Properties

Label 546.8.a.m
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 197669 x^{3} - 12910499 x^{2} + 9274302080 x + 1050512243200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( 60 - \beta_{1} ) q^{5} + 216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( 60 - \beta_{1} ) q^{5} + 216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} + ( 480 - 8 \beta_{1} ) q^{10} + ( 369 - 7 \beta_{1} + \beta_{3} ) q^{11} + 1728 q^{12} -2197 q^{13} -2744 q^{14} + ( 1620 - 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( 11286 + 11 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{17} + 5832 q^{18} + ( -4114 - 69 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{19} + ( 3840 - 64 \beta_{1} ) q^{20} -9261 q^{21} + ( 2952 - 56 \beta_{1} + 8 \beta_{3} ) q^{22} + ( 14891 - 53 \beta_{1} - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{23} + 13824 q^{24} + ( 4523 - 23 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{25} -17576 q^{26} + 19683 q^{27} -21952 q^{28} + ( 26010 + 7 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{29} + ( 12960 - 216 \beta_{1} ) q^{30} + ( 65975 + 219 \beta_{1} - \beta_{2} - 10 \beta_{3} + 10 \beta_{4} ) q^{31} + 32768 q^{32} + ( 9963 - 189 \beta_{1} + 27 \beta_{3} ) q^{33} + ( 90288 + 88 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} ) q^{34} + ( -20580 + 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( 82175 - 245 \beta_{1} + 16 \beta_{2} - 49 \beta_{3} + 14 \beta_{4} ) q^{37} + ( -32912 - 552 \beta_{1} - 16 \beta_{2} - 16 \beta_{4} ) q^{38} -59319 q^{39} + ( 30720 - 512 \beta_{1} ) q^{40} + ( 34668 - 748 \beta_{1} - 8 \beta_{2} + 44 \beta_{3} + 26 \beta_{4} ) q^{41} -74088 q^{42} + ( 177194 + 570 \beta_{1} - 53 \beta_{2} + 35 \beta_{3} - 10 \beta_{4} ) q^{43} + ( 23616 - 448 \beta_{1} + 64 \beta_{3} ) q^{44} + ( 43740 - 729 \beta_{1} ) q^{45} + ( 119128 - 424 \beta_{1} - 32 \beta_{2} - 8 \beta_{3} + 24 \beta_{4} ) q^{46} + ( 152652 + 817 \beta_{1} - 93 \beta_{2} - 39 \beta_{3} + 15 \beta_{4} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( 36184 - 184 \beta_{1} + 16 \beta_{2} + 48 \beta_{3} ) q^{50} + ( 304722 + 297 \beta_{1} + 54 \beta_{2} - 27 \beta_{4} ) q^{51} -140608 q^{52} + ( 343183 - 1495 \beta_{1} + 77 \beta_{2} - 222 \beta_{3} - 78 \beta_{4} ) q^{53} + 157464 q^{54} + ( 540231 - 1967 \beta_{1} + 146 \beta_{2} - 117 \beta_{3} + 16 \beta_{4} ) q^{55} -175616 q^{56} + ( -111078 - 1863 \beta_{1} - 54 \beta_{2} - 54 \beta_{4} ) q^{57} + ( 208080 + 56 \beta_{2} - 24 \beta_{3} - 16 \beta_{4} ) q^{58} + ( 120783 - 306 \beta_{1} - 52 \beta_{2} - 179 \beta_{3} - 23 \beta_{4} ) q^{59} + ( 103680 - 1728 \beta_{1} ) q^{60} + ( 1233775 + 3078 \beta_{1} + 201 \beta_{2} - 30 \beta_{3} - 114 \beta_{4} ) q^{61} + ( 527800 + 1752 \beta_{1} - 