Properties

Label 546.8.a.i
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: \(x^{5} - 2 x^{4} - 86504 x^{3} - 9117228 x^{2} + 89606664 x + 21810067776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5\cdot 13 \)
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.

\(f(q)\) \(=\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( -68 - \beta_{1} ) q^{5} -216 q^{6} + 343 q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( -68 - \beta_{1} ) q^{5} -216 q^{6} + 343 q^{7} -512 q^{8} + 729 q^{9} + ( 544 + 8 \beta_{1} ) q^{10} + ( -261 - 6 \beta_{1} - \beta_{2} ) q^{11} + 1728 q^{12} + 2197 q^{13} -2744 q^{14} + ( -1836 - 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( -850 + 6 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{17} -5832 q^{18} + ( -3395 + 45 \beta_{1} - \beta_{3} + 9 \beta_{4} ) q^{19} + ( -4352 - 64 \beta_{1} ) q^{20} + 9261 q^{21} + ( 2088 + 48 \beta_{1} + 8 \beta_{2} ) q^{22} + ( -15613 + 95 \beta_{1} + 7 \beta_{2} - 8 \beta_{3} - 7 \beta_{4} ) q^{23} -13824 q^{24} + ( -15910 + 101 \beta_{1} + 10 \beta_{2} - 9 \beta_{3} - 15 \beta_{4} ) q^{25} -17576 q^{26} + 19683 q^{27} + 21952 q^{28} + ( -42617 - 83 \beta_{1} + 24 \beta_{2} + 23 \beta_{3} + 9 \beta_{4} ) q^{29} + ( 14688 + 216 \beta_{1} ) q^{30} + ( -37205 + 195 \beta_{1} - 9 \beta_{2} + 38 \beta_{3} + 20 \beta_{4} ) q^{31} -32768 q^{32} + ( -7047 - 162 \beta_{1} - 27 \beta_{2} ) q^{33} + ( 6800 - 48 \beta_{1} - 16 \beta_{3} + 24 \beta_{4} ) q^{34} + ( -23324 - 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( 20214 - 304 \beta_{1} + 19 \beta_{2} - 87 \beta_{3} + 7 \beta_{4} ) q^{37} + ( 27160 - 360 \beta_{1} + 8 \beta_{3} - 72 \beta_{4} ) q^{38} + 59319 q^{39} + ( 34816 + 512 \beta_{1} ) q^{40} + ( -4805 + 30 \beta_{1} - 20 \beta_{2} - 143 \beta_{3} - 9 \beta_{4} ) q^{41} -74088 q^{42} + ( -11097 + 1893 \beta_{1} + 12 \beta_{2} + 85 \beta_{3} + 19 \beta_{4} ) q^{43} + ( -16704 - 384 \beta_{1} - 64 \beta_{2} ) q^{44} + ( -49572 - 729 \beta_{1} ) q^{45} + ( 124904 - 760 \beta_{1} - 56 \beta_{2} + 64 \beta_{3} + 56 \beta_{4} ) q^{46} + ( -197170 + 693 \beta_{1} + 38 \beta_{2} + 272 \beta_{3} + 55 \beta_{4} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( 127280 - 808 \beta_{1} - 80 \beta_{2} + 72 \beta_{3} + 120 \beta_{4} ) q^{50} + ( -22950 + 162 \beta_{1} + 54 \beta_{3} - 81 \beta_{4} ) q^{51} + 140608 q^{52} + ( -378421 + 2903 \beta_{1} - 103 \beta_{2} + 34 \beta_{3} - 242 \beta_{4} ) q^{53} -157464 q^{54} + ( 349438 + 3572 \beta_{1} + 145 \beta_{2} - 99 \beta_{3} - 55 \beta_{4} ) q^{55} -175616 q^{56} + ( -91665 + 1215 \beta_{1} - 27 \beta_{3} + 243 \beta_{4} ) q^{57} + ( 340936 + 664 \beta_{1} - 192 \beta_{2} - 184 \beta_{3} - 72 \beta_{4} ) q^{58} + ( -560499 + 5352 \beta_{1} - 129 \beta_{2} - 336 \beta_{3} - 29 \beta_{4} ) q^{59} + ( -117504 - 1728 \beta_{1} ) q^{60} + ( 228260 + 626 \beta_{1} + 463 \beta_{2} + 179 \beta_{3} - 217 \beta_{4} ) q^{61} + ( 297640 - 1560 \beta_{1} + 72 \beta_{2} - 304 \beta_{3} - 160 \beta_{4} ) q^{62} + 250047 q^{63} + 262144 q^{64} + ( -149396 - 2197 \beta_{1} ) q^{65} + ( 56376 + 1296 \beta_{1} + 216 \beta_{2} ) q^{66} + ( 53046 + 1658 \beta_{1} - 122 \beta_{2} - 300 \beta_{3} + 306 \beta_{4} ) q^{67} + ( -54400 + 384 \beta_{1} + 128 \beta_{3} - 192 \beta_{4} ) q^{68} + ( -421551 + 2565 \beta_{1} + 189 \beta_{2} - 