768.6.d.k 2 112.x odd 12 2
867.6.a.a 1 119.h odd 6 1
1083.6.a.c 1 133.o even 6 1
1089.6.a.b 1 231.k odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 6T_{2} + 36 \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 564T + 318096 \) Copy content Toggle raw display
$13$ \( (T + 638)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 882T + 777924 \) Copy content Toggle raw display
$19$ \( T^{2} + 556T + 309136 \) Copy content Toggle raw display
$23$ \( T^{2} - 840T + 705600 \) Copy content Toggle raw display
$29$ \( (T - 4638)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4400 T + 19360000 \) Copy content Toggle raw display
$37$ \( T^{2} - 2410 T + 5808100 \) Copy content Toggle raw display
$41$ \( (T - 6870)^{2} \) Copy content Toggle raw display
$43$ \( (T - 9644)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 18672 T + 348643584 \) Copy content Toggle raw display
$53$ \( T^{2} + 33750 T + 1139062500 \) Copy content Toggle raw display
$59$ \( T^{2} + 18084 T + 327031056 \) Copy content Toggle raw display
$61$ \( T^{2} - 39758 T + 1580698564 \) Copy content Toggle raw display
$67$ \( T^{2} - 23068 T + 532132624 \) Copy content Toggle raw display
$71$ \( (T + 4248)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 41110 T + 1690032100 \) Copy content Toggle raw display
$79$ \( T^{2} + 21920 T + 480486400 \) Copy content Toggle raw display
$83$ \( (T + 82452)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 94086 T + 8852175396 \) Copy content Toggle raw display
$97$ \( (T + 49442)^{2} \) Copy content Toggle raw display
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