Properties

Label 570.4.i.h
Level $570$
Weight $4$
Character orbit 570.i
Analytic conductor $33.631$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 570.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.6310887033\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{385})\)
Defining polynomial: \(x^{4} - x^{3} + 97 x^{2} + 96 x + 9216\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \beta_{2} ) q^{2} + ( 3 - 3 \beta_{2} ) q^{3} -4 \beta_{2} q^{4} + ( 5 - 5 \beta_{2} ) q^{5} + 6 \beta_{2} q^{6} + ( 8 + 2 \beta_{3} ) q^{7} + 8 q^{8} -9 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -2 + 2 \beta_{2} ) q^{2} + ( 3 - 3 \beta_{2} ) q^{3} -4 \beta_{2} q^{4} + ( 5 - 5 \beta_{2} ) q^{5} + 6 \beta_{2} q^{6} + ( 8 + 2 \beta_{3} ) q^{7} + 8 q^{8} -9 \beta_{2} q^{9} + 10 \beta_{2} q^{10} + ( -6 + \beta_{3} ) q^{11} -12 q^{12} + 46 \beta_{2} q^{13} + ( -16 - 4 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{14} -15 \beta_{2} q^{15} + ( -16 + 16 \beta_{2} ) q^{16} + ( -51 - 3 \beta_{1} + 54 \beta_{2} - 3 \beta_{3} ) q^{17} + 18 q^{18} + ( 43 - 8 \beta_{1} - 13 \beta_{2} - \beta_{3} ) q^{19} -20 q^{20} + ( 24 + 6 \beta_{1} - 30 \beta_{2} + 6 \beta_{3} ) q^{21} + ( 12 - 2 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{22} + ( -16 \beta_{1} - 20 \beta_{2} ) q^{23} + ( 24 - 24 \beta_{2} ) q^{24} -25 \beta_{2} q^{25} -92 q^{26} -27 q^{27} + ( 8 \beta_{1} - 40 \beta_{2} ) q^{28} + ( 15 \beta_{1} + 69 \beta_{2} ) q^{29} + 30 q^{30} + ( 107 + 8 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + ( -18 + 3 \beta_{1} + 15 \beta_{2} + 3 \beta_{3} ) q^{33} + ( 6 \beta_{1} - 108 \beta_{2} ) q^{34} + ( 40 + 10 \beta_{1} - 50 \beta_{2} + 10 \beta_{3} ) q^{35} + ( -36 + 36 \beta_{2} ) q^{36} + ( -292 + 2 \beta_{3} ) q^{37} + ( -44 + 2 \beta_{1} + 84 \beta_{2} - 14 \beta_{3} ) q^{38} + 138 q^{39} + ( 40 - 40 \beta_{2} ) q^{40} + ( -126 + 24 \beta_{1} + 102 \beta_{2} + 24 \beta_{3} ) q^{41} + ( -12 \beta_{1} + 60 \beta_{2} ) q^{42} + ( -110 + 8 \beta_{1} + 102 \beta_{2} + 8 \beta_{3} ) q^{43} + ( 4 \beta_{1} + 20 \beta_{2} ) q^{44} -45 q^{45} + ( 72 - 32 \beta_{3} ) q^{46} + ( -9 \beta_{1} - 132 \beta_{2} ) q^{47} + 48 \beta_{2} q^{48} + ( 105 + 36 \beta_{3} ) q^{49} + 50 q^{50} + ( -9 \beta_{1} + 162 \beta_{2} ) q^{51} + ( 184 - 184 \beta_{2} ) q^{52} + ( 31 \beta_{1} + 134 \beta_{2} ) q^{53} + ( 54 - 54 \beta_{2} ) q^{54} + ( -30 + 5 \beta_{1} + 25 \beta_{2} + 5 \beta_{3} ) q^{55} + ( 64 + 16 \beta_{3} ) q^{56} + ( 66 - 3 \beta_{1} - 126 \beta_{2} + 21 \beta_{3} ) q^{57} + ( -168 + 30 \beta_{3} ) q^{58} + ( 234 - 15 \beta_{1} - 219 \beta_{2} - 15 \beta_{3} ) q^{59} + ( -60 + 60 \beta_{2} ) q^{60} + ( -29 \beta_{1} + 243 \beta_{2} ) q^{61} + ( -214 - 16 \beta_{1} + 230 \beta_{2} - 16 \beta_{3} ) q^{62} + ( 18 \beta_{1} - 90 \beta_{2} ) q^{63} + 64 q^{64} + 230 q^{65} + ( -6 \beta_{1} - 30 \beta_{2} ) q^{66} + ( -8 \beta_{1} + 828 \beta_{2} ) q^{67} + ( 204 + 12 \beta_{3} ) q^{68} + ( -108 + 48 \beta_{3} ) q^{69} + ( -20 \beta_{1} + 100 \beta_{2} ) q^{70} + ( -642 + 45 \beta_{1} + 597 \beta_{2} + 45 \beta_{3} ) q^{71} -72 \beta_{2} q^{72} + ( -182 + 72 \beta_{1} + 110 \beta_{2} + 72 \beta_{3} ) q^{73} + ( 584 - 4 \beta_{1} - 580 \beta_{2} - 4 \beta_{3} ) q^{74} -75 q^{75} + ( -84 + 28 \beta_{1} - 116 \beta_{2} + 32 \beta_{3} ) q^{76} + ( 144 - 2 \beta_{3} ) q^{77} + ( -276 + 276 \beta_{2} ) q^{78} + ( -56 + 97 \beta_{1} - 41 \beta_{2} + 97 \beta_{3} ) q^{79} + 80 \beta_{2} q^{80} + ( -81 + 81 \beta_{2} ) q^{81} + ( -48 \beta_{1} - 204 \beta_{2} ) q^{82} + ( 159 - 5 \beta_{3} ) q^{83} + ( -96 - 24 \beta_{3} ) q^{84} + ( -15 \beta_{1} + 270 \beta_{2} ) q^{85} + ( -16 \beta_{1} - 204 \beta_{2} ) q^{86} + ( 252 - 45 \beta_{3} ) q^{87} + ( -48 + 8 \beta_{3} ) q^{88} + ( 49 \beta_{1} - 139 \beta_{2} ) q^{89} + ( 90 - 90 \beta_{2} ) q^{90} + ( -92 \beta_{1} + 460 \beta_{2} ) q^{91} + ( -144 + 64 \beta_{1} + 80 \beta_{2} + 64 \beta_{3} ) q^{92} + ( 321 + 24 \beta_{1} - 345 \beta_{2} + 24 \beta_{3} ) q^{93} + ( 282 - 18 \beta_{3} ) q^{94} + ( 110 - 5 \beta_{1} - 210 \beta_{2} + 35 \beta_{3} ) q^{95} -96 q^{96} + ( -518 - 14 \beta_{1} + 532 \beta_{2} - 14 \beta_{3} ) q^{97} + ( -210 - 72 \beta_{1} + 282 \beta_{2} - 72 \beta_{3} ) q^{98} + ( 9 \beta_{1} + 45 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} + 12q^{6} + 36q^{7} + 32q^{8} - 18q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} + 12q^{6} + 36q^{7} + 32q^{8} - 18q^{9} + 20q^{10} - 22q^{11} - 48q^{12} + 92q^{13} - 36q^{14} - 30q^{15} - 32q^{16} - 105q^{17} + 72q^{18} + 136q^{19} - 80q^{20} + 54q^{21} + 22q^{22} - 56q^{23} + 48q^{24} - 50q^{25} - 368q^{26} - 108q^{27} - 72q^{28} + 153q^{29} + 120q^{30} + 444q^{31} - 64q^{32} - 33q^{33} - 210q^{34} + 90q^{35} - 72q^{36} - 1164q^{37} - 34q^{38} + 552q^{39} + 80q^{40} - 228q^{41} + 108q^{42} - 212q^{43} + 44q^{44} - 180q^{45} + 224q^{46} - 273q^{47} + 96q^{48} + 492q^{49} + 200q^{50} + 315q^{51} + 368q^{52} + 299q^{53} + 108q^{54} - 55q^{55} + 288q^{56} + 51q^{57} - 612q^{58} + 453q^{59} - 120q^{60} + 457q^{61} - 444q^{62} - 162q^{63} + 256q^{64} + 920q^{65} - 66q^{66} + 1648q^{67} + 840q^{68} - 336q^{69} + 180q^{70} - 1239q^{71} - 144q^{72} - 292q^{73} + 1164q^{74} - 300q^{75} - 476q^{76} + 572q^{77} - 552q^{78} - 15q^{79} + 160q^{80} - 162q^{81} - 456q^{82} + 626q^{83} - 432q^{84} + 525q^{85} - 424q^{86} + 918q^{87} - 176q^{88} - 229q^{89} + 180q^{90} + 828q^{91} - 224q^{92} + 666q^{93} + 1092q^{94} + 85q^{95} - 384q^{96} - 1050q^{97} - 492q^{98} + 99q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 97 x^{2} + 96 x + 9216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 97 \nu^{2} - 97 \nu + 9216 \)\()/9312\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 193 \)\()/97\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 96 \beta_{2} + \beta_{1} - 97\)
