Properties

Label 147.6.e.p
Level $147$
Weight $6$
Character orbit 147.e
Analytic conductor $23.576$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} + 63 x^{10} - 126 x^{9} + 2784 x^{8} - 5290 x^{7} + 62015 x^{6} - 99530 x^{5} + 973971 x^{4} - 1176024 x^{3} + 5644794 x^{2} + 4339328 x + 5466244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - 9 \beta_{2} - 9) q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} - 25 \beta_{2} - 25) q^{4} + (\beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_{3} + 16 \beta_{2} - 2 \beta_1) q^{5} + (9 \beta_{6} + 9 \beta_1) q^{6} + ( - 2 \beta_{11} - 3 \beta_{8} + 2 \beta_{7} + 25 \beta_{6} - 9 \beta_{3} + \cdots - 28) q^{8}+ \cdots + 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - 9 \beta_{2} - 9) q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} - 25 \beta_{2} - 25) q^{4} + (\beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_{3} + 16 \beta_{2} - 2 \beta_1) q^{5} + (9 \beta_{6} + 9 \beta_1) q^{6} + ( - 2 \beta_{11} - 3 \beta_{8} + 2 \beta_{7} + 25 \beta_{6} - 9 \beta_{3} + \cdots - 28) q^{8}+ \cdots + (567 \beta_{11} - 243 \beta_{8} + 648 \beta_{7} - 972 \beta_{6} + \cdots + 8667) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 54 q^{3} - 150 q^{4} - 100 q^{5} + 36 q^{6} - 228 q^{8} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 54 q^{3} - 150 q^{4} - 100 q^{5} + 36 q^{6} - 228 q^{8} - 486 q^{9} - 864 q^{10} - 604 q^{11} - 1350 q^{12} + 2704 q^{13} + 1800 q^{15} - 4578 q^{16} - 3028 q^{17} - 162 q^{18} - 1728 q^{19} + 904 q^{20} - 8232 q^{22} + 4484 q^{23} + 1026 q^{24} - 4806 q^{25} - 14172 q^{26} + 8748 q^{27} - 10640 q^{29} - 7776 q^{30} - 3976 q^{31} + 37326 q^{32} - 5436 q^{33} - 32672 q^{34} + 24300 q^{36} - 22680 q^{37} - 52744 q^{38} - 12168 q^{39} - 100600 q^{40} + 57512 q^{41} - 13536 q^{43} + 64940 q^{44} - 8100 q^{45} - 540 q^{46} - 51552 q^{47} + 82404 q^{48} - 81244 q^{50} - 27252 q^{51} - 119296 q^{52} - 80884 q^{53} - 1458 q^{54} + 23312 q^{55} + 31104 q^{57} + 70464 q^{58} - 8872 q^{59} - 4068 q^{60} - 50896 q^{61} + 23648 q^{62} + 399180 q^{64} - 3492 q^{65} + 37044 q^{66} - 6480 q^{67} - 37348 q^{68} - 80712 q^{69} - 221704 q^{71} + 9234 q^{72} - 64232 q^{73} + 27464 q^{74} - 43254 q^{75} - 389728 q^{76} + 255096 q^{78} - 111696 q^{79} + 308940 q^{80} - 39366 q^{81} + 189640 q^{82} + 202256 q^{83} - 46584 q^{85} - 3824 q^{86} + 47880 q^{87} + 97788 q^{88} + 35012 q^{89} + 139968 q^{90} - 898520 q^{92} - 35784 q^{93} + 121016 q^{94} + 119080 q^{95} + 335934 q^{96} + 141904 q^{97} + 97848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 63 x^{10} - 126 x^{9} + 2784 x^{8} - 5290 x^{7} + 62015 x^{6} - 99530 x^{5} + 973971 x^{4} - 1176024 x^{3} + 5644794 x^{2} + 4339328 x + 5466244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 22\!\cdots\!63 \nu^{11} + \cdots - 53\!\cdots\!20 ) / 82\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 74\!\cdots\!16 \nu^{11} + \cdots - 25\!\cdots\!72 ) / 42\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1293273694 \nu^{11} - 2504891733 \nu^{10} - 58398312001 \nu^{9} - 116805928632 \nu^{8} - 3055159408682 \nu^{7} + \cdots - 19\!\cdots\!72 ) / 18\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 63\!\cdots\!97 \nu^{11} + \cdots + 13\!\cdots\!02 ) / 58\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!82 \nu^{11} + \cdots - 23\!\cdots\!16 ) / 91\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!34 \nu^{11} + \cdots - 12\!\cdots\!62 ) / 82\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!52 \nu^{11} + \cdots + 13\!\cdots\!64 ) / 54\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!60 \nu^{11} + \cdots - 39\!\cdots\!52 ) / 49\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!85 \nu^{11} + \cdots + 11\!\cdots\!66 ) / 41\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 86\!\cdots\!74 \nu^{11} + \cdots + 21\!\cdots\!48 ) / 82\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!56 \nu^{11} + \cdots - 38\!\cdots\!66 ) / 49\!\cdots\!06 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} + \beta_{9} + 12\beta_{6} - 4\beta_{4} + 5\beta_{2} + 5 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{10} - 5\beta_{9} - 5\beta_{8} - 4\beta_{5} - 4\beta_{4} - 4\beta_{3} + 567\beta_{2} - 4\beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{11} + 28\beta_{8} + 4\beta_{7} - 302\beta_{6} - 25\beta_{3} - 302\beta _1 + 129 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 99 \beta_{11} + 99 \beta_{10} + 155 \beta_{9} + 176 \beta_{7} + 80 \beta_{6} + 176 \beta_{5} - 74 \beta_{4} - 14843 \beta_{2} - 14843 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 862 \beta_{10} - 737 \beta_{9} - 737 \beta_{8} + 192 \beta_{5} - 643 \beta_{4} - 643 \beta_{3} + 9528 \beta_{2} + 8710 \beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1530\beta_{11} + 2263\beta_{8} - 3258\beta_{7} - 2818\beta_{6} - 2521\beta_{3} - 2818\beta _1 + 206808 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25517 \beta_{11} - 25517 \beta_{10} + 20051 \beta_{9} - 6548 \beta_{7} + 263676 \beta_{6} - 6548 \beta_{5} + 45742 \beta_{4} - 360335 \beta_{2} - 360335 ) / 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 94089 \beta_{10} - 130713 \beta_{9} - 130713 \beta_{8} - 225276 \beta_{5} + 185130 \beta_{4} + 185130 \beta_{3} + 11964049 \beta_{2} + 222648 \beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 760765 \beta_{11} + 558022 \beta_{8} + 197268 \beta_{7} - 8160666 \beta_{6} + 1927745 \beta_{3} - 8160666 \beta _1 + 11579965 ) / 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2911581 \beta_{11} + 2911581 \beta_{10} + 3777797 \beta_{9} + 7523248 \beta_{7} + 7701176 \beta_{6} + 7523248 \beta_{5} - 5819534 \beta_{4} - 354486141 \beta_{2} + \cdots - 354486141 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22912658 \beta_{10} - 15769549 \beta_{9} - 15769549 \beta_{8} + 5582560 \beta_{5} - 70608227 \beta_{4} - 70608227 \beta_{3} + 349389564 \beta_{2} + 255024446 \beta_1 ) / 28 \) Copy content Toggle raw display

