Properties

Label 546.8.a.k
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: \(x^{5} - x^{4} - 148556 x^{3} - 20997404 x^{2} - 256427072 x + 44264019648\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( 102 - \beta_{1} ) q^{5} -216 q^{6} -343 q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( 102 - \beta_{1} ) q^{5} -216 q^{6} -343 q^{7} -512 q^{8} + 729 q^{9} + ( -816 + 8 \beta_{1} ) q^{10} + ( 192 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{11} + 1728 q^{12} + 2197 q^{13} + 2744 q^{14} + ( 2754 - 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( 1580 - 40 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{17} -5832 q^{18} + ( 12029 + 24 \beta_{1} + 4 \beta_{2} + \beta_{3} - 15 \beta_{4} ) q^{19} + ( 6528 - 64 \beta_{1} ) q^{20} -9261 q^{21} + ( -1536 + 32 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{22} + ( 24564 + 55 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} + 14 \beta_{4} ) q^{23} -13824 q^{24} + ( 14768 - 144 \beta_{1} + 26 \beta_{2} + 25 \beta_{3} + 7 \beta_{4} ) q^{25} -17576 q^{26} + 19683 q^{27} -21952 q^{28} + ( -3499 + 155 \beta_{1} + 23 \beta_{2} + 58 \beta_{3} - 28 \beta_{4} ) q^{29} + ( -22032 + 216 \beta_{1} ) q^{30} + ( -35516 - 144 \beta_{1} + 59 \beta_{2} - 58 \beta_{3} + 47 \beta_{4} ) q^{31} -32768 q^{32} + ( 5184 - 108 \beta_{1} + 54 \beta_{2} + 27 \beta_{3} ) q^{33} + ( -12640 + 320 \beta_{1} - 32 \beta_{2} + 24 \beta_{3} + 32 \beta_{4} ) q^{34} + ( -34986 + 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( -10905 - 555 \beta_{1} - 56 \beta_{2} + 42 \beta_{3} + 71 \beta_{4} ) q^{37} + ( -96232 - 192 \beta_{1} - 32 \beta_{2} - 8 \beta_{3} + 120 \beta_{4} ) q^{38} + 59319 q^{39} + ( -52224 + 512 \beta_{1} ) q^{40} + ( -47569 + 333 \beta_{1} - 58 \beta_{2} - 123 \beta_{3} - 137 \beta_{4} ) q^{41} + 74088 q^{42} + ( -193921 - 961 \beta_{1} - 35 \beta_{2} + 30 \beta_{3} - 194 \beta_{4} ) q^{43} + ( 12288 - 256 \beta_{1} + 128 \beta_{2} + 64 \beta_{3} ) q^{44} + ( 74358 - 729 \beta_{1} ) q^{45} + ( -196512 - 440 \beta_{1} + 48 \beta_{2} + 64 \beta_{3} - 112 \beta_{4} ) q^{46} + ( -76626 - 608 \beta_{1} - 297 \beta_{2} + 14 \beta_{3} + 9 \beta_{4} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( -118144 + 1152 \beta_{1} - 208 \beta_{2} - 200 \beta_{3} - 56 \beta_{4} ) q^{50} + ( 42660 - 1080 \beta_{1} + 108 \beta_{2} - 81 \beta_{3} - 108 \beta_{4} ) q^{51} + 140608 q^{52} + ( -284 - 366 \beta_{1} - 191 \beta_{2} - 496 \beta_{3} + 535 \beta_{4} ) q^{53} -157464 q^{54} + ( 304843 - 4099 \beta_{1} - 102 \beta_{2} + 200 \beta_{3} + 411 \beta_{4} ) q^{55} + 175616 q^{56} + ( 324783 + 648 \beta_{1} + 108 \beta_{2} + 27 \beta_{3} - 405 \beta_{4} ) q^{57} + ( 27992 - 1240 \beta_{1} - 184 \beta_{2} - 464 \beta_{3} + 224 \beta_{4} ) q^{58} + ( 411286 + 966 \beta_{1} + 540 \beta_{2} + 886 \beta_{3} - 722 \beta_{4} ) q^{59} + ( 176256 - 1728 \beta_{1} ) q^{60} + ( -143953 - 4426 \beta_{1} + 