Properties

Label 570.6.a.h
Level $570$
Weight $6$
Character orbit 570.a
Self dual yes
Analytic conductor $91.419$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 12189 x^{2} - 95210 x + 4841400\)
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.

\(f(q)\) \(=\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} -25 q^{5} + 36 q^{6} + ( -22 - \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} -25 q^{5} + 36 q^{6} + ( -22 - \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} + 100 q^{10} + ( 236 + 2 \beta_{2} + \beta_{3} ) q^{11} -144 q^{12} + ( -6 - 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 88 + 4 \beta_{3} ) q^{14} + 225 q^{15} + 256 q^{16} + ( -32 + 7 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} ) q^{17} -324 q^{18} + 361 q^{19} -400 q^{20} + ( 198 + 9 \beta_{3} ) q^{21} + ( -944 - 8 \beta_{2} - 4 \beta_{3} ) q^{22} + ( 124 - 14 \beta_{1} + 5 \beta_{2} - 13 \beta_{3} ) q^{23} + 576 q^{24} + 625 q^{25} + ( 24 + 8 \beta_{1} - 20 \beta_{2} - 8 \beta_{3} ) q^{26} -729 q^{27} + ( -352 - 16 \beta_{3} ) q^{28} + ( 822 - 28 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{29} -900 q^{30} + ( -1666 + 29 \beta_{1} + 11 \beta_{2} - 21 \beta_{3} ) q^{31} -1024 q^{32} + ( -2124 - 18 \beta_{2} - 9 \beta_{3} ) q^{33} + ( 128 - 28 \beta_{1} - 20 \beta_{2} + 28 \beta_{3} ) q^{34} + ( 550 + 25 \beta_{3} ) q^{35} + 1296 q^{36} + ( -3542 + 14 \beta_{1} + 25 \beta_{2} - 38 \beta_{3} ) q^{37} -1444 q^{38} + ( 54 + 18 \beta_{1} - 45 \beta_{2} - 18 \beta_{3} ) q^{39} + 1600 q^{40} + ( 320 + 42 \beta_{1} - 31 \beta_{2} - 50 \beta_{3} ) q^{41} + ( -792 - 36 \beta_{3} ) q^{42} + ( -6588 + 36 \beta_{1} - 11 \beta_{2} - 8 \beta_{3} ) q^{43} + ( 3776 + 32 \beta_{2} + 16 \beta_{3} ) q^{44} -2025 q^{45} + ( -496 + 56 \beta_{1} - 20 \beta_{2} + 52 \beta_{3} ) q^{46} + ( -1716 - 44 \beta_{1} + 73 \beta_{2} - \beta_{3} ) q^{47} -2304 q^{48} + ( 85 - 41 \beta_{1} + 74 \beta_{2} + 148 \beta_{3} ) q^{49} -2500 q^{50} + ( 288 - 63 \beta_{1} - 45 \beta_{2} + 63 \beta_{3} ) q^{51} + ( -96 - 32 \beta_{1} + 80 \beta_{2} + 32 \beta_{3} ) q^{52} + ( -5722 - 4 \beta_{1} - 53 \beta_{2} - 35 \beta_{3} ) q^{53} + 2916 q^{54} + ( -5900 - 50 \beta_{2} - 25 \beta_{3} ) q^{55} + ( 1408 + 64 \beta_{3} ) q^{56} -3249 q^{57} + ( -3288 + 112 \beta_{1} + 16 \beta_{2} + 28 \beta_{3} ) q^{58} + ( -3226 - 66 \beta_{1} + 227 \beta_{2} - 11 \beta_{3} ) q^{59} + 3600 q^{60} + ( 4002 - 13 \beta_{1} - 10 \beta_{3} ) q^{61} + ( 6664 - 116 \beta_{1} - 44 \beta_{2} + 84 \beta_{3} ) q^{62} + ( -1782 - 81 \beta_{3} ) q^{63} + 4096 q^{64} + ( 150 + 50 \beta_{1} - 125 \beta_{2} - 50 \beta_{3} ) q^{65} + ( 8496 + 72 \beta_{2} + 36 \beta_{3} ) q^{66} + ( -5708 - 34 \beta_{1} - 68 \beta_{2} + 384 \beta_{3} ) q^{67} + ( -512 + 112 \beta_{1} + 80 \beta_{2} - 112 \beta_{3} ) q^{68} + ( -1116 + 126 \beta_{1} - 45 \beta_{2} + 117 \beta_{3} ) q^{69} + ( -2200 - 100 \beta_{3} ) q^{70} + ( 13144 - 56 \beta_{1} - 