# Properties

 Label 507.4.a.i.1.1 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2x^{3} - 25x^{2} + 24x + 78$$ x^4 - 2*x^3 - 25*x^2 + 24*x + 78 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.33039$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} +16.4131 q^{5} +15.9912 q^{6} -9.67968 q^{7} -66.1667 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} +16.4131 q^{5} +15.9912 q^{6} -9.67968 q^{7} -66.1667 q^{8} +9.00000 q^{9} -87.4882 q^{10} -27.5882 q^{11} -61.2393 q^{12} +51.5965 q^{14} -49.2393 q^{15} +189.390 q^{16} +107.928 q^{17} -47.9735 q^{18} +2.24723 q^{19} +335.042 q^{20} +29.0391 q^{21} +147.056 q^{22} +41.8090 q^{23} +198.500 q^{24} +144.390 q^{25} -27.0000 q^{27} -197.592 q^{28} +61.6213 q^{29} +262.465 q^{30} -191.932 q^{31} -480.187 q^{32} +82.7645 q^{33} -575.300 q^{34} -158.874 q^{35} +183.718 q^{36} -98.4236 q^{37} -11.9786 q^{38} -1086.00 q^{40} +30.7452 q^{41} -154.790 q^{42} +238.325 q^{43} -563.160 q^{44} +147.718 q^{45} -222.858 q^{46} +511.482 q^{47} -568.169 q^{48} -249.304 q^{49} -769.653 q^{50} -323.785 q^{51} +492.825 q^{53} +143.921 q^{54} -452.807 q^{55} +640.472 q^{56} -6.74170 q^{57} -328.466 q^{58} -484.179 q^{59} -1005.13 q^{60} -444.021 q^{61} +1023.07 q^{62} -87.1172 q^{63} +1044.47 q^{64} -441.167 q^{66} -190.114 q^{67} +2203.15 q^{68} -125.427 q^{69} +846.858 q^{70} -484.785 q^{71} -595.500 q^{72} +957.780 q^{73} +524.636 q^{74} -433.169 q^{75} +45.8729 q^{76} +267.045 q^{77} -375.216 q^{79} +3108.47 q^{80} +81.0000 q^{81} -163.884 q^{82} +715.765 q^{83} +592.777 q^{84} +1771.43 q^{85} -1270.37 q^{86} -184.864 q^{87} +1825.42 q^{88} +1038.15 q^{89} -787.394 q^{90} +853.451 q^{92} +575.796 q^{93} -2726.40 q^{94} +36.8840 q^{95} +1440.56 q^{96} -65.5636 q^{97} +1328.89 q^{98} -248.293 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 12 q^{3} + 22 q^{4} + 6 q^{5} + 6 q^{6} + 14 q^{7} - 54 q^{8} + 36 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 12 * q^3 + 22 * q^4 + 6 * q^5 + 6 * q^6 + 14 * q^7 - 54 * q^8 + 36 * q^9 $$4 q - 2 q^{2} - 12 q^{3} + 22 q^{4} + 6 q^{5} + 6 q^{6} + 14 q^{7} - 54 q^{8} + 36 q^{9} - 62 q^{10} - 40 q^{11} - 66 q^{12} + 40 q^{14} - 18 q^{15} + 122 q^{16} + 98 q^{17} - 18 q^{18} - 124 q^{19} + 466 q^{20} - 42 q^{21} + 220 q^{22} + 104 q^{23} + 162 q^{24} - 58 q^{25} - 108 q^{27} + 144 q^{28} + 194 q^{29} + 186 q^{30} - 26 q^{31} - 654 q^{32} + 120 q^{33} - 1062 q^{34} + 88 q^{35} + 198 q^{36} - 102 q^{37} + 332 q^{38} - 998 q^{40} + 1054 q^{41} - 120 q^{42} + 450 q^{43} + 44 q^{44} + 54 q^{45} + 172 q^{46} + 96 q^{47} - 366 q^{48} + 1070 q^{49} - 996 q^{50} - 294 q^{51} + 262 q^{53} + 54 q^{54} + 204 q^{55} + 2164 q^{56} + 372 q^{57} - 722 q^{58} - 308 q^{59} - 1398 q^{60} - 928 q^{61} + 2780 q^{62} + 126 q^{63} + 1026 q^{64} - 660 q^{66} + 1134 q^{67} + 1786 q^{68} - 312 q^{69} + 2324 q^{70} - 1064 q^{71} - 486 q^{72} - 952 q^{73} + 1158 q^{74} + 174 q^{75} + 1708 q^{76} + 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 324 q^{81} + 1734 q^{82} + 404 q^{83} - 432 q^{84} + 1394 q^{85} - 3168 q^{86} - 582 q^{87} + 3060 q^{88} - 1620 q^{89} - 558 q^{90} + 332 q^{92} + 78 q^{93} - 772 q^{94} + 2204 q^{95} + 1962 q^{96} - 2166 q^{97} + 1906 q^{98} - 360 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 12 * q^3 + 22 * q^4 + 6 * q^5 + 6 * q^6 + 14 * q^7 - 54 * q^8 + 36 * q^9 - 62 * q^10 - 40 * q^11 - 66 * q^12 + 40 * q^14 - 18 * q^15 + 122 * q^16 + 98 * q^17 - 18 * q^18 - 124 * q^19 + 466 * q^20 - 42 * q^21 + 220 * q^22 + 104 * q^23 + 162 * q^24 - 58 * q^25 - 108 * q^27 + 144 * q^28 + 194 * q^29 + 186 * q^30 - 26 * q^31 - 654 * q^32 + 120 * q^33 - 1062 * q^34 + 88 * q^35 + 198 * q^36 - 102 * q^37 + 332 * q^38 - 998 * q^40 + 1054 * q^41 - 120 * q^42 + 450 * q^43 + 44 * q^44 + 54 * q^45 + 172 * q^46 + 96 * q^47 - 366 * q^48 + 1070 * q^49 - 996 * q^50 - 294 * q^51 + 262 * q^53 + 54 * q^54 + 204 * q^55 + 2164 * q^56 + 372 * q^57 - 722 * q^58 - 308 * q^59 - 1398 * q^60 - 928 * q^61 + 2780 * q^62 + 126 * q^63 + 1026 * q^64 - 660 * q^66 + 1134 * q^67 + 1786 * q^68 - 312 * q^69 + 2324 * q^70 - 1064 * q^71 - 486 * q^72 - 952 * q^73 + 1158 * q^74 + 174 * q^75 + 1708 * q^76 + 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 324 * q^81 + 1734 * q^82 + 404 * q^83 - 432 * q^84 + 1394 * q^85 - 3168 * q^86 - 582 * q^87 + 3060 * q^88 - 1620 * q^89 - 558 * q^90 + 332 * q^92 + 78 * q^93 - 772 * q^94 + 2204 * q^95 + 1962 * q^96 - 2166 * q^97 + 1906 * q^98 - 360 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.33039 −1.88458 −0.942289 0.334800i $$-0.891331\pi$$
−0.942289 + 0.334800i $$0.891331\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 20.4131 2.55164
$$5$$ 16.4131 1.46803 0.734016 0.679132i $$-0.237644\pi$$
0.734016 + 0.679132i $$0.237644\pi$$
$$6$$ 15.9912 1.08806
$$7$$ −9.67968 −0.522654 −0.261327 0.965250i $$-0.584160\pi$$
−0.261327 + 0.965250i $$0.584160\pi$$
$$8$$ −66.1667 −2.92418
$$9$$ 9.00000 0.333333
$$10$$ −87.4882 −2.76662
$$11$$ −27.5882 −0.756195 −0.378098 0.925766i $$-0.623422\pi$$
−0.378098 + 0.925766i $$0.623422\pi$$
$$12$$ −61.2393 −1.47319
$$13$$ 0 0
$$14$$ 51.5965 0.984982
$$15$$ −49.2393 −0.847568
$$16$$ 189.390 2.95921
$$17$$ 107.928 1.53979 0.769895 0.638171i $$-0.220309\pi$$
0.769895 + 0.638171i $$0.220309\pi$$
$$18$$ −47.9735 −0.628193