435 \beta_{2} - 311 \beta_{3} - 300 \beta_{4} ) q^{61} + ( 284128 + 1152 \beta_{1} - 472 \beta_{2} + 464 \beta_{3} - 376 \beta_{4} ) q^{62} -250047 q^{63} + 262144 q^{64} + ( 224094 - 2197 \beta_{1} ) q^{65} + ( -41472 + 864 \beta_{1} - 432 \beta_{2} - 216 \beta_{3} ) q^{66} + ( 539366 - 2696 \beta_{1} + 884 \beta_{2} - 898 \beta_{3} + 1390 \beta_{4} ) q^{67} + ( 101120 - 2560 \beta_{1} + 256 \beta_{2} - 192 \beta_{3} - 256 \beta_{4} ) q^{68} + ( 663228 + 1485 \beta_{1} - 162 \beta_{2} - 216 \beta_{3} + 378 \beta_{4} ) q^{69} + ( 279888 - 2744 \beta_{1} ) q^{70} + ( 1478771 - 517 \beta_{1} - 422 \beta_{2} + 553 \beta_{3} - 155 \beta_{4} ) q^{71} -373248 q^{72} + ( 1745255 - 5253 \beta_{1} - 581 \beta_{2} + 1214 \beta_{3} + 90 \beta_{4} ) q^{73} + ( 87240 + 4440 \beta_{1} + 448 \beta_{2} - 336 \beta_{3} - 568 \beta_{4} ) q^{74} + ( 398736 - 3888 \beta_{1} + 702 \beta_{2} + 675 \beta_{3} + 189 \beta_{4} ) q^{75} + ( 769856 + 1536 \beta_{1} + 256 \beta_{2} + 64 \beta_{3} - 960 \beta_{4} ) q^{76} + ( -65856 + 1372 \beta_{1} - 686 \beta_{2} - 343 \beta_{3} ) q^{77} -474552 q^{78} + ( 930036 - 3940 \beta_{1} + 179 \beta_{2} + 512 \beta_{3} - 1749 \beta_{4} ) q^{79} + ( 417792 - 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( 380552 - 2664 \beta_{1} + 464 \beta_{2} + 984 \beta_{3} + 1096 \beta_{4} ) q^{82} + ( 3552620 + 4466 \beta_{1} - 833 \beta_{2} - 754 \beta_{3} + 3335 \beta_{4} ) q^{83} -592704 q^{84} + ( 3299219 - 1015 \beta_{1} + 240 \beta_{2} - 50 \beta_{3} - 223 \beta_{4} ) q^{85} + ( 1551368 + 7688 \beta_{1} + 280 \beta_{2} - 240 \beta_{3} + 1552 \beta_{4} ) q^{86} + ( -94473 + 4185 \beta_{1} + 621 \beta_{2} + 1566 \beta_{3} - 756 \beta_{4} ) q^{87} + ( -98304 + 2048 \beta_{1} - 1024 \beta_{2} - 512 \beta_{3} ) q^{88} + ( 937899 + 5541 \beta_{1} - 2715 \beta_{2} - 1079 \beta_{3} + 2078 \beta_{4} ) q^{89} + ( -594864 + 5832 \beta_{1} ) q^{90} -753571 q^{91} + ( 1572096 + 3520 \beta_{1} - 384 \beta_{2} - 512 \beta_{3} + 896 \beta_{4} ) q^{92} + ( -958932 - 3888 \beta_{1} + 1593 \beta_{2} - 1566 \beta_{3} + 1269 \beta_{4} ) q^{93} + ( 613008 + 4864 \beta_{1} + 2376 \beta_{2} - 112 \beta_{3} - 72 \beta_{4} ) q^{94} + ( -470183 - 3178 \beta_{1} - 1852 \beta_{2} - 1005 \beta_{3} - 3291 \beta_{4} ) q^{95} -884736 q^{96} + ( 3132448 + 20806 \beta_{1} - 2741 \beta_{2} - 684 \beta_{3} - 2279 \beta_{4} ) q^{97} -941192 q^{98} + ( 139968 - 2916 \beta_{1} + 1458 \beta_{2} + 729 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 4072 q^{10} + 958 q^{11} + 8640 q^{12} + 10985 q^{13} + 13720 q^{14} + 13743 q^{15} + 20480 q^{16} + 7864 q^{17} - 29160 q^{18} + 60173 q^{19} + 32576 q^{20} - 46305 q^{21} - 7664 q^{22} + 122869 q^{23} - 69120 q^{24} + 73722 q^{25} - 87880 q^{26} + 98415 q^{27} - 109760 q^{28} - 17317 q^{29} - 109944 q^{30} - 177665 q^{31} - 163840 q^{32} + 25866 q^{33} - 62912 q^{34} - 174587 q^{35} + 233280 q^{36} - 55136 q^{37} - 481384 q^{38} + 