4 \beta_{2} - 42 \beta_{3} ) q^{71} -5184 q^{72} + ( -25274 - 278 \beta_{1} - 24 \beta_{2} ) q^{73} + ( 14168 - 56 \beta_{1} - 100 \beta_{2} + 152 \beta_{3} ) q^{74} -5625 q^{75} + 5776 q^{76} + ( -25936 + 87 \beta_{1} + 14 \beta_{2} - 590 \beta_{3} ) q^{77} + ( -216 - 72 \beta_{1} + 180 \beta_{2} + 72 \beta_{3} ) q^{78} + ( -28330 - 40 \beta_{1} - 115 \beta_{2} + 145 \beta_{3} ) q^{79} -6400 q^{80} + 6561 q^{81} + ( -1280 - 168 \beta_{1} + 124 \beta_{2} + 200 \beta_{3} ) q^{82} + ( -16386 + 496 \beta_{1} - 80 \beta_{2} - 156 \beta_{3} ) q^{83} + ( 3168 + 144 \beta_{3} ) q^{84} + ( 800 - 175 \beta_{1} - 125 \beta_{2} + 175 \beta_{3} ) q^{85} + ( 26352 - 144 \beta_{1} + 44 \beta_{2} + 32 \beta_{3} ) q^{86} + ( -7398 + 252 \beta_{1} + 36 \beta_{2} + 63 \beta_{3} ) q^{87} + ( -15104 - 128 \beta_{2} - 64 \beta_{3} ) q^{88} + ( 24348 + 232 \beta_{1} + 19 \beta_{2} + 470 \beta_{3} ) q^{89} + 8100 q^{90} + ( -31364 + 165 \beta_{1} + 72 \beta_{2} - 796 \beta_{3} ) q^{91} + ( 1984 - 224 \beta_{1} + 80 \beta_{2} - 208 \beta_{3} ) q^{92} + ( 14994 - 261 \beta_{1} - 99 \beta_{2} + 189 \beta_{3} ) q^{93} + ( 6864 + 176 \beta_{1} - 292 \beta_{2} + 4 \beta_{3} ) q^{94} -9025 q^{95} + 9216 q^{96} + ( -49574 + 344 \beta_{1} + 319 \beta_{2} - 472 \beta_{3} ) q^{97} + ( -340 + 164 \beta_{1} - 296 \beta_{2} - 592 \beta_{3} ) q^{98} + ( 19116 + 162 \beta_{2} + 81 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} - 36q^{3} + 64q^{4} - 100q^{5} + 144q^{6} - 88q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} - 36q^{3} + 64q^{4} - 100q^{5} + 144q^{6} - 88q^{7} - 256q^{8} + 324q^{9} + 400q^{10} + 940q^{11} - 576q^{12} - 34q^{13} + 352q^{14} + 900q^{15} + 1024q^{16} - 138q^{17} - 1296q^{18} + 1444q^{19} - 1600q^{20} + 792q^{21} - 3760q^{22} + 486q^{23} + 2304q^{24} + 2500q^{25} + 136q^{26} - 2916q^{27} - 1408q^{28} + 3296q^{29} - 3600q^{30} - 6686q^{31} - 4096q^{32} - 8460q^{33} + 552q^{34} + 2200q^{35} + 5184q^{36} - 14218q^{37} - 5776q^{38} + 306q^{39} + 6400q^{40} + 1342q^{41} - 3168q^{42} - 26330q^{43} + 15040q^{44} - 8100q^{45} - 1944q^{46} - 7010q^{47} - 9216q^{48} + 192q^{49} - 10000q^{50} + 1242q^{51} - 544q^{52} - 22782q^{53} + 11664q^{54} - 23500q^{55} + 5632q^{56} - 12996q^{57} - 13184q^{58} - 13358q^{59} + 14400q^{60} + 16008q^{61} + 26744q^{62} - 7128q^{63} + 16384q^{64} + 850q^{65} + 33840q^{66} - 22696q^{67} - 2208q^{68} - 4374q^{69} - 8800q^{70} + 52584q^{71} - 20736q^{72} - 101048q^{73} + 56872q^{74} - 22500q^{75} + 23104q^{76} - 103772q^{77} - 1224q^{78} - 113090q^{79} - 25600q^{80} + 26244q^{81} - 5368q^{82} - 65384q^{83} + 12672q^{84} + 3450q^{85} + 105320q^{86} - 29664q^{87} - 60160q^{88} + 97354q^{89} + 32400q^{90} - 125600q^{91} + 7776q^{92} + 60174q^{93} + 28040q^{94} - 36100q^{95} + 36864q^{96} - 198934q^{97} - 768q^{98} + 76140q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 12189 x^{2} - 95210 x + 4841400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 14 \nu^{3} - 1355 \nu^{2} - 126121 \nu + 7231980 \)\()/52725\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} - 5 \nu^{2} + 12164 \nu + 101880 \)\()/1425\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(-28 \beta_{3} - 74 \beta_{2} + 31 \beta_{1} + 12152\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2710 \beta_{3} + 370 \beta_{2} + 12009 \beta_{1} + 143000\)\()/2\)

