Properties

Label 546.8.a.e
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 33506 x + 97248\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( -126 + \beta_{2} ) q^{5} + 216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( -126 + \beta_{2} ) q^{5} + 216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} + ( -1008 + 8 \beta_{2} ) q^{10} + ( -2023 - 13 \beta_{1} + 4 \beta_{2} ) q^{11} + 1728 q^{12} + 2197 q^{13} -2744 q^{14} + ( -3402 + 27 \beta_{2} ) q^{15} + 4096 q^{16} + ( -4077 - 9 \beta_{1} - 44 \beta_{2} ) q^{17} + 5832 q^{18} + ( -1612 + 148 \beta_{1} - 109 \beta_{2} ) q^{19} + ( -8064 + 64 \beta_{2} ) q^{20} -9261 q^{21} + ( -16184 - 104 \beta_{1} + 32 \beta_{2} ) q^{22} + ( 37202 + 34 \beta_{1} - 85 \beta_{2} ) q^{23} + 13824 q^{24} + ( 13973 - 10 \beta_{1} - 443 \beta_{2} ) q^{25} + 17576 q^{26} + 19683 q^{27} -21952 q^{28} + ( 35138 - 168 \beta_{1} + 77 \beta_{2} ) q^{29} + ( -27216 + 216 \beta_{2} ) q^{30} + ( -87687 + 755 \beta_{1} - 215 \beta_{2} ) q^{31} + 32768 q^{32} + ( -54621 - 351 \beta_{1} + 108 \beta_{2} ) q^{33} + ( -32616 - 72 \beta_{1} - 352 \beta_{2} ) q^{34} + ( 43218 - 343 \beta_{2} ) q^{35} + 46656 q^{36} + ( -195323 + 129 \beta_{1} - 864 \beta_{2} ) q^{37} + ( -12896 + 1184 \beta_{1} - 872 \beta_{2} ) q^{38} + 59319 q^{39} + ( -64512 + 512 \beta_{2} ) q^{40} + ( -216536 + 386 \beta_{1} + 536 \beta_{2} ) q^{41} -74088 q^{42} + ( -25394 - 14 \beta_{1} + 527 \beta_{2} ) q^{43} + ( -129472 - 832 \beta_{1} + 256 \beta_{2} ) q^{44} + ( -91854 + 729 \beta_{2} ) q^{45} + ( 297616 + 272 \beta_{1} - 680 \beta_{2} ) q^{46} + ( -423073 - 1755 \beta_{1} + 1063 \beta_{2} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( 111784 - 80 \beta_{1} - 3544 \beta_{2} ) q^{50} + ( -110079 - 243 \beta_{1} - 1188 \beta_{2} ) q^{51} + 140608 q^{52} + ( 108837 - 5635 \beta_{1} + 2983 \beta_{2} ) q^{53} + 157464 q^{54} + ( 158437 - 885 \beta_{1} - 1172 \beta_{2} ) q^{55} -175616 q^{56} + ( -43524 + 3996 \beta_{1} - 2943 \beta_{2} ) q^{57} + ( 281104 - 1344 \beta_{1} + 616 \beta_{2} ) q^{58} + ( -1144772 - 4680 \beta_{1} + 3524 \beta_{2} ) q^{59} + ( -217728 + 1728 \beta_{2} ) q^{60} + ( -483499 + 4897 \beta_{1} - 2470 \beta_{2} ) q^{61} + ( -701496 + 6040 \beta_{1} - 1720 \beta_{2} ) q^{62} -250047 q^{63} + 262144 q^{64} + ( -276822 + 2197 \beta_{2} ) q^{65} + ( -436968 - 2808 \beta_{1} + 864 \beta_{2} ) q^{66} + ( -434124 - 9488 \beta_{1} + 4866 \beta_{2} ) q^{67} + ( -260928 - 576 \beta_{1} - 2816 \beta_{2} ) q^{68} + ( 1004454 + 918 \beta_{1} - 2295 \beta_{2} ) q^{69} + ( 345744 - 2744 \beta_{2} ) q^{70} + ( -1728692 + 1100 \beta_{1} - 13388 \beta_{2} ) q^{71} + 373248 