Properties

Label 570.6.a.i
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: \(x^{4} - x^{3} - 2696 x^{2} + 13833 x + 894635\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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} + ( -115 + \beta_{2} + \beta_{3} ) q^{11} -144 q^{12} + ( 74 + \beta_{1} - 2 \beta_{3} ) q^{13} + ( -108 - 4 \beta_{2} ) q^{14} + 225 q^{15} + 256 q^{16} + ( -103 - 3 \beta_{1} - 3 \beta_{2} ) q^{17} -324 q^{18} -361 q^{19} -400 q^{20} + ( -243 - 9 \beta_{2} ) q^{21} + ( 460 - 4 \beta_{2} - 4 \beta_{3} ) q^{22} + ( -192 + 2 \beta_{1} - 14 \beta_{2} + 11 \beta_{3} ) q^{23} + 576 q^{24} + 625 q^{25} + ( -296 - 4 \beta_{1} + 8 \beta_{3} ) q^{26} -729 q^{27} + ( 432 + 16 \beta_{2} ) q^{28} + ( -457 - 6 \beta_{1} - 7 \beta_{2} - 19 \beta_{3} ) q^{29} -900 q^{30} + ( 964 + 8 \beta_{1} - 8 \beta_{2} ) q^{31} -1024 q^{32} + ( 1035 - 9 \beta_{2} - 9 \beta_{3} ) q^{33} + ( 412 + 12 \beta_{1} + 12 \beta_{2} ) q^{34} + ( -675 - 25 \beta_{2} ) q^{35} + 1296 q^{36} + ( 2864 + 5 \beta_{1} - 34 \beta_{2} - 52 \beta_{3} ) q^{37} + 1444 q^{38} + ( -666 - 9 \beta_{1} + 18 \beta_{3} ) q^{39} + 1600 q^{40} + ( -3226 + 19 \beta_{1} + 18 \beta_{2} - 13 \beta_{3} ) q^{41} + ( 972 + 36 \beta_{2} ) q^{42} + ( 3774 - 5 \beta_{1} + 34 \beta_{2} + 76 \beta_{3} ) q^{43} + ( -1840 + 16 \beta_{2} + 16 \beta_{3} ) q^{44} -2025 q^{45} + ( 768 - 8 \beta_{1} + 56 \beta_{2} - 44 \beta_{3} ) q^{46} + ( 4260 - 10 \beta_{1} + 6 \beta_{2} + 127 \beta_{3} ) q^{47} -2304 q^{48} + ( 8427 - 56 \beta_{1} + 42 \beta_{2} + 20 \beta_{3} ) q^{49} -2500 q^{50} + ( 927 + 27 \beta_{1} + 27 \beta_{2} ) q^{51} + ( 1184 + 16 \beta_{1} - 32 \beta_{3} ) q^{52} + ( 4182 + 14 \beta_{1} - 66 \beta_{2} - 65 \beta_{3} ) q^{53} + 2916 q^{54} + ( 2875 - 25 \beta_{2} - 25 \beta_{3} ) q^{55} + ( -1728 - 64 \beta_{2} ) q^{56} + 3249 q^{57} + ( 1828 + 24 \beta_{1} + 28 \beta_{2} + 76 \beta_{3} ) q^{58} + ( -4940 + 14 \beta_{1} + 44 \beta_{2} - 92 \beta_{3} ) q^{59} + 3600 q^{60} + ( 11792 + 20 \beta_{1} - 90 \beta_{2} + 10 \beta_{3} ) q^{61} + ( -3856 - 32 \beta_{1} + 32 \beta_{2} ) q^{62} + ( 2187 + 81 \beta_{2} ) q^{63} + 4096 q^{64} + ( -1850 - 25 \beta_{1} + 50 \beta_{3} ) q^{65} + ( -4140 + 36 \beta_{2} + 36 \beta_{3} ) q^{66} + ( 26145 - 9 \beta_{1} - 89 \beta_{2} - 140 \beta_{3} ) q^{67} + ( -1648 - 48 \beta_{1} - 48 \beta_{2} ) q^{68} + ( 1728 - 18 \beta_{1} + 126 \beta_{2} - 99 \beta_{3} ) q^{69} + ( 2700 + 100 \beta_{2} ) q^{70} + ( -9191 - 3 \beta_{1} + 15 \beta_{2} - 146 \beta_{3} ) q^{71} -5184 q^{72} + ( 18589 - 19 \beta_{1} + 93 \beta_{2} - 10 \beta_{3} ) q^{73} + ( -11456 - 20 \beta_{1} + 136 \beta_{2} + 208 \beta_{3} ) q^{74} -5625 q^{75} -5776 q^{76} + ( 18839 - 39 \beta_{1} - 103 \beta_{2} + 48 \beta_{3} ) q^{77} + ( 2664 + 36 \beta_{1} - 72 \beta_{3} ) q^{78} + ( 20230 + 10 \beta_{1} + 