8 \beta_{2} - 80 \beta_{3} + 80 \beta_{4} ) q^{62} -250047 q^{63} + 262144 q^{64} + ( -131820 + 2197 \beta_{1} ) q^{65} + ( 79704 - 1512 \beta_{1} + 216 \beta_{3} ) q^{66} + ( 1097536 + 3634 \beta_{1} - 320 \beta_{2} + 36 \beta_{3} + 160 \beta_{4} ) q^{67} + ( 722304 + 704 \beta_{1} + 128 \beta_{2} - 64 \beta_{4} ) q^{68} + ( 402057 - 1431 \beta_{1} - 108 \beta_{2} - 27 \beta_{3} + 81 \beta_{4} ) q^{69} + ( -164640 + 2744 \beta_{1} ) q^{70} + ( -315502 - 3530 \beta_{1} - 124 \beta_{2} + 12 \beta_{3} + 36 \beta_{4} ) q^{71} + 373248 q^{72} + ( 420070 - 26 \beta_{1} + \beta_{2} + 563 \beta_{3} + 182 \beta_{4} ) q^{73} + ( 657400 - 1960 \beta_{1} + 128 \beta_{2} - 392 \beta_{3} + 112 \beta_{4} ) q^{74} + ( 122121 - 621 \beta_{1} + 54 \beta_{2} + 162 \beta_{3} ) q^{75} + ( -263296 - 4416 \beta_{1} - 128 \beta_{2} - 128 \beta_{4} ) q^{76} + ( -126567 + 2401 \beta_{1} - 343 \beta_{3} ) q^{77} -474552 q^{78} + ( 1271477 - 653 \beta_{1} + 161 \beta_{2} + 230 \beta_{3} - 244 \beta_{4} ) q^{79} + ( 245760 - 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( 277344 - 5984 \beta_{1} - 64 \beta_{2} + 352 \beta_{3} + 208 \beta_{4} ) q^{82} + ( 1058572 - 337 \beta_{1} - 377 \beta_{2} - 171 \beta_{3} + 13 \beta_{4} ) q^{83} -592704 q^{84} + ( -164559 - 14981 \beta_{1} + 512 \beta_{2} + 211 \beta_{3} - 356 \beta_{4} ) q^{85} + ( 1417552 + 4560 \beta_{1} - 424 \beta_{2} + 280 \beta_{3} - 80 \beta_{4} ) q^{86} + ( 702270 + 189 \beta_{2} - 81 \beta_{3} - 54 \beta_{4} ) q^{87} + ( 188928 - 3584 \beta_{1} + 512 \beta_{3} ) q^{88} + ( 2645547 + 1703 \beta_{1} + 769 \beta_{2} + 32 \beta_{3} + 488 \beta_{4} ) q^{89} + ( 349920 - 5832 \beta_{1} ) q^{90} + 753571 q^{91} + ( 953024 - 3392 \beta_{1} - 256 \beta_{2} - 64 \beta_{3} + 192 \beta_{4} ) q^{92} + ( 1781325 + 5913 \beta_{1} - 27 \beta_{2} - 270 \beta_{3} + 270 \beta_{4} ) q^{93} + ( 1221216 + 6536 \beta_{1} - 744 \beta_{2} - 312 \beta_{3} + 120 \beta_{4} ) q^{94} + ( 5159032 + 10765 \beta_{1} + 108 \beta_{2} - 676 \beta_{3} - 400 \beta_{4} ) q^{95} + 884736 q^{96} + ( 4462697 + 8965 \beta_{1} - 719 \beta_{2} + 584 \beta_{3} + 348 \beta_{4} ) q^{97} + 941192 q^{98} + ( 269001 - 5103 \beta_{1} + 729 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 299 q^{5} + 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 299 q^{5} + 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} + 2392 q^{10} + 1838 q^{11} + 8640 q^{12} - 10985 q^{13} - 13720 q^{14} + 8073 q^{15} + 20480 q^{16} + 56444 q^{17} + 29160 q^{18} - 20639 q^{19} + 19136 q^{20} - 46305 q^{21} + 14704 q^{22} + 74395 q^{23} + 69120 q^{24} + 22594 q^{25} - 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 130059 q^{29} + 64584 q^{30} + 330083 q^{31} + 163840 q^{32} + 49626 q^{33} + 451552 q^{34} - 102557 q^{35} + 233280 