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
2.79840 4.84697i
1.87676 3.25065i
−2.55045 + 4.41750i
2.13607 3.69977i
−0.455061 + 0.788188i
−2.80572 + 4.85966i
2.79840 + 4.84697i
1.87676 + 3.25065i
−2.55045 4.41750i
2.13607 + 3.69977i
−0.455061 0.788188i
−2.80572 4.85966i
−5.00445 8.66795i −4.50000 + 7.79423i −34.0890 + 59.0438i −35.1756 60.9260i 90.0800 0 362.101 −40.5000 70.1481i −352.069 + 609.801i
67.2 −4.10431 7.10888i −4.50000 + 7.79423i −17.6908 + 30.6414i −14.6130 25.3104i 73.8777 0 27.7583 −40.5000 70.1481i −119.952 + 207.764i
67.3 −1.69017 2.92745i −4.50000 + 7.79423i 10.2867 17.8171i 27.2626 + 47.2203i 30.4230 0 −177.715 −40.5000 70.1481i 92.1567 159.620i
67.4 1.54581 + 2.67743i −4.50000 + 7.79423i 11.2209 19.4352i −6.89630 11.9448i −27.8246 0 168.314 −40.5000 70.1481i 21.3208 36.9287i
67.5 2.65908 + 4.60566i −4.50000 + 7.79423i 1.85862 3.21922i −51.7353 89.6081i −47.8634 0 189.950 −40.5000 70.1481i 275.136 476.550i
67.6 5.59404 + 9.68915i −4.50000 + 7.79423i −46.5865 + 80.6901i 31.1575 + 53.9664i −100.693 0 −684.407 −40.5000 70.1481i −348.592 + 603.780i
79.1 −5.00445 + 8.66795i −4.50000 7.79423i −34.0890 59.0438i −35.1756 + 60.9260i 90.0800 0 362.101 −40.5000 + 70.1481i −352.069 609.801i
79.2 −4.10431 + 7.10888i −4.50000 7.79423i −17.6908 30.6414i −14.6130 + 25.3104i 73.8777 0 27.7583 −40.5000 + 70.1481i −119.952 207.764i
79.3 −1.69017 + 2.92745i −4.50000 7.79423i 10.2867 + 17.8171i 27.2626 47.2203i 30.4230 0 −177.715 −40.5000 + 70.1481i 92.1567 + 159.620i
79.4 1.54581 2.67743i −4.50000 7.79423i 11.2209 + 19.4352i −6.89630 + 11.9448i −27.8246 0 168.314 −40.5000 + 70.1481i 21.3208 + 36.9287i
79.5 2.65908 4.60566i −4.50000 7.79423i 1.85862 + 3.21922i −51.7353 + 89.6081i −47.8634 0 189.950 −40.5000 + 70.1481i 275.136 + 476.550i
79.6 5.59404 9.68915i −4.50000 7.79423i −46.5865 80.6901i 31.1575 53.9664i −100.693 0 −684.407 −40.5000 + 70.1481i −348.592 603.780i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
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.p 12
7.b odd 2 1 147.6.e.q 12
7.c even 3 1 147.6.a.o yes 6
7.c even 3 1 inner 147.6.e.p 12
7.d odd 6 1 147.6.a.n 6
7.d odd 6 1 147.6.e.q 12
21.g even 6 1 441.6.a.bb 6
21.h odd 6 1 441.6.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n 6 7.d odd 6 1
147.6.a.o yes 6 7.c even 3 1
147.6.e.p 12 1.a even 1 1 trivial
147.6.e.p 12 7.c even 3 1 inner
147.6.e.q 12 7.b odd 2 1
147.6.e.q 12 7.d odd 6 1
441.6.a.ba 6 21.h odd 6 1
441.6.a.bb 6 21.g even 6 1