216 \beta_{3} - 189 \beta_{4} ) q^{69} + ( 186592 + 2744 \beta_{1} ) q^{70} + ( -896737 - 3506 \beta_{1} - 410 \beta_{2} - 147 \beta_{3} + 411 \beta_{4} ) q^{71} -373248 q^{72} + ( -470217 + 8955 \beta_{1} - 402 \beta_{2} - 539 \beta_{3} + 297 \beta_{4} ) q^{73} + ( -161712 + 2432 \beta_{1} - 152 \beta_{2} + 696 \beta_{3} - 56 \beta_{4} ) q^{74} + ( -429570 + 2727 \beta_{1} + 270 \beta_{2} - 243 \beta_{3} - 405 \beta_{4} ) q^{75} + ( -217280 + 2880 \beta_{1} - 64 \beta_{3} + 576 \beta_{4} ) q^{76} + ( -89523 - 2058 \beta_{1} - 343 \beta_{2} ) q^{77} -474552 q^{78} + ( -816049 + 3639 \beta_{1} - 291 \beta_{2} + 1428 \beta_{3} + 226 \beta_{4} ) q^{79} + ( -278528 - 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( 38440 - 240 \beta_{1} + 160 \beta_{2} + 1144 \beta_{3} + 72 \beta_{4} ) q^{82} + ( -1746006 - 9603 \beta_{1} + 366 \beta_{2} + 1164 \beta_{3} + 809 \beta_{4} ) q^{83} + 592704 q^{84} + ( -343108 - 4526 \beta_{1} + 95 \beta_{2} + 45 \beta_{3} + 165 \beta_{4} ) q^{85} + ( 88776 - 15144 \beta_{1} - 96 \beta_{2} - 680 \beta_{3} - 152 \beta_{4} ) q^{86} + ( -1150659 - 2241 \beta_{1} + 648 \beta_{2} + 621 \beta_{3} + 243 \beta_{4} ) q^{87} + ( 133632 + 3072 \beta_{1} + 512 \beta_{2} ) q^{88} + ( -4015464 - 6533 \beta_{1} + 257 \beta_{2} - 437 \beta_{3} + 45 \beta_{4} ) q^{89} + ( 396576 + 5832 \beta_{1} ) q^{90} + 753571 q^{91} + ( -999232 + 6080 \beta_{1} + 448 \beta_{2} - 512 \beta_{3} - 448 \beta_{4} ) q^{92} + ( -1004535 + 5265 \beta_{1} - 243 \beta_{2} + 1026 \beta_{3} + 540 \beta_{4} ) q^{93} + ( 1577360 - 5544 \beta_{1} - 304 \beta_{2} - 2176 \beta_{3} - 440 \beta_{4} ) q^{94} + ( -2316419 + 23459 \beta_{1} - 790 \beta_{2} - 63 \beta_{3} + 225 \beta_{4} ) q^{95} -884736 q^{96} + ( 755557 - 11139 \beta_{1} + 549 \beta_{2} + 140 \beta_{3} - 3522 \beta_{4} ) q^{97} -941192 q^{98} + ( -190269 - 4374 \beta_{1} - 729 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 86504 x^{3} - 9117228 x^{2} + 89606664 x + 21810067776\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 69 \nu^{4} - 1741 \nu^{3} - 6302648 \nu^{2} - 371941044 \nu + 25331552820 \)\()/68034708\)
\(\beta_{2}\)\(=\)\((\)\( 283 \nu^{4} + 38508 \nu^{3} - 32715544 \nu^{2} - 7562728596 \nu + 89369189784 \)\()/ 136069416 \)
\(\beta_{3}\)\(=\)\((\)\( -2173 \nu^{4} + 283072 \nu^{3} + 164159976 \nu^{2} - 783684612 \nu - 877198268520 \)\()/ 136069416 \)
\(\beta_{4}\)\(=\)\((\)\( -3887 \nu^{4} + 308060 \nu^{3} + 296253744 \nu^{2} + 13428544068 \nu - 560044524816 \)\()/ 136069416 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - 5 \beta_{2} + 26 \beta_{1} + 50\)\()/130\)
\(\nu^{2}\)\(=\)\((\)\(-325 \beta_{4} + 372 \beta_{3} - 365 \beta_{2} - 2548 \beta_{1} + 2248945\)\()/65\)
\(\nu^{3}\)\(=\)\((\)\(-48880 \beta_{4} + 122076 \beta_{3} - 191780 \beta_{2} + 938756 \beta_{1} + 362232820\)\()/65\)
\(\nu^{4}\)\(=\)\((\)\(-30919720 \beta_{4} + 39754926 \beta_{3} - 51655190 \beta_{2} - 74888164 \beta_{1} + 190836284620\)\()/65\)

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
−65.0947
−108.730
336.062
44.7350
−204.972
−8.00000 27.0000 64.0000 −428.929 −216.000 343.000 −512.000 729.000 3431.43
1.2 −8.00000 27.0000 64.0000 −114.198 −216.000 343.000 −512.000 729.000 913.581
1.3 −8.00000 27.0000 64.0000 −105.351 −216.000 343.000 −512.000 729.000 842.807
