Properties

Label 546.8.a.n
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 309949 x^{4} - 14548431 x^{3} + 25221499020 x^{2} + 1862570808000 x - 308009568384000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8 q^{2} -27 q^{3} + 64 q^{4} + ( -30 - \beta_{1} ) q^{5} + 216 q^{6} + 343 q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})\) \( q -8 q^{2} -27 q^{3} + 64 q^{4} + ( -30 - \beta_{1} ) q^{5} + 216 q^{6} + 343 q^{7} -512 q^{8} + 729 q^{9} + ( 240 + 8 \beta_{1} ) q^{10} + ( -1021 - 3 \beta_{1} - \beta_{2} ) q^{11} -1728 q^{12} + 2197 q^{13} -2744 q^{14} + ( 810 + 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( -5768 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{17} -5832 q^{18} + ( -687 + 31 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{19} + ( -1920 - 64 \beta_{1} ) q^{20} -9261 q^{21} + ( 8168 + 24 \beta_{1} + 8 \beta_{2} ) q^{22} + ( 257 + 3 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{23} + 13824 q^{24} + ( 26078 + 133 \beta_{1} - 5 \beta_{2} - 7 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} -17576 q^{26} -19683 q^{27} + 21952 q^{28} + ( -9893 - 21 \beta_{1} - 23 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} - 7 \beta_{5} ) q^{29} + ( -6480 - 216 \beta_{1} ) q^{30} + ( 79703 + 18 \beta_{1} - 13 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} ) q^{31} -32768 q^{32} + ( 27567 + 81 \beta_{1} + 27 \beta_{2} ) q^{33} + ( 46144 - 8 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} + 8 \beta_{4} ) q^{34} + ( -10290 - 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( 95734 - 47 \beta_{1} + 24 \beta_{2} + 29 \beta_{3} - 15 \beta_{4} + \beta_{5} ) q^{37} + ( 5496 - 248 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 24 \beta_{5} ) q^{38} -59319 q^{39} + ( 15360 + 512 \beta_{1} ) q^{40} + ( 33597 - 34 \beta_{1} - 69 \beta_{2} - 43 \beta_{3} + \beta_{4} + 9 \beta_{5} ) q^{41} + 74088 q^{42} + ( 121547 - 679 \beta_{1} - 29 \beta_{2} - 4 \beta_{3} - 18 \beta_{4} + 27 \beta_{5} ) q^{43} + ( -65344 - 192 \beta_{1} - 64 \beta_{2} ) q^{44} + ( -21870 - 729 \beta_{1} ) q^{45} + ( -2056 - 24 \beta_{1} + 88 \beta_{2} - 32 \beta_{3} + 24 \beta_{4} + 16 \beta_{5} ) q^{46} + ( 38068 - 684 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} - 58 \beta_{4} + 10 \beta_{5} ) q^{47} -110592 q^{48} + 117649 q^{49} + ( -208624 - 1064 \beta_{1} + 40 \beta_{2} + 56 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{50} + ( 155736 - 27 \beta_{1} + 54 \beta_{2} + 54 \beta_{3} + 27 \beta_{4} ) q^{51} + 140608 q^{52} + ( 4739 - 2070 \beta_{1} + 57 \beta_{2} + 79 \beta_{3} - 59 \beta_{4} + 52 \beta_{5} ) q^{53} + 157464 q^{54} + ( 352326 + 845 \beta_{1} + 134 \beta_{2} - 153 \beta_{3} + 25 \beta_{4} - 51 \beta_{5} ) q^{55} -175616 q^{56} + ( 18549 - 837 \beta_{1} - 27 \beta_{2} - 27 \beta_{3} + 27 \beta_{4} - 81 \beta_{5} ) q^{57} + ( 79144 + 168 \beta_{1} + 184 \beta_{2} - 16 \beta_{3} - 80 \beta_{4} + 56 \beta_{5} ) q^{58} + ( 80203 - 3182 \beta_{1} + 47 \beta_{2} + 28 \beta_{3} + 103 \beta_{4} - 28 \beta_{5} ) q^{59} + ( 51840 + 1728 \beta_{1} ) q^{60} + ( -82956 - 3853 \beta_{1} - 42 \beta_{2} + 78 \beta_{3} + 22 \beta_{4} + 79 \beta_{5} ) q^{61} + ( -637624 - 144 \beta_{1} + 104 \beta_{2} + 24 \beta_{3} - 56 \beta_{4} + 16 \beta_{5} ) q^{62} + 250047 q^{63} + 262144 q^{64} + ( -65910 - 2197 \beta_{1} ) q^{65} + ( -220536 - 648 \beta_{1} - 216 \beta_{2} ) q^{66} + ( -525750 - 1774 \beta_{1} - 194 \beta_{2} + 114 \beta_{3} + 150 \beta_{4} - 78 \beta_{5} ) q^{67} + ( -369152 + 64 \beta_{1} - 128 \beta_{2} - 128 \beta_{3} - 64 \beta_{4} ) q^{68} + ( -6939 - 81 \beta_{1} + 297 \beta_{2} - 108 \beta_{3} + 81 \beta_{4} + 54 \beta_{5} ) q^{69} + ( 82320 + 2744 \beta_{1} ) q^{70} + ( -333605 - 1800 \beta_{1} - 485 \beta_{2} - 391 \beta_{3} - 17 \beta_{4} - 23 \beta_{5} ) q^{71} -373248 q^{72} + ( 609883 + 499 \beta_{1} + 29 \beta_{2} + 112 \beta_{3} - 100 \beta_{4} - 101 \beta_{5} ) q^{73} + ( -765872 + 376 \beta_{1} - 192 \beta_{2} - 232 \beta_{3} + 120 \beta_{4} - 8 \beta_{5} ) q^{74} + ( -704106 - 3591 \beta_{1} + 135 \beta_{2} + 189 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} ) q^{75} + ( -43968 + 1984 \beta_{1} + 64 \beta_{2} + 64 \beta_{3} - 64 \beta_{4} + 192 \beta_{5} ) q^{76} + ( -350203 - 1029 \beta_{1} - 343 \beta_{2} ) q^{77} + 474552 q^{78} + ( 188211 + 3092 \beta_{1} - 433 \beta_{2} + 197 \beta_{3} - 27 \beta_{4} - 194 \beta_{5} ) q^{79} + ( -122880 - 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( -268776 + 272 \beta_{1} + 552 \beta_{2} + 344 \beta_{3} - 8 \beta_{4} - 72 \beta_{5} ) q^{82} + ( -1605580 + 3438 \beta_{1} - 188 \beta_{2} - 381 \beta_{3} + 172 \beta_{4} + 12 \beta_{5} ) q^{83} -592704 q^{84} + ( 149888 - 2635 \beta_{1} - 364 \beta_{2} + 67 \beta_{3} - 157 \beta_{4} - 241 \beta_{5} ) q^{85} + ( -972376 + 5432 \beta_{1} + 232 \beta_{2} + 32 \beta_{3} + 144 \beta_{4} - 216 \beta_{5} ) q^{86} + ( 267111 + 567 \beta_{1} + 621 \beta_{2} - 54 \beta_{3} - 270 \beta_{4} + 189 \beta_{5} ) q^{87} + ( 522752 + 1536 \beta_{1} + 512 \beta_{2} ) q^{88} + ( -3661466 - 8934 \beta_{1} - 200 \beta_{2} + 404 \beta_{3} + 4 \beta_{4} + 315 \beta_{5} ) q^{89} + ( 174960 + 5832 \beta_{1} ) q^{90} + 753571 q^{91} + ( 16448 + 192 \beta_{1} - 704 \beta_{2} + 256 \beta_{3} - 192 \beta_{4} - 128 \beta_{5} ) q^{92} + ( -2151981 - 486 \beta_{1} + 351 \beta_{2} + 81 \beta_{3} - 189 \beta_{4} + 54 \beta_{5} ) q^{93} + ( -304544 + 5472 \beta_{1} - 256 \beta_{2} + 24 \beta_{3} + 464 \beta_{4} - 80 \beta_{5} ) q^{94} + ( -3220183 - 6201 \beta_{1} + 1805 \beta_{2} + 457 \beta_{3} + 149 \beta_{4} - \beta_{5} ) q^{95} + 884736 q^{96} + ( -4397499 - 942 \beta_{1} - 929 \beta_{2} + 201 \beta_{3} - 285 \beta_{4} + 424 \beta_{5} ) q^{97} -941192 q^{98} + ( -744309 - 2187 \beta_{1} - 729 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 309949 x^{4} - 14548431 x^{3} + 25221499020 x^{2} + 1862570808000 x - 308009568384000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2024389 \nu^{5} + 1548086629 \nu^{4} + 626570347621 \nu^{3} - 319874939700201 \nu^{2} - 53542834336954620 \nu + 9427731423802360200\)\()/ 1236949910141400 \)
