Properties

Label 546.8.a.j
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 5672 x^{3} - 117684 x^{2} + 1695035 x + 39011360\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( 11 - \beta_{2} ) q^{5} -216 q^{6} -343 q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( 11 - \beta_{2} ) q^{5} -216 q^{6} -343 q^{7} -512 q^{8} + 729 q^{9} + ( -88 + 8 \beta_{2} ) q^{10} + ( -735 - 7 \beta_{1} - 4 \beta_{3} + 5 \beta_{4} ) q^{11} + 1728 q^{12} -2197 q^{13} + 2744 q^{14} + ( 297 - 27 \beta_{2} ) q^{15} + 4096 q^{16} + ( 86 - 25 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} ) q^{17} -5832 q^{18} + ( 6750 + 66 \beta_{1} + 33 \beta_{2} + 31 \beta_{3} - 9 \beta_{4} ) q^{19} + ( 704 - 64 \beta_{2} ) q^{20} -9261 q^{21} + ( 5880 + 56 \beta_{1} + 32 \beta_{3} - 40 \beta_{4} ) q^{22} + ( -28400 - 88 \beta_{1} + 37 \beta_{2} + 16 \beta_{3} - 15 \beta_{4} ) q^{23} -13824 q^{24} + ( 30761 + 210 \beta_{1} + 39 \beta_{2} + 15 \beta_{3} - 35 \beta_{4} ) q^{25} + 17576 q^{26} + 19683 q^{27} -21952 q^{28} + ( 17558 - 30 \beta_{1} - 353 \beta_{2} + 43 \beta_{3} - \beta_{4} ) q^{29} + ( -2376 + 216 \beta_{2} ) q^{30} + ( 48806 + 169 \beta_{1} + 69 \beta_{2} - 104 \beta_{3} - 205 \beta_{4} ) q^{31} -32768 q^{32} + ( -19845 - 189 \beta_{1} - 108 \beta_{3} + 135 \beta_{4} ) q^{33} + ( -688 + 200 \beta_{1} - 160 \beta_{2} + 96 \beta_{3} ) q^{34} + ( -3773 + 343 \beta_{2} ) q^{35} + 46656 q^{36} + ( 146592 - 225 \beta_{1} + 38 \beta_{2} + 867 \beta_{3} ) q^{37} + ( -54000 - 528 \beta_{1} - 264 \beta_{2} - 248 \beta_{3} + 72 \beta_{4} ) q^{38} -59319 q^{39} + ( -5632 + 512 \beta_{2} ) q^{40} + ( 95587 + 604 \beta_{1} - 162 \beta_{2} - 411 \beta_{3} + 11 \beta_{4} ) q^{41} + 74088 q^{42} + ( -81420 + 560 \beta_{1} + 755 \beta_{2} - 451 \beta_{3} + 143 \beta_{4} ) q^{43} + ( -47040 - 448 \beta_{1} - 256 \beta_{3} + 320 \beta_{4} ) q^{44} + ( 8019 - 729 \beta_{2} ) q^{45} + ( 227200 + 704 \beta_{1} - 296 \beta_{2} - 128 \beta_{3} + 120 \beta_{4} ) q^{46} + ( 228569 - 1469 \beta_{1} + 2119 \beta_{2} + 332 \beta_{3} - 352 \beta_{4} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( -246088 - 1680 \beta_{1} - 312 \beta_{2} - 120 \beta_{3} + 280 \beta_{4} ) q^{50} + ( 2322 - 675 \beta_{1} + 540 \beta_{2} - 324 \beta_{3} ) q^{51} -140608 q^{52} + ( 59508 + 927 \beta_{1} - 1469 \beta_{2} + 908 \beta_{3} + 525 \beta_{4} ) q^{53} -157464 q^{54} + ( -367068 - 665 \beta_{1} + 4338 \beta_{2} - 2305 \beta_{3} - 1310 \beta_{4} ) q^{55} + 175616 q^{56} + ( 