$$19$$ 2.24723 0.0271342 0.0135671 0.999908i $$-0.495681\pi$$
0.0135671 + 0.999908i $$0.495681\pi$$
$$20$$ 335.042 3.74588
$$21$$ 29.0391 0.301754
$$22$$ 147.056 1.42511
$$23$$ 41.8090 0.379034 0.189517 0.981877i $$-0.439308\pi$$
0.189517 + 0.981877i $$0.439308\pi$$
$$24$$ 198.500 1.68828
$$25$$ 144.390 1.15512
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ −197.592 −1.33362
$$29$$ 61.6213 0.394579 0.197289 0.980345i $$-0.436786\pi$$
0.197289 + 0.980345i $$0.436786\pi$$
$$30$$ 262.465 1.59731
$$31$$ −191.932 −1.11200 −0.556000 0.831182i $$-0.687665\pi$$
−0.556000 + 0.831182i $$0.687665\pi$$
$$32$$ −480.187 −2.65269
$$33$$ 82.7645 0.436589
$$34$$ −575.300 −2.90185
$$35$$ −158.874 −0.767272
$$36$$ 183.718 0.850545
$$37$$ −98.4236 −0.437317 −0.218659 0.975801i $$-0.570168\pi$$
−0.218659 + 0.975801i $$0.570168\pi$$
$$38$$ −11.9786 −0.0511366
$$39$$ 0 0
$$40$$ −1086.00 −4.29279
$$41$$ 30.7452 0.117112 0.0585561 0.998284i $$-0.481350\pi$$
0.0585561 + 0.998284i $$0.481350\pi$$
$$42$$ −154.790 −0.568680
$$43$$ 238.325 0.845216 0.422608 0.906313i $$-0.361115\pi$$
0.422608 + 0.906313i $$0.361115\pi$$
$$44$$ −563.160 −1.92953
$$45$$ 147.718 0.489344
$$46$$ −222.858 −0.714319
$$47$$ 511.482 1.58739 0.793695 0.608316i $$-0.208155\pi$$
0.793695 + 0.608316i $$0.208155\pi$$
$$48$$ −568.169 −1.70850
$$49$$ −249.304 −0.726833
$$50$$ −769.653 −2.17691
$$51$$ −323.785 −0.888998
$$52$$ 0 0
$$53$$ 492.825 1.27726 0.638630 0.769514i $$-0.279502\pi$$
0.638630 + 0.769514i $$0.279502\pi$$
$$54$$ 143.921 0.362687
$$55$$ −452.807 −1.11012
$$56$$ 640.472 1.52833
$$57$$ −6.74170 −0.0156660
$$58$$ −328.466 −0.743615
$$59$$ −484.179 −1.06838 −0.534192 0.845363i $$-0.679384\pi$$
−0.534192 + 0.845363i $$0.679384\pi$$
$$60$$ −1005.13 −2.16269
$$61$$ −444.021 −0.931985 −0.465993 0.884789i $$-0.654303\pi$$
−0.465993 + 0.884789i $$0.654303\pi$$
$$62$$ 1023.07 2.09565
$$63$$ −87.1172 −0.174218
$$64$$ 1044.47 2.03998
$$65$$ 0 0
$$66$$ −441.167 −0.822787
$$67$$ −190.114 −0.346658 −0.173329 0.984864i $$-0.555452\pi$$
−0.173329 + 0.984864i $$0.555452\pi$$
$$68$$ 2203.15 3.92898
$$69$$ −125.427 −0.218835
$$70$$ 846.858 1.44598
$$71$$ −484.785 −0.810329 −0.405164 0.914244i $$-0.632786\pi$$
−0.405164 + 0.914244i $$0.632786\pi$$
$$72$$ −595.500 −0.974727
$$73$$ 957.780 1.53561 0.767806 0.640683i $$-0.221349\pi$$
0.767806 + 0.640683i $$0.221349\pi$$
$$74$$ 524.636 0.824159
$$75$$ −433.169 −0.666907
$$76$$ 45.8729 0.0692367
$$77$$ 267.045 0.395228
$$78$$ 0 0
$$79$$ −375.216 −0.534368 −0.267184 0.963646i $$-0.586093\pi$$
−0.267184 + 0.963646i $$0.586093\pi$$
$$80$$ 3108.47 4.34422
$$81$$ 81.0000 0.111111
$$82$$ −163.884 −0.220707
$$83$$ 715.765 0.946571 0.473286 0.880909i $$-0.343068\pi$$
0.473286 + 0.880909i $$0.343068\pi$$
$$84$$ 592.777 0.769967
$$85$$ 1771.43 2.26046
$$86$$ −1270.37 −1.59288
$$87$$ −184.864 −0.227810
$$88$$ 1825.42 2.21125
$$89$$ 1038.15 1.23645 0.618224 0.786002i $$-0.287852\pi$$
0.618224 + 0.786002i $$0.287852\pi$$
$$90$$ −787.394 −0.922207
$$91$$ 0 0
$$92$$ 853.451 0.967157
$$93$$ 575.796 0.642013
$$94$$ −2726.40 −2.99156
$$95$$ 36.8840 0.0398339
$$96$$ 1440.56 1.53153
$$97$$ −65.5636 −0.0686286 −0.0343143 0.999411i $$-0.510925\pi$$
−0.0343143 + 0.999411i $$0.510925\pi$$
$$98$$ 1328.89 1.36977
$$99$$ −248.293 −0.252065
$$100$$ 2947.44 2.94744
$$101$$ 531.798 0.523920 0.261960 0.965079i $$-0.415631\pi$$
0.261960 + 0.965079i $$0.415631\pi$$
$$102$$ 1725.90 1.67539
$$103$$ −735.984 −0.704064 −0.352032 0.935988i $$-0.614509\pi$$
−0.352032 + 0.935988i $$0.614509\pi$$
$$104$$ 0 0
$$105$$ 476.621 0.442985
$$106$$ −2626.95 −2.40710
$$107$$ −783.265 −0.707673 −0.353837 0.935307i $$-0.615123\pi$$
−0.353837 + 0.935307i $$0.615123\pi$$
$$108$$ −551.153 −0.491063
$$109$$ 532.339 0.467788 0.233894 0.972262i $$-0.424853\pi$$
0.233894 + 0.972262i $$0.424853\pi$$
$$110$$ 2413.64 2.09210
$$111$$ 295.271 0.252485
$$112$$ −1833.23 −1.54664
$$113$$ −180.589 −0.150340 −0.0751699 0.997171i $$-0.523950\pi$$
−0.0751699 + 0.997171i $$0.523950\pi$$
$$114$$ 35.9359 0.0295237
$$115$$ 686.215 0.556434
$$116$$ 1257.88 1.00682
$$117$$ 0 0
$$118$$ 2580.86 2.01346
$$119$$ −1044.71 −0.804777
$$120$$ 3258.00 2.47844
$$121$$ −569.893 −0.428169
$$122$$ 2366.81 1.75640
$$123$$ −92.2357 −0.0676147
$$124$$ −3917.92 −2.83742
$$125$$ 318.242 0.227716
$$126$$ 464.369 0.328327
$$127$$ 1431.63 1.00029 0.500146 0.865941i $$-0.333280\pi$$
0.500146 + 0.865941i $$0.333280\pi$$
$$128$$ −1725.94 −1.19182
$$129$$ −714.976 −0.487986
$$130$$ 0 0
$$131$$ 2067.32 1.37880 0.689400 0.724381i $$-0.257874\pi$$
0.689400 + 0.724381i $$0.257874\pi$$
$$132$$ 1689.48 1.11402
$$133$$ −21.7525 −0.0141818
$$134$$ 1013.38 0.653304
$$135$$ −443.153 −0.282523
$$136$$ −7141.25 −4.50262
$$137$$ 387.512 0.241660 0.120830 0.992673i $$-0.461444\pi$$
0.120830 + 0.992673i $$0.461444\pi$$
$$138$$ 668.575 0.412412
$$139$$ 752.568 0.459223 0.229611 0.973282i $$-0.426254\pi$$
0.229611 + 0.973282i $$0.426254\pi$$
$$140$$ −3243.10 −1.95780
$$141$$ −1534.45 −0.916480
$$142$$ 2584.09 1.52713
$$143$$ 0 0
$$144$$ 1704.51 0.986404
$$145$$ 1011.40 0.579254
$$146$$ −5105.34 −2.89398
$$147$$ 747.911 0.419637
$$148$$ −2009.13 −1.11587
$$149$$ 2636.72 1.44972 0.724862 0.688895i $$-0.241904\pi$$
0.724862 + 0.688895i $$0.241904\pi$$
$$150$$ 2308.96 1.25684
