Properties

Label 570.6.a.j
Level $570$
Weight $6$
Character orbit 570.a
Self dual yes
Analytic conductor $91.419$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.4187772934\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 25872 x^{2} - 1407374 x - 6356280\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} + 25 q^{5} + 36 q^{6} + ( -27 - \beta_{2} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} + 25 q^{5} + 36 q^{6} + ( -27 - \beta_{2} ) q^{7} -64 q^{8} + 81 q^{9} -100 q^{10} + ( -61 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} -144 q^{12} + ( 159 - \beta_{2} - 2 \beta_{3} ) q^{13} + ( 108 + 4 \beta_{2} ) q^{14} -225 q^{15} + 256 q^{16} + ( -152 + 5 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{17} -324 q^{18} + 361 q^{19} + 400 q^{20} + ( 243 + 9 \beta_{2} ) q^{21} + ( 244 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{22} + ( -312 - 7 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} ) q^{23} + 576 q^{24} + 625 q^{25} + ( -636 + 4 \beta_{2} + 8 \beta_{3} ) q^{26} -729 q^{27} + ( -432 - 16 \beta_{2} ) q^{28} + ( -1563 - \beta_{1} + 31 \beta_{2} - 11 \beta_{3} ) q^{29} + 900 q^{30} + ( -2830 + 21 \beta_{1} + 6 \beta_{2} + 20 \beta_{3} ) q^{31} -1024 q^{32} + ( 549 + 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{33} + ( 608 - 20 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} ) q^{34} + ( -675 - 25 \beta_{2} ) q^{35} + 1296 q^{36} + ( -1203 - 32 \beta_{1} - 23 \beta_{2} - 10 \beta_{3} ) q^{37} -1444 q^{38} + ( -1431 + 9 \beta_{2} + 18 \beta_{3} ) q^{39} -1600 q^{40} + ( 2305 - 9 \beta_{1} + 23 \beta_{2} + 25 \beta_{3} ) q^{41} + ( -972 - 36 \beta_{2} ) q^{42} + ( -2885 + 22 \beta_{1} + 69 \beta_{2} - 64 \beta_{3} ) q^{43} + ( -976 - 16 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{44} + 2025 q^{45} + ( 1248 + 28 \beta_{1} - 40 \beta_{2} + 12 \beta_{3} ) q^{46} + ( -7436 + 7 \beta_{1} - 2 \beta_{2} + 57 \beta_{3} ) q^{47} -2304 q^{48} + ( 7761 - 57 \beta_{1} - 24 \beta_{3} ) q^{49} -2500 q^{50} + ( 1368 - 45 \beta_{1} - 54 \beta_{2} - 18 \beta_{3} ) q^{51} + ( 2544 - 16 \beta_{2} - 32 \beta_{3} ) q^{52} + ( 6894 + 55 \beta_{1} - 10 \beta_{2} + 39 \beta_{3} ) q^{53} + 2916 q^{54} + ( -1525 - 25 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} ) q^{55} + ( 1728 + 64 \beta_{2} ) q^{56} -3249 q^{57} + ( 6252 + 4 \beta_{1} - 124 \beta_{2} + 44 \beta_{3} ) q^{58} + ( 13738 + 26 \beta_{1} - 6 \beta_{2} - 16 \beta_{3} ) q^{59} -3600 q^{60} + ( 5166 + 83 \beta_{1} + 132 \beta_{2} - 102 \beta_{3} ) q^{61} + ( 11320 - 84 \beta_{1} - 24 \beta_{2} - 80 \beta_{3} ) q^{62} + ( -2187 - 81 \beta_{2} ) q^{63} + 4096 q^{64} + ( 3975 - 25 \beta_{2} - 50 \beta_{3} ) q^{65} + ( -2196 - 36 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{66} + ( -5000 + 64 \beta_{1} - 144 \beta_{2} + 122 \beta_{3} ) q^{67} + ( -2432 + 80 \beta_{1} + 96 \beta_{2} + 32 \beta_{3} ) q^{68} + ( 2808 + 63 \beta_{1} - 90 \beta_{2} + 27 \beta_{3} ) q^{69} + ( 2700 + 100 \beta_{2} ) q^{70} + ( 