182250 + 1782 \beta_{1} + 891 \beta_{2} + 837 \beta_{3} - 243 \beta_{4} ) q^{57} + ( -140464 + 240 \beta_{1} + 2824 \beta_{2} - 344 \beta_{3} + 8 \beta_{4} ) q^{58} + ( 188085 + 436 \beta_{1} + 1980 \beta_{2} - 1566 \beta_{3} + 361 \beta_{4} ) q^{59} + ( 19008 - 1728 \beta_{2} ) q^{60} + ( -596038 + 1393 \beta_{1} + 1380 \beta_{2} + 1661 \beta_{3} + 1760 \beta_{4} ) q^{61} + ( -390448 - 1352 \beta_{1} - 552 \beta_{2} + 832 \beta_{3} + 1640 \beta_{4} ) q^{62} -250047 q^{63} + 262144 q^{64} + ( -24167 + 2197 \beta_{2} ) q^{65} + ( 158760 + 1512 \beta_{1} + 864 \beta_{3} - 1080 \beta_{4} ) q^{66} + ( -465438 + 1532 \beta_{1} + 3614 \beta_{2} + 1988 \beta_{3} + 256 \beta_{4} ) q^{67} + ( 5504 - 1600 \beta_{1} + 1280 \beta_{2} - 768 \beta_{3} ) q^{68} + ( -766800 - 2376 \beta_{1} + 999 \beta_{2} + 432 \beta_{3} - 405 \beta_{4} ) q^{69} + ( 30184 - 2744 \beta_{2} ) q^{70} + ( -2262179 - 1554 \beta_{1} + 7958 \beta_{2} + 1873 \beta_{3} - 1881 \beta_{4} ) q^{71} -373248 q^{72} + ( -1325158 - 5538 \beta_{1} + 803 \beta_{2} - 197 \beta_{3} - 133 \beta_{4} ) q^{73} + ( -1172736 + 1800 \beta_{1} - 304 \beta_{2} - 6936 \beta_{3} ) q^{74} + ( 830547 + 5670 \beta_{1} + 1053 \beta_{2} + 405 \beta_{3} - 945 \beta_{4} ) q^{75} + ( 432000 + 4224 \beta_{1} + 2112 \beta_{2} + 1984 \beta_{3} - 576 \beta_{4} ) q^{76} + ( 252105 + 2401 \beta_{1} + 1372 \beta_{3} - 1715 \beta_{4} ) q^{77} + 474552 q^{78} + ( -2116324 - 6783 \beta_{1} - 97 \beta_{2} + 5770 \beta_{3} + 419 \beta_{4} ) q^{79} + ( 45056 - 4096 \beta_{2} ) q^{80} + 531441 q^{81} + ( -764696 - 4832 \beta_{1} + 1296 \beta_{2} + 3288 \beta_{3} - 88 \beta_{4} ) q^{82} + ( -490101 + 9657 \beta_{1} - 5269 \beta_{2} - 6686 \beta_{3} + 3022 \beta_{4} ) q^{83} -592704 q^{84} + ( -2452910 - 8375 \beta_{1} + 5820 \beta_{2} - 5795 \beta_{3} + 550 \beta_{4} ) q^{85} + ( 651360 - 4480 \beta_{1} - 6040 \beta_{2} + 3608 \beta_{3} - 1144 \beta_{4} ) q^{86} + ( 474066 - 810 \beta_{1} - 9531 \beta_{2} + 1161 \beta_{3} - 27 \beta_{4} ) q^{87} + ( 376320 + 3584 \beta_{1} + 2048 \beta_{3} - 2560 \beta_{4} ) q^{88} + ( -320667 + 8601 \beta_{1} + 3765 \beta_{2} - 10573 \beta_{3} + 4484 \beta_{4} ) q^{89} + ( -64152 + 5832 \beta_{2} ) q^{90} + 753571 q^{91} + ( -1817600 - 5632 \beta_{1} + 2368 \beta_{2} + 1024 \beta_{3} - 960 \beta_{4} ) q^{92} + ( 1317762 + 4563 \beta_{1} + 1863 \beta_{2} - 2808 \beta_{3} - 5535 \beta_{4} ) q^{93} + ( -1828552 + 11752 \beta_{1} - 16952 \beta_{2} - 2656 \beta_{3} + 2816 \beta_{4} ) q^{94} + ( -2301288 + 2910 \beta_{1} - 29567 \beta_{2} + 14915 \beta_{3} + 3925 \beta_{4} ) q^{95} -884736 q^{96} + ( 1053714 + 11239 \beta_{1} + 14895 \beta_{2} - 14500 \beta_{3} - 4493 \beta_{4} ) q^{97} -941192 q^{98} + ( -535815 - 5103 \beta_{1} - 2916 \beta_{3} + 3645 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 5672 x^{3} - 117684 x^{2} + 1695035 x + 39011360\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 151 \nu^{4} - 2753 \nu^{3} - 769713 \nu^{2} - 3662645 \nu + 202349920 \)\()/190400\)
\(\beta_{2}\)\(=\)\((\)\( 209 \nu^{4} - 4567 \nu^{3} - 1085287 \nu^{2} - 1378755 \nu + 378627040 \)\()/76160\)
\(\beta_{3}\)\(=\)\((\)\( 209 \nu^{4} - 4567 \nu^{3} - 1085287 \nu^{2} - 693315 \nu + 378627040 \)\()/38080\)
\(\beta_{4}\)\(=\)\((\)\( 9 \nu^{4} - 183 \nu^{3} - 47327 \nu^{2} - 93667 \nu + 16683560 \)\()/952\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - 2 \beta_{2}\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(-9 \beta_{4} + 17 \beta_{3} - 16 \beta_{2} + 45 \beta_{1} + 20412\)\()/9\)
\(\nu^{3}\)\(=\)\((\)\(474 \beta_{4} + 3983 \beta_{3} - 10726 \beta_{2} + 3900 \beta_{1} + 1269750\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(-13852 \beta_{4} + 45031 \beta_{3} - 67864 \beta_{2} + 92095 \beta_{1} + 34521067\)\()/3\)

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
−57.6599
−20.7333
18.0269
−21.9822
82.3485
−8.00000 27.0000 64.0000 −455.940 −216.000 −343.000 −512.000 729.000 3647.52
1.2 −8.00000 27.0000 64.0000 −251.667 −216.000 −343.000 −512.000 729.000 2013.33
1.3 −8.00000 27.0000 64.0000 58.2086 −216.000 −343.000 −512.000 729.000 −465.669
1.4 −8.00000 27.0000 64.0000 249.727 −216.000 −343.000 −512.000 729.000 −1997.82
1.5 −8.00000 27.0000 64.0000 455.671 −216.000 −343.000 −512.000 729.000 −3645.37
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 546.8.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.j 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 56 T_{5}^{4} - 270735 T_{5}^{3} + 15331750 T_{5}^{2} + 13081672000 T_{5} - 760043700000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T )^{5} \)
$3$ \( ( -27 + T )^{5} \)
$5$ \( -760043700000 + 13081672000 T + 15331750 T^{2} - 270735 T^{3} - 56 T^{4} + T^{5} \)
$7$ \( ( 343 + T )^{5} \)
$11$ \( 128332751156309984 + 1014607061306696 T - 170113710772 T^{2} - 70559798 T^{3} + 3679 T^{4} + T^{5} \)
$13$ \( ( 2197 + T )^{5} \)
$17$ \( 39014855835681024000 - 27298508783052600 T + 6310789288140 T^{2} - 494966346 T^{3} - 409 T^{4} + T^{5} \)