$$151$$ 3332.42 1.79595 0.897975 0.440046i $$-0.145038\pi$$
0.897975 + 0.440046i $$0.145038\pi$$
$$152$$ −148.692 −0.0793454
$$153$$ 971.354 0.513263
$$154$$ −1423.45 −0.744839
$$155$$ −3150.20 −1.63245
$$156$$ 0 0
$$157$$ −1625.26 −0.826179 −0.413089 0.910690i $$-0.635550\pi$$
−0.413089 + 0.910690i $$0.635550\pi$$
$$158$$ 2000.05 1.00706
$$159$$ −1478.48 −0.737426
$$160$$ −7881.36 −3.89423
$$161$$ −404.698 −0.198104
$$162$$ −431.762 −0.209398
$$163$$ 1835.37 0.881944 0.440972 0.897521i $$-0.354634\pi$$
0.440972 + 0.897521i $$0.354634\pi$$
$$164$$ 627.605 0.298828
$$165$$ 1358.42 0.640927
$$166$$ −3815.31 −1.78389
$$167$$ −1945.00 −0.901248 −0.450624 0.892714i $$-0.648798\pi$$
−0.450624 + 0.892714i $$0.648798\pi$$
$$168$$ −1921.42 −0.882384
$$169$$ 0 0
$$170$$ −9442.44 −4.26001
$$171$$ 20.2251 0.00904474
$$172$$ 4864.96 2.15668
$$173$$ 2531.63 1.11258 0.556289 0.830989i $$-0.312225\pi$$
0.556289 + 0.830989i $$0.312225\pi$$
$$174$$ 985.397 0.429326
$$175$$ −1397.65 −0.603726
$$176$$ −5224.91 −2.23774
$$177$$ 1452.54 0.616832
$$178$$ −5533.76 −2.33018
$$179$$ 4263.01 1.78007 0.890035 0.455892i $$-0.150680\pi$$
0.890035 + 0.455892i $$0.150680\pi$$
$$180$$ 3015.38 1.24863
$$181$$ 3944.61 1.61989 0.809946 0.586504i $$-0.199496\pi$$
0.809946 + 0.586504i $$0.199496\pi$$
$$182$$ 0 0
$$183$$ 1332.06 0.538082
$$184$$ −2766.36 −1.10836
$$185$$ −1615.43 −0.641995
$$186$$ −3069.22 −1.20992
$$187$$ −2977.54 −1.16438
$$188$$ 10440.9 4.05044
$$189$$ 261.351 0.100585
$$190$$ −196.606 −0.0750701
$$191$$ −214.109 −0.0811119 −0.0405559 0.999177i $$-0.512913\pi$$
−0.0405559 + 0.999177i $$0.512913\pi$$
$$192$$ −3133.42 −1.17779
$$193$$ 1207.19 0.450234 0.225117 0.974332i $$-0.427724\pi$$
0.225117 + 0.974332i $$0.427724\pi$$
$$194$$ 349.480 0.129336
$$195$$ 0 0
$$196$$ −5089.06 −1.85461
$$197$$ 927.631 0.335487 0.167744 0.985831i $$-0.446352\pi$$
0.167744 + 0.985831i $$0.446352\pi$$
$$198$$ 1323.50 0.475036
$$199$$ 478.951 0.170613 0.0853064 0.996355i $$-0.472813\pi$$
0.0853064 + 0.996355i $$0.472813\pi$$
$$200$$ −9553.77 −3.37777
$$201$$ 570.341 0.200143
$$202$$ −2834.69 −0.987368
$$203$$ −596.474 −0.206228
$$204$$ −6609.44 −2.26840
$$205$$ 504.624 0.171924
$$206$$ 3923.08 1.32686
$$207$$ 376.281 0.126345
$$208$$ 0 0
$$209$$ −61.9970 −0.0205188
$$210$$ −2540.58 −0.834840
$$211$$ −1450.95 −0.473402 −0.236701 0.971583i $$-0.576066\pi$$
−0.236701 + 0.971583i $$0.576066\pi$$
$$212$$ 10060.1 3.25910
$$213$$ 1454.35 0.467843
$$214$$ 4175.11 1.33367
$$215$$ 3911.66 1.24080
$$216$$ 1786.50 0.562759
$$217$$ 1857.84 0.581191
$$218$$ −2837.58 −0.881583
$$219$$ −2873.34 −0.886586
$$220$$ −9243.19 −2.83262
$$221$$ 0 0
$$222$$ −1573.91 −0.475828
$$223$$ 2059.79 0.618536 0.309268 0.950975i $$-0.399916\pi$$
0.309268 + 0.950975i $$0.399916\pi$$
$$224$$ 4648.06 1.38644
$$225$$ 1299.51 0.385039
$$226$$ 962.612 0.283327
$$227$$ −4482.46 −1.31062 −0.655311 0.755359i $$-0.727463\pi$$
−0.655311 + 0.755359i $$0.727463\pi$$
$$228$$ −137.619 −0.0399738
$$229$$ 1630.39 0.470477 0.235239 0.971938i $$-0.424413\pi$$
0.235239 + 0.971938i $$0.424413\pi$$
$$230$$ −3657.80 −1.04864
$$231$$ −801.134 −0.228185
$$232$$ −4077.27 −1.15382
$$233$$ −1903.69 −0.535258 −0.267629 0.963522i $$-0.586240\pi$$
−0.267629 + 0.963522i $$0.586240\pi$$
$$234$$ 0 0
$$235$$ 8395.00 2.33034
$$236$$ −9883.58 −2.72613
$$237$$ 1125.65 0.308517
$$238$$ 5568.72 1.51667
$$239$$ −3763.79 −1.01866 −0.509328 0.860572i $$-0.670106\pi$$
−0.509328 + 0.860572i $$0.670106\pi$$
$$240$$ −9325.40 −2.50813
$$241$$ 3614.74 0.966166 0.483083 0.875575i $$-0.339517\pi$$
0.483083 + 0.875575i $$0.339517\pi$$
$$242$$ 3037.75 0.806918
$$243$$ −243.000 −0.0641500
$$244$$ −9063.85 −2.37809
$$245$$ −4091.84 −1.06701
$$246$$ 491.652 0.127425
$$247$$ 0 0
$$248$$ 12699.5 3.25169
$$249$$ −2147.30 −0.546503
$$250$$ −1696.36 −0.429148
$$251$$ −5729.77 −1.44088 −0.720438 0.693520i $$-0.756059\pi$$
−0.720438 + 0.693520i $$0.756059\pi$$
$$252$$ −1778.33 −0.444541
$$253$$ −1153.43 −0.286624
$$254$$ −7631.18 −1.88513
$$255$$ −5314.30 −1.30508
$$256$$ 844.191 0.206101
$$257$$ −5525.79 −1.34120 −0.670602 0.741818i $$-0.733964\pi$$
−0.670602 + 0.741818i $$0.733964\pi$$
$$258$$ 3811.10 0.919647
$$259$$ 952.709 0.228565
$$260$$ 0 0
$$261$$ 554.591 0.131526
$$262$$ −11019.6 −2.59846
$$263$$ 5223.21 1.22463 0.612313 0.790615i $$-0.290239\pi$$
0.612313 + 0.790615i $$0.290239\pi$$
$$264$$ −5476.25 −1.27667
$$265$$ 8088.78 1.87506
$$266$$ 115.949 0.0267267
$$267$$ −3114.46 −0.713864
$$268$$ −3880.81 −0.884545
$$269$$ 7203.88 1.63282 0.816410 0.577473i $$-0.195961\pi$$
0.816410 + 0.577473i $$0.195961\pi$$
$$270$$ 2362.18 0.532436
$$271$$ 8577.69 1.92272 0.961360 0.275293i $$-0.0887750\pi$$
0.961360 + 0.275293i $$0.0887750\pi$$
$$272$$ 20440.5 4.55656
$$273$$ 0 0
$$274$$ −2065.59 −0.455427
$$275$$ −3983.44 −0.873493
$$276$$ −2560.35 −0.558388
$$277$$ 7169.19 1.55507 0.777536 0.628838i $$-0.216469\pi$$
0.777536 + 0.628838i $$0.216469\pi$$
$$278$$ −4011.48 −0.865442
$$279$$ −1727.39 −0.370667
$$280$$ 10512.1 2.24364
$$281$$ −849.157 −0.180272 −0.0901360 0.995929i $$-0.528730\pi$$
−0.0901360 + 0.995929i $$0.528730\pi$$
$$282$$ 8179.20 1.72718
$$283$$ 1115.37 0.234283 0.117141 0.993115i $$-0.462627\pi$$
0.117141 + 0.993115i $$0.462627\pi$$
$$284$$ −9895.95 −2.06766
$$285$$ −110.652 −0.0229981
$$286$$ 0 0
