Properties

Label 570.8.a.c
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: \(x^{4} - x^{3} - 2087 x^{2} + 44517 x - 205110\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + 125 q^{5} + 216 q^{6} + ( -327 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + 125 q^{5} + 216 q^{6} + ( -327 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{7} + 512 q^{8} + 729 q^{9} + 1000 q^{10} + ( -1642 + 13 \beta_{1} + 15 \beta_{2} + 3 \beta_{3} ) q^{11} + 1728 q^{12} + ( -4436 - 51 \beta_{1} + 23 \beta_{2} - 9 \beta_{3} ) q^{13} + ( -2616 + 24 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{14} + 3375 q^{15} + 4096 q^{16} + ( -7064 + 2 \beta_{1} - 52 \beta_{2} + 51 \beta_{3} ) q^{17} + 5832 q^{18} -6859 q^{19} + 8000 q^{20} + ( -8829 + 81 \beta_{1} - 54 \beta_{2} - 27 \beta_{3} ) q^{21} + ( -13136 + 104 \beta_{1} + 120 \beta_{2} + 24 \beta_{3} ) q^{22} + ( -8034 - 160 \beta_{1} - 202 \beta_{2} - 42 \beta_{3} ) q^{23} + 13824 q^{24} + 15625 q^{25} + ( -35488 - 408 \beta_{1} + 184 \beta_{2} - 72 \beta_{3} ) q^{26} + 19683 q^{27} + ( -20928 + 192 \beta_{1} - 128 \beta_{2} - 64 \beta_{3} ) q^{28} + ( -39478 + 221 \beta_{1} - 195 \beta_{2} + 189 \beta_{3} ) q^{29} + 27000 q^{30} + ( -17580 - 1000 \beta_{1} + 228 \beta_{2} + 55 \beta_{3} ) q^{31} + 32768 q^{32} + ( -44334 + 351 \beta_{1} + 405 \beta_{2} + 81 \beta_{3} ) q^{33} + ( -56512 + 16 \beta_{1} - 416 \beta_{2} + 408 \beta_{3} ) q^{34} + ( -40875 + 375 \beta_{1} - 250 \beta_{2} - 125 \beta_{3} ) q^{35} + 46656 q^{36} + ( -205160 + 391 \beta_{1} - 1371 \beta_{2} - 231 \beta_{3} ) q^{37} -54872 q^{38} + ( -119772 - 1377 \beta_{1} + 621 \beta_{2} - 243 \beta_{3} ) q^{39} + 64000 q^{40} + ( -196607 - 629 \beta_{1} + 1804 \beta_{2} + 181 \beta_{3} ) q^{41} + ( -70632 + 648 \beta_{1} - 432 \beta_{2} - 216 \beta_{3} ) q^{42} + ( -276402 + 2467 \beta_{1} - 1079 \beta_{2} + 1469 \beta_{3} ) q^{43} + ( -105088 + 832 \beta_{1} + 960 \beta_{2} + 192 \beta_{3} ) q^{44} + 91125 q^{45} + ( -64272 - 1280 \beta_{1} - 1616 \beta_{2} - 336 \beta_{3} ) q^{46} + ( -55934 - 3460 \beta_{1} + 3622 \beta_{2} - 206 \beta_{3} ) q^{47} + 110592 q^{48} + ( -292354 - 3626 \beta_{1} + 1085 \beta_{2} - 343 \beta_{3} ) q^{49} + 125000 q^{50} + ( -190728 + 54 \beta_{1} - 1404 \beta_{2} + 1377 \beta_{3} ) q^{51} + ( -283904 - 3264 \beta_{1} + 1472 \beta_{2} - 576 \beta_{3} ) q^{52} + ( -213296 - 724 \beta_{1} + 174 \beta_{2} - 1482 \beta_{3} ) q^{53} + 157464 q^{54} + ( -205250 + 1625 \beta_{1} + 1875 \beta_{2} + 375 \beta_{3} ) q^{55} + ( -167424 + 1536 \beta_{1} - 1024 \beta_{2} - 512 \beta_{3} ) q^{56} -185193 q^{57} + ( -315824 + 1768 \beta_{1} - 1560 \beta_{2} + 1512 \beta_{3} ) q^{58} + ( -922277 - 14 \beta_{1} + 3351 \beta_{2} - 2274 \beta_{3} ) q^{59} + 216000 q^{60} + ( -292043 + 2550 \beta_{1} + 7435 \beta_{2} + 1335 \beta_{3} ) q^{61} + ( -140640 - 8000 \beta_{1} + 1824 \beta_{2} + 440 \beta_{3} ) q^{62} + ( -238383 + 2187 \beta_{1} - 1458 \beta_{2} - 729 \beta_{3} ) q^{63} + 262144 q^{64} + ( -554500 - 6375 \beta_{1} + 2875 \beta_{2} - 1125 \beta_{3} ) q^{65} + ( -354672 + 2808 \beta_{1} + 3240 \beta_{2} + 648 \beta_{3} ) q^{66} + ( -1535320 + 10926 \beta_{1} - 646 \beta_{2} - 4842 \beta_{3} ) q^{67} + ( -452096 + 128 \beta_{1} - 3328 \beta_{2} + 3264 \beta_{3} ) q^{68} + ( -216918 - 4320 \beta_{1} - 5454 \beta_{2} - 1134 \beta_{3} ) q^{69} + ( -327000 + 3000 \beta_{1} - 2000 \beta_{2} - 1000 \beta_{3} ) q^{70} + ( -1107438 + 12180 \beta_{1} - 2230 \beta_{2} - 1174 \beta_{3} ) q^{71} + 373248 q^{72} + ( 276584 + 15438 \beta_{1} - 2440 \beta_{2} + 6394 \beta_{3} ) q^{73} + ( -1641280 + 3128 \beta_{1} - 10968 \beta_{2} - 1848 \beta_{3} ) q^{74} + 421875 q^{75} -438976 q^{76} + ( 591385 - 18358 \beta_{1} + 2813 \beta_{2} + 2747 \beta_{3} ) q^{77} + ( -958176 - 11016 \beta_{1} + 4968 \beta_{2} - 1944 \beta_{3} ) q^{78} + ( 733155 + 7816 \beta_{1} - 2127 \beta_{2} + 1322 \beta_{3} ) q^{79} + 512000 q^{80} + 531441 q^{81} + ( -1572856 - 5032 \beta_{1} + 14432 \beta_{2} + 1448 \beta_{3} ) q^{82} + ( -436049 - 3374 \beta_{1} - 18209 \beta_{2} - 3334 \beta_{3} ) q^{83} + ( -565056 + 5184 \beta_{1} - 3456 \beta_{2} - 1728 \beta_{3} ) q^{84} + ( -883000 + 250 \beta_{1} - 6500 \beta_{2} + 6375 \beta_{3} ) q^{85} + ( -2211216 + 19736 \beta_{1} - 8632 \beta_{2} + 11752 \beta_{3} ) q^{86} + ( -1065906 + 5967 \beta_{1} - 5265 \beta_{2} + 5103 \beta_{3} ) q^{87} + ( -840704 + 6656 \beta_{1} + 7680 \beta_{2} + 1536 \beta_{3} ) q^{88} + ( -917521 + 5659 \beta_{1} - 15514 \beta_{2} - 17535 \beta_{3} ) q^{89} + 729000 q^{90} + ( -1654401 + 36820 \beta_{1} - 9703 \beta_{2} + 8541 \beta_{3} ) q^{91} + ( -514176 - 10240 \beta_{1} - 12928 \beta_{2} - 2688 \beta_{3} ) q^{92} + ( -474660 - 27000 \beta_{1} + 6156 \beta_{2} + 1485 \beta_{3} ) q^{93} + ( -447472 - 27680 \beta_{1} + 28976 \beta_{2} - 1648 \beta_{3} ) q^{94} -857375 q^{95} + 884736 q^{96} + ( -2596742 + 7587 \beta_{1} - 16165 \beta_{2} - 7689 \beta_{3} ) q^{97} + ( -2338832 - 29008 \beta_{1} + 8680 \beta_{2} - 2744 \beta_{3} ) q^{98} + ( -1197018 + 9477 \beta_{1} + 10935 \beta_{2} + 2187 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + 4000q^{10} - 6570q^{11} + 6912q^{12} - 17578q^{13} - 10528q^{14} + 13500q^{15} + 16384q^{16} - 28466q^{17} + 23328q^{18} - 27436q^{19} + 32000q^{20} - 35532q^{21} - 52560q^{22} - 32136q^{23} + 55296q^{24} + 62500q^{25} - 140624q^{26} + 78732q^{27} - 84224q^{28} - 159122q^{29} + 108000q^{30} - 67974q^{31} + 131072q^{32} - 177390q^{33} - 227728q^{34} - 164500q^{35} + 186624q^{36} - 823702q^{37} - 219488q^{38} - 474606q^{39} + 256000q^{40} - 781924q^{41} - 284256q^{42} - 1115638q^{43} - 420480q^{44} + 364500q^{45} - 257088q^{46} - 209160q^{47} + 442368q^{48} - 1159308q^{49} + 500000q^{50} - 768582q^{51} - 1124992q^{52} - 848424q^{53} + 629856q^{54} - 821250q^{55} - 673792q^{56} - 740772q^{57} - 1272976q^{58} - 3677830q^{59} + 864000q^{60} - 1161072q^{61} - 543792q^{62} - 959364q^{63} + 1048576q^{64} - 2197250q^{65} - 1419120q^{66} - 6154740q^{67} - 1821824q^{68} - 867672q^{69} - 1316000q^{70} - 4456224q^{71} + 1492992q^{72} + 1057792q^{73} - 6589616q^{74} + 1687500q^{75} - 1755904q^{76} + 2402388q^{77} - 3796848q^{78} + 2910090q^{79} + 2048000q^{80} + 2125764q^{81} - 6255392q^{82} - 1767198q^{83} - 2274048q^{84} - 3558250q^{85} - 8925104q^{86} - 4296294q^{87} - 3363840q^{88} - 3677360q^{89} + 2916000q^{90} - 6727732q^{91} - 2056704q^{92} - 1835298q^{93} - 1673280q^{94} - 3429500q^{95} + 3538944q^{96} - 10419094q^{97} - 9274464q^{98} - 4789530q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2087 x^{2} + 44517 x - 205110\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 31 \nu - 1053 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 22 \nu^{2} - 1533 \nu + 9252 \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} - 22 \nu^{2} + 1749 \nu - 9306 \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 3\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-31 \beta_{3} - 31 \beta_{2} + 36 \beta_{1} + 12543\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(2215 \beta_{3} + 2431 \beta_{2} - 792 \beta_{1} - 382371\)\()/12\)

