Properties

Label 570.6.a.n
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: \(x^{4} - x^{3} - 6167 x^{2} - 254912 x - 2938616\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + 4 q^{2} -9 q^{3} + 16 q^{4} -25 q^{5} -36 q^{6} + ( 2 - \beta_{3} ) q^{7} + 64 q^{8} + 81 q^{9} +O(q^{10})\) \( q + 4 q^{2} -9 q^{3} + 16 q^{4} -25 q^{5} -36 q^{6} + ( 2 - \beta_{3} ) q^{7} + 64 q^{8} + 81 q^{9} -100 q^{10} + ( 26 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} -144 q^{12} + ( -32 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{13} + ( 8 - 4 \beta_{3} ) q^{14} + 225 q^{15} + 256 q^{16} + ( -89 - 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{17} + 324 q^{18} -361 q^{19} -400 q^{20} + ( -18 + 9 \beta_{3} ) q^{21} + ( 104 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{22} + ( -99 + 7 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{23} -576 q^{24} + 625 q^{25} + ( -128 - 8 \beta_{1} + 8 \beta_{2} - 12 \beta_{3} ) q^{26} -729 q^{27} + ( 32 - 16 \beta_{3} ) q^{28} + ( -2 + \beta_{1} + 31 \beta_{2} - \beta_{3} ) q^{29} + 900 q^{30} + ( -2672 + 18 \beta_{1} + 2 \beta_{2} ) q^{31} + 1024 q^{32} + ( -234 - 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{33} + ( -356 - 8 \beta_{1} - 20 \beta_{2} - 12 \beta_{3} ) q^{34} + ( -50 + 25 \beta_{3} ) q^{35} + 1296 q^{36} + ( 2646 + 4 \beta_{1} - 30 \beta_{2} + 13 \beta_{3} ) q^{37} -1444 q^{38} + ( 288 + 18 \beta_{1} - 18 \beta_{2} + 27 \beta_{3} ) q^{39} -1600 q^{40} + ( -942 - 41 \beta_{1} - 39 \beta_{2} + 27 \beta_{3} ) q^{41} + ( -72 + 36 \beta_{3} ) q^{42} + ( 120 + 20 \beta_{1} - 58 \beta_{2} + 3 \beta_{3} ) q^{43} + ( 416 + 16 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{44} -2025 q^{45} + ( -396 + 28 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} ) q^{46} + ( 2409 - 21 \beta_{1} + 28 \beta_{2} - 39 \beta_{3} ) q^{47} -2304 q^{48} + ( 10618 - 50 \beta_{1} + 55 \beta_{2} + 15 \beta_{3} ) q^{49} + 2500 q^{50} + ( 801 + 18 \beta_{1} + 45 \beta_{2} + 27 \beta_{3} ) q^{51} + ( -512 - 32 \beta_{1} + 32 \beta_{2} - 48 \beta_{3} ) q^{52} + ( 3411 - 85 \beta_{1} - 4 \beta_{2} - 103 \beta_{3} ) q^{53} -2916 q^{54} + ( -650 - 25 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} ) q^{55} + ( 128 - 64 \beta_{3} ) q^{56} + 3249 q^{57} + ( -8 + 4 \beta_{1} + 124 \beta_{2} - 4 \beta_{3} ) q^{58} + ( 5531 + 16 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{59} + 3600 q^{60} + ( 5147 + 88 \beta_{1} + 53 \beta_{2} + 83 \beta_{3} ) q^{61} + ( -10688 + 72 \beta_{1} + 8 \beta_{2} ) q^{62} + ( 162 - 81 \beta_{3} ) q^{63} + 4096 q^{64} + ( 800 + 50 \beta_{1} - 50 \beta_{2} + 75 \beta_{3} ) q^{65} + ( -936 - 36 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{66} + ( 9293 + 44 \beta_{1} + 137 \beta_{2} + 93 \beta_{3} ) q^{67} + ( -1424 - 32 \beta_{1} - 80 \beta_{2} - 48 \beta_{3} ) q^{68} + ( 891 - 63 \beta_{1} + 36 \beta_{2} + 27 \beta_{3} ) q^{69} + ( -200 + 100 \beta_{3} ) q^{70} + ( 10553 + 82 \beta_{1} - 77 \beta_{2} - 17 \beta_{3} ) q^{71} + 5184 q^{72} + ( 9639 + 38 \beta_{1} - 153 \beta_{2} + 113 \beta_{3} ) q^{73} + ( 10584 + 16 \beta_{1} - 120 \beta_{2} + 52 \beta_{3} ) q^{74} -5625 q^{75} -5776 q^{76} + ( 30322 - 70 \beta_{1} + 300 \beta_{2} - 6 \beta_{3} ) q^{77} + ( 1152 + 72 \beta_{1} - 72 \beta_{2} + 108 \beta_{3} ) q^{78} + ( 47713 + 68 \beta_{1} + 97 \beta_{2} + 181 \beta_{3} ) q^{79} -6400 q^{80} + 6561 q^{81} + ( -3768 - 164 \beta_{1} - 156 \beta_{2} + 108 \beta_{3} ) q^{82} + ( 29430 - 177 \beta_{1} + 73 \beta_{2} - 264 \beta_{3} ) q^{83} + ( -288 + 144 \beta_{3} ) q^{84} + ( 2225 + 50 \beta_{1} + 125 \beta_{2} + 75 \beta_{3} ) q^{85} + ( 480 + 80 \beta_{1} - 232 \beta_{2} + 12 \beta_{3} ) q^{86} + ( 18 - 9 \beta_{1} - 279 \beta_{2} + 9 \beta_{3} ) q^{87} + ( 1664 + 64 \beta_{1} + 64 \beta_{2} - 64 \beta_{3} ) q^{88} + ( 62086 + 211 \beta_{1} + 33 \beta_{2} - 17 \beta_{3} ) q^{89} -8100 q^{90} + ( 57521 + 170 \beta_{1} + 135 \beta_{2} - 203 \beta_{3} ) q^{91} + ( -1584 + 112 \beta_{1} - 64 \beta_{2} - 48 \beta_{3} ) q^{92} + ( 24048 - 162 \beta_{1} - 18 \beta_{2} ) q^{93} + ( 9636 - 84 \beta_{1} + 112 \beta_{2} - 156 \beta_{3} ) q^{94} + 9025 q^{95} -9216 q^{96} + ( -38649 - 284 \beta_{1} - 149 \beta_{2} - 64 \beta_{3} ) q^{97} + ( 42472 - 200 \beta_{1} + 220 \beta_{2} + 60 \beta_{3} ) q^{98} + ( 2106 + 81 \beta_{1} + 81 \beta_{2} - 81 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 16q^{2} - 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 10q^{7} + 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q + 16q^{2} - 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 10q^{7} + 256q^{8} + 324q^{9} - 400q^{10} + 102q^{11} - 576q^{12} - 122q^{13} + 40q^{14} + 900q^{15} + 1024q^{16} - 336q^{17} + 1296q^{18} - 1444q^{19} - 1600q^{20} - 90q^{21} + 408q^{22} - 396q^{23} - 2304q^{24} + 2500q^{25} - 488q^{26} - 2916q^{27} + 160q^{28} - 70q^{29} + 3600q^{30} - 10728q^{31} + 4096q^{32} - 918q^{33} - 1344q^{34} - 250q^{35} + 5184q^{36} + 10610q^{37} - 5776q^{38} + 1098q^{39} - 6400q^{40} - 3662q^{41} - 360q^{42} + 550q^{43} + 1632q^{44} - 8100q^{45} - 1584q^{46} + 9700q^{47} - 9216q^{48} + 42432q^{49} + 10000q^{50} + 3024q^{51} - 1952q^{52} + 14028q^{53} - 11664q^{54} - 2550q^{55} + 640q^{56} + 12996q^{57} - 280q^{58} + 22092q^{59} + 14400q^{60} + 20140q^{61} - 42912q^{62} + 810q^{63} + 16384q^{64} + 3050q^{65} - 3672q^{66} + 36624q^{67} - 5376q^{68} + 3564q^{69} - 1000q^{70} + 42236q^{71} + 20736q^{72} + 38560q^{73} + 42440q^{74} - 22500q^{75} - 23104q^{76} + 120840q^{77} + 4392q^{78} + 190160q^{79} - 25600q^{80} + 26244q^{81} - 14648q^{82} + 118456q^{83} - 1440q^{84} + 8400q^{85} + 2200q^{86} + 630q^{87} + 6528q^{88} + 247890q^{89} - 32400q^{90} + 229880q^{91} - 6336q^{92} + 96552q^{93} + 38800q^{94} + 36100q^{95} - 36864q^{96} - 153602q^{97} + 169728q^{98} + 8262q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6167 x^{2} - 254912 x - 2938616\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 19 \nu^{2} - 5975 \nu - 135737 \)\()/25\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 29 \nu^{2} + 5365 \nu + 105037 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 29 \nu^{2} + 5345 \nu + 105042 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-61 \beta_{3} + 63 \beta_{2} + 10 \beta_{1} + 12341\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-3567 \beta_{3} + 3586 \beta_{2} + 145 \beta_{1} + 391701\)\()/2\)