$$287$$ −297.604 −0.0612091
$$288$$ −4321.69 −0.884229
$$289$$ 6735.49 1.37095
$$290$$ −5391.14 −1.09165
$$291$$ 196.691 0.0396227
$$292$$ 19551.2 3.91832
$$293$$ 1863.53 0.371565 0.185782 0.982591i $$-0.440518\pi$$
0.185782 + 0.982591i $$0.440518\pi$$
$$294$$ −3986.66 −0.790839
$$295$$ −7946.87 −1.56842
$$296$$ 6512.36 1.27879
$$297$$ 744.880 0.145530
$$298$$ −14054.8 −2.73212
$$299$$ 0 0
$$300$$ −8842.31 −1.70170
$$301$$ −2306.91 −0.441755
$$302$$ −17763.1 −3.38461
$$303$$ −1595.39 −0.302485
$$304$$ 425.602 0.0802959
$$305$$ −7287.76 −1.36818
$$306$$ −5177.70 −0.967285
$$307$$ 6387.50 1.18747 0.593736 0.804660i $$-0.297652\pi$$
0.593736 + 0.804660i $$0.297652\pi$$
$$308$$ 5451.21 1.00848
$$309$$ 2207.95 0.406492
$$310$$ 16791.8 3.07648
$$311$$ −3492.59 −0.636806 −0.318403 0.947955i $$-0.603147\pi$$
−0.318403 + 0.947955i $$0.603147\pi$$
$$312$$ 0 0
$$313$$ −5912.01 −1.06762 −0.533812 0.845603i $$-0.679241\pi$$
−0.533812 + 0.845603i $$0.679241\pi$$
$$314$$ 8663.29 1.55700
$$315$$ −1429.86 −0.255757
$$316$$ −7659.31 −1.36351
$$317$$ 1677.54 0.297224 0.148612 0.988896i $$-0.452519\pi$$
0.148612 + 0.988896i $$0.452519\pi$$
$$318$$ 7880.86 1.38974
$$319$$ −1700.02 −0.298378
$$320$$ 17143.0 2.99476
$$321$$ 2349.79 0.408575
$$322$$ 2157.20 0.373342
$$323$$ 242.540 0.0417810
$$324$$ 1653.46 0.283515
$$325$$ 0 0
$$326$$ −9783.22 −1.66209
$$327$$ −1597.02 −0.270077
$$328$$ −2034.31 −0.342457
$$329$$ −4950.98 −0.829655
$$330$$ −7240.92 −1.20788
$$331$$ −2010.31 −0.333827 −0.166913 0.985972i $$-0.553380\pi$$
−0.166913 + 0.985972i $$0.553380\pi$$
$$332$$ 14611.0 2.41531
$$333$$ −885.812 −0.145772
$$334$$ 10367.6 1.69847
$$335$$ −3120.35 −0.508905
$$336$$ 5499.69 0.892955
$$337$$ 7139.24 1.15400 0.577002 0.816743i $$-0.304222\pi$$
0.577002 + 0.816743i $$0.304222\pi$$
$$338$$ 0 0
$$339$$ 541.768 0.0867988
$$340$$ 36160.5 5.76787
$$341$$ 5295.05 0.840889
$$342$$ −107.808 −0.0170455
$$343$$ 5733.31 0.902536
$$344$$ −15769.2 −2.47156
$$345$$ −2058.64 −0.321257
$$346$$ −13494.6 −2.09674
$$347$$ 1.13990 0.000176349 0 8.81743e−5 1.00000i $$-0.499972\pi$$
8.81743e−5 1.00000i $$0.499972\pi$$
$$348$$ −3773.64 −0.581289
$$349$$ −12199.1 −1.87107 −0.935535 0.353235i $$-0.885082\pi$$
−0.935535 + 0.353235i $$0.885082\pi$$
$$350$$ 7450.00 1.13777
$$351$$ 0 0
$$352$$ 13247.5 2.00595
$$353$$ 10892.3 1.64232 0.821160 0.570698i $$-0.193327\pi$$
0.821160 + 0.570698i $$0.193327\pi$$
$$354$$ −7742.59 −1.16247
$$355$$ −7956.81 −1.18959
$$356$$ 21191.9 3.15497
$$357$$ 3134.13 0.464638
$$358$$ −22723.5 −3.35468
$$359$$ 3525.78 0.518339 0.259169 0.965832i $$-0.416551\pi$$
0.259169 + 0.965832i $$0.416551\pi$$
$$360$$ −9773.99 −1.43093
$$361$$ −6853.95 −0.999264
$$362$$ −21026.3 −3.05281
$$363$$ 1709.68 0.247204
$$364$$ 0 0
$$365$$ 15720.1 2.25433
$$366$$ −7100.42 −1.01406
$$367$$ 2383.75 0.339049 0.169525 0.985526i $$-0.445777\pi$$
0.169525 + 0.985526i $$0.445777\pi$$
$$368$$ 7918.19 1.12164
$$369$$ 276.707 0.0390374
$$370$$ 8610.90 1.20989
$$371$$ −4770.39 −0.667564
$$372$$ 11753.8 1.63818
$$373$$ −13282.2 −1.84377 −0.921885 0.387463i $$-0.873352\pi$$
−0.921885 + 0.387463i $$0.873352\pi$$
$$374$$ 15871.5 2.19437
$$375$$ −954.727 −0.131472
$$376$$ −33843.1 −4.64181
$$377$$ 0 0
$$378$$ −1393.11 −0.189560
$$379$$ 4436.73 0.601318 0.300659 0.953732i $$-0.402793\pi$$
0.300659 + 0.953732i $$0.402793\pi$$
$$380$$ 752.917 0.101642
$$381$$ −4294.90 −0.577519
$$382$$ 1141.28 0.152862
$$383$$ 810.412 0.108120 0.0540602 0.998538i $$-0.482784\pi$$
0.0540602 + 0.998538i $$0.482784\pi$$
$$384$$ 5177.83 0.688100
$$385$$ 4383.03 0.580207
$$386$$ −6434.78 −0.848501
$$387$$ 2144.93 0.281739
$$388$$ −1338.35 −0.175115
$$389$$ 3463.79 0.451469 0.225734 0.974189i $$-0.427522\pi$$
0.225734 + 0.974189i $$0.427522\pi$$
$$390$$ 0 0
$$391$$ 4512.37 0.583633
$$392$$ 16495.6 2.12539
$$393$$ −6201.96 −0.796050
$$394$$ −4944.64 −0.632252
$$395$$ −6158.45 −0.784469
$$396$$ −5068.44 −0.643178
$$397$$ −425.405 −0.0537796 −0.0268898 0.999638i $$-0.508560\pi$$
−0.0268898 + 0.999638i $$0.508560\pi$$
$$398$$ −2553.00 −0.321533
$$399$$ 65.2575 0.00818787
$$400$$ 27345.9 3.41823
$$401$$ 1186.85 0.147801 0.0739007 0.997266i $$-0.476455\pi$$
0.0739007 + 0.997266i $$0.476455\pi$$
$$402$$ −3040.14 −0.377185
$$403$$ 0 0
$$404$$ 10855.6 1.33685
$$405$$ 1329.46 0.163115
$$406$$ 3179.44 0.388653
$$407$$ 2715.33 0.330697
$$408$$ 21423.7 2.59959
$$409$$ −8007.42 −0.968071 −0.484036 0.875048i $$-0.660830\pi$$
−0.484036 + 0.875048i $$0.660830\pi$$
$$410$$ −2689.84 −0.324005
$$411$$ −1162.54 −0.139522
$$412$$ −15023.7 −1.79652
$$413$$ 4686.70 0.558395
$$414$$ −2005.73 −0.238106
$$415$$ 11747.9 1.38960
$$416$$ 0 0
$$417$$ −2257.70 −0.265132
$$418$$ 330.468 0.0386692
$$419$$ 6832.46 0.796629 0.398314 0.917249i $$-0.369595\pi$$
0.398314 + 0.917249i $$0.369595\pi$$
$$420$$ 9729.30 1.13034
$$421$$ −10739.6 −1.24326 −0.621632 0.783309i $$-0.713530\pi$$
−0.621632 + 0.783309i $$0.713530\pi$$
$$422$$ 7734.16 0.892163
$$423$$ 4603.34 0.529130
$$424$$ −32608.6 −3.73494
$$425$$ 15583.7 1.77864
$$426$$ −7752.28 −0.881688
$$427$$ 4297.99 0.487106
$$428$$ −15988.9 −1.80573
$$429$$ 0 0
$$430$$ −20850.7 −2.33839
$$431$$ −5214.45 −0.582763 −0.291382 0.956607i $$-0.594115\pi$$
−0.291382 + 0.956607i $$0.594115\pi$$
$$432$$ −5113.52 −0.569501
$$433$$ 8642.24 0.959168 0.479584 0.877496i $$-0.340788\pi$$