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
6.63333
28.1583
20.3008
−54.0924
8.00000 27.0000 64.0000 125.000 216.000 −1226.02 512.000 729.000 1000.00
1.2 8.00000 27.0000 64.0000 125.000 216.000 −374.399 512.000 729.000 1000.00
1.3 8.00000 27.0000 64.0000 125.000 216.000 −332.718 512.000 729.000 1000.00
1.4 8.00000 27.0000 64.0000 125.000 216.000 617.139 512.000 729.000 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.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.8.a.c 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 1316 T_{7}^{3} - 201504 T_{7}^{2} - 459174632 T_{7} - 94252384288 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{4} \)
$3$ \( ( -27 + T )^{4} \)
$5$ \( ( -125 + T )^{4} \)
$7$ \( -94252384288 - 459174632 T - 201504 T^{2} + 1316 T^{3} + T^{4} \)
$11$ \( -423058023937152 - 235458938448 T - 22146612 T^{2} + 6570 T^{3} + T^{4} \)
$13$ \( -4842750112877520 - 1432806270216 T - 29575296 T^{2} + 17578 T^{3} + T^{4} \)
$17$ \( -121018490428929184 - 26676162141224 T - 804302700 T^{2} + 28466 T^{3} + T^{4} \)
$19$ \( ( 6859 + T )^{4} \)
$23$ \( -1619477845251802368 + 182304523438976 T - 6131174592 T^{2} + 32136 T^{3} + T^{4} \)
$29$ \( -56934720065249450480 - 1580035366651816 T - 4280338008 T^{2} + 159122 T^{3} + T^{4} \)
$31$ \( -\)\(22\!\cdots\!64\)\( - 7416643637848864 T - 51823985904 T^{2} + 67974 T^{3} + T^{4} \)
$37$ \( \)\(30\!\cdots\!68\)\( - 18458706942796472 T + 110391039576 T^{2} + 823702 T^{3} + T^{4} \)
$41$ \( \)\(12\!\cdots\!96\)\( - 74225172802141880 T - 11510315136 T^{2} + 781924 T^{3} + T^{4} \)
$43$ \( -\)\(55\!\cdots\!52\)\( - 553514682870214512 T - 350007791436 T^{2} + 1115638 T^{3} + T^{4} \)
$47$ \( \)\(20\!\cdots\!68\)\( - 120347484451159680 T - 1162804291584 T^{2} + 209160 T^{3} + T^{4} \)
$53$ \( -\)\(17\!\cdots\!08\)\( - 210297208854686688 T - 321153360648 T^{2} + 848424 T^{3} + T^{4} \)
$59$ \( \)\(81\!\cdots\!20\)\( - 448544568005494208 T + 2189822588496 T^{2} + 3677830 T^{3} + T^{4} \)
$61$ \( -\)\(47\!\cdots\!24\)\( - 10633822134484215872 T - 5501250875256 T^{2} + 1161072 T^{3} + T^{4} \)
$67$ \( \)\(15\!\cdots\!64\)\( - 35347938155315065024 T - 1448987553024 T^{2} + 6154740 T^{3} + T^{4} \)
$71$ \( \)\(18\!\cdots\!40\)\( - 4298123098477914624 T - 1156264675200 T^{2} + 4456224 T^{3} + T^{4} \)
$73$ \( \)\(76\!\cdots\!04\)\( + 11373711384864065280 T - 19138014643224 T^{2} - 1057792 T^{3} + T^{4} \)
$79$ \( -\)\(29\!\cdots\!40\)\( + 5212972389288197632 T - 9284106240 T^{2} - 2910090 T^{3} + T^{4} \)
$83$ \( \)\(36\!\cdots\!64\)\( + 48670054885194127328 T - 30820854048360 T^{2} + 1767198 T^{3} + T^{4} \)
$89$ \( -\)\(11\!\cdots\!80\)\( - \)\(62\!\cdots\!68\)\( T - 85778238387984 T^{2} + 3677360 T^{3} + T^{4} \)
$97$ \( -\)\(30\!\cdots\!12\)\( - \)\(15\!\cdots\!12\)\( T + 9673301793168 T^{2} + 10419094 T^{3} + T^{4} \)
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