Properties

Label 570.6.a.k
Level $570$
Weight $6$
Character orbit 570.a
Self dual yes
Analytic conductor $91.419$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(x^{4} - x^{3} - 26355 x^{2} - 7203 x + 128450070\)
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.

\(f(q)\) \(=\) \( q -4 q^{2} + 9 q^{3} + 16 q^{4} -25 q^{5} -36 q^{6} + ( 6 - \beta_{2} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} + 9 q^{3} + 16 q^{4} -25 q^{5} -36 q^{6} + ( 6 - \beta_{2} ) q^{7} -64 q^{8} + 81 q^{9} + 100 q^{10} + ( -117 + 2 \beta_{2} + \beta_{3} ) q^{11} + 144 q^{12} + ( -120 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{13} + ( -24 + 4 \beta_{2} ) q^{14} -225 q^{15} + 256 q^{16} + ( 457 - \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{17} -324 q^{18} -361 q^{19} -400 q^{20} + ( 54 - 9 \beta_{2} ) q^{21} + ( 468 - 8 \beta_{2} - 4 \beta_{3} ) q^{22} + ( -107 - 2 \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{23} -576 q^{24} + 625 q^{25} + ( 480 + 8 \beta_{1} + 4 \beta_{2} - 16 \beta_{3} ) q^{26} + 729 q^{27} + ( 96 - 16 \beta_{2} ) q^{28} + ( -2609 - 20 \beta_{2} - 13 \beta_{3} ) q^{29} + 900 q^{30} + ( 107 + 27 \beta_{1} - 23 \beta_{2} - 11 \beta_{3} ) q^{31} -1024 q^{32} + ( -1053 + 18 \beta_{2} + 9 \beta_{3} ) q^{33} + ( -1828 + 4 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} ) q^{34} + ( -150 + 25 \beta_{2} ) q^{35} + 1296 q^{36} + ( 1424 - 18 \beta_{1} - 57 \beta_{2} + 12 \beta_{3} ) q^{37} + 1444 q^{38} + ( -1080 - 18 \beta_{1} - 9 \beta_{2} + 36 \beta_{3} ) q^{39} + 1600 q^{40} + ( 2425 - 14 \beta_{1} + 26 \beta_{2} - 19 \beta_{3} ) q^{41} + ( -216 + 36 \beta_{2} ) q^{42} + ( 5306 - 76 \beta_{1} - 13 \beta_{2} - 16 \beta_{3} ) q^{43} + ( -1872 + 32 \beta_{2} + 16 \beta_{3} ) q^{44} -2025 q^{45} + ( 428 + 8 \beta_{1} + 28 \beta_{2} - 4 \beta_{3} ) q^{46} + ( -4769 + 68 \beta_{1} - 41 \beta_{2} + 7 \beta_{3} ) q^{47} + 2304 q^{48} + ( 530 + 55 \beta_{1} + 41 \beta_{2} - 13 \beta_{3} ) q^{49} -2500 q^{50} + ( 4113 - 9 \beta_{1} + 45 \beta_{2} + 45 \beta_{3} ) q^{51} + ( -1920 - 32 \beta_{1} - 16 \beta_{2} + 64 \beta_{3} ) q^{52} + ( 181 + 92 \beta_{1} + 43 \beta_{2} - 5 \beta_{3} ) q^{53} -2916 q^{54} + ( 2925 - 50 \beta_{2} - 25 \beta_{3} ) q^{55} + ( -384 + 64 \beta_{2} ) q^{56} -3249 q^{57} + ( 10436 + 80 \beta_{2} + 52 \beta_{3} ) q^{58} + ( 7276 - 74 \beta_{1} + 142 \beta_{2} - 24 \beta_{3} ) q^{59} -3600 q^{60} + ( 19429 + 15 \beta_{1} - \beta_{2} - 79 \beta_{3} ) q^{61} + ( -428 - 108 \beta_{1} + 92 \beta_{2} + 44 \beta_{3} ) q^{62} + ( 486 - 81 \beta_{2} ) q^{63} + 4096 q^{64} + ( 3000 + 50 \beta_{1} + 25 \beta_{2} - 100 \beta_{3} ) q^{65} + ( 4212 - 72 \beta_{2} - 36 \beta_{3} ) q^{66} + ( 20550 - 34 \beta_{1} - 134 \beta_{2} + 54 \beta_{3} ) q^{67} + ( 7312 - 16 \beta_{1} + 80 \beta_{2} + 80 \beta_{3} ) q^{68} + ( -963 - 18 \beta_{1} - 63 \beta_{2} + 9 \beta_{3} ) q^{69} + ( 600 - 100 \beta_{2} ) q^{70} + ( 358 - 172 \beta_{1} + 80 \beta_{2} + 66 \beta_{3} ) q^{71} -5184 q^{72} + ( 30246 + 38 \beta_{1} - 20 \beta_{2} - 16 \beta_{3} ) q^{73} + ( -5696 + 72 \beta_{1} + 228 \beta_{2} - 48 \beta_{3} ) q^{74} + 5625 q^{75} -5776 q^{76} + ( -30011 - 25 \beta_{1} + 63 \beta_{2} + 83 \beta_{3} ) q^{77} + ( 4320 + 72 \beta_{1} + 36 \beta_{2} - 144 \beta_{3} ) q^{78} + ( 36174 + 52 \beta_{1} + 258 \beta_{2} + 218 \beta_{3} ) q^{79} -6400 q^{80} + 6561 q^{81} + ( -9700 + 56 \beta_{1} - 104 \beta_{2} + 76 \beta_{3} ) q^{82} + ( -1811 + 40 \beta_{1} - 541 \beta_{2} - 191 \beta_{3} ) q^{83} + ( 864 - 144 \beta_{2} ) q^{84} + ( -11425 + 25 \beta_{1} - 125 \beta_{2} - 125 \beta_{3} ) q^{85} + ( -21224 + 304 \beta_{1} + 52 \beta_{2} + 64 \beta_{3} ) q^{86} + ( -23481 - 180 \beta_{2} - 117 \beta_{3} ) q^{87} + ( 7488 - 128 \beta_{2} - 64 \beta_{3} ) q^{88} + ( 10937 - 224 \beta_{1} + 20 \beta_{2} - 11 \beta_{3} ) q^{89} + 8100 q^{90} + ( 33889 + 375 \beta_{1} + 579 \beta_{2} + 47 \beta_{3} ) q^{91} + ( -1712 - 32 \beta_{1} - 112 \beta_{2} + 16 \beta_{3} ) q^{92} + ( 963 + 243 \beta_{1} - 207 \beta_{2} - 99 \beta_{3} ) q^{93} + ( 19076 - 272 \beta_{1} + 164 \beta_{2} - 28 \beta_{3} ) q^{94} + 9025 q^{95} -9216 q^{96} + ( -1436 - 180 \beta_{1} + 379 \beta_{2} - 308 \beta_{3} ) q^{97} + ( -2120 - 220 \beta_{1} - 164 \beta_{2} + 52 \beta_{3} ) q^{98} + ( -9477 + 162 \beta_{2} + 81 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + 400q^{10} - 472q^{11} + 576q^{12} - 482q^{13} - 104q^{14} - 900q^{15} + 1024q^{16} + 1816q^{17} - 1296q^{18} - 1444q^{19} - 1600q^{20} + 234q^{21} + 1888q^{22} - 418q^{23} - 2304q^{24} + 2500q^{25} + 1928q^{26} + 2916q^{27} + 416q^{28} - 10396q^{29} + 3600q^{30} + 528q^{31} - 4096q^{32} - 4248q^{33} - 7264q^{34} - 650q^{35} + 5184q^{36} + 5774q^{37} + 5776q^{38} - 4338q^{39} + 6400q^{40} + 9620q^{41} - 936q^{42} + 21098q^{43} - 7552q^{44} - 8100q^{45} + 1672q^{46} - 18858q^{47} + 9216q^{48} + 2148q^{49} - 10000q^{50} + 16344q^{51} - 7712q^{52} + 822q^{53} - 11664q^{54} + 11800q^{55} - 1664q^{56} - 12996q^{57} + 41584q^{58} + 28672q^{59} - 14400q^{60} + 77748q^{61} - 2112q^{62} + 2106q^{63} + 16384q^{64} + 12050q^{65} + 16992q^{66} + 82400q^{67} + 29056q^{68} - 3762q^{69} + 2600q^{70} + 928q^{71} - 20736q^{72} + 121100q^{73} - 23096q^{74} + 22500q^{75} - 23104q^{76} - 120220q^{77} + 17352q^{78} + 144284q^{79} - 25600q^{80} + 26244q^{81} - 38480q^{82} - 6082q^{83} + 3744q^{84} - 45400q^{85} - 84392q^{86} - 93564q^{87} + 30208q^{88} + 43260q^{89} + 32400q^{90} + 135148q^{91} - 6688q^{92} + 4752q^{93} + 75432q^{94} + 36100q^{95} - 36864q^{96} - 6862q^{97} - 8592q^{98} - 38232q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 26355 x^{2} - 7203 x + 128450070\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 41 \nu^{2} + 16254 \nu + 558600 \)\()/5586\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 358 \nu^{2} + 21441 \nu - 4697826 \)\()/11172\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(56 \beta_{3} - 28 \beta_{2} - 13 \beta_{1} + 26348\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2296 \beta_{3} - 10024 \beta_{2} + 16787 \beta_{1} + 36932\)\()/2\)

