# Properties

 Label 2160.4.a.bs.1.2 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1257.1 Defining polynomial: $$x^{3} - x^{2} - 8x + 9$$ x^3 - x^2 - 8*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.14974$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} +3.85875 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} +3.85875 q^{7} +42.2783 q^{11} +4.96700 q^{13} +25.8876 q^{17} -28.9958 q^{19} -191.469 q^{23} +25.0000 q^{25} -287.595 q^{29} -52.6915 q^{31} +19.2937 q^{35} -225.550 q^{37} -73.9907 q^{41} +275.370 q^{43} -192.271 q^{47} -328.110 q^{49} -275.204 q^{53} +211.392 q^{55} +497.084 q^{59} +44.0473 q^{61} +24.8350 q^{65} -761.667 q^{67} +264.284 q^{71} +728.781 q^{73} +163.141 q^{77} -664.347 q^{79} +1491.33 q^{83} +129.438 q^{85} -106.783 q^{89} +19.1664 q^{91} -144.979 q^{95} -924.560 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} + 10 q^{7}+O(q^{10})$$ 3 * q + 15 * q^5 + 10 * q^7 $$3 q + 15 q^{5} + 10 q^{7} - 28 q^{11} - 78 q^{13} + 11 q^{17} + 71 q^{19} + 25 q^{23} + 75 q^{25} - 118 q^{29} + 107 q^{31} + 50 q^{35} - 410 q^{37} - 592 q^{41} - 52 q^{43} + 580 q^{47} + 479 q^{49} - 169 q^{53} - 140 q^{55} - 234 q^{59} - 673 q^{61} - 390 q^{65} - 386 q^{67} - 16 q^{71} - 892 q^{73} - 1800 q^{77} - 1263 q^{79} + 1815 q^{83} + 55 q^{85} - 1800 q^{89} - 1284 q^{91} + 355 q^{95} - 840 q^{97}+O(q^{100})$$ 3 * q + 15 * q^5 + 10 * q^7 - 28 * q^11 - 78 * q^13 + 11 * q^17 + 71 * q^19 + 25 * q^23 + 75 * q^25 - 118 * q^29 + 107 * q^31 + 50 * q^35 - 410 * q^37 - 592 * q^41 - 52 * q^43 + 580 * q^47 + 479 * q^49 - 169 * q^53 - 140 * q^55 - 234 * q^59 - 673 * q^61 - 390 * q^65 - 386 * q^67 - 16 * q^71 - 892 * q^73 - 1800 * q^77 - 1263 * q^79 + 1815 * q^83 + 55 * q^85 - 1800 * q^89 - 1284 * q^91 + 355 * q^95 - 840 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 3.85875 0.208353 0.104176 0.994559i $$-0.466779\pi$$
0.104176 + 0.994559i $$0.466779\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 42.2783 1.15885 0.579427 0.815024i $$-0.303276\pi$$
0.579427 + 0.815024i $$0.303276\pi$$
$$12$$ 0 0
$$13$$ 4.96700 0.105969 0.0529845 0.998595i $$-0.483127\pi$$
0.0529845 + 0.998595i $$0.483127\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 25.8876 0.369333 0.184666 0.982801i $$-0.440880\pi$$
0.184666 + 0.982801i $$0.440880\pi$$
$$18$$ 0 0
$$19$$ −28.9958 −0.350110 −0.175055 0.984559i $$-0.556010\pi$$
−0.175055 + 0.984559i $$0.556010\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −191.469 −1.73583 −0.867914 0.496715i $$-0.834539\pi$$
−0.867914 + 0.496715i $$0.834539\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −287.595 −1.84155 −0.920776 0.390092i $$-0.872443\pi$$
−0.920776 + 0.390092i $$0.872443\pi$$
$$30$$ 0 0
$$31$$ −52.6915 −0.305280 −0.152640 0.988282i $$-0.548777\pi$$
−0.152640 + 0.988282i $$0.548777\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 19.2937 0.0931782
$$36$$ 0 0
$$37$$ −225.550 −1.00217 −0.501084 0.865398i $$-0.667065\pi$$
−0.501084 + 0.865398i $$0.667065\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −73.9907 −0.281839 −0.140920 0.990021i $$-0.545006\pi$$
−0.140920 + 0.990021i $$0.545006\pi$$
$$42$$ 0 0
$$43$$ 275.370 0.976593 0.488297 0.872678i $$-0.337618\pi$$
0.488297 + 0.872678i $$0.337618\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −192.271 −0.596715 −0.298357 0.954454i $$-0.596439\pi$$
−0.298357 + 0.954454i $$0.596439\pi$$
$$48$$ 0 0
$$49$$ −328.110 −0.956589
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −275.204 −0.713249 −0.356624 0.934248i $$-0.616072\pi$$
−0.356624 + 0.934248i $$0.616072\pi$$
$$54$$ 0 0
$$55$$ 211.392 0.518255
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 497.084 1.09686 0.548431 0.836196i $$-0.315225\pi$$
0.548431 + 0.836196i $$0.315225\pi$$
$$60$$ 0 0
$$61$$ 44.0473 0.0924538 0.0462269 0.998931i $$-0.485280\pi$$
0.0462269 + 0.998931i $$0.485280\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 24.8350 0.0473908
$$66$$ 0 0
$$67$$ −761.667 −1.38884 −0.694422 0.719568i $$-0.744340\pi$$
−0.694422 + 0.719568i $$0.744340\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 264.284 0.441757 0.220879 0.975301i $$-0.429108\pi$$
0.220879 + 0.975301i $$0.429108\pi$$
$$72$$ 0 0
$$73$$ 728.781 1.16846 0.584229 0.811589i $$-0.301397\pi$$
0.584229 + 0.811589i $$0.301397\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 163.141 0.241450
