Properties

Label 147.6.a.n
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 9 q^{3} + (\beta_{4} - \beta_1 + 25) q^{4} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 16) q^{5} + 9 \beta_{2} q^{6} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 25 \beta_{2} + 9 \beta_1 - 28) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 9 q^{3} + (\beta_{4} - \beta_1 + 25) q^{4} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 16) q^{5} + 9 \beta_{2} q^{6} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 25 \beta_{2} + 9 \beta_1 - 28) q^{8} + 81 q^{9} + ( - 3 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 19 \beta_1 - 138) q^{10} + ( - 7 \beta_{5} - 8 \beta_{4} - 3 \beta_{3} + 12 \beta_{2} + 6 \beta_1 + 107) q^{11} + ( - 9 \beta_{4} + 9 \beta_1 - 225) q^{12} + (3 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 34 \beta_{2} - 12 \beta_1 - 215) q^{13} + ( - 9 \beta_{3} - 18 \beta_{2} + 18 \beta_1 + 144) q^{15} + (12 \beta_{5} + 33 \beta_{4} + 18 \beta_{3} + 126 \beta_{2} - 27 \beta_1 + 801) q^{16} + (17 \beta_{5} + 8 \beta_{4} + 7 \beta_{3} - 32 \beta_{2} + 31 \beta_1 - 521) q^{17} - 81 \beta_{2} q^{18} + (3 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} + 174 \beta_{2} + 55 \beta_1 - 231) q^{19} + ( - 14 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} + 335 \beta_{2} - 100 \beta_1 + 41) q^{20} + ( - 36 \beta_{5} - 32 \beta_{4} + 18 \beta_{3} + 30 \beta_{2} - 238 \beta_1 - 664) q^{22} + (51 \beta_{5} - 24 \beta_{4} + 45 \beta_{3} + 16 \beta_{2} - 759) q^{23} + ( - 18 \beta_{5} + 18 \beta_{4} + 27 \beta_{3} + 225 \beta_{2} + \cdots + 252) q^{24}+ \cdots + ( - 567 \beta_{5} - 648 \beta_{4} - 243 \beta_{3} + 972 \beta_{2} + \cdots + 8667) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9} - 864 q^{10} + 604 q^{11} - 1350 q^{12} - 1352 q^{13} + 900 q^{15} + 4578 q^{16} - 3028 q^{17} + 162 q^{18} - 1728 q^{19} - 452 q^{20} - 4116 q^{22} - 4484 q^{23} + 1026 q^{24} + 4806 q^{25} - 14172 q^{26} - 4374 q^{27} - 5320 q^{29} + 7776 q^{30} - 3976 q^{31} - 37326 q^{32} - 5436 q^{33} + 16336 q^{34} + 12150 q^{36} + 22680 q^{37} - 52744 q^{38} + 12168 q^{39} - 100600 q^{40} - 28756 q^{41} - 6768 q^{43} - 64940 q^{44} - 8100 q^{45} + 540 q^{46} - 51552 q^{47} - 41202 q^{48} - 40622 q^{50} + 27252 q^{51} - 119296 q^{52} + 80884 q^{53} - 1458 q^{54} - 11656 q^{55} + 15552 q^{57} - 70464 q^{58} - 8872 q^{59} + 4068 q^{60} - 50896 q^{61} - 11824 q^{62} + 199590 q^{64} + 3492 q^{65} + 37044 q^{66} + 6480 q^{67} - 37348 q^{68} + 40356 q^{69} - 110852 q^{71} - 9234 q^{72} - 64232 q^{73} - 27464 q^{74} - 43254 q^{75} + 194864 q^{76} + 127548 q^{78} + 111696 q^{79} + 308940 q^{80} + 39366 q^{81} + 189640 q^{82} - 101128 q^{83} - 23292 q^{85} + 3824 q^{86} + 47880 q^{87} - 97788 q^{88} + 35012 q^{89} - 69984 q^{90} - 449260 q^{92} + 35784 q^{93} + 121016 q^{94} - 119080 q^{95} + 335934 q^{96} - 70952 q^{97} + 48924 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28\nu^{5} - 35\nu^{4} - 546\nu^{3} + 742\nu^{2} - 7063\nu + 11802 ) / 1941 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -87\nu^{5} - 53\nu^{4} + 3961\nu^{3} + 2547\nu^{2} - 42269\nu - 27289 ) / 1941 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 158\nu^{5} - 521\nu^{4} - 6316\nu^{3} + 21656\nu^{2} + 33579\nu - 143678 ) / 1941 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 108\nu^{5} + 512\nu^{4} - 5988\nu^{3} - 17195\nu^{2} + 81453\nu + 71402 ) / 647 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -774\nu^{5} - 1297\nu^{4} + 39032\nu^{3} + 55188\nu^{2} - 447553\nu - 433643 ) / 1941 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - 12\beta_{2} - 4\beta _1 + 5 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 4\beta_{4} + 5\beta_{3} - 4\beta_{2} - 4\beta _1 + 567 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{5} + 4\beta_{4} - 28\beta_{3} - 302\beta_{2} - 25\beta _1 - 129 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 99\beta_{5} + 176\beta_{4} + 155\beta_{3} + 80\beta_{2} + 74\beta _1 + 14843 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 862\beta_{5} + 192\beta_{4} - 737\beta_{3} - 8710\beta_{2} + 643\beta _1 - 9528 ) / 28 \) 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
