Properties

Label 570.4.a.s
Level $570$
Weight $4$
Character orbit 570.a
Self dual yes
Analytic conductor $33.631$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.6310887033\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 410 x^{2} + 4362 x - 12540\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} + ( 9 + \beta_{3} ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} + ( 9 + \beta_{3} ) q^{7} + 8 q^{8} + 9 q^{9} + 10 q^{10} + ( 13 + \beta_{1} - \beta_{3} ) q^{11} + 12 q^{12} + ( 12 - \beta_{1} - \beta_{2} ) q^{13} + ( 18 + 2 \beta_{3} ) q^{14} + 15 q^{15} + 16 q^{16} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{17} + 18 q^{18} + 19 q^{19} + 20 q^{20} + ( 27 + 3 \beta_{3} ) q^{21} + ( 26 + 2 \beta_{1} - 2 \beta_{3} ) q^{22} + ( 25 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{23} + 24 q^{24} + 25 q^{25} + ( 24 - 2 \beta_{1} - 2 \beta_{2} ) q^{26} + 27 q^{27} + ( 36 + 4 \beta_{3} ) q^{28} + ( 3 + \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{29} + 30 q^{30} + ( 10 - 5 \beta_{1} - 2 \beta_{2} ) q^{31} + 32 q^{32} + ( 39 + 3 \beta_{1} - 3 \beta_{3} ) q^{33} + ( 8 - 2 \beta_{1} + 4 \beta_{2} ) q^{34} + ( 45 + 5 \beta_{3} ) q^{35} + 36 q^{36} + ( 10 - 5 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{37} + 38 q^{38} + ( 36 - 3 \beta_{1} - 3 \beta_{2} ) q^{39} + 40 q^{40} + ( -6 - 6 \beta_{1} - 3 \beta_{2} - 12 \beta_{3} ) q^{41} + ( 54 + 6 \beta_{3} ) q^{42} + ( 22 + 7 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} ) q^{43} + ( 52 + 4 \beta_{1} - 4 \beta_{3} ) q^{44} + 45 q^{45} + ( 50 + 4 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{46} + ( 95 + 14 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} ) q^{47} + 48 q^{48} + ( 118 + 4 \beta_{1} - 7 \beta_{2} + 17 \beta_{3} ) q^{49} + 50 q^{50} + ( 12 - 3 \beta_{1} + 6 \beta_{2} ) q^{51} + ( 48 - 4 \beta_{1} - 4 \beta_{2} ) q^{52} + ( -43 - 2 \beta_{1} + \beta_{2} - 21 \beta_{3} ) q^{53} + 54 q^{54} + ( 65 + 5 \beta_{1} - 5 \beta_{3} ) q^{55} + ( 72 + 8 \beta_{3} ) q^{56} + 57 q^{57} + ( 6 + 2 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} ) q^{58} + ( 19 - 5 \beta_{1} + 3 \beta_{2} + 17 \beta_{3} ) q^{59} + 60 q^{60} + ( 17 - 4 \beta_{1} + 7 \beta_{2} + 23 \beta_{3} ) q^{61} + ( 20 - 10 \beta_{1} - 4 \beta_{2} ) q^{62} + ( 81 + 9 \beta_{3} ) q^{63} + 64 q^{64} + ( 60 - 5 \beta_{1} - 5 \beta_{2} ) q^{65} + ( 78 + 6 \beta_{1} - 6 \beta_{3} ) q^{66} + ( -40 + 4 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} ) q^{67} + ( 16 - 4 \beta_{1} + 8 \beta_{2} ) q^{68} + ( 75 + 6 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} ) q^{69} + ( 90 + 10 \beta_{3} ) q^{70} + ( 40 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{71} + 