Properties

Label 570.8.a.b
Level $570$
Weight $8$
Character orbit 570.a
Self dual yes
Analytic conductor $178.059$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(178.059464526\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 3046 x^{2} + 50476 x + 497070\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} -125 q^{5} + 216 q^{6} + ( -186 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} -125 q^{5} + 216 q^{6} + ( -186 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{7} + 512 q^{8} + 729 q^{9} -1000 q^{10} + ( -86 + 5 \beta_{1} + 11 \beta_{2} + 4 \beta_{3} ) q^{11} + 1728 q^{12} + ( -1596 - 9 \beta_{1} - 10 \beta_{2} - 24 \beta_{3} ) q^{13} + ( -1488 - 8 \beta_{1} - 24 \beta_{2} + 16 \beta_{3} ) q^{14} -3375 q^{15} + 4096 q^{16} + ( -4110 - 14 \beta_{1} + 85 \beta_{2} - 39 \beta_{3} ) q^{17} + 5832 q^{18} + 6859 q^{19} -8000 q^{20} + ( -5022 - 27 \beta_{1} - 81 \beta_{2} + 54 \beta_{3} ) q^{21} + ( -688 + 40 \beta_{1} + 88 \beta_{2} + 32 \beta_{3} ) q^{22} + ( -16896 + 278 \beta_{1} - 23 \beta_{2} + 68 \beta_{3} ) q^{23} + 13824 q^{24} + 15625 q^{25} + ( -12768 - 72 \beta_{1} - 80 \beta_{2} - 192 \beta_{3} ) q^{26} + 19683 q^{27} + ( -11904 - 64 \beta_{1} - 192 \beta_{2} + 128 \beta_{3} ) q^{28} + ( -30216 - 189 \beta_{1} - 57 \beta_{2} + 54 \beta_{3} ) q^{29} -27000 q^{30} + ( -55712 + 240 \beta_{1} - 243 \beta_{2} + 101 \beta_{3} ) q^{31} + 32768 q^{32} + ( -2322 + 135 \beta_{1} + 297 \beta_{2} + 108 \beta_{3} ) q^{33} + ( -32880 - 112 \beta_{1} + 680 \beta_{2} - 312 \beta_{3} ) q^{34} + ( 23250 + 125 \beta_{1} + 375 \beta_{2} - 250 \beta_{3} ) q^{35} + 46656 q^{36} + ( -102528 - 91 \beta_{1} + 1002 \beta_{2} - 820 \beta_{3} ) q^{37} + 54872 q^{38} + ( -43092 - 243 \beta_{1} - 270 \beta_{2} - 648 \beta_{3} ) q^{39} -64000 q^{40} + ( 52220 - 151 \beta_{1} + 450 \beta_{2} - 60 \beta_{3} ) q^{41} + ( -40176 - 216 \beta_{1} - 648 \beta_{2} + 432 \beta_{3} ) q^{42} + ( -245490 + 803 \beta_{1} + 614 \beta_{2} + 160 \beta_{3} ) q^{43} + ( -5504 + 320 \beta_{1} + 704 \beta_{2} + 256 \beta_{3} ) q^{44} -91125 q^{45} + ( -135168 + 2224 \beta_{1} - 184 \beta_{2} + 544 \beta_{3} ) q^{46} + ( -94432 - 3154 \beta_{1} + 273 \beta_{2} - 636 \beta_{3} ) q^{47} + 110592 q^{48} + ( -208795 - 1274 \beta_{1} + 938 \beta_{2} - 987 \beta_{3} ) q^{49} + 125000 q^{50} + ( -110970 - 378 \beta_{1} + 2295 \beta_{2} - 1053 \beta_{3} ) q^{51} + ( -102144 - 576 \beta_{1} - 640 \beta_{2} - 1536 \beta_{3} ) q^{52} + ( -313838 - 330 \beta_{1} - 6531 \beta_{2} - 2240 \beta_{3} ) q^{53} + 157464 q^{54} + ( 10750 - 625 \beta_{1} - 1375 \beta_{2} - 500 \beta_{3} ) q^{55} + ( -95232 - 512 \beta_{1} - 1536 \beta_{2} + 1024 \beta_{3} ) q^{56} + 185193 q^{57} + ( -241728 - 1512 \beta_{1} - 456 \beta_{2} + 432 \beta_{3} ) q^{58} + ( -196644 + 5254 \beta_{1} + 4735 \beta_{2} + 690 \beta_{3} ) q^{59} -216000 q^{60} + ( -853294 + 5738 \beta_{1} + 4560 \beta_{2} + 4399 \beta_{3} ) q^{61} + ( -445696 + 1920 \beta_{1} - 1944 \beta_{2} + 808 \beta_{3} ) q^{62} + ( -135594 - 729 \beta_{1} - 2187 \beta_{2} + 1458 \beta_{3} ) q^{63} + 262144 q^{64} + ( 199500 + 1125 \beta_{1} + 1250 \beta_{2} + 3000 \beta_{3} ) q^{65} + ( -18576 + 1080 \beta_{1} + 2376 \beta_{2} + 864 \beta_{3} ) q^{66} + ( -270032 - 3578 \beta_{1} + 1980 \beta_{2} + 7162 \beta_{3} ) q^{67} + ( -263040 - 896 \beta_{1} + 5440 \beta_{2} - 2496 \beta_{3} ) q^{68} + ( -456192 + 7506 \beta_{1} - 621 \beta_{2} + 1836 \beta_{3} ) q^{69} + ( 186000 + 1000 \beta_{1} + 3000 \beta_{2} - 2000 \beta_{3} ) q^{70} + ( -522400 - 6180 \beta_{1} - 15146 \beta_{2} - 9960 \beta_{3} ) q^{71} + 373248 q^{72} + ( -63034 + 2822 \beta_{1} - 958 \beta_{2} + 3246 \beta_{3} ) q^{73} + ( -820224 - 728 \beta_{1} + 8016 \beta_{2} - 6560 \beta_{3} ) q^{74} + 421875 q^{75} + 438976 q^{76} + ( -605788 + 6442 \beta_{1} - 390 \beta_{2} - 185 \beta_{3} ) q^{77} + ( -344736 - 1944 \beta_{1} - 2160 \beta_{2} - 5184 \beta_{3} ) q^{78} + ( -531336 - 6556 \beta_{1} + 14781 \beta_{2} - 2560 \beta_{3} ) q^{79} -512000 q^{80} + 531441 q^{81} + ( 417760 - 1208 \beta_{1} + 3600 \beta_{2} - 480 \beta_{3} ) q^{82} + ( -2193040 - 14428 \beta_{1} - 7070 \beta_{2} + 10072 \beta_{3} ) q^{83} + ( -321408 - 1728 \beta_{1} - 5184 \beta_{2} + 3456 \beta_{3} ) q^{84} + ( 513750 + 1750 \beta_{1} - 10625 \beta_{2} + 4875 \beta_{3} ) q^{85} + ( -1963920 + 6424 \beta_{1} + 4912 \beta_{2} + 1280 \beta_{3} ) q^{86} + ( -815832 - 5103 \beta_{1} - 1539 \beta_{2} + 1458 \beta_{3} ) q^{87} + ( -44032 + 2560 \beta_{1} + 5632 \beta_{2} + 2048 \beta_{3} ) q^{88} + ( -4594196 - 12307 \beta_{1} - 12562 \beta_{2} + 17440 \beta_{3} ) q^{89} -729000 q^{90} + ( -1757744 - 6272 \beta_{1} + 2012 \beta_{2} + 757 \beta_{3} ) q^{91} + ( -1081344 + 17792 \beta_{1} - 1472 \beta_{2} + 4352 \beta_{3} ) q^{92} + ( -1504224 + 6480 \beta_{1} - 6561 \beta_{2} + 2727 \beta_{3} ) q^{93} + ( -755456 - 25232 \beta_{1} + 2184 \beta_{2} - 5088 \beta_{3} ) q^{94} -857375 q^{95} + 884736 q^{96} + ( 6624 + 4173 \beta_{1} - 20960 \beta_{2} + 24956 \beta_{3} ) q^{97} + ( -1670360 - 10192 \beta_{1} + 7504 \beta_{2} - 7896 \beta_{3} ) q^{98} + ( -62694 + 3645 \beta_{1} + 8019 \beta_{2} + 2916 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} - 4000q^{10} - 354q^{11} + 6912q^{12} - 6366q^{13} - 5936q^{14} - 13500q^{15} + 16384q^{16} - 16412q^{17} + 23328q^{18} + 27436q^{19} - 32000q^{20} - 20034q^{21} - 2832q^{22} - 68140q^{23} + 55296q^{24} + 62500q^{25} - 50928q^{26} + 78732q^{27} - 47488q^{28} - 120486q^{29} - 108000q^{30} - 223328q^{31} + 131072q^{32} - 9558q^{33} - 131296q^{34} + 92750q^{35} + 186624q^{36} - 409930q^{37} + 219488q^{38} - 171882q^{39} - 256000q^{40} + 209182q^{41} - 160272q^{42} - 983566q^{43} - 22656q^{44} - 364500q^{45} - 545120q^{46} - 371420q^{47} + 442368q^{48} - 832632q^{49} + 500000q^{50} - 443124q^{51} - 407424q^{52} - 1254692q^{53} + 629856q^{54} + 44250q^{55} - 379904q^{56} + 740772q^{57} - 963888q^{58} - 797084q^{59} - 864000q^{60} - 3424652q^{61} - 1786624q^{62} - 540918q^{63} + 1048576q^{64} + 795750q^{65} - 76464q^{66} - 1072972q^{67} - 1050368q^{68} - 1839780q^{69} + 742000q^{70} - 2077240q^{71} + 1492992q^{72} - 257780q^{73} - 3279440q^{74} + 1687500q^{75} + 1755904q^{76} - 2436036q^{77} - 1375056q^{78} - 2112232q^{79} - 2048000q^{80} + 2125764q^{81} + 1673456q^{82} - 8743304q^{83} - 1282176q^{84} + 2051500q^{85} - 7868528q^{86} - 3253122q^{87} - 181248q^{88} - 18352170q^{89} - 2916000q^{90} - 7018432q^{91} - 4360960q^{92} - 6029856q^{93} - 2971360q^{94} - 3429500q^{95} + 3538944q^{96} + 18150q^{97} - 6661056q^{98} - 258066q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 3046 x^{2} + 50476 x + 497070\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -14 \nu^{3} - 330 \nu^{2} + 38354 \nu - 5670 \)\()/1485\)