$19$ \( -\)\(78\!\cdots\!84\)\( + 1311285841761427488 T + 52492299209076 T^{2} - 2228868831 T^{3} - 33730 T^{4} + T^{5} \)
$23$ \( -\)\(98\!\cdots\!52\)\( - 7302534284923527496 T - 81992827232180 T^{2} + 5066478205 T^{3} + 142142 T^{4} + T^{5} \)
$29$ \( -\)\(89\!\cdots\!00\)\( + \)\(13\!\cdots\!80\)\( T + 2233303588363714 T^{2} - 32932499967 T^{3} - 88028 T^{4} + T^{5} \)
$31$ \( \)\(28\!\cdots\!00\)\( - \)\(14\!\cdots\!80\)\( T + 19977481497131120 T^{2} - 65294089728 T^{3} - 244543 T^{4} + T^{5} \)
$37$ \( \)\(59\!\cdots\!52\)\( - \)\(47\!\cdots\!84\)\( T + 254771163335109756 T^{2} - 209555221662 T^{3} - 730963 T^{4} + T^{5} \)
$41$ \( \)\(52\!\cdots\!00\)\( - \)\(69\!\cdots\!44\)\( T + 60086810086133808 T^{2} - 78539612508 T^{3} - 479512 T^{4} + T^{5} \)
$43$ \( -\)\(14\!\cdots\!60\)\( + \)\(34\!\cdots\!24\)\( T - 63661420625834788 T^{2} - 335455695037 T^{3} + 406536 T^{4} + T^{5} \)
$47$ \( -\)\(12\!\cdots\!28\)\( + \)\(23\!\cdots\!08\)\( T + 1443900561732994628 T^{2} - 1390190461104 T^{3} - 1138945 T^{4} + T^{5} \)
$53$ \( \)\(24\!\cdots\!40\)\( + \)\(69\!\cdots\!52\)\( T - 368824218890445528 T^{2} - 1980410611536 T^{3} - 297595 T^{4} + T^{5} \)
$59$ \( -\)\(13\!\cdots\!20\)\( + \)\(10\!\cdots\!60\)\( T + 2911697407019761792 T^{2} - 2684931754036 T^{3} - 941652 T^{4} + T^{5} \)
$61$ \( -\)\(19\!\cdots\!80\)\( - \)\(19\!\cdots\!60\)\( T - 25759046686456815328 T^{2} - 6149167703540 T^{3} + 2985259 T^{4} + T^{5} \)
$67$ \( \)\(16\!\cdots\!60\)\( + \)\(52\!\cdots\!48\)\( T - 4607042935915083040 T^{2} - 4961827778148 T^{3} + 2333504 T^{4} + T^{5} \)
$71$ \( -\)\(48\!\cdots\!00\)\( - \)\(18\!\cdots\!64\)\( T - 40536882598285456432 T^{2} + 30621448855284 T^{3} + 11322272 T^{4} + T^{5} \)
$73$ \( -\)\(16\!\cdots\!80\)\( - \)\(41\!\cdots\!60\)\( T - 22027711930815149738 T^{2} + 6942592379845 T^{3} + 6631604 T^{4} + T^{5} \)
$79$ \( -\)\(10\!\cdots\!88\)\( - \)\(23\!\cdots\!08\)\( T - 93634777223162163008 T^{2} + 17770461747538 T^{3} + 10600265 T^{4} + T^{5} \)
$83$ \( -\)\(26\!\cdots\!12\)\( + \)\(19\!\cdots\!68\)\( T + 67660052929538178708 T^{2} - 57072525823194 T^{3} + 2425229 T^{4} + T^{5} \)
$89$ \( -\)\(38\!\cdots\!48\)\( + \)\(26\!\cdots\!36\)\( T + 78646915459971645156 T^{2} - 112954533784032 T^{3} + 1581837 T^{4} + T^{5} \)
$97$ \( \)\(37\!\cdots\!32\)\( + \)\(11\!\cdots\!24\)\( T - 49855463099836603040 T^{2} - 219873587471940 T^{3} - 5298407 T^{4} + T^{5} \)
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