Properties

Label 570.8.a.a
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: \(x^{4} - 2 x^{3} - 4122 x^{2} - 49773 x + 620550\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + 125 q^{5} -216 q^{6} + ( -374 + 2 \beta_{1} + \beta_{2} ) q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + 125 q^{5} -216 q^{6} + ( -374 + 2 \beta_{1} + \beta_{2} ) q^{7} + 512 q^{8} + 729 q^{9} + 1000 q^{10} + ( -728 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{11} -1728 q^{12} + ( -924 - 24 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{13} + ( -2992 + 16 \beta_{1} + 8 \beta_{2} ) q^{14} -3375 q^{15} + 4096 q^{16} + ( 4 + 68 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} ) q^{17} + 5832 q^{18} + 6859 q^{19} + 8000 q^{20} + ( 10098 - 54 \beta_{1} - 27 \beta_{2} ) q^{21} + ( -5824 + 8 \beta_{1} - 24 \beta_{2} + 16 \beta_{3} ) q^{22} + ( -18254 - 177 \beta_{1} + 6 \beta_{2} - 44 \beta_{3} ) q^{23} -13824 q^{24} + 15625 q^{25} + ( -7392 - 192 \beta_{1} - 24 \beta_{2} + 32 \beta_{3} ) q^{26} -19683 q^{27} + ( -23936 + 128 \beta_{1} + 64 \beta_{2} ) q^{28} + ( -34446 - 51 \beta_{1} - 133 \beta_{2} - 104 \beta_{3} ) q^{29} -27000 q^{30} + ( 49518 - 280 \beta_{1} - 218 \beta_{2} + 53 \beta_{3} ) q^{31} + 32768 q^{32} + ( 19656 - 27 \beta_{1} + 81 \beta_{2} - 54 \beta_{3} ) q^{33} + ( 32 + 544 \beta_{1} + 96 \beta_{2} - 120 \beta_{3} ) q^{34} + ( -46750 + 250 \beta_{1} + 125 \beta_{2} ) q^{35} + 46656 q^{36} + ( 51814 - 941 \beta_{1} + 273 \beta_{2} + 246 \beta_{3} ) q^{37} + 54872 q^{38} + ( 24948 + 648 \beta_{1} + 81 \beta_{2} - 108 \beta_{3} ) q^{39} + 64000 q^{40} + ( -126014 + 841 \beta_{1} - 223 \beta_{2} - 252 \beta_{3} ) q^{41} + ( 80784 - 432 \beta_{1} - 216 \beta_{2} ) q^{42} + ( -62592 + 1495 \beta_{1} + 15 \beta_{2} + 326 \beta_{3} ) q^{43} + ( -46592 + 64 \beta_{1} - 192 \beta_{2} + 128 \beta_{3} ) q^{44} + 91125 q^{45} + ( -146032 - 1416 \beta_{1} + 48 \beta_{2} - 352 \beta_{3} ) q^{46} + ( -250094 - 797 \beta_{1} + 186 \beta_{2} + 56 \beta_{3} ) q^{47} -110592 q^{48} + ( -235227 - 931 \beta_{1} - 882 \beta_{2} - 49 \beta_{3} ) q^{49} + 125000 q^{50} + ( -108 - 1836 \beta_{1} - 324 \beta_{2} + 405 \beta_{3} ) q^{51} + ( -59136 - 1536 \beta_{1} - 192 \beta_{2} + 256 \beta_{3} ) q^{52} + ( -544672 + 1955 \beta_{1} + 1290 \beta_{2} + 220 \beta_{3} ) q^{53} -157464 q^{54} + ( -91000 + 125 \beta_{1} - 375 \beta_{2} + 250 \beta_{3} ) q^{55} + ( -191488 + 1024 \beta_{1} + 512 \beta_{2} ) q^{56} -185193 q^{57} + ( -275568 - 408 \beta_{1} - 1064 \beta_{2} - 832 \beta_{3} ) q^{58} + ( 81994 - 6033 \beta_{1} - 600 \beta_{2} - 788 \beta_{3} ) q^{59} -216000 q^{60} + ( 143234 + 431 \beta_{1} - 1914 \beta_{2} + 2405 \beta_{3} ) q^{61} + ( 396144 - 2240 \beta_{1} - 1744 \beta_{2} + 424 \beta_{3} ) q^{62} + ( -272646 + 1458 \beta_{1} + 729 \beta_{2} ) q^{63} + 262144 q^{64} + ( -115500 - 3000 \beta_{1} - 375 \beta_{2} + 500 \beta_{3} ) q^{65} + ( 157248 - 216 \beta_{1} + 648 \beta_{2} - 432 \beta_{3} ) q^{66} + ( 504288 - 6324 \beta_{1} - 1154 \beta_{2} - 984 \beta_{3} ) q^{67} + ( 256 + 4352 \beta_{1} + 768 \beta_{2} - 960 \beta_{3} ) q^{68} + ( 492858 + 4779 \beta_{1} - 162 \beta_{2} + 1188 \beta_{3} ) q^{69} + ( -374000 + 2000 \beta_{1} + 1000 \beta_{2} ) q^{70} + ( 707240 + 18406 \beta_{1} - 1248 \beta_{2} - 