Hecke kernels

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

\( T_{2}^{12} + 2 T_{2}^{11} + 173 T_{2}^{10} + 334 T_{2}^{9} + 22761 T_{2}^{8} + 35152 T_{2}^{7} + 1095976 T_{2}^{6} - 838240 T_{2}^{5} + 34682928 T_{2}^{4} - 6786304 T_{2}^{3} + 348755072 T_{2}^{2} + \cdots + 2609983744 \) Copy content Toggle raw display
\( T_{5}^{12} + 100 T_{5}^{11} + 16778 T_{5}^{10} + 624824 T_{5}^{9} + 101403192 T_{5}^{8} + 3439946000 T_{5}^{7} + 413764540744 T_{5}^{6} + 7702343637280 T_{5}^{5} + \cdots + 99\!\cdots\!44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 2609983744 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 100 T^{11} + \cdots + 99\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 604 T^{11} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{6} - 1352 T^{5} + \cdots + 88\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 3028 T^{11} + \cdots + 60\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{12} + 1728 T^{11} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} - 4484 T^{11} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + 5320 T^{5} + \cdots - 45\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 3976 T^{11} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{12} + 22680 T^{11} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} - 28756 T^{5} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6768 T^{5} + \cdots - 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 51552 T^{11} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + 80884 T^{11} + \cdots + 87\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + 8872 T^{11} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{12} + 50896 T^{11} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} + 6480 T^{11} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{6} + 110852 T^{5} + \cdots - 29\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 64232 T^{11} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + 111696 T^{11} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{6} - 101128 T^{5} + \cdots - 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 35012 T^{11} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{6} - 70952 T^{5} + \cdots - 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
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