q^{72} + ( -1220980 + 9738 \beta_{1} + 6511 \beta_{2} ) q^{73} + ( -1562584 + 1032 \beta_{1} - 6912 \beta_{2} ) q^{74} + ( 377271 - 270 \beta_{1} - 11961 \beta_{2} ) q^{75} + ( -103168 + 9472 \beta_{1} - 6976 \beta_{2} ) q^{76} + ( 693889 + 4459 \beta_{1} - 1372 \beta_{2} ) q^{77} + 474552 q^{78} + ( -983417 - 3531 \beta_{1} + 11521 \beta_{2} ) q^{79} + ( -516096 + 4096 \beta_{2} ) q^{80} + 531441 q^{81} + ( -1732288 + 3088 \beta_{1} + 4288 \beta_{2} ) q^{82} + ( -4550075 - 11333 \beta_{1} - 5169 \beta_{2} ) q^{83} -592704 q^{84} + ( -3117923 - 145 \beta_{1} + 11338 \beta_{2} ) q^{85} + ( -203152 - 112 \beta_{1} + 4216 \beta_{2} ) q^{86} + ( 948726 - 4536 \beta_{1} + 2079 \beta_{2} ) q^{87} + ( -1035776 - 6656 \beta_{1} + 2048 \beta_{2} ) q^{88} + ( 2126511 + 13335 \beta_{1} + 11553 \beta_{2} ) q^{89} + ( -734832 + 5832 \beta_{2} ) q^{90} -753571 q^{91} + ( 2380928 + 2176 \beta_{1} - 5440 \beta_{2} ) q^{92} + ( -2367549 + 20385 \beta_{1} - 5805 \beta_{2} ) q^{93} + ( -3384584 - 14040 \beta_{1} + 8504 \beta_{2} ) q^{94} + ( -3535882 + 10710 \beta_{1} + 8817 \beta_{2} ) q^{95} + 884736 q^{96} + ( -6010745 + 19319 \beta_{1} - 38671 \beta_{2} ) q^{97} + 941192 q^{98} + ( -1474767 - 9477 \beta_{1} + 2916 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9} + O(q^{10}) \) \( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9} - 3024 q^{10} - 6069 q^{11} + 5184 q^{12} + 6591 q^{13} - 8232 q^{14} - 10206 q^{15} + 12288 q^{16} - 12231 q^{17} + 17496 q^{18} - 4836 q^{19} - 24192 q^{20} - 27783 q^{21} - 48552 q^{22} + 111606 q^{23} + 41472 q^{24} + 41919 q^{25} + 52728 q^{26} + 59049 q^{27} - 65856 q^{28} + 105414 q^{29} - 81648 q^{30} - 263061 q^{31} + 98304 q^{32} - 163863 q^{33} - 97848 q^{34} + 129654 q^{35} + 139968 q^{36} - 585969 q^{37} - 38688 q^{38} + 177957 q^{39} - 193536 q^{40} - 649608 q^{41} - 222264 q^{42} - 76182 q^{43} - 388416 q^{44} - 275562 q^{45} + 892848 q^{46} - 1269219 q^{47} + 331776 q^{48} + 352947 q^{49} + 335352 q^{50} - 330237 q^{51} + 421824 q^{52} + 326511 q^{53} + 472392 q^{54} + 475311 q^{55} - 526848 q^{56} - 130572 q^{57} + 843312 q^{58} - 3434316 q^{59} - 653184 q^{60} - 1450497 q^{61} - 2104488 q^{62} - 750141 q^{63} + 786432 q^{64} - 830466 q^{65} - 1310904 q^{66} - 1302372 q^{67} - 782784 q^{68} + 3013362 q^{69} + 1037232 q^{70} - 5186076 q^{71} + 1119744 q^{72} - 3662940 q^{73} - 4687752 q^{74} + 1131813 q^{75} - 309504 q^{76} + 2081667 q^{77} + 1423656 q^{78} - 2950251 q^{79} - 1548288 q^{80} + 1594323 q^{81} - 5196864 q^{82} - 13650225 q^{83} - 1778112 q^{84} - 9353769 q^{85} - 609456 q^{86} + 2846178 q^{87} - 3107328 q^{88} + 6379533 q^{89} - 2204496 q^{90} - 2260713 q^{91} + 