21734 - 46 \beta_{1} - 46 \beta_{2} + 36 \beta_{3} ) q^{71} -5184 q^{72} + ( -818 + 206 \beta_{1} + 44 \beta_{2} + 228 \beta_{3} ) q^{73} + ( 4812 + 128 \beta_{1} + 92 \beta_{2} + 40 \beta_{3} ) q^{74} -5625 q^{75} + 5776 q^{76} + ( 4574 - 29 \beta_{1} + 78 \beta_{2} - 250 \beta_{3} ) q^{77} + ( 5724 - 36 \beta_{2} - 72 \beta_{3} ) q^{78} + ( 40992 - 266 \beta_{1} - 8 \beta_{2} - 112 \beta_{3} ) q^{79} + 6400 q^{80} + 6561 q^{81} + ( -9220 + 36 \beta_{1} - 92 \beta_{2} - 100 \beta_{3} ) q^{82} + ( -15246 - 165 \beta_{1} + 228 \beta_{2} - 239 \beta_{3} ) q^{83} + ( 3888 + 144 \beta_{2} ) q^{84} + ( -3800 + 125 \beta_{1} + 150 \beta_{2} + 50 \beta_{3} ) q^{85} + ( 11540 - 88 \beta_{1} - 276 \beta_{2} + 256 \beta_{3} ) q^{86} + ( 14067 + 9 \beta_{1} - 279 \beta_{2} + 99 \beta_{3} ) q^{87} + ( 3904 + 64 \beta_{1} + 64 \beta_{2} - 64 \beta_{3} ) q^{88} + ( 28409 - 57 \beta_{1} + 499 \beta_{2} - 81 \beta_{3} ) q^{89} -8100 q^{90} + ( 40018 - 193 \beta_{1} - 434 \beta_{2} + 272 \beta_{3} ) q^{91} + ( -4992 - 112 \beta_{1} + 160 \beta_{2} - 48 \beta_{3} ) q^{92} + ( 25470 - 189 \beta_{1} - 54 \beta_{2} - 180 \beta_{3} ) q^{93} + ( 29744 - 28 \beta_{1} + 8 \beta_{2} - 228 \beta_{3} ) q^{94} + 9025 q^{95} + 9216 q^{96} + ( 19891 + 176 \beta_{1} + 119 \beta_{2} + 62 \beta_{3} ) q^{97} + ( -31044 + 228 \beta_{1} + 96 \beta_{3} ) q^{98} + ( -4941 - 81 \beta_{1} - 81 \beta_{2} + 81 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} - 36q^{3} + 64q^{4} + 100q^{5} + 144q^{6} - 108q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} - 36q^{3} + 64q^{4} + 100q^{5} + 144q^{6} - 108q^{7} - 256q^{8} + 324q^{9} - 400q^{10} - 246q^{11} - 576q^{12} + 640q^{13} + 432q^{14} - 900q^{15} + 1024q^{16} - 612q^{17} - 1296q^{18} + 1444q^{19} + 1600q^{20} + 972q^{21} + 984q^{22} - 1242q^{23} + 2304q^{24} + 2500q^{25} - 2560q^{26} - 2916q^{27} - 1728q^{28} - 6230q^{29} + 3600q^{30} - 11360q^{31} - 4096q^{32} + 2214q^{33} + 2448q^{34} - 2700q^{35} + 5184q^{36} - 4792q^{37} - 5776q^{38} - 5760q^{39} - 6400q^{40} + 9170q^{41} - 3888q^{42} - 11412q^{43} - 3936q^{44} + 8100q^{45} + 4968q^{46} - 29858q^{47} - 9216q^{48} + 31092q^{49} - 10000q^{50} + 5508q^{51} + 10240q^{52} + 27498q^{53} + 11664q^{54} - 6150q^{55} + 6912q^{56} - 12996q^{57} + 24920q^{58} + 54984q^{59} - 14400q^{60} + 20868q^{61} + 45440q^{62} - 8748q^{63} + 16384q^{64} + 16000q^{65} - 8856q^{66} - 20244q^{67} - 9792q^{68} + 11178q^{69} + 10800q^{70} + 86864q^{71} - 20736q^{72} - 3728q^{73} + 19168q^{74} - 22500q^{75} + 23104q^{76} + 18796q^{77} + 23040q^{78} + 164192q^{79} + 25600q^{80} + 26244q^{81} - 36680q^{82} - 60506q^{83} + 15552q^{84} - 15300q^{85} + 45648q^{86} + 56070q^{87} + 15744q^{88} + 113798q^{89} - 32400q^{90} + 159528q^{91} - 19872q^{92} + 102240q^{93} + 119432q^{94} + 36100q^{95} + 36864q^{96} + 79440q^{97} - 124368q^{98} - 19926q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 25872 x^{2} - 1407374 x - 6356280\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -14 \nu^{3} + 677 \nu^{2} + 301432 \nu + 6019755 \)\()/30195\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 288 \nu^{2} - 3392 \nu - 2675070 \)\()/10065\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(42 \beta_{3} - 9 \beta_{2} + 52 \beta_{1} + 12957\)
