Properties

Label 147.6.a.m
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 9 \beta_1 + 9) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 27 \beta_1 + 38) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 9 \beta_1 + 9) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 27 \beta_1 + 38) q^{8} + 81 q^{9} + ( - \beta_{2} - 23 \beta_1 + 76) q^{10} + (3 \beta_{3} + 9 \beta_{2} - 8 \beta_1 + 107) q^{11} + (9 \beta_{2} - 9 \beta_1 + 162) q^{12} + (9 \beta_{3} - \beta_{2} + 76 \beta_1 + 96) q^{13} + (9 \beta_{3} + 9 \beta_{2} - 18 \beta_1 + 9) q^{15} + ( - 6 \beta_{3} + \beta_{2} - 85 \beta_1 + 840) q^{16} + (4 \beta_{3} - 8 \beta_{2} - 108 \beta_1 + 92) q^{17} + ( - 81 \beta_1 + 81) q^{18} + ( - 15 \beta_{3} - \beta_{2} + 220 \beta_1 + 72) q^{19} + ( - 30 \beta_{3} - 9 \beta_{2} + 29 \beta_1 + 1168) q^{20} + ( - 12 \beta_{3} - \beta_{2} - 419 \beta_1 + 448) q^{22} + (20 \beta_{3} + 8 \beta_{2} - 300 \beta_1 + 1804) q^{23} + ( - 18 \beta_{3} + 9 \beta_{2} - 243 \beta_1 + 342) q^{24} + ( - 45 \beta_{3} - 95 \beta_{2} + 80 \beta_1 + 636) q^{25} + (20 \beta_{3} - 103 \beta_{2} + 116 \beta_1 - 3865) q^{26} + 729 q^{27} + (5 \beta_{3} - 103 \beta_{2} - 254 \beta_1 + 147) q^{29} + ( - 9 \beta_{2} - 207 \beta_1 + 684) q^{30} + (24 \beta_{3} + 94 \beta_{2} - 130 \beta_1 - 1523) q^{31} + (50 \beta_{3} + 71 \beta_{2} - 131 \beta_1 + 3948) q^{32} + (27 \beta_{3} + 81 \beta_{2} - 72 \beta_1 + 963) q^{33} + (24 \beta_{3} + 96 \beta_{2} + 312 \beta_1 + 5256) q^{34} + (81 \beta_{2} - 81 \beta_1 + 1458) q^{36} + ( - 9 \beta_{3} - 95 \beta_{2} + 1028 \beta_1 + 3508) q^{37} + ( - 28 \beta_{3} - 175 \beta_{2} - 316 \beta_1 - 10321) q^{38} + (81 \beta_{3} - 9 \beta_{2} + 684 \beta_1 + 864) q^{39} + ( - 42 \beta_{3} + 93 \beta_{2} - 633 \beta_1 - 1932) q^{40} + (142 \beta_{3} - 62 \beta_{2} - 328 \beta_1 + 1128) q^{41} + ( - 33 \beta_{3} + 93 \beta_{2} + 816 \beta_1 + 7142) q^{43} + ( - 118 \beta_{3} + 167 \beta_{2} - 379 \beta_1 + 17864) q^{44} + (81 \beta_{3} + 81 \beta_{2} - 162 \beta_1 + 81) q^{45} + (24 \beta_{3} + 240 \beta_{2} - 1752 \beta_1 + 16008) q^{46} + ( - 28 \beta_{3} + 56 \beta_{2} + 324 \beta_1 - 3818) q^{47} + ( - 54 \beta_{3} + 9 \beta_{2} - 765 \beta_1 + 7560) q^{48} + (100 \beta_{3} + 55 \beta_{2} + 2404 \beta_1 - 2399) q^{50} + (36 \beta_{3} - 72 \beta_{2} - 972 \beta_1 + 828) q^{51} + ( - 42 \beta_{3} - 144 \beta_{2} + 6036 \beta_1 - 13450) q^{52} + ( - 239 \beta_{3} + 13 \beta_{2} + 1338 \beta_1 + 3095) q^{53} + ( - 729 \beta_1 + 729) q^{54} + ( - 315 \beta_{3} - 335 \beta_{2} + 506 \beta_1 + 17987) q^{55} + ( - 135 \beta_{3} - 9 \beta_{2} + 1980 \beta_1 + 648) q^{57} + (216 \beta_{3} + 239 \beta_{2} + 4171 \beta_1 + 12154) q^{58} + ( - 71 \beta_{3} + 163 \beta_{2} - 888 \beta_1 + 9011) q^{59} + ( - 270 \beta_{3} - 81 \beta_{2} + 261 \beta_1 + 10512) q^{60} + (240 \beta_{3} - 52 \beta_{2} + 796 \beta_1 - 1566) q^{61} + ( - 140 \beta_{3} + 58 \beta_{2} - 1875 \beta_1 + 4505) q^{62} + (150 \beta_{3} - 51 \beta_{2} - 