7142784 q^{92} - 7102647 q^{93} - 10153752 q^{94} - 10607646 q^{95} + 2654208 q^{96} - 18032235 q^{97} + 2823576 q^{98} - 4424301 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 33506 x + 97248\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 163 \nu - 22392 \)\()/114\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{2} + 179 \nu + 22278 \)\()/114\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-163 \beta_{2} + 179 \beta_{1} + 67013\)\()/3\)

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
−183.985
182.082
2.90288
8.00000 27.0000 64.0000 −516.403 216.000 −343.000 512.000 729.000 −4131.22
1.2 8.00000 27.0000 64.0000 64.4978 216.000 −343.000 512.000 729.000 515.983
1.3 8.00000 27.0000 64.0000 73.9052 216.000 −343.000 512.000 729.000 591.241
\(n\): e.g. 2-40 or 990-1000
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.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.e 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 378 T_{5}^{2} - 66705 T_{5} + 2461550 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{3} \)
$3$ \( ( -27 + T )^{3} \)
$5$ \( 2461550 - 66705 T + 378 T^{2} + T^{3} \)
$7$ \( ( 343 + T )^{3} \)
$11$ \( -11019191384 - 723798 T + 6069 T^{2} + T^{3} \)
$13$ \( ( -2197 + T )^{3} \)
$17$ \( -2540015942296 - 215823030 T + 12231 T^{2} + T^{3} \)
$19$ \( 26372444700364 - 1928692953 T + 4836 T^{2} + T^{3} \)
$23$ \( -29333382239720 + 3484216017 T - 111606 T^{2} + T^{3} \)
$29$ \( 26262769167062 + 1554144639 T - 105414 T^{2} + T^{3} \)
$31$ \( -4854922162860672 - 21110848968 T + 263061 T^{2} + T^{3} \)
$37$ \( -14934462179679712 + 37852717278 T + 585969 T^{2} + T^{3} \)
$41$ \( -2018235088188896 + 74557837116 T + 649608 T^{2} + T^{3} \)
$43$ \( 1324624289254652 - 29154223647 T + 76182 T^{2} + T^{3} \)
$47$ \( -59068196171942400 + 289180543920 T + 1269219 T^{2} + T^{3} \)
$53$ \( -330345946605923100 - 2428993691760 T - 326511 T^{2} + T^{3} \)
$59$ \( -1724136331477029120 + 1967081380224 T + 3434316 T^{2} + T^{3} \)
$61$ \( -473622024839505860 - 1144624414932 T + 1450497 T^{2} + T^{3} \)
$67$ \( -5371743788767397776 - 6382235429988 T + 1302372 T^{2} + T^{3} \)
$71$ \( -59588707830950060800 - 10278242919360 T + 5186076 T^{2} + T^{3} \)
$73$ \( -33260598587214439434 - 15218246092755 T + 3662940 T^{2} + T^{3} \)
$79$ \( -405115690620691232 - 9686215757622 T + 2950251 T^{2} + T^{3} \)
$83$ \( 22446568528090507864 + 41478315722754 T + 13650225 T^{2} + T^{3} \)
$89$ \( 77220703793201299380 - 32785770974004 T - 6379533 T^{2} + T^{3} \)
$97$ \( -\)\(67\!\cdots\!20\)\( - 28706252463252 T + 18032235 T^{2} + T^{3} \)
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