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
79.8406
−140.053
−80.8568
142.069
−4.00000 9.00000 16.0000 −25.0000 −36.0000 −188.420 −64.0000 81.0000 100.000
1.2 −4.00000 9.00000 16.0000 −25.0000 −36.0000 −34.2967 −64.0000 81.0000 100.000
1.3 −4.00000 9.00000 16.0000 −25.0000 −36.0000 94.6269 −64.0000 81.0000 100.000
1.4 −4.00000 9.00000 16.0000 −25.0000 −36.0000 154.089 −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.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.k 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 26 T_{7}^{3} - 34350 T_{7}^{2} + 1640180 T_{7} + 94224800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{4} \)
$3$ \( ( -9 + T )^{4} \)
$5$ \( ( 25 + T )^{4} \)
$7$ \( 94224800 + 1640180 T - 34350 T^{2} - 26 T^{3} + T^{4} \)
$11$ \( 2545136640 - 13655376 T - 115562 T^{2} + 472 T^{3} + T^{4} \)
$13$ \( 513734238192 - 644891828 T - 1680082 T^{2} + 482 T^{3} + T^{4} \)
$17$ \( -1237206516448 + 3416167936 T - 1566588 T^{2} - 1816 T^{3} + T^{4} \)
$19$ \( ( 361 + T )^{4} \)
$23$ \( -211185000000 + 1238933000 T - 2089564 T^{2} + 418 T^{3} + T^{4} \)
$29$ \( -274001753102280 - 116099412672 T + 14772022 T^{2} + 10396 T^{3} + T^{4} \)
$31$ \( -130309549171200 + 218285366960 T - 91915764 T^{2} - 528 T^{3} + T^{4} \)
$37$ \( -3764723695413200 + 1485659189740 T - 144233202 T^{2} - 5774 T^{3} + T^{4} \)
$41$ \( -2392876839448520 + 989074551520 T - 73641834 T^{2} - 9620 T^{3} + T^{4} \)
$43$ \( 76601036657708784 + 7725261130308 T - 524211750 T^{2} - 21098 T^{3} + T^{4} \)
$47$ \( -20473016766955200 - 9104678917560 T - 470078124 T^{2} + 18858 T^{3} + T^{4} \)
$53$ \( 186768509234310720 + 758471896080 T - 908018304 T^{2} - 822 T^{3} + T^{4} \)
$59$ \( -436698693074482560 + 51681909961728 T - 1273527224 T^{2} - 28672 T^{3} + T^{4} \)
$61$ \( 3360069517707264 - 6908698264832 T + 1661090580 T^{2} - 77748 T^{3} + T^{4} \)
$67$ \( -528258807043473408 + 21348869743616 T + 1478518256 T^{2} - 82400 T^{3} + T^{4} \)
$71$ \( 2007471434526720 - 53019284473728 T - 3205930664 T^{2} - 928 T^{3} + T^{4} \)
$73$ \( 672352249412903472 - 100605905126608 T + 5338824944 T^{2} - 121100 T^{3} + T^{4} \)
$79$ \( -21755281114556835840 + 602305812951168 T + 1035645600 T^{2} - 144284 T^{3} + T^{4} \)
$83$ \( 24464955609002178816 - 152080367600872 T - 11632644436 T^{2} + 6082 T^{3} + T^{4} \)
$89$ \( 4081303956239428920 + 136817195506368 T - 4780795498 T^{2} - 43260 T^{3} + T^{4} \)
$97$ \( -9408704726970676800 + 1474725040179060 T - 24025430730 T^{2} + 6862 T^{3} + T^{4} \)
show more
show less