3189 \beta_1 - 17600) q^{64} + (246 \beta_{3} - 330 \beta_{2} + 1660 \beta_1 + 16136) q^{65} + ( - 108 \beta_{3} - 9 \beta_{2} - 3771 \beta_1 + 4032) q^{66} + (465 \beta_{3} - 193 \beta_{2} + 2764 \beta_1 - 2286) q^{67} + ( - 272 \beta_{3} - 128 \beta_{2} - 5280 \beta_1 - 13312) q^{68} + (180 \beta_{3} + 72 \beta_{2} - 2700 \beta_1 + 16236) q^{69} + ( - 180 \beta_{3} - 24 \beta_{2} + 3660 \beta_1 + 21390) q^{71} + ( - 162 \beta_{3} + 81 \beta_{2} - 2187 \beta_1 + 3078) q^{72} + ( - 93 \beta_{3} - 143 \beta_{2} + 3056 \beta_1 + 14074) q^{73} + (172 \beta_{3} - 1001 \beta_{2} + 216 \beta_1 - 46915) q^{74} + ( - 405 \beta_{3} - 855 \beta_{2} + 720 \beta_1 + 5724) q^{75} + (774 \beta_{3} + 432 \beta_{2} + 9924 \beta_1 + 3062) q^{76} + (180 \beta_{3} - 927 \beta_{2} + 1044 \beta_1 - 34785) q^{78} + ( - 786 \beta_{3} - 96 \beta_{2} + 2286 \beta_1 - 11635) q^{79} + (690 \beta_{3} + 1047 \beta_{2} - 3607 \beta_1 - 6920) q^{80} + 6561 q^{81} + (408 \beta_{3} - 98 \beta_{2} + 4112 \beta_1 + 13322) q^{82} + ( - 129 \beta_{3} + 1005 \beta_{2} - 6432 \beta_1 - 50001) q^{83} + (372 \beta_{3} - 192 \beta_{2} - 2988 \beta_1 + 9732) q^{85} + ( - 252 \beta_{3} - 717 \beta_{2} - 11582 \beta_1 - 31705) q^{86} + (45 \beta_{3} - 927 \beta_{2} - 2286 \beta_1 + 1323) q^{87} + ( - 186 \beta_{3} + 765 \beta_{2} - 13545 \beta_1 + 25668) q^{88} + (622 \beta_{3} + 1582 \beta_{2} + 9508 \beta_1 - 20966) q^{89} + ( - 81 \beta_{2} - 1863 \beta_1 + 6156) q^{90} + ( - 1072 \beta_{3} + 1424 \beta_{2} - 15792 \beta_1 + 44224) q^{92} + (216 \beta_{3} + 846 \beta_{2} - 1170 \beta_1 - 13707) q^{93} + ( - 168 \beta_{3} - 240 \beta_{2} + 990 \beta_1 - 18798) q^{94} + (294 \beta_{3} + 1542 \beta_{2} + 3820 \beta_1 - 55528) q^{95} + (450 \beta_{3} + 639 \beta_{2} - 1179 \beta_1 + 35532) q^{96} + ( - 669 \beta_{3} - 863 \beta_{2} + 464 \beta_1 - 47705) q^{97} + (243 \beta_{3} + 729 \beta_{2} - 648 \beta_1 + 8667) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22} + 6900 q^{23} + 1107 q^{24} + 2814 q^{25} - 15138 q^{26} + 2916 q^{27} + 540 q^{29} + 2547 q^{30} - 6410 q^{31} + 15519 q^{32} + 3618 q^{33} + 21144 q^{34} + 5589 q^{36} + 15250 q^{37} - 41250 q^{38} + 4158 q^{39} - 8547 q^{40} + 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 29457 q^{48} - 7302 q^{50} + 2484 q^{51} - 47476 q^{52} + 13692 q^{53} + 2187 q^{54} + 73124 q^{55} + 4590 q^{57} + 52309 q^{58} + 34830 q^{59} + 42471 q^{60} - 5364 q^{61} + 16029 q^{62} - 73487 q^{64} + 66864 q^{65} + 12375 q^{66} - 5994 q^{67} - 58272 q^{68} + 62100 q^{69} + 89268 q^{71} + 9963 q^{72} + 59638 q^{73} - 185442 q^{74} + 25326 q^{75} + 21308 q^{76} - 136242 q^{78} - 44062 q^{79} - 33381 q^{80} + 26244 q^{81} + 57596 q^{82} - 208446 q^{83} + 36324 q^{85} - 136968 q^{86} + 4860 q^{87} + 87597 q^{88} - 77520 q^{89} + 22923 q^{90} + 158256 q^{92} - 57690 q^{93} - 73722 q^{94} - 221376 q^{95} + 139671 q^{96} - 188630 q^{97} + 32562 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 89\nu + 52 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 91\beta _1 + 46 \) 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
