Properties

Label 570.6.a.l
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: \(x^{4} - 1748 x^{2} - 8028 x + 111960\)
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.

\(f(q)\) \(=\) \( q -4 q^{2} + 9 q^{3} + 16 q^{4} + 25 q^{5} -36 q^{6} + ( -7 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} + 9 q^{3} + 16 q^{4} + 25 q^{5} -36 q^{6} + ( -7 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} -100 q^{10} + ( -85 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{11} + 144 q^{12} + ( -183 - 10 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 28 - 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{14} + 225 q^{15} + 256 q^{16} + ( 123 + 8 \beta_{1} + 17 \beta_{2} + 6 \beta_{3} ) q^{17} -324 q^{18} -361 q^{19} + 400 q^{20} + ( -63 + 18 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{21} + ( 340 - 12 \beta_{1} + 20 \beta_{2} + 8 \beta_{3} ) q^{22} + ( -470 - 23 \beta_{1} - 2 \beta_{2} + 41 \beta_{3} ) q^{23} -576 q^{24} + 625 q^{25} + ( 732 + 40 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{26} + 729 q^{27} + ( -112 + 32 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{28} + ( -2205 - 159 \beta_{1} - 23 \beta_{2} - 50 \beta_{3} ) q^{29} -900 q^{30} + ( -17 - 64 \beta_{1} - 61 \beta_{2} + 86 \beta_{3} ) q^{31} -1024 q^{32} + ( -765 + 27 \beta_{1} - 45 \beta_{2} - 18 \beta_{3} ) q^{33} + ( -492 - 32 \beta_{1} - 68 \beta_{2} - 24 \beta_{3} ) q^{34} + ( -175 + 50 \beta_{1} + 25 \beta_{2} - 25 \beta_{3} ) q^{35} + 1296 q^{36} + ( -3589 + 70 \beta_{1} - 151 \beta_{2} + 135 \beta_{3} ) q^{37} + 1444 q^{38} + ( -1647 - 90 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{39} -1600 q^{40} + ( -2819 - 45 \beta_{1} + 51 \beta_{2} - 92 \beta_{3} ) q^{41} + ( 252 - 72 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{42} + ( -4205 - 220 \beta_{1} + 155 \beta_{2} - 21 \beta_{3} ) q^{43} + ( -1360 + 48 \beta_{1} - 80 \beta_{2} - 32 \beta_{3} ) q^{44} + 2025 q^{45} + ( 1880 + 92 \beta_{1} + 8 \beta_{2} - 164 \beta_{3} ) q^{46} + ( -8760 + 545 \beta_{1} + 116 \beta_{2} - 115 \beta_{3} ) q^{47} + 2304 q^{48} + ( -8526 + 256 \beta_{1} - 11 \beta_{2} - 36 \beta_{3} ) q^{49} -2500 q^{50} + ( 1107 + 72 \beta_{1} + 153 \beta_{2} + 54 \beta_{3} ) q^{51} + ( -2928 - 160 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{52} + ( -4528 - 643 \beta_{1} + 294 \beta_{2} + 37 \beta_{3} ) q^{53} -2916 q^{54} + ( -2125 + 75 \beta_{1} - 125 \beta_{2} - 50 \beta_{3} ) q^{55} + ( 448 - 128 \beta_{1} - 64 \beta_{2} + 64 \beta_{3} ) q^{56} -3249 q^{57} + ( 8820 + 636 \beta_{1} + 92 \beta_{2} + 200 \beta_{3} ) q^{58} + ( -9352 + 216 \beta_{1} - 468 \beta_{2} + 646 \beta_{3} ) q^{59} + 3600 q^{60} + ( -3191 + 1006 \beta_{1} + 111 \beta_{2} - 342 \beta_{3} ) q^{61} + ( 68 + 256 \beta_{1} + 244 \beta_{2} - 344 \beta_{3} ) q^{62} + ( -567 + 162 \beta_{1} + 81 \beta_{2} - 81 \beta_{3} ) q^{63} + 4096 q^{64} + ( -4575 - 250 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} ) q^{65} + ( 3060 - 108 \beta_{1} + 180 \beta_{2} + 72 \beta_{3} ) q^{66} + ( -21276 - 1384 \beta_{1} + 92 \beta_{2} - 