Properties

Label 570.6.a.m
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: \(x^{4} - x^{3} - 39154 x^{2} - 3172892 x - 35506440\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q -4 q^{2} + 9 q^{3} + 16 q^{4} + 25 q^{5} -36 q^{6} + ( 67 + \beta_{1} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} + 9 q^{3} + 16 q^{4} + 25 q^{5} -36 q^{6} + ( 67 + \beta_{1} ) q^{7} -64 q^{8} + 81 q^{9} -100 q^{10} + ( 84 - \beta_{1} + \beta_{2} ) q^{11} + 144 q^{12} + ( 326 - \beta_{2} + \beta_{3} ) q^{13} + ( -268 - 4 \beta_{1} ) q^{14} + 225 q^{15} + 256 q^{16} + ( 135 + \beta_{2} + 2 \beta_{3} ) q^{17} -324 q^{18} + 361 q^{19} + 400 q^{20} + ( 603 + 9 \beta_{1} ) q^{21} + ( -336 + 4 \beta_{1} - 4 \beta_{2} ) q^{22} + ( -24 + \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{23} -576 q^{24} + 625 q^{25} + ( -1304 + 4 \beta_{2} - 4 \beta_{3} ) q^{26} + 729 q^{27} + ( 1072 + 16 \beta_{1} ) q^{28} + ( 1254 + 21 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{29} -900 q^{30} + ( 1243 + 16 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{31} -1024 q^{32} + ( 756 - 9 \beta_{1} + 9 \beta_{2} ) q^{33} + ( -540 - 4 \beta_{2} - 8 \beta_{3} ) q^{34} + ( 1675 + 25 \beta_{1} ) q^{35} + 1296 q^{36} + ( 2006 + 10 \beta_{1} - \beta_{2} - 9 \beta_{3} ) q^{37} -1444 q^{38} + ( 2934 - 9 \beta_{2} + 9 \beta_{3} ) q^{39} -1600 q^{40} + ( 695 - 48 \beta_{1} - 8 \beta_{2} - 9 \beta_{3} ) q^{41} + ( -2412 - 36 \beta_{1} ) q^{42} + ( 6340 + 44 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{43} + ( 1344 - 16 \beta_{1} + 16 \beta_{2} ) q^{44} + 2025 q^{45} + ( 96 - 4 \beta_{1} - 16 \beta_{2} + 20 \beta_{3} ) q^{46} + ( 2504 - 81 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{47} + 2304 q^{48} + ( 9313 + 135 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} ) q^{49} -2500 q^{50} + ( 1215 + 9 \beta_{2} + 18 \beta_{3} ) q^{51} + ( 5216 - 16 \beta_{2} + 16 \beta_{3} ) q^{52} + ( -66 - 115 \beta_{1} + 20 \beta_{2} - 25 \beta_{3} ) q^{53} -2916 q^{54} + ( 2100 - 25 \beta_{1} + 25 \beta_{2} ) q^{55} + ( -4288 - 64 \beta_{1} ) q^{56} + 3249 q^{57} + ( -5016 - 84 \beta_{1} + 4 \beta_{2} - 16 \beta_{3} ) q^{58} + ( 7419 - 3 \beta_{1} - 11 \beta_{2} + 35 \beta_{3} ) q^{59} + 3600 q^{60} + ( 6830 - 43 \beta_{1} - 22 \beta_{2} - 19 \beta_{3} ) q^{61} + ( -4972 - 64 \beta_{1} + 20 \beta_{2} - 8 \beta_{3} ) q^{62} + ( 5427 + 81 \beta_{1} ) q^{63} + 4096 q^{64} + ( 8150 - 25 \beta_{2} + 25 \beta_{3} ) q^{65} + ( -3024 + 36 \beta_{1} - 36 \beta_{2} ) q^{66} + ( 6492 - 100 \beta_{1} + 28 \beta_{2} - 16 \beta_{3} ) q^{67} + ( 2160 + 16 \beta_{2} + 32 \beta_{3} ) q^{68} + ( -216 + 9 \beta_{1} + 36 \beta_{2} - 45 \beta_{3} ) q^{69} + ( -6700 - 100 \beta_{1} ) q^{70} + ( -6704 + 60 \beta_{1} - 62 \beta_{2} - 38 \beta_{3} ) q^{71} -5184 q^{72} + ( -15268 - 120 \beta_{1} + 78 \beta_{2} - 48 \beta_{3} ) q^{73} + ( -8024 - 40 \beta_{1} + 4 \beta_{2} + 36 \beta_{3} ) q^{74} + 5625 q^{75} + 5776 q^{76} + ( -5318 + 65 \beta_{1} + 22 \beta_{2} - 109 \beta_{3} ) q^{77} + ( -11736 + 36 \beta_{2} - 36 \beta_{3} ) q^{78} + ( 8733 + 49 \beta_{1} - 15 \beta_{2} - 35 \beta_{3} ) q^{79} + 6400 q^{80} + 6561 q^{81} + ( -2780 + 192 \beta_{1} + 32 \beta_{2} + 36 \beta_{3} ) q^{82} + ( -12103 - 124 \beta_{1} - 9 \beta_{2} + 116 \beta_{3} ) q^{83} + ( 9648 + 144 \beta_{1} ) q^{84} + ( 3375 + 25 \beta_{2} + 50 \beta_{3} ) q^{85} + ( -25360 - 176 \beta_{1} + 28 \beta_{2} - 12 \beta_{3} ) q^{86} + ( 11286 + 189 \beta_{1} - 9 \beta_{2} + 36 \beta_{3} ) q^{87} + ( -5376 + 64 \beta_{1} - 64 \beta_{2} ) q^{88} + ( -9587 + 344 \beta_{1} + 9 \beta_{3} ) q^{89} -8100 q^{90} + ( 17420 + 449 \beta_{1} - 200 \beta_{2} + 215 \beta_{3} ) q^{91} + ( -384 + 16 \beta_{1} + 64 \beta_{2} - 80 \beta_{3} ) q^{92} + ( 11187 + 144 \beta_{1} - 45 \beta_{2} + 18 \beta_{3} ) q^{93} + ( -10016 + 324 \beta_{1} + 16 \beta_{2} - 20 \beta_{3} ) q^{94} + 9025 q^{95} -9216 q^{96} + ( 11446 - 434 \beta_{1} + 79 \beta_{2} - 25 \beta_{3} ) q^{97} + ( -37252 - 540 \beta_{1} + 8 \beta_{2} - 28 \beta_{3} ) q^{98} + ( 6804 - 81 \beta_{1} + 81 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} + 100q^{5} - 144q^{6} + 268q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} + 100q^{5} - 144q^{6} + 268q^{7} - 256q^{8} + 324q^{9} - 400q^{10} + 338q^{11} + 576q^{12} + 1302q^{13} - 1072q^{14} + 900q^{15} + 1024q^{16} + 542q^{17} - 1296q^{18} + 1444q^{19} + 1600q^{20} + 2412q^{21} - 1352q^{22} - 88q^{23} - 2304q^{24} + 2500q^{25} - 5208q^{26} + 2916q^{27} + 4288q^{28} + 5014q^{29} - 3600q^{30} + 4962q^{31} - 4096q^{32} + 3042q^{33} - 2168q^{34} + 6700q^{35} + 5184q^{36} + 8022q^{37} - 5776q^{38} + 11718q^{39} - 6400q^{40} + 2764q^{41} - 9648q^{42} + 25346q^{43} + 5408q^{44} + 8100q^{45} + 352q^{46} + 10008q^{47} + 9216q^{48} + 37248q^{49} - 10000q^{50} + 4878q^{51} + 20832q^{52} - 224q^{53} - 11664q^{54} + 8450q^{55} - 17152q^{56} + 12996q^{57} - 20056q^{58} + 29654q^{59} + 14400q^{60} + 27276q^{61} - 19848q^{62} + 21708q^{63} + 16384q^{64} + 32550q^{65} - 12168q^{66} + 26024q^{67} + 8672q^{68} - 792q^{69} - 26800q^{70} - 26940q^{71} - 20736q^{72} - 60916q^{73} - 32088q^{74} + 22500q^{75} + 23104q^{76} - 21228q^{77} - 46872q^{78} + 34902q^{79} + 25600q^{80} + 26244q^{81} - 11056q^{82} - 48430q^{83} + 38592q^{84} + 13550q^{85} - 101384q^{86} + 45126q^{87} - 21632q^{88} - 38348q^{89} - 32400q^{90} + 69280q^{91} - 1408q^{92} + 44658q^{93} - 40032q^{94} + 36100q^{95} - 36864q^{96} + 45942q^{97} - 148992q^{98} + 27378q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 39154 x^{2} - 3172892 x - 35506440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 7 \nu^{3} - 1173 \nu^{2} - 164020 \nu + 6141876 \)\()/49776\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} - 187 \nu^{2} - 55307 \nu - 1142260 \)\()/2074\)