q^{36} + 410632 q^{37} - 165112 q^{38} - 296595 q^{39} + 153088 q^{40} + 172558 q^{41} - 370440 q^{42} + 886497 q^{43} + 117632 q^{44} + 217971 q^{45} + 595160 q^{46} + 763969 q^{47} + 552960 q^{48} + 588245 q^{49} + 180752 q^{50} + 1523988 q^{51} - 703040 q^{52} + 1714575 q^{53} + 787320 q^{54} + 2699318 q^{55} - 878080 q^{56} - 557253 q^{57} + 1040472 q^{58} + 603580 q^{59} + 516672 q^{60} + 6172268 q^{61} + 2640664 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 656903 q^{65} + 397008 q^{66} + 5490834 q^{67} + 3612416 q^{68} + 2008665 q^{69} - 820456 q^{70} - 1581200 q^{71} + 1866240 q^{72} + 2100143 q^{73} + 3285056 q^{74} + 610038 q^{75} - 1320896 q^{76} - 630434 q^{77} - 2372760 q^{78} + 6357137 q^{79} + 1224704 q^{80} + 2657205 q^{81} + 1380464 q^{82} + 5292133 q^{83} - 2963520 q^{84} - 836908 q^{85} + 7091976 q^{86} + 3511593 q^{87} + 941056 q^{88} + 13229719 q^{89} + 1743768 q^{90} + 3767855 q^{91} + 4761280 q^{92} + 8912241 q^{93} + 6111752 q^{94} + 25806433 q^{95} + 4423680 q^{96} + 22321383 q^{97} + 4705960 q^{98} + 1339902 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 197669 x^{3} - 12910499 x^{2} + 9274302080 x + 1050512243200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} - 8374 \nu^{3} + 2071899 \nu^{2} + 681717104 \nu - 89873684320 \)\()/11077920\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 8374 \nu^{3} + 3467061 \nu^{2} - 1218996224 \nu - 347970025760 \)\()/33233760\)
\(\beta_{4}\)\(=\)\((\)\( -97 \nu^{4} + 18566 \nu^{3} + 14311251 \nu^{2} - 1474509076 \nu - 479920256704 \)\()/6646752\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(6 \beta_{3} + 2 \beta_{2} + 97 \beta_{1} + 79048\)
\(\nu^{3}\)\(=\)\(8 \beta_{4} + 1348 \beta_{3} - 844 \beta_{2} + 103157 \beta_{1} + 7844440\)
\(\nu^{4}\)\(=\)\(-66992 \beta_{4} + 1143242 \beta_{3} + 133534 \beta_{2} + 18854589 \beta_{1} + 8216447272\)

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
390.906
280.784
−136.219
−233.276
−301.195
8.00000 27.0000 64.0000 −330.906 216.000 −343.000 512.000 729.000 −2647.25
1.2 8.00000 27.0000 64.0000 −220.784 216.000 −343.000 512.000 729.000 −1766.27
1.3 8.00000 27.0000 64.0000 196.219 216.000 −343.000 512.000 729.000 1569.76
1.4 8.00000 27.0000 64.0000 293.276 216.000 −343.000 512.000 729.000 2346.21
1.5 8.00000 27.0000 64.0000 361.195 216.000 −343.000 512.000 729.000 2889.56
\(n\): 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.m 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.m 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 299 T_{5}^{4} - 161909 T_{5}^{3} + 46352519 T_{5}^{2} + 5654153000 T_{5} - \)\(15\!\cdots\!00\)\( \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{5} \)
$3$ \( ( -27 + T )^{5} \)
$5$ \( -1518560707600 + 5654153000 T + 46352519 T^{2} - 161909 T^{3} - 299 T^{4} + T^{5} \)