444 \beta_{3} ) q^{67} + ( 1968 + 128 \beta_{1} + 272 \beta_{2} + 96 \beta_{3} ) q^{68} + ( -4230 - 207 \beta_{1} - 18 \beta_{2} + 369 \beta_{3} ) q^{69} + ( 700 - 200 \beta_{1} - 100 \beta_{2} + 100 \beta_{3} ) q^{70} + ( 770 + 682 \beta_{1} - 258 \beta_{2} - 256 \beta_{3} ) q^{71} -5184 q^{72} + ( -10454 - 442 \beta_{1} + 28 \beta_{2} - 318 \beta_{3} ) q^{73} + ( 14356 - 280 \beta_{1} + 604 \beta_{2} - 540 \beta_{3} ) q^{74} + 5625 q^{75} -5776 q^{76} + ( -1697 - 108 \beta_{1} + 351 \beta_{2} - 46 \beta_{3} ) q^{77} + ( 6588 + 360 \beta_{1} + 36 \beta_{2} - 36 \beta_{3} ) q^{78} + ( 3998 - 952 \beta_{1} - 570 \beta_{2} + 360 \beta_{3} ) q^{79} + 6400 q^{80} + 6561 q^{81} + ( 11276 + 180 \beta_{1} - 204 \beta_{2} + 368 \beta_{3} ) q^{82} + ( 33582 + 1757 \beta_{1} - 418 \beta_{2} + 611 \beta_{3} ) q^{83} + ( -1008 + 288 \beta_{1} + 144 \beta_{2} - 144 \beta_{3} ) q^{84} + ( 3075 + 200 \beta_{1} + 425 \beta_{2} + 150 \beta_{3} ) q^{85} + ( 16820 + 880 \beta_{1} - 620 \beta_{2} + 84 \beta_{3} ) q^{86} + ( -19845 - 1431 \beta_{1} - 207 \beta_{2} - 450 \beta_{3} ) q^{87} + ( 5440 - 192 \beta_{1} + 320 \beta_{2} + 128 \beta_{3} ) q^{88} + ( 33015 - 235 \beta_{1} + 421 \beta_{2} - 822 \beta_{3} ) q^{89} -8100 q^{90} + ( -21423 - 1100 \beta_{1} - 379 \beta_{2} + 482 \beta_{3} ) q^{91} + ( -7520 - 368 \beta_{1} - 32 \beta_{2} + 656 \beta_{3} ) q^{92} + ( -153 - 576 \beta_{1} - 549 \beta_{2} + 774 \beta_{3} ) q^{93} + ( 35040 - 2180 \beta_{1} - 464 \beta_{2} + 460 \beta_{3} ) q^{94} -9025 q^{95} -9216 q^{96} + ( -39961 + 1572 \beta_{1} - 931 \beta_{2} + 875 \beta_{3} ) q^{97} + ( 34104 - 1024 \beta_{1} + 44 \beta_{2} + 144 \beta_{3} ) q^{98} + ( -6885 + 243 \beta_{1} - 405 \beta_{2} - 162 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} + 100q^{5} - 144q^{6} - 26q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} + 100q^{5} - 144q^{6} - 26q^{7} - 256q^{8} + 324q^{9} - 400q^{10} - 336q^{11} + 576q^{12} - 734q^{13} + 104q^{14} + 900q^{15} + 1024q^{16} + 480q^{17} - 1296q^{18} - 1444q^{19} + 1600q^{20} - 234q^{21} + 1344q^{22} - 1962q^{23} - 2304q^{24} + 2500q^{25} + 2936q^{26} + 2916q^{27} - 416q^{28} - 8720q^{29} - 3600q^{30} - 240q^{31} - 4096q^{32} - 3024q^{33} - 1920q^{34} - 650q^{35} + 5184q^{36} - 14626q^{37} + 5776q^{38} - 6606q^{39} - 6400q^{40} - 11092q^{41} + 936q^{42} - 16778q^{43} - 5376q^{44} + 8100q^{45} + 7848q^{46} - 34810q^{47} + 9216q^{48} - 34032q^{49} - 10000q^{50} + 4320q^{51} - 11744q^{52} - 18186q^{53} - 11664q^{54} - 8400q^{55} + 1664q^{56} - 12996q^{57} + 34880q^{58} - 38700q^{59} + 14400q^{60} - 12080q^{61} + 960q^{62} - 2106q^{63} + 16384q^{64} - 18350q^{65} + 12096q^{66} - 84216q^{67} + 7680q^{68} - 17658q^{69} + 2600q^{70} + 3592q^{71} - 20736q^{72} - 41180q^{73} + 58504q^{74} + 22500q^{75} - 23104q^{76} - 6696q^{77} + 26424q^{78} + 15272q^{79} + 25600q^{80} + 26244q^{81} + 44368q^{82} + 133106q^{83} - 3744q^{84} + 12000q^{85} + 67112q^{86} - 78480q^{87} + 21504q^{88} + 133704q^{89} - 32400q^{90} - 86656q^{91} - 31392q^{92} - 2160q^{93} + 139240q^{94} - 36100q^{95} - 36864q^{96} - 161594q^{97} + 136128q^{98} - 27216q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 1748 x^{2} - 8028 x + 111960\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 1592 \nu - 3399 \)\()/147\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 24 \nu^{2} - 1466 \nu + 14808 \)\()/294\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-14 \beta_{3} + 7 \beta_{2} + 6 \beta_{1} + 867\)