6 \beta_{3} ) q^{71} + 373248 q^{72} + ( -33098 - 11788 \beta_{1} + 2834 \beta_{2} - 1372 \beta_{3} ) q^{73} + ( 414512 - 7528 \beta_{1} + 2184 \beta_{2} + 1968 \beta_{3} ) q^{74} -421875 q^{75} + 438976 q^{76} + ( -576176 + 3329 \beta_{1} - 446 \beta_{2} - 271 \beta_{3} ) q^{77} + ( 199584 + 5184 \beta_{1} + 648 \beta_{2} - 864 \beta_{3} ) q^{78} + ( 854602 - 16551 \beta_{1} + 7602 \beta_{2} - 250 \beta_{3} ) q^{79} + 512000 q^{80} + 531441 q^{81} + ( -1008112 + 6728 \beta_{1} - 1784 \beta_{2} - 2016 \beta_{3} ) q^{82} + ( -800440 + 19594 \beta_{1} - 2838 \beta_{2} - 3498 \beta_{3} ) q^{83} + ( 646272 - 3456 \beta_{1} - 1728 \beta_{2} ) q^{84} + ( 500 + 8500 \beta_{1} + 1500 \beta_{2} - 1875 \beta_{3} ) q^{85} + ( -500736 + 11960 \beta_{1} + 120 \beta_{2} + 2608 \beta_{3} ) q^{86} + ( 930042 + 1377 \beta_{1} + 3591 \beta_{2} + 2808 \beta_{3} ) q^{87} + ( -372736 + 512 \beta_{1} - 1536 \beta_{2} + 1024 \beta_{3} ) q^{88} + ( -347348 + 10322 \beta_{1} - 5673 \beta_{2} - 246 \beta_{3} ) q^{89} + 729000 q^{90} + ( -1966320 + 9541 \beta_{1} - 1692 \beta_{2} - 3825 \beta_{3} ) q^{91} + ( -1168256 - 11328 \beta_{1} + 384 \beta_{2} - 2816 \beta_{3} ) q^{92} + ( -1336986 + 7560 \beta_{1} + 5886 \beta_{2} - 1431 \beta_{3} ) q^{93} + ( -2000752 - 6376 \beta_{1} + 1488 \beta_{2} + 448 \beta_{3} ) q^{94} + 857375 q^{95} -884736 q^{96} + ( -5265286 + 24657 \beta_{1} + 1809 \beta_{2} - 576 \beta_{3} ) q^{97} + ( -1881816 - 7448 \beta_{1} - 7056 \beta_{2} - 392 \beta_{3} ) q^{98} + ( -530712 + 729 \beta_{1} - 2187 \beta_{2} + 1458 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} - 108q^{3} + 256q^{4} + 500q^{5} - 864q^{6} - 1496q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} - 108q^{3} + 256q^{4} + 500q^{5} - 864q^{6} - 1496q^{7} + 2048q^{8} + 2916q^{9} + 4000q^{10} - 2912q^{11} - 6912q^{12} - 3696q^{13} - 11968q^{14} - 13500q^{15} + 16384q^{16} + 16q^{17} + 23328q^{18} + 27436q^{19} + 32000q^{20} + 40392q^{21} - 23296q^{22} - 73016q^{23} - 55296q^{24} + 62500q^{25} - 29568q^{26} - 78732q^{27} - 95744q^{28} - 137784q^{29} - 108000q^{30} + 198072q^{31} + 131072q^{32} + 78624q^{33} + 128q^{34} - 187000q^{35} + 186624q^{36} + 207256q^{37} + 219488q^{38} + 99792q^{39} + 256000q^{40} - 504056q^{41} + 323136q^{42} - 250368q^{43} - 186368q^{44} + 364500q^{45} - 584128q^{46} - 1000376q^{47} - 442368q^{48} - 940908q^{49} + 500000q^{50} - 432q^{51} - 236544q^{52} - 2178688q^{53} - 629856q^{54} - 364000q^{55} - 765952q^{56} - 740772q^{57} - 1102272q^{58} + 327976q^{59} - 864000q^{60} + 572936q^{61} + 1584576q^{62} - 1090584q^{63} + 1048576q^{64} - 462000q^{65} + 628992q^{66} + 2017152q^{67} + 1024q^{68} + 1971432q^{69} - 1496000q^{70} + 2828960q^{71} + 1492992q^{72} - 132392q^{73} + 1658048q^{74} - 1687500q^{75} + 1755904q^{76} - 2304704q^{77} + 798336q^{78} + 3418408q^{79} + 2048000q^{80} + 2125764q^{81} - 4032448q^{82} - 3201760q^{83} + 2585088q^{84} + 2000q^{85} - 2002944q^{86} + 3720168q^{87} - 1490944q^{88} - 1389392q^{89} + 2916000q^{90} - 7865280q^{91} - 4673024q^{92} - 5347944q^{93} - 8003008q^{94} + 3429500q^{95} - 3538944q^{96} - 21061144q^{97} - 7527264q^{98} - 2122848q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4122 x^{2} - 49773 x + 620550\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 2 \)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{3} - 40 \nu^{2} - 14770 \nu - 84194 \)\()/233\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{3} + 1012 \nu^{2} + 8104 \nu - 1742678 \)\()/1631\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{3} + 2 \beta_{2} + 23 \beta_{1} + 8248\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(140 \beta_{3} + 506 \beta_{2} + 7845 \beta_{1} + 348118\)\()/8\)