10.1812
1.79080
−1.74818
−9.22385
−9.18123 9.00000 52.2950 22.0716 −82.6311 0 −186.333 81.0000 −202.644
1.2 −0.790805 9.00000 −31.3746 −104.192 −7.11724 0 50.1170 81.0000 82.3953
1.3 2.74818 9.00000 −24.4475 58.3673 24.7336 0 −155.128 81.0000 160.404
1.4 10.2239 9.00000 72.5272 23.7528 92.0147 0 414.344 81.0000 242.845
\(n\): 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.m 4
3.b odd 2 1 441.6.a.w 4
7.b odd 2 1 147.6.a.l 4
7.c even 3 2 21.6.e.c 8
7.d odd 6 2 147.6.e.o 8
21.c even 2 1 441.6.a.v 4
21.h odd 6 2 63.6.e.e 8
28.g odd 6 2 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 7.c even 3 2
63.6.e.e 8 21.h odd 6 2
147.6.a.l 4 7.b odd 2 1
147.6.a.m 4 1.a even 1 1 trivial
147.6.e.o 8 7.d odd 6 2
336.6.q.j 8 28.g odd 6 2
441.6.a.v 4 21.c even 2 1
441.6.a.w 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{4} - 3T_{2}^{3} - 94T_{2}^{2} + 186T_{2} + 204 \) Copy content Toggle raw display
\( T_{5}^{4} - 7657T_{5}^{2} + 302700T_{5} - 3188244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} - 94 T^{2} + 186 T + 204 \) Copy content Toggle raw display
$3$ \( (T - 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7657 T^{2} + \cdots - 3188244 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 402 T^{3} + \cdots - 1682132124 \) Copy content Toggle raw display
$13$ \( T^{4} - 462 T^{3} + \cdots + 149501563456 \) Copy content Toggle raw display
$17$ \( T^{4} - 276 T^{3} + \cdots - 50104147968 \) Copy content Toggle raw display
$19$ \( T^{4} - 510 T^{3} + \cdots + 7391138416576 \) Copy content Toggle raw display
$23$ \( T^{4} - 6900 T^{3} + \cdots + 3007939608576 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 408027025117872 \) Copy content Toggle raw display
$31$ \( T^{4} + 6410 T^{3} + \cdots + 86716089209547 \) Copy content Toggle raw display
$37$ \( T^{4} - 15250 T^{3} + \cdots - 50\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} - 4308 T^{3} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 991662745581932 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 270685655359056 \) Copy content Toggle raw display
$53$ \( T^{4} - 13692 T^{3} + \cdots - 85\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{4} - 34830 T^{3} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + 5364 T^{3} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + 5994 T^{3} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} - 89268 T^{3} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{4} - 59638 T^{3} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + 44062 T^{3} + \cdots + 16\!\cdots\!59 \) Copy content Toggle raw display
$83$ \( T^{4} + 208446 T^{3} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{4} + 77520 T^{3} + \cdots - 47\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{4} + 188630 T^{3} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
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