$7$ \( ( 343 + T )^{5} \)
$11$ \( -1680310994634492792 + 498447609120772 T + 226620352918 T^{2} - 68463141 T^{3} - 1838 T^{4} + T^{5} \)
$13$ \( ( 2197 + T )^{5} \)
$17$ \( \)\(25\!\cdots\!12\)\( - 178190191695663180 T + 12161044769988 T^{2} + 543538251 T^{3} - 56444 T^{4} + T^{5} \)
$19$ \( \)\(82\!\cdots\!40\)\( + 33068065870809648 T - 77431758519027 T^{2} - 2937554661 T^{3} + 20639 T^{4} + T^{5} \)
$23$ \( \)\(60\!\cdots\!08\)\( + 1185342480789876064 T + 67731120388511 T^{2} - 2553020301 T^{3} - 74395 T^{4} + T^{5} \)
$29$ \( \)\(18\!\cdots\!16\)\( + 4735320682486698288 T + 289407123405215 T^{2} - 392082817 T^{3} - 130059 T^{4} + T^{5} \)
$31$ \( -\)\(13\!\cdots\!20\)\( + 26556785386422135488 T + 6107428616536016 T^{2} - 4203250004 T^{3} - 330083 T^{4} + T^{5} \)
$37$ \( -\)\(26\!\cdots\!72\)\( + \)\(31\!\cdots\!28\)\( T + 78407595389644260 T^{2} - 214782035313 T^{3} - 410632 T^{4} + T^{5} \)
$41$ \( \)\(43\!\cdots\!68\)\( + \)\(12\!\cdots\!32\)\( T - 36526621827422496 T^{2} - 382938698424 T^{3} - 172558 T^{4} + T^{5} \)
$43$ \( -\)\(27\!\cdots\!72\)\( + \)\(53\!\cdots\!56\)\( T + 224139622148637905 T^{2} - 294615085081 T^{3} - 886497 T^{4} + T^{5} \)
$47$ \( \)\(32\!\cdots\!04\)\( - \)\(40\!\cdots\!44\)\( T + 1375607927131856164 T^{2} - 1254700667408 T^{3} - 763969 T^{4} + T^{5} \)
$53$ \( -\)\(60\!\cdots\!40\)\( + \)\(31\!\cdots\!32\)\( T + 6976879717292036496 T^{2} - 3930977984460 T^{3} - 1714575 T^{4} + T^{5} \)
$59$ \( \)\(19\!\cdots\!32\)\( + \)\(11\!\cdots\!84\)\( T - 10491323485251424 T^{2} - 2568894816948 T^{3} - 603580 T^{4} + T^{5} \)
$61$ \( \)\(12\!\cdots\!40\)\( - \)\(48\!\cdots\!52\)\( T + 26067506848129299086 T^{2} + 5723150832361 T^{3} - 6172268 T^{4} + T^{5} \)
$67$ \( \)\(71\!\cdots\!24\)\( - \)\(15\!\cdots\!80\)\( T + 92312795600378653544 T^{2} - 8567300340292 T^{3} - 5490834 T^{4} + T^{5} \)
$71$ \( \)\(30\!\cdots\!00\)\( + \)\(28\!\cdots\!24\)\( T - 4925939118742662128 T^{2} - 3620691633548 T^{3} + 1581200 T^{4} + T^{5} \)
$73$ \( -\)\(17\!\cdots\!88\)\( + \)\(12\!\cdots\!16\)\( T + 53029697226594055219 T^{2} - 25855840218633 T^{3} - 2100143 T^{4} + T^{5} \)
$79$ \( -\)\(11\!\cdots\!80\)\( - \)\(38\!\cdots\!72\)\( T + 78806907109103988844 T^{2} - 7621070034984 T^{3} - 6357137 T^{4} + T^{5} \)
$83$ \( -\)\(98\!\cdots\!52\)\( - \)\(17\!\cdots\!76\)\( T + 53768506824228391704 T^{2} - 11809863226356 T^{3} - 5292133 T^{4} + T^{5} \)
$89$ \( -\)\(39\!\cdots\!00\)\( - \)\(78\!\cdots\!80\)\( T + \)\(23\!\cdots\!56\)\( T^{2} - 123941645551692 T^{3} - 13229719 T^{4} + T^{5} \)
$97$ \( \)\(15\!\cdots\!20\)\( - \)\(36\!\cdots\!96\)\( T + \)\(78\!\cdots\!32\)\( T^{2} + 82162078645724 T^{3} - 22321383 T^{4} + T^{5} \)
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