0.479584 + 0.877496i $$0.340788\pi$$
$$434$$ −9903.02 −1.09530
$$435$$ −3034.19 −0.334432
$$436$$ 10866.7 1.19362
$$437$$ 93.9545 0.0102848
$$438$$ 15316.0 1.67084
$$439$$ −13026.2 −1.41619 −0.708097 0.706116i $$-0.750446\pi$$
−0.708097 + 0.706116i $$0.750446\pi$$
$$440$$ 29960.7 3.24619
$$441$$ −2243.73 −0.242278
$$442$$ 0 0
$$443$$ −11533.0 −1.23690 −0.618450 0.785824i $$-0.712239\pi$$
−0.618450 + 0.785824i $$0.712239\pi$$
$$444$$ 6027.39 0.644250
$$445$$ 17039.3 1.81515
$$446$$ −10979.5 −1.16568
$$447$$ −7910.17 −0.836998
$$448$$ −10110.2 −1.06621
$$449$$ −9882.75 −1.03874 −0.519372 0.854548i $$-0.673834\pi$$
−0.519372 + 0.854548i $$0.673834\pi$$
$$450$$ −6926.88 −0.725636
$$451$$ −848.204 −0.0885596
$$452$$ −3686.38 −0.383613
$$453$$ −9997.26 −1.03689
$$454$$ 23893.3 2.46997
$$455$$ 0 0
$$456$$ 446.075 0.0458101
$$457$$ 15628.1 1.59967 0.799836 0.600218i $$-0.204920\pi$$
0.799836 + 0.600218i $$0.204920\pi$$
$$458$$ −8690.63 −0.886652
$$459$$ −2914.06 −0.296333
$$460$$ 14007.8 1.41982
$$461$$ 7747.46 0.782723 0.391361 0.920237i $$-0.372004\pi$$
0.391361 + 0.920237i $$0.372004\pi$$
$$462$$ 4270.36 0.430033
$$463$$ 333.422 0.0334675 0.0167337 0.999860i $$-0.494673\pi$$
0.0167337 + 0.999860i $$0.494673\pi$$
$$464$$ 11670.4 1.16764
$$465$$ 9450.59 0.942496
$$466$$ 10147.4 1.00874
$$467$$ 8198.33 0.812363 0.406182 0.913792i $$-0.366860\pi$$
0.406182 + 0.913792i $$0.366860\pi$$
$$468$$ 0 0
$$469$$ 1840.24 0.181182
$$470$$ −44748.7 −4.39171
$$471$$ 4875.79 0.476995
$$472$$ 32036.5 3.12415
$$473$$ −6574.96 −0.639148
$$474$$ −6000.14 −0.581425
$$475$$ 324.477 0.0313432
$$476$$ −21325.8 −2.05350
$$477$$ 4435.43 0.425753
$$478$$ 20062.5 1.91974
$$479$$ 6435.88 0.613910 0.306955 0.951724i $$-0.400690\pi$$
0.306955 + 0.951724i $$0.400690\pi$$
$$480$$ 23644.1 2.24833
$$481$$ 0 0
$$482$$ −19268.0 −1.82082
$$483$$ 1214.09 0.114375
$$484$$ −11633.3 −1.09253
$$485$$ −1076.10 −0.100749
$$486$$ 1295.29 0.120896
$$487$$ −8095.37 −0.753257 −0.376629 0.926364i $$-0.622917\pi$$
−0.376629 + 0.926364i $$0.622917\pi$$
$$488$$ 29379.4 2.72529
$$489$$ −5506.10 −0.509191
$$490$$ 21811.1 2.01087
$$491$$ 5116.46 0.470270 0.235135 0.971963i $$-0.424447\pi$$
0.235135 + 0.971963i $$0.424447\pi$$
$$492$$ −1882.82 −0.172528
$$493$$ 6650.67 0.607568
$$494$$ 0 0
$$495$$ −4075.26 −0.370039
$$496$$ −36349.9 −3.29064
$$497$$ 4692.56 0.423521
$$498$$ 11445.9 1.02993
$$499$$ 18050.7 1.61936 0.809682 0.586870i $$-0.199640\pi$$
0.809682 + 0.586870i $$0.199640\pi$$
$$500$$ 6496.31 0.581047
$$501$$ 5834.99 0.520336
$$502$$ 30541.9 2.71544
$$503$$ 10531.1 0.933512 0.466756 0.884386i $$-0.345423\pi$$
0.466756 + 0.884386i $$0.345423\pi$$
$$504$$ 5764.25 0.509445
$$505$$ 8728.45 0.769131
$$506$$ 6148.25 0.540165
$$507$$ 0 0
$$508$$ 29224.1 2.55238
$$509$$ 1963.31 0.170967 0.0854834 0.996340i $$-0.472757\pi$$
0.0854834 + 0.996340i $$0.472757\pi$$
$$510$$ 28327.3 2.45952
$$511$$ −9271.00 −0.802593
$$512$$ 9307.69 0.803409
$$513$$ −60.6753 −0.00522198
$$514$$ 29454.6 2.52760
$$515$$ −12079.8 −1.03359
$$516$$ −14594.9 −1.24516
$$517$$ −14110.9 −1.20038
$$518$$ −5078.31 −0.430750
$$519$$ −7594.89 −0.642348
$$520$$ 0 0
$$521$$ −7044.93 −0.592407 −0.296203 0.955125i $$-0.595721\pi$$
−0.296203 + 0.955125i $$0.595721\pi$$
$$522$$ −2956.19 −0.247872
$$523$$ −3213.29 −0.268657 −0.134328 0.990937i $$-0.542888\pi$$
−0.134328 + 0.990937i $$0.542888\pi$$
$$524$$ 42200.4 3.51819
$$525$$ 4192.94 0.348561
$$526$$ −27841.7 −2.30790
$$527$$ −20714.9 −1.71225
$$528$$ 15674.7 1.29196
$$529$$ −10419.0 −0.856333
$$530$$ −43116.4 −3.53369
$$531$$ −4357.61 −0.356128
$$532$$ −444.036 −0.0361868
$$533$$ 0 0
$$534$$ 16601.3 1.34533
$$535$$ −12855.8 −1.03889
$$536$$ 12579.2 1.01369
$$537$$ −12789.0 −1.02772
$$538$$ −38399.5 −3.07718
$$539$$ 6877.83 0.549627
$$540$$ −9046.13 −0.720895
$$541$$ −11251.4 −0.894150 −0.447075 0.894497i $$-0.647534\pi$$
−0.447075 + 0.894497i $$0.647534\pi$$
$$542$$ −45722.4 −3.62352
$$543$$ −11833.8 −0.935245
$$544$$ −51825.8 −4.08458
$$545$$ 8737.33 0.686727
$$546$$ 0 0
$$547$$ 1533.54 0.119871 0.0599353 0.998202i $$-0.480911\pi$$
0.0599353 + 0.998202i $$0.480911\pi$$
$$548$$ 7910.31 0.616628
$$549$$ −3996.19 −0.310662
$$550$$ 21233.3 1.64617
$$551$$ 138.477 0.0107066
$$552$$ 8299.08 0.639914
$$553$$ 3631.97 0.279289
$$554$$ −38214.6 −2.93066
$$555$$ 4846.30 0.370656
$$556$$ 15362.2 1.17177
$$557$$ −16845.7 −1.28146 −0.640731 0.767766i $$-0.721369\pi$$
−0.640731 + 0.767766i $$0.721369\pi$$
$$558$$ 9207.65 0.698550
$$559$$ 0 0
$$560$$ −30089.0 −2.27052
$$561$$ 8932.62 0.672256
$$562$$ 4526.34 0.339737
$$563$$ −20820.1 −1.55855 −0.779273 0.626685i $$-0.784411\pi$$
−0.779273 + 0.626685i $$0.784411\pi$$
$$564$$ −31322.8 −2.33852
$$565$$ −2964.03 −0.220704
$$566$$ −5945.37 −0.441524
$$567$$ −784.054 −0.0580726
$$568$$ 32076.6 2.36955
$$569$$ 23636.6 1.74147 0.870735 0.491752i $$-0.163643\pi$$
0.870735 + 0.491752i $$0.163643\pi$$
$$570$$ 589.819 0.0433417
$$571$$ −26955.1 −1.97554 −0.987771 0.155913i $$-0.950168\pi$$
−0.987771 + 0.155913i $$0.950168\pi$$
$$572$$ 0 0
$$573$$ 642.326 0.0468300
$$574$$ 1586.35 0.115353
$$575$$ 6036.78 0.437828
$$576$$ 9400.25 0.679995
$$577$$ −23499.8 −1.69551 −0.847755 0.530388i $$-0.822046\pi$$
−0.847755 + 0.530388i $$0.822046\pi$$
$$578$$ −35902.8 −2.58367