\(\nu^{3}\)\(=\)\(2031 \beta_{3} - 2592 \beta_{2} + 13280 \beta_{1} + 1056546\)

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
−4.97007
−121.221
183.641
−57.4501
−4.00000 −9.00000 16.0000 25.0000 36.0000 −177.358 −64.0000 81.0000 −100.000
1.2 −4.00000 −9.00000 16.0000 25.0000 36.0000 −171.600 −64.0000 81.0000 −100.000
1.3 −4.00000 −9.00000 16.0000 25.0000 36.0000 55.7215 −64.0000 81.0000 −100.000
1.4 −4.00000 −9.00000 16.0000 25.0000 36.0000 185.237 −64.0000 81.0000 −100.000
\(n\): 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 570.6.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.j 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 108 T_{7}^{3} - 43328 T_{7}^{2} - 3731652 T_{7} + 314136496 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{4} \)
$3$ \( ( 9 + T )^{4} \)
$5$ \( ( -25 + T )^{4} \)
$7$ \( 314136496 - 3731652 T - 43328 T^{2} + 108 T^{3} + T^{4} \)
$11$ \( 8188152720 - 38645044 T - 222744 T^{2} + 246 T^{3} + T^{4} \)
$13$ \( -49310893800 + 351352860 T - 428216 T^{2} - 640 T^{3} + T^{4} \)
$17$ \( -120474316368 + 1223221152 T - 3185512 T^{2} + 612 T^{3} + T^{4} \)
$19$ \( ( -361 + T )^{4} \)
$23$ \( 16334769204576 + 103106232 T - 10687828 T^{2} + 1242 T^{3} + T^{4} \)
$29$ \( -174680673460200 - 221088231180 T - 31816836 T^{2} + 6230 T^{3} + T^{4} \)
$31$ \( 130163285128320 - 323246995664 T - 16741672 T^{2} + 11360 T^{3} + T^{4} \)
$37$ \( -83005416323416 - 213756895324 T - 91376328 T^{2} + 4792 T^{3} + T^{4} \)
$41$ \( -143674860375000 - 278456676684 T - 119502572 T^{2} - 9170 T^{3} + T^{4} \)
$43$ \( -11804776595069472 - 7215209343852 T - 538169464 T^{2} + 11412 T^{3} + T^{4} \)
$47$ \( -14306351435330400 - 4496485545864 T - 8954820 T^{2} + 29858 T^{3} + T^{4} \)
$53$ \( 2589535817589600 + 1034864267808 T - 75973096 T^{2} - 27498 T^{3} + T^{4} \)
$59$ \( 3377648620978560 - 6152984356704 T + 995764496 T^{2} - 54984 T^{3} + T^{4} \)
$61$ \( 731193669364696080 + 13633388993344 T - 2088650160 T^{2} - 20868 T^{3} + T^{4} \)
$67$ \( -15887064923136000 + 12161351009280 T - 2118746176 T^{2} + 20244 T^{3} + T^{4} \)
$71$ \( 69325299093058560 - 23229813393504 T + 2413832016 T^{2} - 86864 T^{3} + T^{4} \)
$73$ \( 11164529439783360464 - 49647416235776 T - 7363860888 T^{2} + 3728 T^{3} + T^{4} \)
$79$ \( -23228463978697600000 + 560418393276800 T + 3334776000 T^{2} - 164192 T^{3} + T^{4} \)
$83$ \( -3365021019210801408 - 362624306350616 T - 7235410708 T^{2} + 60506 T^{3} + T^{4} \)
$89$ \( 22597405996755257640 + 561966298009908 T - 7368484676 T^{2} - 113798 T^{3} + T^{4} \)
$97$ \( -888776289607009400 + 81244417227900 T - 657003224 T^{2} - 79440 T^{3} + T^{4} \)
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