\(\nu^{3}\)\(=\)\(97 \beta_{3} - 193\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
5.15535 8.92934i
−4.65535 + 8.06331i
5.15535 + 8.92934i
−4.65535 8.06331i
−1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i 3.00000 5.19615i −10.6214 8.00000 −4.50000 + 7.79423i 5.00000 8.66025i
121.2 −1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i 3.00000 5.19615i 28.6214 8.00000 −4.50000 + 7.79423i 5.00000 8.66025i
391.1 −1.00000 + 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.50000 4.33013i 3.00000 + 5.19615i −10.6214 8.00000 −4.50000 7.79423i 5.00000 + 8.66025i
391.2 −1.00000 + 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.50000 4.33013i 3.00000 + 5.19615i 28.6214 8.00000 −4.50000 7.79423i 5.00000 + 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.4.i.h 4
19.c even 3 1 inner 570.4.i.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.i.h 4 1.a even 1 1 trivial
570.4.i.h 4 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 18 T_{7} - 304 \) acting on \(S_{4}^{\mathrm{new}}(570, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{2} \)
$3$ \( ( 9 - 3 T + T^{2} )^{2} \)
$5$ \( ( 25 - 5 T + T^{2} )^{2} \)
$7$ \( ( -304 - 18 T + T^{2} )^{2} \)
$11$ \( ( -66 + 11 T + T^{2} )^{2} \)
$13$ \( ( 2116 - 46 T + T^{2} )^{2} \)
$17$ \( 3572100 + 198450 T + 9135 T^{2} + 105 T^{3} + T^{4} \)
$19$ \( 47045881 - 932824 T + 14877 T^{2} - 136 T^{3} + T^{4} \)
$23$ \( 569108736 - 1335936 T + 26992 T^{2} + 56 T^{3} + T^{4} \)
$29$ \( 249766416 + 2418012 T + 39213 T^{2} - 153 T^{3} + T^{4} \)
$31$ \( ( 6161 - 222 T + T^{2} )^{2} \)
$37$ \( ( 84296 + 582 T + T^{2} )^{2} \)
$41$ \( 1801493136 - 9677232 T + 94428 T^{2} + 228 T^{3} + T^{4} \)
$43$ \( 25765776 + 1076112 T + 39868 T^{2} + 212 T^{3} + T^{4} \)
$47$ \( 117418896 + 2958228 T + 63693 T^{2} + 273 T^{3} + T^{4} \)
$53$ \( 4920461316 + 20973654 T + 159547 T^{2} - 299 T^{3} + T^{4} \)
$59$ \( 878885316 - 13429638 T + 175563 T^{2} - 453 T^{3} + T^{4} \)
$61$ \( 825642756 + 13131438 T + 237583 T^{2} - 457 T^{3} + T^{4} \)
$67$ \( 452681369856 - 1108800768 T + 2043088 T^{2} - 1648 T^{3} + T^{4} \)
$71$ \( 35673387876 + 234014886 T + 1346247 T^{2} + 1239 T^{3} + T^{4} \)
$73$ \( 228143790736 - 139472048 T + 562908 T^{2} + 292 T^{3} + T^{4} \)
$79$ \( 820038913600 - 13583400 T + 905785 T^{2} + 15 T^{3} + T^{4} \)
$83$ \( ( 22086 - 313 T + T^{2} )^{2} \)
$89$ \( 47517896196 - 49918794 T + 270427 T^{2} + 229 T^{3} + T^{4} \)
$97$ \( 65925697600 + 269598000 T + 845740 T^{2} + 1050 T^{3} + T^{4} \)
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