296595 q^{39} - 260608 q^{40} - 237570 q^{41} + 370440 q^{42} - 970601 q^{43} + 61312 q^{44} + 371061 q^{45} - 982952 q^{46} - 384035 q^{47} + 552960 q^{48} + 588245 q^{49} - 589776 q^{50} + 212328 q^{51} + 703040 q^{52} - 1977 q^{53} - 787320 q^{54} + 1520014 q^{55} + 878080 q^{56} + 1624671 q^{57} + 138536 q^{58} + 2057936 q^{59} + 879552 q^{60} - 723756 q^{61} + 1421320 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 1118273 q^{65} - 206928 q^{66} + 2695018 q^{67} + 503296 q^{68} + 3317463 q^{69} + 1396696 q^{70} + 7392916 q^{71} - 1866240 q^{72} + 8720441 q^{73} + 441088 q^{74} + 1990494 q^{75} + 3851072 q^{76} - 328594 q^{77} - 2372760 q^{78} + 4646419 q^{79} + 2084864 q^{80} + 2657205 q^{81} + 1900560 q^{82} + 17766733 q^{83} - 2963520 q^{84} + 16495320 q^{85} + 7764808 q^{86} - 467559 q^{87} - 490496 q^{88} + 4692321 q^{89} - 2968488 q^{90} - 3767855 q^{91} + 7863616 q^{92} - 4796955 q^{93} + 3072280 q^{94} - 2355945 q^{95} - 4423680 q^{96} + 15680305 q^{97} - 4705960 q^{98} + 698382 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 148556 x^{3} - 20997404 x^{2} - 256427072 x + 44264019648\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -67 \nu^{4} + 4229 \nu^{3} + 9590106 \nu^{2} + 834162828 \nu - 17355924152 \)\()/58323928\)
\(\beta_{2}\)\(=\)\((\)\( 57 \nu^{4} - 7461 \nu^{3} - 7789168 \nu^{2} - 221698300 \nu + 41732843296 \)\()/8972912\)
\(\beta_{3}\)\(=\)\((\)\( 917 \nu^{4} - 74621 \nu^{3} - 132645092 \nu^{2} - 7627377340 \nu + 531906698624 \)\()/ 116647856 \)
\(\beta_{4}\)\(=\)\((\)\( 2039 \nu^{4} - 212403 \nu^{3} - 280855608 \nu^{2} - 13688812132 \nu + 934627614848 \)\()/ 116647856 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - 2 \beta_{2} + 11 \beta_{1} + 3\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(70 \beta_{4} - 47 \beta_{3} - 101 \beta_{2} + 185 \beta_{1} + 178251\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(44971 \beta_{4} + 22585 \beta_{3} - 96416 \beta_{2} + 305687 \beta_{1} + 76083603\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(12995059 \beta_{4} - 4458338 \beta_{3} - 20612135 \beta_{2} + 50635028 \beta_{1} + 27142832862\)\()/3\)

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
36.4029
443.950
−280.509
−110.406
−88.4385
−8.00000 27.0000 64.0000 −340.441 −216.000 −343.000 −512.000 729.000 2723.53
1.2 −8.00000 27.0000 64.0000 −78.1553 −216.000 −343.000 −512.000 729.000 625.242
1.3 −8.00000 27.0000 64.0000 186.173 −216.000 −343.000 −512.000 729.000 −1489.38
1.4 −8.00000 27.0000 64.0000 242.602 −216.000 −343.000 −512.000 729.000 −1940.82
1.5 −8.00000 27.0000 64.0000 498.821 −216.000 −343.000 −512.000 729.000 −3990.57
\(n\): 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.k 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.k 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 509 T_{5}^{4} - 102633 T_{5}^{3} + 61226049 T_{5}^{2} - 2538310720 T_{5} - 599455015900 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T )^{5} \)