1.4 −8.00000 27.0000 64.0000 −12.1505 −216.000 343.000 −512.000 729.000 97.2036
1.5 −8.00000 27.0000 64.0000 320.628 −216.000 343.000 −512.000 729.000 −2565.02
\(n\): 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.i 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.i 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 340 T_{5}^{4} - 97735 T_{5}^{3} - 30126750 T_{5}^{2} - 2005595100 T_{5} - 20103633000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T )^{5} \)
$3$ \( ( -27 + T )^{5} \)
$5$ \( -20103633000 - 2005595100 T - 30126750 T^{2} - 97735 T^{3} + 340 T^{4} + T^{5} \)
$7$ \( ( -343 + T )^{5} \)
$11$ \( 472759058028177504 + 353916337614912 T - 126164977512 T^{2} - 55327618 T^{3} + 1303 T^{4} + T^{5} \)
$13$ \( ( -2197 + T )^{5} \)
$17$ \( -40332956705849892384 + 11992463195101272 T + 1178443110732 T^{2} - 438002818 T^{3} + 4247 T^{4} + T^{5} \)
$19$ \( \)\(15\!\cdots\!56\)\( + 1984144016501690128 T - 36515968273244 T^{2} - 2889729133 T^{3} + 16984 T^{4} + T^{5} \)
$23$ \( -\)\(39\!\cdots\!64\)\( - 9524971936250323488 T - 412822777067508 T^{2} - 3451779413 T^{3} + 78072 T^{4} + T^{5} \)
$29$ \( \)\(68\!\cdots\!28\)\( - \)\(11\!\cdots\!32\)\( T - 5768015571958758 T^{2} - 22024243957 T^{3} + 213142 T^{4} + T^{5} \)
$31$ \( \)\(31\!\cdots\!40\)\( + \)\(21\!\cdots\!36\)\( T - 5412265351896992 T^{2} - 33934766976 T^{3} + 186027 T^{4} + T^{5} \)
$37$ \( -\)\(92\!\cdots\!00\)\( + \)\(77\!\cdots\!00\)\( T + 28894910820168100 T^{2} - 222188508590 T^{3} - 101025 T^{4} + T^{5} \)
$41$ \( -\)\(92\!\cdots\!00\)\( + \)\(22\!\cdots\!00\)\( T + 2410001635624560 T^{2} - 428236884080 T^{3} + 23976 T^{4} + T^{5} \)
$43$ \( -\)\(13\!\cdots\!48\)\( + \)\(89\!\cdots\!68\)\( T + 81398594748079588 T^{2} - 685178858837 T^{3} + 55528 T^{4} + T^{5} \)
$47$ \( \)\(81\!\cdots\!00\)\( - \)\(13\!\cdots\!24\)\( T - 1373663362803969744 T^{2} - 1256669183264 T^{3} + 985981 T^{4} + T^{5} \)
$53$ \( \)\(36\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T - 5856003863804457720 T^{2} - 2774786348360 T^{3} + 1891657 T^{4} + T^{5} \)
$59$ \( \)\(19\!\cdots\!44\)\( + \)\(42\!\cdots\!92\)\( T - 9232347605260851792 T^{2} - 3674020647808 T^{3} + 2802208 T^{4} + T^{5} \)
$61$ \( -\)\(40\!\cdots\!00\)\( + \)\(27\!\cdots\!00\)\( T + 14326472252990949760 T^{2} - 10781014404500 T^{3} - 1140591 T^{4} + T^{5} \)
$67$ \( \)\(42\!\cdots\!00\)\( - \)\(46\!\cdots\!00\)\( T - 5150253795222589520 T^{2} - 6062377254220 T^{3} - 265168 T^{4} + T^{5} \)
$71$ \( \)\(10\!\cdots\!32\)\( - \)\(21\!\cdots\!56\)\( T - 34492496878117683984 T^{2} - 5701398079496 T^{3} + 4483276 T^{4} + T^{5} \)
$73$ \( -\)\(25\!\cdots\!60\)\( + \)\(87\!\cdots\!96\)\( T - 12748271846092850738 T^{2} - 24083522996261 T^{3} + 2350578 T^{4} + T^{5} \)
$79$ \( \)\(19\!\cdots\!00\)\( + \)\(50\!\cdots\!76\)\( T - \)\(18\!\cdots\!36\)\( T^{2} - 46094274278174 T^{3} + 4079889 T^{4} + T^{5} \)
$83$ \( \)\(65\!\cdots\!32\)\( - \)\(64\!\cdots\!96\)\( T - \)\(36\!\cdots\!44\)\( T^{2} - 29764324900846 T^{3} + 8731571 T^{4} + T^{5} \)
$89$ \( \)\(19\!\cdots\!00\)\( + \)\(58\!\cdots\!00\)\( T + \)\(46\!\cdots\!40\)\( T^{2} + 145541767034120 T^{3} + 20077879 T^{4} + T^{5} \)
$97$ \( -\)\(19\!\cdots\!92\)\( + \)\(33\!\cdots\!12\)\( T + \)\(18\!\cdots\!04\)\( T^{2} - 402455205190672 T^{3} - 3780209 T^{4} + T^{5} \)
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