$$579$$ −3621.56 −0.259943
$$580$$ 20645.7 1.47805
$$581$$ −6928.38 −0.494729
$$582$$ −1048.44 −0.0746721
$$583$$ −13596.1 −0.965857
$$584$$ −63373.1 −4.49040
$$585$$ 0 0
$$586$$ −9933.33 −0.700243
$$587$$ −4637.50 −0.326082 −0.163041 0.986619i $$-0.552130\pi$$
−0.163041 + 0.986619i $$0.552130\pi$$
$$588$$ 15267.2 1.07076
$$589$$ −431.316 −0.0301733
$$590$$ 42359.9 2.95582
$$591$$ −2782.89 −0.193694
$$592$$ −18640.4 −1.29411
$$593$$ −12633.5 −0.874869 −0.437434 0.899250i $$-0.644113\pi$$
−0.437434 + 0.899250i $$0.644113\pi$$
$$594$$ −3970.51 −0.274262
$$595$$ −17146.9 −1.18144
$$596$$ 53823.7 3.69917
$$597$$ −1436.85 −0.0985033
$$598$$ 0 0
$$599$$ −18757.1 −1.27946 −0.639730 0.768600i $$-0.720954\pi$$
−0.639730 + 0.768600i $$0.720954\pi$$
$$600$$ 28661.3 1.95016
$$601$$ −3632.98 −0.246576 −0.123288 0.992371i $$-0.539344\pi$$
−0.123288 + 0.992371i $$0.539344\pi$$
$$602$$ 12296.8 0.832522
$$603$$ −1711.02 −0.115553
$$604$$ 68025.0 4.58261
$$605$$ −9353.71 −0.628566
$$606$$ 8504.08 0.570057
$$607$$ −12700.0 −0.849219 −0.424610 0.905377i $$-0.639589\pi$$
−0.424610 + 0.905377i $$0.639589\pi$$
$$608$$ −1079.09 −0.0719786
$$609$$ 1789.42 0.119066
$$610$$ 38846.6 2.57845
$$611$$ 0 0
$$612$$ 19828.3 1.30966
$$613$$ −21640.1 −1.42584 −0.712918 0.701248i $$-0.752627\pi$$
−0.712918 + 0.701248i $$0.752627\pi$$
$$614$$ −34047.9 −2.23788
$$615$$ −1513.87 −0.0992605
$$616$$ −17669.5 −1.15572
$$617$$ 16541.7 1.07933 0.539663 0.841881i $$-0.318552\pi$$
0.539663 + 0.841881i $$0.318552\pi$$
$$618$$ −11769.2 −0.766066
$$619$$ 21138.9 1.37261 0.686303 0.727316i $$-0.259233\pi$$
0.686303 + 0.727316i $$0.259233\pi$$
$$620$$ −64305.2 −4.16542
$$621$$ −1128.84 −0.0729451
$$622$$ 18616.9 1.20011
$$623$$ −10049.0 −0.646235
$$624$$ 0 0
$$625$$ −12825.4 −0.820823
$$626$$ 31513.3 2.01202
$$627$$ 185.991 0.0118465
$$628$$ −33176.6 −2.10811
$$629$$ −10622.7 −0.673377
$$630$$ 7621.73 0.481995
$$631$$ −5489.80 −0.346348 −0.173174 0.984891i $$-0.555402\pi$$
−0.173174 + 0.984891i $$0.555402\pi$$
$$632$$ 24826.8 1.56259
$$633$$ 4352.86 0.273319
$$634$$ −8941.94 −0.560141
$$635$$ 23497.5 1.46846
$$636$$ −30180.3 −1.88164
$$637$$ 0 0
$$638$$ 9061.76 0.562318
$$639$$ −4363.06 −0.270110
$$640$$ −28328.1 −1.74963
$$641$$ 4297.04 0.264778 0.132389 0.991198i $$-0.457735\pi$$
0.132389 + 0.991198i $$0.457735\pi$$
$$642$$ −12525.3 −0.769993
$$643$$ −25696.9 −1.57603 −0.788016 0.615655i $$-0.788892\pi$$
−0.788016 + 0.615655i $$0.788892\pi$$
$$644$$ −8261.14 −0.505488
$$645$$ −11735.0 −0.716378
$$646$$ −1292.83 −0.0787396
$$647$$ 2174.98 0.132160 0.0660798 0.997814i $$-0.478951\pi$$
0.0660798 + 0.997814i $$0.478951\pi$$
$$648$$ −5359.50 −0.324909
$$649$$ 13357.6 0.807907
$$650$$ 0 0
$$651$$ −5573.52 −0.335551
$$652$$ 37465.5 2.25040
$$653$$ −15454.5 −0.926160 −0.463080 0.886316i $$-0.653256\pi$$
−0.463080 + 0.886316i $$0.653256\pi$$
$$654$$ 8512.73 0.508982
$$655$$ 33931.1 2.02412
$$656$$ 5822.82 0.346560
$$657$$ 8620.02 0.511870
$$658$$ 26390.7 1.56355
$$659$$ −3148.77 −0.186129 −0.0930643 0.995660i $$-0.529666\pi$$
−0.0930643 + 0.995660i $$0.529666\pi$$
$$660$$ 27729.6 1.63541
$$661$$ 2099.70 0.123553 0.0617767 0.998090i $$-0.480323\pi$$
0.0617767 + 0.998090i $$0.480323\pi$$
$$662$$ 10715.7 0.629122
$$663$$ 0 0
$$664$$ −47359.8 −2.76795
$$665$$ −357.026 −0.0208193
$$666$$ 4721.73 0.274720
$$667$$ 2576.32 0.149559
$$668$$ −39703.4 −2.29966
$$669$$ −6179.36 −0.357112
$$670$$ 16632.7 0.959071
$$671$$ 12249.7 0.704763
$$672$$ −13944.2 −0.800460
$$673$$ 30970.8 1.77390 0.886950 0.461865i $$-0.152819\pi$$
0.886950 + 0.461865i $$0.152819\pi$$
$$674$$ −38055.0 −2.17481
$$675$$ −3898.52 −0.222302
$$676$$ 0 0
$$677$$ 14640.6 0.831141 0.415570 0.909561i $$-0.363582\pi$$
0.415570 + 0.909561i $$0.363582\pi$$
$$678$$ −2887.83 −0.163579
$$679$$ 634.635 0.0358690
$$680$$ −117210. −6.60999
$$681$$ 13447.4 0.756688
$$682$$ −28224.7 −1.58472
$$683$$ 6685.83 0.374563 0.187281 0.982306i $$-0.440032\pi$$
0.187281 + 0.982306i $$0.440032\pi$$
$$684$$ 412.856 0.0230789
$$685$$ 6360.27 0.354764
$$686$$ −30560.8 −1.70090
$$687$$ −4891.18 −0.271630
$$688$$ 45136.3 2.50117
$$689$$ 0 0
$$690$$ 10973.4 0.605434
$$691$$ 30194.1 1.66228 0.831141 0.556062i $$-0.187688\pi$$
0.831141 + 0.556062i $$0.187688\pi$$
$$692$$ 51678.4 2.83890
$$693$$ 2403.40 0.131743
$$694$$ −6.07611 −0.000332343 0
$$695$$ 12352.0 0.674154
$$696$$ 12231.8 0.666158
$$697$$ 3318.28 0.180328
$$698$$ 65026.0 3.52618
$$699$$ 5711.08 0.309031
$$700$$ −28530.3 −1.54049
$$701$$ −30300.9 −1.63260 −0.816298 0.577631i $$-0.803977\pi$$
−0.816298 + 0.577631i $$0.803977\pi$$
$$702$$ 0 0
$$703$$ −221.181 −0.0118663
$$704$$ −28815.1 −1.54263
$$705$$ −25185.0 −1.34542
$$706$$ −58060.3 −3.09508
$$707$$ −5147.64 −0.273829
$$708$$ 29650.8 1.57393
$$709$$ −26123.2 −1.38375 −0.691875 0.722017i $$-0.743215\pi$$
−0.691875 + 0.722017i $$0.743215\pi$$
$$710$$ 42412.9 2.24187
$$711$$ −3376.94 −0.178123
$$712$$ −68691.1 −3.61560
$$713$$ −8024.48 −0.421486
$$714$$ −16706.2 −0.875647
$$715$$ 0 0
$$716$$ 87021.3 4.54209
$$717$$ 11291.4 0.588122
$$718$$ −18793.8 −0.976850
$$719$$ 19325.7 1.00240 0.501200 0.865331i $$-0.332892\pi$$
0.501200 + 0.865331i $$0.332892\pi$$
$$720$$ 27976.2 1.44807
$$721$$ 7124.09 0.367982
$$722$$ 36534.2 1.88319
$$723$$ −10844.2 −0.557816
$$724$$ 80521.7 4.13338