\(\beta_{3}\)\(=\)\((\)\( -9 \nu^{3} + 323 \nu^{2} + 377988 \nu + 15263364 \)\()/16592\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 3 \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(19 \beta_{3} + 47 \beta_{2} - 249 \beta_{1} + 39131\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(32983 \beta_{3} + 47059 \beta_{2} - 152805 \beta_{1} + 14430679\)\()/6\)

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
231.807
−128.214
−89.1990
−13.3932
−4.00000 9.00000 16.0000 25.0000 −36.0000 −88.0477 −64.0000 81.0000 −100.000
1.2 −4.00000 9.00000 16.0000 25.0000 −36.0000 −70.9208 −64.0000 81.0000 −100.000
1.3 −4.00000 9.00000 16.0000 25.0000 −36.0000 197.010 −64.0000 81.0000 −100.000
1.4 −4.00000 9.00000 16.0000 25.0000 −36.0000 229.958 −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.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.m 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 268 T_{7}^{3} - 16326 T_{7}^{2} + 4535764 T_{7} + 282897872 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{4} \)
$3$ \( ( -9 + T )^{4} \)
$5$ \( ( -25 + T )^{4} \)
$7$ \( 282897872 + 4535764 T - 16326 T^{2} - 268 T^{3} + T^{4} \)
$11$ \( 31621155840 + 52998456 T - 374282 T^{2} - 338 T^{3} + T^{4} \)
$13$ \( -129277370856 + 555202480 T - 216054 T^{2} - 1302 T^{3} + T^{4} \)
$17$ \( 1667650550400 + 1398361248 T - 3158760 T^{2} - 542 T^{3} + T^{4} \)
$19$ \( ( -361 + T )^{4} \)
$23$ \( -939061848960 - 16990398576 T - 18558972 T^{2} + 88 T^{3} + T^{4} \)
$29$ \( 13968516390408 + 20948416488 T - 20431950 T^{2} - 5014 T^{3} + T^{4} \)
$31$ \( -854008501952 + 18319828008 T - 10262276 T^{2} - 4962 T^{3} + T^{4} \)
$37$ \( 372689310336664 + 193054532160 T - 29887934 T^{2} - 8022 T^{3} + T^{4} \)
$41$ \( 180884133450000 - 671835716100 T - 214102110 T^{2} - 2764 T^{3} + T^{4} \)
$43$ \( -810042253597440 + 17716075088 T + 144928334 T^{2} - 25346 T^{3} + T^{4} \)
$47$ \( 18200330224295040 + 1647418838256 T - 268557372 T^{2} - 10008 T^{3} + T^{4} \)
$53$ \( -69690045404421504 + 16295261674464 T - 1006785684 T^{2} + 224 T^{3} + T^{4} \)
$59$ \( 43003981339149888 + 10016295559344 T - 408618024 T^{2} - 29654 T^{3} + T^{4} \)
$61$ \( 3987724707017376 - 274141475312 T - 364526460 T^{2} - 27276 T^{3} + T^{4} \)
$67$ \( -106177793698855680 + 17021551031168 T - 501130912 T^{2} - 26024 T^{3} + T^{4} \)
$71$ \( -73400824763596800 - 30128968137600 T - 2602095912 T^{2} + 26940 T^{3} + T^{4} \)
$73$ \( 772217898434858224 - 39708420581840 T - 2244576576 T^{2} + 60916 T^{3} + T^{4} \)
$79$ \( 35296611629028352 + 18933806123136 T - 524427056 T^{2} - 34902 T^{3} + T^{4} \)
$83$ \( 12649042580869992960 - 141630005565096 T - 7667621132 T^{2} + 48430 T^{3} + T^{4} \)
$89$ \( 5649768230451455520 - 109089090185868 T - 4694457582 T^{2} + 38348 T^{3} + T^{4} \)
$97$ \( -929934014391091880 + 337374183362784 T - 8783920982 T^{2} - 45942 T^{3} + T^{4} \)
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