\(\beta_{3}\)\(=\)\((\)\(-27588299 \nu^{5} + 2827655399 \nu^{4} + 5325442304951 \nu^{3} - 1059348303204831 \nu^{2} - 93918776326732980 \nu + 89343994329715910400\)\()/ 7421699460848400 \)
\(\beta_{4}\)\(=\)\((\)\( -1384243 \nu^{5} + 593013523 \nu^{4} + 248055256927 \nu^{3} - 92775859244187 \nu^{2} - 8954762572396740 \nu + 1854087064738915200 \)\()/ 117804753346800 \)
\(\beta_{5}\)\(=\)\((\)\(-83321227 \nu^{5} + 14438167357 \nu^{4} + 20223862688443 \nu^{3} - 1872553860103833 \nu^{2} - 1120675241488354020 \nu + 12374299269754128000\)\()/ 3710849730424200 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} - 7 \beta_{3} - 5 \beta_{2} + 73 \beta_{1} + 103303\)
\(\nu^{3}\)\(=\)\(425 \beta_{5} - 771 \beta_{4} - 1055 \beta_{3} + 2101 \beta_{2} + 147337 \beta_{1} + 7404281\)
\(\nu^{4}\)\(=\)\(105869 \beta_{5} + 316233 \beta_{4} - 1485023 \beta_{3} - 349969 \beta_{2} + 22069185 \beta_{1} + 15214288683\)
\(\nu^{5}\)\(=\)\(54491425 \beta_{5} - 154814415 \beta_{4} - 356083435 \beta_{3} + 561683925 \beta_{2} + 24495440641 \beta_{1} + 2260450105145\)

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
457.447
348.888
83.4678
−193.695
−309.836
−385.273
−8.00000 −27.0000 64.0000 −487.447 216.000 343.000 −512.000 729.000 3899.58
1.2 −8.00000 −27.0000 64.0000 −378.888 216.000 343.000 −512.000 729.000 3031.11
1.3 −8.00000 −27.0000 64.0000 −113.468 216.000 343.000 −512.000 729.000 907.742
1.4 −8.00000 −27.0000 64.0000 163.695 216.000 343.000 −512.000 729.000 −1309.56
1.5 −8.00000 −27.0000 64.0000 279.836 216.000 343.000 −512.000 729.000 −2238.69
1.6 −8.00000 −27.0000 64.0000 355.273 216.000 343.000 −512.000 729.000 −2842.18
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.n 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 181 T_{5}^{5} - 296299 T_{5}^{4} - 22096449 T_{5}^{3} + 24869553210 T_{5}^{2} - 343324745100 T_{5} - \)\(34\!\cdots\!00\)\( \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T )^{6} \)
$3$ \( ( 27 + T )^{6} \)
$5$ \( -341044841259000 - 343324745100 T + 24869553210 T^{2} - 22096449 T^{3} - 296299 T^{4} + 181 T^{5} + T^{6} \)
$7$ \( ( -343 + T )^{6} \)
$11$ \( -\)\(31\!\cdots\!36\)\( + 1736123897206401984 T + 531250914534696 T^{2} - 233699671890 T^{3} - 43044145 T^{4} + 6130 T^{5} + T^{6} \)
$13$ \( ( -2197 + T )^{6} \)
$17$ \( \)\(35\!\cdots\!08\)\( + \)\(22\!\cdots\!68\)\( T - 5845188076735428 T^{2} - 29198969866242 T^{3} - 829139125 T^{4} + 34610 T^{5} + T^{6} \)
$19$ \( -\)\(37\!\cdots\!40\)\( + \)\(87\!\cdots\!88\)\( T + 5343265149887319716 T^{2} - 9492878171013 T^{3} - 4734740997 T^{4} + 4085 T^{5} + T^{6} \)