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
16.5544
112.485
−104.052
−24.9870
−4.00000 −9.00000 16.0000 −25.0000 36.0000 −230.660 −64.0000 81.0000 100.000
1.2 −4.00000 −9.00000 16.0000 −25.0000 36.0000 −10.5154 −64.0000 81.0000 100.000
1.3 −4.00000 −9.00000 16.0000 −25.0000 36.0000 42.1343 −64.0000 81.0000 100.000
1.4 −4.00000 −9.00000 16.0000 −25.0000 36.0000 111.041 −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.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.h 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 88 T_{7}^{3} - 29838 T_{7}^{2} + 756848 T_{7} + 11347976 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{4} \)
$3$ \( ( 9 + T )^{4} \)
$5$ \( ( 25 + T )^{4} \)
$7$ \( 11347976 + 756848 T - 29838 T^{2} + 88 T^{3} + T^{4} \)
$11$ \( -3999659040 + 56577888 T + 115962 T^{2} - 940 T^{3} + T^{4} \)
$13$ \( 337501772640 - 8076324 T - 1185726 T^{2} + 34 T^{3} + T^{4} \)
$17$ \( 2817171947136 + 661924208 T - 4085208 T^{2} + 138 T^{3} + T^{4} \)
$19$ \( ( -361 + T )^{4} \)
$23$ \( 85007443762272 + 4857792776 T - 18751908 T^{2} - 486 T^{3} + T^{4} \)
$29$ \( 188791634756520 + 33645222336 T - 43243886 T^{2} - 3296 T^{3} + T^{4} \)
$31$ \( 236745118093056 - 115839667496 T - 31997692 T^{2} + 6686 T^{3} + T^{4} \)
$37$ \( -776580628218880 - 430260044404 T + 8813250 T^{2} + 14218 T^{3} + T^{4} \)
$41$ \( 64312502899680 + 16966239372 T - 148306986 T^{2} - 1342 T^{3} + T^{4} \)
$43$ \( -2121460557297456 + 180908073580 T + 202347950 T^{2} + 26330 T^{3} + T^{4} \)
$47$ \( 5032254978269280 - 822079607928 T - 244235988 T^{2} + 7010 T^{3} + T^{4} \)
$53$ \( -2797568849689920 - 1437323681072 T + 14673024 T^{2} + 22782 T^{3} + T^{4} \)
$59$ \( 632149509794788800 - 26149461195600 T - 2028334608 T^{2} + 13358 T^{3} + T^{4} \)
$61$ \( 77830571168320 - 141293991456 T + 81420460 T^{2} - 16008 T^{3} + T^{4} \)
$67$ \( -571308474795171840 - 101857712412672 T - 4372045488 T^{2} + 22696 T^{3} + T^{4} \)
$71$ \( 1708491829224960 - 2204258660864 T + 763277736 T^{2} - 52584 T^{3} + T^{4} \)
$73$ \( -1830046095326758576 - 139096202858336 T - 70952088 T^{2} + 101048 T^{3} + T^{4} \)
$79$ \( 143226864632000000 + 40970310760000 T + 3697244400 T^{2} + 113090 T^{3} + T^{4} \)
$83$ \( -13418603575178849904 - 737421343219424 T - 9037432072 T^{2} + 65384 T^{3} + T^{4} \)
$89$ \( 24594473964587363280 + 416314910569476 T - 9119587314 T^{2} - 97354 T^{3} + T^{4} \)
$97$ \( 6827971109421322000 - 877015075516220 T + 1073723754 T^{2} + 198934 T^{3} + T^{4} \)
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