\(\beta_{2}\)\(=\)\((\)\( 28 \nu^{3} + 660 \nu^{2} - 58888 \nu + 5400 \)\()/1485\)
\(\beta_{3}\)\(=\)\((\)\( 76 \nu^{3} + 2640 \nu^{2} - 143716 \nu - 1281960 \)\()/1485\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 4\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(21 \beta_{3} - 76 \beta_{2} - 38 \beta_{1} + 18260\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-495 \beta_{3} + 4531 \beta_{2} + 5102 \beta_{1} - 424316\)\()/12\)

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
−6.96971
32.6184
36.0320
−60.6808
8.00000 27.0000 64.0000 −125.000 216.000 −1119.70 512.000 729.000 −1000.00
1.2 8.00000 27.0000 64.0000 −125.000 216.000 −677.988 512.000 729.000 −1000.00
1.3 8.00000 27.0000 64.0000 −125.000 216.000 218.883 512.000 729.000 −1000.00
1.4 8.00000 27.0000 64.0000 −125.000 216.000 836.803 512.000 729.000 −1000.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 570.8.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.8.a.b 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 742 T_{7}^{3} - 955488 T_{7}^{2} - 472147984 T_{7} + 139045611104 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{4} \)
$3$ \( ( -27 + T )^{4} \)
$5$ \( ( 125 + T )^{4} \)
$7$ \( 139045611104 - 472147984 T - 955488 T^{2} + 742 T^{3} + T^{4} \)
$11$ \( 25168902899040 - 1965232048 T - 10902960 T^{2} + 354 T^{3} + T^{4} \)
$13$ \( 953713968000 - 16454859080 T - 71721396 T^{2} + 6366 T^{3} + T^{4} \)
$17$ \( -76630168465693776 - 15188656076784 T - 636286128 T^{2} + 16412 T^{3} + T^{4} \)
$19$ \( ( -6859 + T )^{4} \)
$23$ \( 17068849843588053504 - 308262078965440 T - 7422352720 T^{2} + 68140 T^{3} + T^{4} \)
$29$ \( -3151909070986914720 - 198319599489096 T + 839594484 T^{2} + 120486 T^{3} + T^{4} \)
$31$ \( 3545509185106513920 - 367559598378368 T + 4934219264 T^{2} + 223328 T^{3} + T^{4} \)
$37$ \( -60295929414942068320 - 39789203160860056 T - 92581678092 T^{2} + 409930 T^{3} + T^{4} \)
$41$ \( -1938111983985651840 + 230326040862792 T - 3651364596 T^{2} - 209182 T^{3} + T^{4} \)
$43$ \( \)\(23\!\cdots\!56\)\( + 26578055592138384 T + 284845083408 T^{2} + 983566 T^{3} + T^{4} \)
$47$ \( \)\(32\!\cdots\!80\)\( - 206049315757703616 T - 1120514000592 T^{2} + 371420 T^{3} + T^{4} \)
$53$ \( \)\(22\!\cdots\!00\)\( - 2571014311375371920 T - 3226860260608 T^{2} + 1254692 T^{3} + T^{4} \)
$59$ \( -\)\(78\!\cdots\!60\)\( + 1063320794001114624 T - 3340001049504 T^{2} + 797084 T^{3} + T^{4} \)
$61$ \( -\)\(95\!\cdots\!80\)\( - 13182905191887719856 T - 1628595036560 T^{2} + 3424652 T^{3} + T^{4} \)
$67$ \( -\)\(11\!\cdots\!00\)\( + 9802800857629874880 T - 9948700066224 T^{2} + 1072972 T^{3} + T^{4} \)
$71$ \( \)\(14\!\cdots\!40\)\( - 14898988407247750656 T - 28137381986112 T^{2} + 2077240 T^{3} + T^{4} \)
$73$ \( -\)\(17\!\cdots\!88\)\( - 1226688063337824848 T - 2150885380656 T^{2} + 257780 T^{3} + T^{4} \)
$79$ \( -\)\(66\!\cdots\!00\)\( - 80951443284989267200 T - 23193044474880 T^{2} + 2112232 T^{3} + T^{4} \)
$83$ \( -\)\(17\!\cdots\!16\)\( - \)\(14\!\cdots\!12\)\( T - 12778501557952 T^{2} + 8743304 T^{3} + T^{4} \)
$89$ \( -\)\(39\!\cdots\!16\)\( - \)\(25\!\cdots\!60\)\( T + 57544799111340 T^{2} + 18352170 T^{3} + T^{4} \)
$97$ \( \)\(19\!\cdots\!00\)\( + 72261009133945342440 T - 112386160877036 T^{2} - 18150 T^{3} + T^{4} \)
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