$23$ \( -\)\(16\!\cdots\!00\)\( + \)\(31\!\cdots\!72\)\( T + \)\(11\!\cdots\!24\)\( T^{2} + 4727534295759 T^{3} - 20521250801 T^{4} - 1515 T^{5} + T^{6} \)
$29$ \( -\)\(21\!\cdots\!16\)\( - \)\(10\!\cdots\!52\)\( T + \)\(91\!\cdots\!98\)\( T^{2} - 216235904101143 T^{3} - 63710108383 T^{4} + 59395 T^{5} + T^{6} \)
$31$ \( -\)\(24\!\cdots\!60\)\( + \)\(17\!\cdots\!96\)\( T - \)\(21\!\cdots\!40\)\( T^{2} - 2857508603742272 T^{3} + 72784985460 T^{4} - 478241 T^{5} + T^{6} \)
$37$ \( \)\(38\!\cdots\!56\)\( - \)\(36\!\cdots\!04\)\( T - \)\(33\!\cdots\!92\)\( T^{2} + 99163911793545626 T^{3} - 117782223737 T^{4} - 574310 T^{5} + T^{6} \)
$41$ \( \)\(41\!\cdots\!08\)\( - \)\(34\!\cdots\!20\)\( T + \)\(47\!\cdots\!16\)\( T^{2} + 245803952696366640 T^{3} - 622802708432 T^{4} - 201552 T^{5} + T^{6} \)
$43$ \( \)\(64\!\cdots\!16\)\( - \)\(27\!\cdots\!80\)\( T - \)\(68\!\cdots\!04\)\( T^{2} + 467557746167504117 T^{3} - 432298077205 T^{4} - 728605 T^{5} + T^{6} \)
$47$ \( \)\(11\!\cdots\!76\)\( + \)\(12\!\cdots\!92\)\( T + \)\(36\!\cdots\!12\)\( T^{2} - 151915489918348752 T^{3} - 1287424320764 T^{4} - 227615 T^{5} + T^{6} \)
$53$ \( -\)\(48\!\cdots\!60\)\( + \)\(24\!\cdots\!68\)\( T + \)\(13\!\cdots\!28\)\( T^{2} - 1189810742698797576 T^{3} - 4243282064810 T^{4} - 26321 T^{5} + T^{6} \)
$59$ \( -\)\(16\!\cdots\!00\)\( + \)\(43\!\cdots\!92\)\( T + \)\(11\!\cdots\!84\)\( T^{2} - 1202809625885360064 T^{3} - 7301321480608 T^{4} - 478280 T^{5} + T^{6} \)
$61$ \( -\)\(13\!\cdots\!20\)\( + \)\(85\!\cdots\!72\)\( T + \)\(18\!\cdots\!92\)\( T^{2} - 2581773418155138100 T^{3} - 9282662906231 T^{4} + 501406 T^{5} + T^{6} \)
$67$ \( \)\(10\!\cdots\!32\)\( + \)\(68\!\cdots\!36\)\( T + \)\(19\!\cdots\!52\)\( T^{2} - 30812333776301144088 T^{3} - 9791651241644 T^{4} + 3156366 T^{5} + T^{6} \)
$71$ \( -\)\(16\!\cdots\!20\)\( - \)\(33\!\cdots\!04\)\( T + \)\(38\!\cdots\!92\)\( T^{2} - 20665613994075126720 T^{3} - 37121399529968 T^{4} + 2003644 T^{5} + T^{6} \)
$73$ \( \)\(39\!\cdots\!28\)\( - \)\(25\!\cdots\!16\)\( T + \)\(67\!\cdots\!22\)\( T^{2} + 36272125819456405523 T^{3} - 9855730520255 T^{4} - 3659111 T^{5} + T^{6} \)
$79$ \( -\)\(28\!\cdots\!00\)\( - \)\(27\!\cdots\!64\)\( T + \)\(67\!\cdots\!56\)\( T^{2} + 20776819780345680956 T^{3} - 47075544906248 T^{4} - 1131065 T^{5} + T^{6} \)
$83$ \( \)\(29\!\cdots\!28\)\( + \)\(14\!\cdots\!08\)\( T - \)\(42\!\cdots\!52\)\( T^{2} - \)\(26\!\cdots\!08\)\( T^{3} - 6623928686524 T^{4} + 9629297 T^{5} + T^{6} \)
$89$ \( -\)\(15\!\cdots\!00\)\( + \)\(40\!\cdots\!40\)\( T - \)\(21\!\cdots\!04\)\( T^{2} - \)\(47\!\cdots\!36\)\( T^{3} + 94721604221654 T^{4} + 21977377 T^{5} + T^{6} \)
$97$ \( -\)\(27\!\cdots\!80\)\( - \)\(23\!\cdots\!96\)\( T - \)\(12\!\cdots\!48\)\( T^{2} - \)\(16\!\cdots\!48\)\( T^{3} + 108642148800462 T^{4} + 26386649 T^{5} + T^{6} \)
show more
show less