−5.61145
−0.910122
4.27213
−5.10089
3.75353
5.59680
−11.1881 −9.00000 93.1729 62.3150 100.693 0 −684.407 81.0000 −697.185
1.2 −5.31815 −9.00000 −3.71724 −103.471 47.8634 0 189.950 81.0000 550.272
1.3 −3.09163 −9.00000 −22.4418 −13.7926 27.8246 0 168.314 81.0000 42.6416
1.4 3.38033 −9.00000 −20.5734 54.5253 −30.4230 0 −177.715 81.0000 184.313
1.5 8.20863 −9.00000 35.3816 −29.2259 −73.8777 0 27.7583 81.0000 −239.905
1.6 10.0089 −9.00000 68.1779 −70.3512 −90.0800 0 362.101 81.0000 −704.138
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.n 6
3.b odd 2 1 441.6.a.bb 6
7.b odd 2 1 147.6.a.o yes 6
7.c even 3 2 147.6.e.q 12
7.d odd 6 2 147.6.e.p 12
21.c even 2 1 441.6.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n 6 1.a even 1 1 trivial
147.6.a.o yes 6 7.b odd 2 1
147.6.e.p 12 7.d odd 6 2
147.6.e.q 12 7.c even 3 2
441.6.a.ba 6 21.c even 2 1
441.6.a.bb 6 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{6} - 2T_{2}^{5} - 169T_{2}^{4} + 336T_{2}^{3} + 6472T_{2}^{2} - 4256T_{2} - 51088 \) Copy content Toggle raw display
\( T_{5}^{6} + 100T_{5}^{5} - 6778T_{5}^{4} - 651312T_{5}^{3} + 9669292T_{5}^{2} + 959211664T_{5} + 9969962312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} - 169 T^{4} + \cdots - 51088 \) Copy content Toggle raw display
$3$ \( (T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 100 T^{5} + \cdots + 9969962312 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 604 T^{5} + \cdots - 55273527989696 \) Copy content Toggle raw display
$13$ \( T^{6} + 1352 T^{5} + \cdots + 88\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{6} + 3028 T^{5} + \cdots - 77\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{6} + 1728 T^{5} + \cdots - 37\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{6} + 4484 T^{5} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{6} + 5320 T^{5} + \cdots - 45\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{6} + 3976 T^{5} + \cdots - 32\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{6} - 22680 T^{5} + \cdots - 79\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} + 28756 T^{5} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{6} + 6768 T^{5} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{6} + 51552 T^{5} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} - 80884 T^{5} + \cdots + 93\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} + 8872 T^{5} + \cdots - 34\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{6} + 50896 T^{5} + \cdots - 18\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{6} - 6480 T^{5} + \cdots - 24\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} + 110852 T^{5} + \cdots - 29\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + 64232 T^{5} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{6} - 111696 T^{5} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} + 101128 T^{5} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{6} - 35012 T^{5} + \cdots + 48\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{6} + 70952 T^{5} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
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