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
7.65975
−53.5671
−21.7209
69.6283
8.00000 −27.0000 64.0000 125.000 −216.000 −1165.98 512.000 729.000 1000.00
1.2 8.00000 −27.0000 64.0000 125.000 −216.000 −903.592 512.000 729.000 1000.00
1.3 8.00000 −27.0000 64.0000 125.000 −216.000 206.862 512.000 729.000 1000.00
1.4 8.00000 −27.0000 64.0000 125.000 −216.000 366.713 512.000 729.000 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.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.8.a.a 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 1496 T_{7}^{3} - 57624 T_{7}^{2} - 447306872 T_{7} + 79922833472 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{4} \)
$3$ \( ( 27 + T )^{4} \)
$5$ \( ( -125 + T )^{4} \)
$7$ \( 79922833472 - 447306872 T - 57624 T^{2} + 1496 T^{3} + T^{4} \)
$11$ \( 293521583520 + 734936376 T - 8652072 T^{2} + 2912 T^{3} + T^{4} \)
$13$ \( -703307956188096 - 440884329560 T - 66970416 T^{2} + 3696 T^{3} + T^{4} \)
$17$ \( -41021953608913200 + 11076769039296 T - 784337400 T^{2} - 16 T^{3} + T^{4} \)
$19$ \( ( -6859 + T )^{4} \)
$23$ \( 1865511609579601920 - 93388753540608 T - 3155765568 T^{2} + 73016 T^{3} + T^{4} \)
$29$ \( -\)\(14\!\cdots\!32\)\( - 4754430509937048 T - 26612523840 T^{2} + 137784 T^{3} + T^{4} \)
$31$ \( \)\(16\!\cdots\!08\)\( + 4011494695070848 T - 24230385936 T^{2} - 198072 T^{3} + T^{4} \)
$37$ \( 71479153462205099744 + 32080385701555960 T - 227424814416 T^{2} - 207256 T^{3} + T^{4} \)
$41$ \( -\)\(29\!\cdots\!00\)\( - 46839876455068200 T - 120680160000 T^{2} + 504056 T^{3} + T^{4} \)
$43$ \( \)\(34\!\cdots\!60\)\( - 89930503893345176 T - 292571703144 T^{2} + 250368 T^{3} + T^{4} \)
$47$ \( \)\(47\!\cdots\!20\)\( + 24245634389140992 T + 297829779072 T^{2} + 1000376 T^{3} + T^{4} \)
$53$ \( \)\(43\!\cdots\!56\)\( - 601524600884159808 T + 329950072104 T^{2} + 2178688 T^{3} + T^{4} \)
$59$ \( -\)\(10\!\cdots\!92\)\( + 3892465284428922816 T - 3540964785504 T^{2} - 327976 T^{3} + T^{4} \)
$61$ \( \)\(91\!\cdots\!56\)\( + 8201033826836996128 T - 11516894322600 T^{2} - 572936 T^{3} + T^{4} \)
$67$ \( -\)\(38\!\cdots\!00\)\( + 8456551735240613056 T - 3571360934208 T^{2} - 2017152 T^{3} + T^{4} \)
$71$ \( \)\(75\!\cdots\!00\)\( + 17869769257315706880 T - 20834284087872 T^{2} - 2828960 T^{3} + T^{4} \)
$73$ \( \)\(13\!\cdots\!44\)\( - 12161359552041493280 T - 17013052319784 T^{2} + 132392 T^{3} + T^{4} \)
$79$ \( \)\(40\!\cdots\!32\)\( + 66256122332163787264 T - 54360756756096 T^{2} - 3418408 T^{3} + T^{4} \)
$83$ \( \)\(16\!\cdots\!40\)\( - 5907763678070515008 T - 54354960075168 T^{2} + 3201760 T^{3} + T^{4} \)
$89$ \( \)\(80\!\cdots\!40\)\( - 28928111635024130472 T - 29503645421472 T^{2} + 1389392 T^{3} + T^{4} \)
$97$ \( -\)\(54\!\cdots\!60\)\( + \)\(11\!\cdots\!88\)\( T + 123950449040352 T^{2} + 21061144 T^{3} + T^{4} \)
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