\(\nu^{3}\)\(=\)\(-42 \beta_{3} + 168 \beta_{2} + 1610 \beta_{1} + 6000\)

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
6.07644
−11.1347
−38.2281
43.2864
−4.00000 9.00000 16.0000 25.0000 −36.0000 −100.821 −64.0000 81.0000 −100.000
1.2 −4.00000 9.00000 16.0000 25.0000 −36.0000 −34.7979 −64.0000 81.0000 −100.000
1.3 −4.00000 9.00000 16.0000 25.0000 −36.0000 −34.1064 −64.0000 81.0000 −100.000
1.4 −4.00000 9.00000 16.0000 25.0000 −36.0000 143.725 −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.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.l 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 26 T_{7}^{3} - 16260 T_{7}^{2} - 1049380 T_{7} - 17197800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{4} \)
$3$ \( ( -9 + T )^{4} \)
$5$ \( ( -25 + T )^{4} \)
$7$ \( -17197800 - 1049380 T - 16260 T^{2} + 26 T^{3} + T^{4} \)
$11$ \( -172946744 + 13554276 T - 266744 T^{2} + 336 T^{3} + T^{4} \)
$13$ \( -928945056 - 15960452 T + 15248 T^{2} + 734 T^{3} + T^{4} \)
$17$ \( 1320406657296 - 397121360 T - 3286832 T^{2} - 480 T^{3} + T^{4} \)
$19$ \( ( 361 + T )^{4} \)
$23$ \( 7620496750080 - 22350849576 T - 11792092 T^{2} + 1962 T^{3} + T^{4} \)
$29$ \( -815189400230688 - 521725532964 T - 45885824 T^{2} + 8720 T^{3} + T^{4} \)
$31$ \( 382581725818880 + 41995687408 T - 59363368 T^{2} + 240 T^{3} + T^{4} \)
$37$ \( -7564800749785200 - 2303276818860 T - 109708872 T^{2} + 14626 T^{3} + T^{4} \)
$41$ \( 61624111290400 - 244817502340 T - 10050800 T^{2} + 11092 T^{3} + T^{4} \)
$43$ \( 4807730613129208 - 1530753008828 T - 145588724 T^{2} + 16778 T^{3} + T^{4} \)
$47$ \( -145044174094922880 - 17149749719688 T - 203051068 T^{2} + 34810 T^{3} + T^{4} \)
$53$ \( 345872420236332960 - 17257403350720 T - 1339365752 T^{2} + 18186 T^{3} + T^{4} \)
$59$ \( 626055853767594240 - 74419034591584 T - 2380647632 T^{2} + 38700 T^{3} + T^{4} \)
$61$ \( -545618844936932368 - 74490644069328 T - 2510443256 T^{2} + 12080 T^{3} + T^{4} \)
$67$ \( -6379181420765724672 - 305418399797504 T - 1868450816 T^{2} + 84216 T^{3} + T^{4} \)
$71$ \( 125090759095628032 + 36417417754528 T - 2261681312 T^{2} - 3592 T^{3} + T^{4} \)
$73$ \( -291565740894678288 - 31114440888048 T - 394851456 T^{2} + 41180 T^{3} + T^{4} \)
$79$ \( 130276964906802176 + 89426852339328 T - 3846242048 T^{2} - 15272 T^{3} + T^{4} \)
$83$ \( -15684714952788366720 + 610612557304152 T - 1002119188 T^{2} - 133106 T^{3} + T^{4} \)
$89$ \( 391477206626933568 + 130942093819420 T + 2424999072 T^{2} - 133704 T^{3} + T^{4} \)
$97$ \( -16063142828025693600 - 702927984032580 T - 1388413040 T^{2} + 161594 T^{3} + T^{4} \)
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