266 \beta_{2} + 284 \beta_{3} ) q^{79} -6400 q^{80} + 6561 q^{81} + ( 12904 - 76 \beta_{1} - 72 \beta_{2} + 52 \beta_{3} ) q^{82} + ( 4854 + 78 \beta_{1} - 112 \beta_{2} - 69 \beta_{3} ) q^{83} + ( -3888 - 144 \beta_{2} ) q^{84} + ( 2575 + 75 \beta_{1} + 75 \beta_{2} ) q^{85} + ( -15096 + 20 \beta_{1} - 136 \beta_{2} - 304 \beta_{3} ) q^{86} + ( 4113 + 54 \beta_{1} + 63 \beta_{2} + 171 \beta_{3} ) q^{87} + ( 7360 - 64 \beta_{2} - 64 \beta_{3} ) q^{88} + ( -1190 + 87 \beta_{1} - 82 \beta_{2} - 57 \beta_{3} ) q^{89} + 8100 q^{90} + ( 8072 + 4 \beta_{1} - 332 \beta_{2} + 24 \beta_{3} ) q^{91} + ( -3072 + 32 \beta_{1} - 224 \beta_{2} + 176 \beta_{3} ) q^{92} + ( -8676 - 72 \beta_{1} + 72 \beta_{2} ) q^{93} + ( -17040 + 40 \beta_{1} - 24 \beta_{2} - 508 \beta_{3} ) q^{94} + 9025 q^{95} + 9216 q^{96} + ( 34893 + 46 \beta_{1} - 387 \beta_{2} + 320 \beta_{3} ) q^{97} + ( -33708 + 224 \beta_{1} - 168 \beta_{2} - 80 \beta_{3} ) q^{98} + ( -9315 + 81 \beta_{2} + 81 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} - 36q^{3} + 64q^{4} - 100q^{5} + 144q^{6} + 108q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} - 36q^{3} + 64q^{4} - 100q^{5} + 144q^{6} + 108q^{7} - 256q^{8} + 324q^{9} + 400q^{10} - 460q^{11} - 576q^{12} + 296q^{13} - 432q^{14} + 900q^{15} + 1024q^{16} - 412q^{17} - 1296q^{18} - 1444q^{19} - 1600q^{20} - 972q^{21} + 1840q^{22} - 768q^{23} + 2304q^{24} + 2500q^{25} - 1184q^{26} - 2916q^{27} + 1728q^{28} - 1828q^{29} - 3600q^{30} + 3856q^{31} - 4096q^{32} + 4140q^{33} + 1648q^{34} - 2700q^{35} + 5184q^{36} + 11456q^{37} + 5776q^{38} - 2664q^{39} + 6400q^{40} - 12904q^{41} + 3888q^{42} + 15096q^{43} - 7360q^{44} - 8100q^{45} + 3072q^{46} + 17040q^{47} - 9216q^{48} + 33708q^{49} - 10000q^{50} + 3708q^{51} + 4736q^{52} + 16728q^{53} + 11664q^{54} + 11500q^{55} - 6912q^{56} + 12996q^{57} + 7312q^{58} - 19760q^{59} + 14400q^{60} + 47168q^{61} - 15424q^{62} + 8748q^{63} + 16384q^{64} - 7400q^{65} - 16560q^{66} + 104580q^{67} - 6592q^{68} + 6912q^{69} + 10800q^{70} - 36764q^{71} - 20736q^{72} + 74356q^{73} - 45824q^{74} - 22500q^{75} - 23104q^{76} + 75356q^{77} + 10656q^{78} + 80920q^{79} - 25600q^{80} + 26244q^{81} + 51616q^{82} + 19416q^{83} - 15552q^{84} + 10300q^{85} - 60384q^{86} + 16452q^{87} + 29440q^{88} - 4760q^{89} + 32400q^{90} + 32288q^{91} - 12288q^{92} - 34704q^{93} - 68160q^{94} + 36100q^{95} + 36864q^{96} + 139572q^{97} - 134832q^{98} - 37260q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2696 x^{2} + 13833 x + 894635\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 7 \nu^{3} + 74 \nu^{2} - 11736 \nu - 38369 \)\()/403\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 46 \nu^{2} + 3076 \nu + 52898 \)\()/403\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{2} + 38 \nu - 2706 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{3} + 7 \beta_{2} + \beta_{1} + 9\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(158 \beta_{3} - 133 \beta_{2} - 19 \beta_{1} + 48537\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(2518 \beta_{3} + 6571 \beta_{2} + 1975 \beta_{1} - 150345\)\()/18\)