$$725$$ 8897.47 0.455784
$$726$$ −9113.26 −0.465875
$$727$$ 26065.8 1.32975 0.664875 0.746954i $$-0.268485\pi$$
0.664875 + 0.746954i $$0.268485\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −83794.4 −4.24845
$$731$$ 25722.0 1.30145
$$732$$ 27191.5 1.37299
$$733$$ −1055.45 −0.0531843 −0.0265921 0.999646i $$-0.508466\pi$$
−0.0265921 + 0.999646i $$0.508466\pi$$
$$734$$ −12706.4 −0.638965
$$735$$ 12275.5 0.616041
$$736$$ −20076.2 −1.00546
$$737$$ 5244.89 0.262141
$$738$$ −1474.96 −0.0735690
$$739$$ −9410.40 −0.468426 −0.234213 0.972185i $$-0.575251\pi$$
−0.234213 + 0.972185i $$0.575251\pi$$
$$740$$ −32976.0 −1.63814
$$741$$ 0 0
$$742$$ 25428.1 1.25808
$$743$$ 7523.70 0.371491 0.185746 0.982598i $$-0.440530\pi$$
0.185746 + 0.982598i $$0.440530\pi$$
$$744$$ −38098.5 −1.87736
$$745$$ 43276.8 2.12824
$$746$$ 70799.4 3.47473
$$747$$ 6441.89 0.315524
$$748$$ −60780.8 −2.97108
$$749$$ 7581.76 0.369868
$$750$$ 5089.07 0.247769
$$751$$ −12984.1 −0.630886 −0.315443 0.948945i $$-0.602153\pi$$
−0.315443 + 0.948945i $$0.602153\pi$$
$$752$$ 96869.3 4.69742
$$753$$ 17189.3 0.831890
$$754$$ 0 0
$$755$$ 54695.3 2.63651
$$756$$ 5334.99 0.256656
$$757$$ −27934.6 −1.34122 −0.670609 0.741811i $$-0.733967\pi$$
−0.670609 + 0.741811i $$0.733967\pi$$
$$758$$ −23649.5 −1.13323
$$759$$ 3460.30 0.165482
$$760$$ −2440.49 −0.116482
$$761$$ 15519.3 0.739255 0.369627 0.929180i $$-0.379485\pi$$
0.369627 + 0.929180i $$0.379485\pi$$
$$762$$ 22893.5 1.08838
$$763$$ −5152.88 −0.244491
$$764$$ −4370.62 −0.206968
$$765$$ 15942.9 0.753487
$$766$$ −4319.82 −0.203761
$$767$$ 0 0
$$768$$ −2532.57 −0.118993
$$769$$ −12885.2 −0.604228 −0.302114 0.953272i $$-0.597692\pi$$
−0.302114 + 0.953272i $$0.597692\pi$$
$$770$$ −23363.3 −1.09345
$$771$$ 16577.4 0.774344
$$772$$ 24642.4 1.14883
$$773$$ −5892.04 −0.274155 −0.137078 0.990560i $$-0.543771\pi$$
−0.137078 + 0.990560i $$0.543771\pi$$
$$774$$ −11433.3 −0.530958
$$775$$ −27713.0 −1.28449
$$776$$ 4338.12 0.200682
$$777$$ −2858.13 −0.131962
$$778$$ −18463.4 −0.850828
$$779$$ 69.0916 0.00317775
$$780$$ 0 0
$$781$$ 13374.3 0.612766
$$782$$ −24052.7 −1.09990
$$783$$ −1663.77 −0.0759367
$$784$$ −47215.5 −2.15085
$$785$$ −26675.6 −1.21286
$$786$$ 33058.9 1.50022
$$787$$ 21020.4 0.952091 0.476045 0.879421i $$-0.342070\pi$$
0.476045 + 0.879421i $$0.342070\pi$$
$$788$$ 18935.8 0.856042
$$789$$ −15669.6 −0.707038
$$790$$ 32827.0 1.47839
$$791$$ 1748.05 0.0785757
$$792$$ 16428.7 0.737084
$$793$$ 0 0
$$794$$ 2267.58 0.101352
$$795$$ −24266.4 −1.08256
$$796$$ 9776.87 0.435342
$$797$$ 31355.6 1.39356 0.696782 0.717283i $$-0.254614\pi$$
0.696782 + 0.717283i $$0.254614\pi$$
$$798$$ −347.848 −0.0154307
$$799$$ 55203.3 2.44425
$$800$$ −69334.0 −3.06416
$$801$$ 9343.37 0.412150
$$802$$ −6326.37 −0.278543
$$803$$ −26423.4 −1.16122
$$804$$ 11642.4 0.510692
$$805$$ −6642.34 −0.290822
$$806$$ 0 0
$$807$$ −21611.7 −0.942709
$$808$$ −35187.3 −1.53204
$$809$$ −18132.5 −0.788017 −0.394009 0.919107i $$-0.628912\pi$$
−0.394009 + 0.919107i $$0.628912\pi$$
$$810$$ −7086.55 −0.307402
$$811$$ 24755.3 1.07186 0.535928 0.844263i $$-0.319962\pi$$
0.535928 + 0.844263i $$0.319962\pi$$
$$812$$ −12175.9 −0.526219
$$813$$ −25733.1 −1.11008
$$814$$ −14473.8 −0.623225
$$815$$ 30124.0 1.29472
$$816$$ −61321.4 −2.63073
$$817$$ 535.572 0.0229343
$$818$$ 42682.7 1.82441
$$819$$ 0 0
$$820$$ 10300.9 0.438688
$$821$$ −4082.65 −0.173551 −0.0867755 0.996228i $$-0.527656\pi$$
−0.0867755 + 0.996228i $$0.527656\pi$$
$$822$$ 6196.77 0.262941
$$823$$ −34327.0 −1.45391 −0.726954 0.686687i $$-0.759065\pi$$
−0.726954 + 0.686687i $$0.759065\pi$$
$$824$$ 48697.6 2.05881
$$825$$ 11950.3 0.504312
$$826$$ −24981.9 −1.05234
$$827$$ −3228.87 −0.135767 −0.0678833 0.997693i $$-0.521625\pi$$
−0.0678833 + 0.997693i $$0.521625\pi$$
$$828$$ 7681.06 0.322386
$$829$$ −10452.4 −0.437908 −0.218954 0.975735i $$-0.570265\pi$$
−0.218954 + 0.975735i $$0.570265\pi$$
$$830$$ −62621.0 −2.61880
$$831$$ −21507.6 −0.897821
$$832$$ 0 0
$$833$$ −26906.9 −1.11917
$$834$$ 12034.5 0.499663
$$835$$ −31923.4 −1.32306
$$836$$ −1265.55 −0.0523564
$$837$$ 5182.16 0.214004
$$838$$ −36419.7 −1.50131
$$839$$ 28289.0 1.16406 0.582028 0.813169i $$-0.302259\pi$$
0.582028 + 0.813169i $$0.302259\pi$$
$$840$$ −31536.4 −1.29537
$$841$$ −20591.8 −0.844308
$$842$$ 57246.1 2.34303
$$843$$ 2547.47 0.104080
$$844$$ −29618.5 −1.20795
$$845$$ 0 0
$$846$$ −24537.6 −0.997187
$$847$$ 5516.39 0.223784
$$848$$ 93335.9 3.77968
$$849$$ −3346.12 −0.135263
$$850$$ −83067.2 −3.35198
$$851$$ −4114.99 −0.165758
$$852$$ 29687.8 1.19377
$$853$$ 26631.8 1.06900 0.534498 0.845170i $$-0.320501\pi$$
0.534498 + 0.845170i $$0.320501\pi$$
$$854$$ −22910.0 −0.917989
$$855$$ 331.956 0.0132780
$$856$$ 51826.0 2.06936
$$857$$ 11796.7 0.470209 0.235104 0.971970i $$-0.424457\pi$$
0.235104 + 0.971970i $$0.424457\pi$$
$$858$$ 0 0
$$859$$ −22672.8 −0.900567 −0.450283 0.892886i $$-0.648677\pi$$
−0.450283 + 0.892886i $$0.648677\pi$$
$$860$$ 79849.0 3.16608
$$861$$ 892.812 0.0353391
$$862$$ 27795.0 1.09826
$$863$$ −21421.1 −0.844940 −0.422470 0.906377i $$-0.638837\pi$$
−0.422470 + 0.906377i $$0.638837\pi$$
$$864$$ 12965.1 0.510510
$$865$$ 41551.9 1.63330
$$866$$ −46066.6 −1.80763
$$867$$ −20206.5 −0.791520
$$868$$ 37924.3 1.48299
$$869$$ 10351.5 0.404086
$$870$$ 16173.4 0.630264
$$871$$ 0 0
$$872$$ −35223.1 −1.36790