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
−27.2566
−39.2598
−28.5777
96.0940
4.00000 −9.00000 16.0000 −25.0000 −36.0000 −227.941 64.0000 81.0000 −100.000
1.2 4.00000 −9.00000 16.0000 −25.0000 −36.0000 −79.8621 64.0000 81.0000 −100.000
1.3 4.00000 −9.00000 16.0000 −25.0000 −36.0000 138.582 64.0000 81.0000 −100.000
1.4 4.00000 −9.00000 16.0000 −25.0000 −36.0000 179.221 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.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.n 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 10 T_{7}^{3} - 54780 T_{7}^{2} + 1859600 T_{7} + 452124000 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{4} \)
$3$ \( ( 9 + T )^{4} \)
$5$ \( ( 25 + T )^{4} \)
$7$ \( 452124000 + 1859600 T - 54780 T^{2} - 10 T^{3} + T^{4} \)
$11$ \( 10200696544 + 4002672 T - 240140 T^{2} - 102 T^{3} + T^{4} \)
$13$ \( -15341802672 + 249056920 T - 962032 T^{2} + 122 T^{3} + T^{4} \)
$17$ \( -763231396848 - 2699221696 T - 2552648 T^{2} + 336 T^{3} + T^{4} \)
$19$ \( ( 361 + T )^{4} \)
$23$ \( -330936269760 + 3390011424 T - 8504236 T^{2} + 396 T^{3} + T^{4} \)
$29$ \( 346386292950672 + 49977126696 T - 59993984 T^{2} + 70 T^{3} + T^{4} \)
$31$ \( -138204792064000 - 171345158144 T + 3515408 T^{2} + 10728 T^{3} + T^{4} \)
$37$ \( -282244606671600 + 344125381320 T - 17568336 T^{2} - 10610 T^{3} + T^{4} \)
$41$ \( 25857081546444400 - 363864343640 T - 338267480 T^{2} + 3662 T^{3} + T^{4} \)
$43$ \( 17389223705925856 + 77189046640 T - 264642092 T^{2} - 550 T^{3} + T^{4} \)
$47$ \( -1590346470582720 + 925523397408 T - 110047948 T^{2} - 9700 T^{3} + T^{4} \)
$53$ \( -60994226974375680 + 20284880568640 T - 1134391484 T^{2} - 14028 T^{3} + T^{4} \)
$59$ \( 132892725089280 - 312947743232 T + 149701936 T^{2} - 22092 T^{3} + T^{4} \)
$61$ \( -56009751601253968 + 17406384947952 T - 1181154896 T^{2} - 20140 T^{3} + T^{4} \)
$67$ \( -843969357030683904 + 77739371137024 T - 1469820032 T^{2} - 36624 T^{3} + T^{4} \)
$71$ \( -97909940891902976 + 26614147986944 T - 656057360 T^{2} - 42236 T^{3} + T^{4} \)
$73$ \( -290408979831904368 + 56240172073728 T - 1375531176 T^{2} - 38560 T^{3} + T^{4} \)
$79$ \( -3162820756742514688 - 110369267722752 T + 10686421024 T^{2} - 190160 T^{3} + T^{4} \)
$83$ \( -10184954747495383296 + 458255245944384 T - 1041919300 T^{2} - 118456 T^{3} + T^{4} \)
$89$ \( 281546735777800560 - 299659116592792 T + 17463331272 T^{2} - 247890 T^{3} + T^{4} \)
$97$ \( 6183856427049630000 - 557519556670440 T - 1979063720 T^{2} + 153602 T^{3} + T^{4} \)
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