72 q^{72} + ( 70 + 2 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{73} + ( 20 - 10 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} ) q^{74} + 75 q^{75} + 76 q^{76} + ( -73 - 4 \beta_{1} + 9 \beta_{2} + 19 \beta_{3} ) q^{77} + ( 72 - 6 \beta_{1} - 6 \beta_{2} ) q^{78} + ( 135 + 7 \beta_{1} + \beta_{2} + 17 \beta_{3} ) q^{79} + 80 q^{80} + 81 q^{81} + ( -12 - 12 \beta_{1} - 6 \beta_{2} - 24 \beta_{3} ) q^{82} + ( -94 - 9 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{83} + ( 108 + 12 \beta_{3} ) q^{84} + ( 20 - 5 \beta_{1} + 10 \beta_{2} ) q^{85} + ( 44 + 14 \beta_{1} + 6 \beta_{2} - 24 \beta_{3} ) q^{86} + ( 9 + 3 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} ) q^{87} + ( 104 + 8 \beta_{1} - 8 \beta_{3} ) q^{88} + ( 2 - 34 \beta_{1} - 13 \beta_{2} + 52 \beta_{3} ) q^{89} + 90 q^{90} + ( 13 - 10 \beta_{1} - 21 \beta_{2} + 29 \beta_{3} ) q^{91} + ( 100 + 8 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{92} + ( 30 - 15 \beta_{1} - 6 \beta_{2} ) q^{93} + ( 190 + 28 \beta_{1} - 14 \beta_{2} - 10 \beta_{3} ) q^{94} + 95 q^{95} + 96 q^{96} + ( -230 - \beta_{1} - 11 \beta_{2} - 62 \beta_{3} ) q^{97} + ( 236 + 8 \beta_{1} - 14 \beta_{2} + 34 \beta_{3} ) q^{98} + ( 117 + 9 \beta_{1} - 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{2} + 12q^{3} + 16q^{4} + 20q^{5} + 24q^{6} + 36q^{7} + 32q^{8} + 36q^{9} + O(q^{10}) \) \( 4q + 8q^{2} + 12q^{3} + 16q^{4} + 20q^{5} + 24q^{6} + 36q^{7} + 32q^{8} + 36q^{9} + 40q^{10} + 54q^{11} + 48q^{12} + 46q^{13} + 72q^{14} + 60q^{15} + 64q^{16} + 14q^{17} + 72q^{18} + 76q^{19} + 80q^{20} + 108q^{21} + 108q^{22} + 104q^{23} + 96q^{24} + 100q^{25} + 92q^{26} + 108q^{27} + 144q^{28} + 14q^{29} + 120q^{30} + 30q^{31} + 128q^{32} + 162q^{33} + 28q^{34} + 180q^{35} + 144q^{36} + 30q^{37} + 152q^{38} + 138q^{39} + 160q^{40} - 36q^{41} + 216q^{42} + 102q^{43} + 216q^{44} + 180q^{45} + 208q^{46} + 408q^{47} + 192q^{48} + 480q^{49} + 200q^{50} + 42q^{51} + 184q^{52} - 176q^{53} + 216q^{54} + 270q^{55} + 288q^{56} + 228q^{57} + 28q^{58} + 66q^{59} + 240q^{60} + 60q^{61} + 60q^{62} + 324q^{63} + 256q^{64} + 230q^{65} + 324q^{66} - 152q^{67} + 56q^{68} + 312q^{69} + 360q^{70} + 172q^{71} + 288q^{72} + 284q^{73} + 60q^{74} + 300q^{75} + 304q^{76} - 300q^{77} + 276q^{78} + 554q^{79} + 320q^{80} + 324q^{81} - 72q^{82} - 394q^{83} + 432q^{84} + 70q^{85} + 204q^{86} + 42q^{87} + 432q^{88} - 60q^{89} + 360q^{90} + 32q^{91} + 416q^{92} + 90q^{93} + 816q^{94} + 380q^{95} + 384q^{96} - 922q^{97} + 960q^{98} + 486q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 410 x^{2} + 4362 x - 12540\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 6 \nu^{2} - 367 \nu + 1824 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} - 11 \nu^{2} + 749 \nu - 3857 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + 4 \beta_{2} - 15 \beta_{1} + 418\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-12 \beta_{3} - 22 \beta_{2} + 457 \beta_{1} - 6156\)\()/2\)