$3$ \( ( -27 + T )^{5} \)
$5$ \( -599455015900 - 2538310720 T + 61226049 T^{2} - 102633 T^{3} - 509 T^{4} + T^{5} \)
$7$ \( ( 343 + T )^{5} \)
$11$ \( 357916927020597472 + 407228264310016 T + 40291016150 T^{2} - 50921545 T^{3} - 958 T^{4} + T^{5} \)
$13$ \( ( -2197 + T )^{5} \)
$17$ \( -\)\(57\!\cdots\!68\)\( + 66475620168764812 T + 7491042157592 T^{2} - 716770853 T^{3} - 7864 T^{4} + T^{5} \)
$19$ \( -\)\(39\!\cdots\!04\)\( + 38309540824651320 T + 36113430634853 T^{2} - 324561217 T^{3} - 60173 T^{4} + T^{5} \)
$23$ \( \)\(88\!\cdots\!16\)\( - 2358236817936976432 T + 36571660396569 T^{2} + 3747208723 T^{3} - 122869 T^{4} + T^{5} \)
$29$ \( -\)\(19\!\cdots\!76\)\( + 45283182588294462664 T + 1390135243053775 T^{2} - 35450633913 T^{3} + 17317 T^{4} + T^{5} \)
$31$ \( -\)\(18\!\cdots\!08\)\( + \)\(98\!\cdots\!56\)\( T - 10551002925989920 T^{2} - 77594991900 T^{3} + 177665 T^{4} + T^{5} \)
$37$ \( \)\(49\!\cdots\!32\)\( + \)\(47\!\cdots\!52\)\( T - 15573323876637988 T^{2} - 170881324873 T^{3} + 55136 T^{4} + T^{5} \)
$41$ \( \)\(81\!\cdots\!16\)\( + \)\(24\!\cdots\!48\)\( T + 11298849230412152 T^{2} - 458213936636 T^{3} + 237570 T^{4} + T^{5} \)
$43$ \( -\)\(72\!\cdots\!00\)\( - \)\(95\!\cdots\!80\)\( T - 352565975202629881 T^{2} - 163981189945 T^{3} + 970601 T^{4} + T^{5} \)
$47$ \( \)\(91\!\cdots\!12\)\( + \)\(10\!\cdots\!56\)\( T - 161045493328442880 T^{2} - 1004072984880 T^{3} + 384035 T^{4} + T^{5} \)
$53$ \( \)\(28\!\cdots\!52\)\( + \)\(13\!\cdots\!96\)\( T - 466005514929715568 T^{2} - 3054374306444 T^{3} + 1977 T^{4} + T^{5} \)
$59$ \( -\)\(20\!\cdots\!48\)\( + \)\(99\!\cdots\!24\)\( T + 15578041828028869504 T^{2} - 7578369568496 T^{3} - 2057936 T^{4} + T^{5} \)
$61$ \( -\)\(55\!\cdots\!12\)\( + \)\(99\!\cdots\!20\)\( T + 1480009860968110810 T^{2} - 7597103851567 T^{3} + 723756 T^{4} + T^{5} \)
$67$ \( -\)\(25\!\cdots\!56\)\( + \)\(11\!\cdots\!76\)\( T + 69291871518313837128 T^{2} - 27518590867812 T^{3} - 2695018 T^{4} + T^{5} \)
$71$ \( \)\(76\!\cdots\!40\)\( - \)\(17\!\cdots\!24\)\( T - 3927436705004738048 T^{2} + 16183381694080 T^{3} - 7392916 T^{4} + T^{5} \)
$73$ \( -\)\(24\!\cdots\!76\)\( - \)\(91\!\cdots\!88\)\( T + \)\(10\!\cdots\!85\)\( T^{2} + 1761824625247 T^{3} - 8720441 T^{4} + T^{5} \)
$79$ \( \)\(21\!\cdots\!00\)\( + \)\(63\!\cdots\!40\)\( T - 5813151801189852428 T^{2} - 16553047064272 T^{3} - 4646419 T^{4} + T^{5} \)
$83$ \( \)\(87\!\cdots\!68\)\( - \)\(49\!\cdots\!20\)\( T + \)\(74\!\cdots\!28\)\( T^{2} + 43802323688476 T^{3} - 17766733 T^{4} + T^{5} \)
$89$ \( \)\(67\!\cdots\!64\)\( + \)\(20\!\cdots\!36\)\( T - \)\(19\!\cdots\!96\)\( T^{2} - 83989557223164 T^{3} - 4692321 T^{4} + T^{5} \)
$97$ \( \)\(11\!\cdots\!64\)\( - \)\(11\!\cdots\!64\)\( T + \)\(29\!\cdots\!80\)\( T^{2} - 128430202207348 T^{3} - 15680305 T^{4} + T^{5} \)
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