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
−50.6958
−16.6309
44.4607
23.8661
−4.00000 −9.00000 16.0000 −25.0000 36.0000 −198.741 −64.0000 81.0000 100.000
1.2 −4.00000 −9.00000 16.0000 −25.0000 36.0000 11.1642 −64.0000 81.0000 100.000
1.3 −4.00000 −9.00000 16.0000 −25.0000 36.0000 53.8996 −64.0000 81.0000 100.000
1.4 −4.00000 −9.00000 16.0000 −25.0000 36.0000 241.678 −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.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.i 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 108 T_{7}^{3} - 44636 T_{7}^{2} + 3099264 T_{7} - 28902608 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{4} \)
$3$ \( ( 9 + T )^{4} \)
$5$ \( ( 25 + T )^{4} \)
$7$ \( -28902608 + 3099264 T - 44636 T^{2} - 108 T^{3} + T^{4} \)
$11$ \( 701793648 - 15079008 T - 9932 T^{2} + 460 T^{3} + T^{4} \)
$13$ \( 11914965360 - 39431472 T - 458804 T^{2} - 296 T^{3} + T^{4} \)
$17$ \( 2073578773936 - 1537808432 T - 3784080 T^{2} + 412 T^{3} + T^{4} \)
$19$ \( ( 361 + T )^{4} \)
$23$ \( 56213471217472 + 5977932912 T - 19403228 T^{2} + 768 T^{3} + T^{4} \)
$29$ \( -60017099567280 + 87199017936 T - 36523028 T^{2} + 1828 T^{3} + T^{4} \)
$31$ \( 41853614930176 + 72220633856 T - 21298464 T^{2} - 3856 T^{3} + T^{4} \)
$37$ \( 3115018277471248 + 524186875264 T - 123580236 T^{2} - 11456 T^{3} + T^{4} \)
$41$ \( 2632223620987216 - 588507899840 T - 90431676 T^{2} + 12904 T^{3} + T^{4} \)
$43$ \( -3038489551438992 + 2993005327536 T - 229056084 T^{2} - 15096 T^{3} + T^{4} \)
$47$ \( -32236059411847872 + 11593812328560 T - 683041596 T^{2} - 17040 T^{3} + T^{4} \)
$53$ \( 17922605392887552 + 2372336961984 T - 309234372 T^{2} - 16728 T^{3} + T^{4} \)
$59$ \( 23400815487909120 - 7041708536832 T - 438473264 T^{2} + 19760 T^{3} + T^{4} \)
$61$ \( -163477708527909104 + 13633040712448 T + 271515384 T^{2} - 47168 T^{3} + T^{4} \)
$67$ \( -543492099741975296 + 8190283055808 T + 2763662464 T^{2} - 104580 T^{3} + T^{4} \)
$71$ \( 42728187302599680 - 13883956124544 T - 625254320 T^{2} + 36764 T^{3} + T^{4} \)
$73$ \( 27581810741693424 - 11274640174320 T + 1498213024 T^{2} - 74356 T^{3} + T^{4} \)
$79$ \( 7631945090504232960 + 207960107094528 T - 4481239040 T^{2} - 80920 T^{3} + T^{4} \)
$83$ \( 287793922567281664 + 51786142040256 T - 2652174980 T^{2} - 19416 T^{3} + T^{4} \)
$89$ \( 1343399662426671120 + 9502557411168 T - 3035256044 T^{2} + 4760 T^{3} + T^{4} \)
$97$ \( 12954908604295659888 + 1113462908705664 T - 7828021788 T^{2} - 139572 T^{3} + T^{4} \)
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