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
6.43713
−24.2310
12.2089
6.58500
2.00000 3.00000 4.00000 5.00000 6.00000 −15.8571 8.00000 9.00000 10.0000
1.2 2.00000 3.00000 4.00000 5.00000 6.00000 −1.42654 8.00000 9.00000 10.0000
1.3 2.00000 3.00000 4.00000 5.00000 6.00000 17.1829 8.00000 9.00000 10.0000
1.4 2.00000 3.00000 4.00000 5.00000 6.00000 36.1008 8.00000 9.00000 10.0000
\(n\): 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 570.4.a.s 4
3.b odd 2 1 1710.4.a.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.a.s 4 1.a even 1 1 trivial
1710.4.a.bc 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(570))\):

\( T_{7}^{4} - 36 T_{7}^{3} - 278 T_{7}^{2} + 9516 T_{7} + 14032 \)
\( T_{11}^{4} - 54 T_{11}^{3} - 570 T_{11}^{2} + 36648 T_{11} + 34560 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{4} \)
$3$ \( ( -3 + T )^{4} \)
$5$ \( ( -5 + T )^{4} \)
$7$ \( 14032 + 9516 T - 278 T^{2} - 36 T^{3} + T^{4} \)
$11$ \( 34560 + 36648 T - 570 T^{2} - 54 T^{3} + T^{4} \)
$13$ \( -3407016 + 250224 T - 3830 T^{2} - 46 T^{3} + T^{4} \)
$17$ \( 41635968 + 183456 T - 14216 T^{2} - 14 T^{3} + T^{4} \)
$19$ \( ( -19 + T )^{4} \)
$23$ \( 1104000 + 82800 T - 7196 T^{2} - 104 T^{3} + T^{4} \)
$29$ \( -181111800 + 7215240 T - 73310 T^{2} - 14 T^{3} + T^{4} \)
$31$ \( 16169536 + 1542360 T - 51908 T^{2} - 30 T^{3} + T^{4} \)
$37$ \( 15035992 + 2480448 T - 47486 T^{2} - 30 T^{3} + T^{4} \)
$41$ \( 12058908048 - 1542564 T - 234846 T^{2} + 36 T^{3} + T^{4} \)
$43$ \( 4956833536 + 13098288 T - 162578 T^{2} - 102 T^{3} + T^{4} \)
$47$ \( 1082908800 + 161761680 T - 377628 T^{2} - 408 T^{3} + T^{4} \)
$53$ \( 17352011904 + 8349216 T - 373940 T^{2} + 176 T^{3} + T^{4} \)
$59$ \( -3721014720 + 55122192 T - 208488 T^{2} - 66 T^{3} + T^{4} \)
$61$ \( -5968735328 + 128324304 T - 458204 T^{2} - 60 T^{3} + T^{4} \)
$67$ \( -21765056256 - 351878784 T - 805088 T^{2} + 152 T^{3} + T^{4} \)
$71$ \( 337920000 + 9648000 T - 86120 T^{2} - 172 T^{3} + T^{4} \)
$73$ \( 52421119600 + 261627760 T - 765216 T^{2} - 284 T^{3} + T^{4} \)
$79$ \( 26204800000 + 96826240 T - 270576 T^{2} - 554 T^{3} + T^{4} \)
$83$ \( 6066401280 - 28798200 T - 127052 T^{2} + 394 T^{3} + T^{4} \)
$89$ \( 2442413711520 - 304480044 T - 3364782 T^{2} + 60 T^{3} + T^{4} \)
$97$ \( 994890105880 - 2264201696 T - 2794182 T^{2} + 922 T^{3} + T^{4} \)
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