$$873$$ −590.072 −0.0228762
$$874$$ −500.814 −0.0193825
$$875$$ −3080.48 −0.119016
$$876$$ −58653.7 −2.26224
$$877$$ 5155.20 0.198493 0.0992466 0.995063i $$-0.468357\pi$$
0.0992466 + 0.995063i $$0.468357\pi$$
$$878$$ 69435.0 2.66893
$$879$$ −5590.58 −0.214523
$$880$$ −85756.9 −3.28507
$$881$$ 23692.2 0.906027 0.453013 0.891504i $$-0.350349\pi$$
0.453013 + 0.891504i $$0.350349\pi$$
$$882$$ 11960.0 0.456591
$$883$$ −14591.5 −0.556108 −0.278054 0.960565i $$-0.589689\pi$$
−0.278054 + 0.960565i $$0.589689\pi$$
$$884$$ 0 0
$$885$$ 23840.6 0.905529
$$886$$ 61475.2 2.33104
$$887$$ −9722.26 −0.368029 −0.184014 0.982924i $$-0.558909\pi$$
−0.184014 + 0.982924i $$0.558909\pi$$
$$888$$ −19537.1 −0.738312
$$889$$ −13857.8 −0.522806
$$890$$ −90826.1 −3.42078
$$891$$ −2234.64 −0.0840217
$$892$$ 42046.6 1.57828
$$893$$ 1149.42 0.0430726
$$894$$ 42164.3 1.57739
$$895$$ 69969.2 2.61320
$$896$$ 16706.6 0.622911
$$897$$ 0 0
$$898$$ 52678.9 1.95759
$$899$$ −11827.1 −0.438771
$$900$$ 26526.9 0.982479
$$901$$ 53189.7 1.96671
$$902$$ 4521.26 0.166898
$$903$$ 6920.74 0.255048
$$904$$ 11949.0 0.439621
$$905$$ 64743.2 2.37805
$$906$$ 53289.3 1.95411
$$907$$ 11799.0 0.431951 0.215975 0.976399i $$-0.430707\pi$$
0.215975 + 0.976399i $$0.430707\pi$$
$$908$$ −91500.9 −3.34423
$$909$$ 4786.18 0.174640
$$910$$ 0 0
$$911$$ −43012.4 −1.56429 −0.782143 0.623099i $$-0.785873\pi$$
−0.782143 + 0.623099i $$0.785873\pi$$
$$912$$ −1276.81 −0.0463589
$$913$$ −19746.6 −0.715793
$$914$$ −83303.8 −3.01471
$$915$$ 21863.3 0.789921
$$916$$ 33281.3 1.20049
$$917$$ −20011.0 −0.720635
$$918$$ 15533.1 0.558462
$$919$$ −4951.41 −0.177728 −0.0888639 0.996044i $$-0.528324\pi$$
−0.0888639 + 0.996044i $$0.528324\pi$$
$$920$$ −45404.5 −1.62711
$$921$$ −19162.5 −0.685587
$$922$$ −41297.0 −1.47510
$$923$$ 0 0
$$924$$ −16353.6 −0.582245
$$925$$ −14211.3 −0.505152
$$926$$ −1777.27 −0.0630721
$$927$$ −6623.85 −0.234688
$$928$$ −29589.8 −1.04669
$$929$$ 8934.86 0.315547 0.157774 0.987475i $$-0.449568\pi$$
0.157774 + 0.987475i $$0.449568\pi$$
$$930$$ −50375.4 −1.77621
$$931$$ −560.243 −0.0197221
$$932$$ −38860.2 −1.36578
$$933$$ 10477.8 0.367660
$$934$$ −43700.3 −1.53096
$$935$$ −48870.6 −1.70935
$$936$$ 0 0
$$937$$ −13182.8 −0.459620 −0.229810 0.973235i $$-0.573811\pi$$
−0.229810 + 0.973235i $$0.573811\pi$$
$$938$$ −9809.20 −0.341452
$$939$$ 17736.0 0.616393
$$940$$ 171368. 5.94618
$$941$$ 21693.7 0.751536 0.375768 0.926714i $$-0.377379\pi$$
0.375768 + 0.926714i $$0.377379\pi$$
$$942$$ −25989.9 −0.898934
$$943$$ 1285.43 0.0443895
$$944$$ −91698.4 −3.16158
$$945$$ 4289.59 0.147662
$$946$$ 35047.1 1.20452
$$947$$ 49790.0 1.70851 0.854254 0.519856i $$-0.174014\pi$$
0.854254 + 0.519856i $$0.174014\pi$$
$$948$$ 22977.9 0.787224
$$949$$ 0 0
$$950$$ −1729.59 −0.0590687
$$951$$ −5032.61 −0.171602
$$952$$ 69125.0 2.35331
$$953$$ 4217.93 0.143371 0.0716853 0.997427i $$-0.477162\pi$$
0.0716853 + 0.997427i $$0.477162\pi$$
$$954$$ −23642.6 −0.802365
$$955$$ −3514.19 −0.119075
$$956$$ −76830.5 −2.59924
$$957$$ 5100.05 0.172269
$$958$$ −34305.8 −1.15696
$$959$$ −3750.99 −0.126304
$$960$$ −51429.0 −1.72903
$$961$$ 7046.87 0.236544
$$962$$ 0 0
$$963$$ −7049.38 −0.235891
$$964$$ 73788.1 2.46530
$$965$$ 19813.7 0.660958
$$966$$ −6471.60 −0.215549
$$967$$ 40927.9 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$968$$ 37707.9 1.25204
$$969$$ −727.619 −0.0241223
$$970$$ 5736.04 0.189869
$$971$$ −17114.8 −0.565645 −0.282822 0.959172i $$-0.591271\pi$$
−0.282822 + 0.959172i $$0.591271\pi$$
$$972$$ −4960.38 −0.163688
$$973$$ −7284.62 −0.240015
$$974$$ 43151.5 1.41957
$$975$$ 0 0
$$976$$ −84093.0 −2.75794
$$977$$ 118.470 0.00387940 0.00193970 0.999998i $$-0.499383\pi$$
0.00193970 + 0.999998i $$0.499383\pi$$
$$978$$ 29349.7 0.959610
$$979$$ −28640.7 −0.934996
$$980$$ −83527.2 −2.72263
$$981$$ 4791.05 0.155929
$$982$$ −27272.8 −0.886261
$$983$$ −26002.8 −0.843705 −0.421852 0.906665i $$-0.638620\pi$$
−0.421852 + 0.906665i $$0.638620\pi$$
$$984$$ 6102.93 0.197718
$$985$$ 15225.3 0.492506
$$986$$ −35450.7 −1.14501
$$987$$ 14853.0 0.479002
$$988$$ 0 0
$$989$$ 9964.14 0.320365
$$990$$ 21722.8 0.697368
$$991$$ −16062.0 −0.514860 −0.257430 0.966297i $$-0.582876\pi$$
−0.257430 + 0.966297i $$0.582876\pi$$
$$992$$ 92163.3 2.94979
$$993$$ 6030.93 0.192735
$$994$$ −25013.2 −0.798159
$$995$$ 7861.07 0.250465
$$996$$ −43832.9 −1.39448
$$997$$ 1361.54 0.0432503 0.0216251 0.999766i $$-0.493116\pi$$
0.0216251 + 0.999766i $$0.493116\pi$$
$$998$$ −96217.5 −3.05182
$$999$$ 2657.44 0.0841617
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.4.a.i.1.1 4
3.2 odd 2 1521.4.a.bb.1.4 4
13.4 even 6 39.4.e.c.16.1 8
13.5 odd 4 507.4.b.h.337.8 8
13.8 odd 4 507.4.b.h.337.1 8
13.10 even 6 39.4.e.c.22.1 yes 8
13.12 even 2 507.4.a.m.1.4 4
39.17 odd 6 117.4.g.e.55.4 8
39.23 odd 6 117.4.g.e.100.4 8
39.38 odd 2 1521.4.a.v.1.1 4
52.23 odd 6 624.4.q.i.529.1 8
52.43 odd 6 624.4.q.i.289.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 13.4 even 6
39.4.e.c.22.1 yes 8 13.10 even 6
117.4.g.e.55.4 8 39.17 odd 6
117.4.g.e.100.4 8 39.23 odd 6
507.4.a.i.1.1 4 1.1 even 1 trivial
507.4.a.m.1.4 4 13.12 even 2
507.4.b.h.337.1 8 13.8 odd 4
507.4.b.h.337.8 8 13.5 odd 4
624.4.q.i.289.1 8 52.43 odd 6
624.4.q.i.529.1 8 52.23 odd 6
1521.4.a.v.1.1 4 39.38 odd 2
1521.4.a.bb.1.4 4 3.2 odd 2