$$78$$ 0 0
$$79$$ −664.347 −0.946137 −0.473069 0.881026i $$-0.656854\pi$$
−0.473069 + 0.881026i $$0.656854\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1491.33 1.97223 0.986113 0.166075i $$-0.0531094\pi$$
0.986113 + 0.166075i $$0.0531094\pi$$
$$84$$ 0 0
$$85$$ 129.438 0.165171
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −106.783 −0.127180 −0.0635899 0.997976i $$-0.520255\pi$$
−0.0635899 + 0.997976i $$0.520255\pi$$
$$90$$ 0 0
$$91$$ 19.1664 0.0220789
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −144.979 −0.156574
$$96$$ 0 0
$$97$$ −924.560 −0.967782 −0.483891 0.875128i $$-0.660777\pi$$
−0.483891 + 0.875128i $$0.660777\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −627.973 −0.618670 −0.309335 0.950953i $$-0.600106\pi$$
−0.309335 + 0.950953i $$0.600106\pi$$
$$102$$ 0 0
$$103$$ 224.878 0.215125 0.107562 0.994198i $$-0.465695\pi$$
0.107562 + 0.994198i $$0.465695\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −70.1076 −0.0633417 −0.0316708 0.999498i $$-0.510083\pi$$
−0.0316708 + 0.999498i $$0.510083\pi$$
$$108$$ 0 0
$$109$$ −235.022 −0.206523 −0.103261 0.994654i $$-0.532928\pi$$
−0.103261 + 0.994654i $$0.532928\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1647.70 −1.37170 −0.685850 0.727743i $$-0.740569\pi$$
−0.685850 + 0.727743i $$0.740569\pi$$
$$114$$ 0 0
$$115$$ −957.344 −0.776286
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 99.8936 0.0769515
$$120$$ 0 0
$$121$$ 456.458 0.342943
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1282.64 0.896189 0.448095 0.893986i $$-0.352103\pi$$
0.448095 + 0.893986i $$0.352103\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1070.64 0.714066 0.357033 0.934092i $$-0.383788\pi$$
0.357033 + 0.934092i $$0.383788\pi$$
$$132$$ 0 0
$$133$$ −111.888 −0.0729465
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2045.70 −1.27574 −0.637869 0.770145i $$-0.720184\pi$$
−0.637869 + 0.770145i $$0.720184\pi$$
$$138$$ 0 0
$$139$$ −784.256 −0.478559 −0.239280 0.970951i $$-0.576911\pi$$
−0.239280 + 0.970951i $$0.576911\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 209.997 0.122803
$$144$$ 0 0
$$145$$ −1437.97 −0.823567
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1300.68 −0.715140 −0.357570 0.933886i $$-0.616395\pi$$
−0.357570 + 0.933886i $$0.616395\pi$$
$$150$$ 0 0
$$151$$ −1511.72 −0.814718 −0.407359 0.913268i $$-0.633550\pi$$
−0.407359 + 0.913268i $$0.633550\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −263.458 −0.136525
$$156$$ 0 0
$$157$$ 167.967 0.0853838 0.0426919 0.999088i $$-0.486407\pi$$
0.0426919 + 0.999088i $$0.486407\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −738.830 −0.361664
$$162$$ 0 0
$$163$$ 67.1490 0.0322670 0.0161335 0.999870i $$-0.494864\pi$$
0.0161335 + 0.999870i $$0.494864\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2568.36 1.19009 0.595046 0.803692i $$-0.297134\pi$$
0.595046 + 0.803692i $$0.297134\pi$$
$$168$$ 0 0
$$169$$ −2172.33 −0.988771
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 3012.46 1.32389 0.661945 0.749553i $$-0.269731\pi$$
0.661945 + 0.749553i $$0.269731\pi$$
$$174$$ 0 0
$$175$$ 96.4687 0.0416705
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2462.80 1.02837 0.514185 0.857679i $$-0.328095\pi$$
0.514185 + 0.857679i $$0.328095\pi$$
$$180$$ 0 0
$$181$$ −3691.10 −1.51578 −0.757892 0.652380i $$-0.773771\pi$$
−0.757892 + 0.652380i $$0.773771\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1127.75 −0.448183
$$186$$ 0 0
$$187$$ 1094.48 0.428003
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1969.02 0.745932 0.372966 0.927845i $$-0.378341\pi$$
0.372966 + 0.927845i $$0.378341\pi$$
$$192$$ 0 0
$$193$$ 519.119 0.193612 0.0968058 0.995303i $$-0.469137\pi$$
0.0968058 + 0.995303i $$0.469137\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3286.33 −1.18853 −0.594267 0.804268i $$-0.702558\pi$$
−0.594267 + 0.804268i $$0.702558\pi$$
$$198$$ 0 0
$$199$$ −3626.47 −1.29183 −0.645914 0.763410i $$-0.723524\pi$$
−0.645914 + 0.763410i $$0.723524\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1109.76 −0.383692
$$204$$ 0 0
$$205$$ −369.954 −0.126042
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1225.90 −0.405727
$$210$$ 0 0
$$211$$ −1410.39 −0.460166 −0.230083 0.973171i $$-0.573900\pi$$
−0.230083 + 0.973171i $$0.573900\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1376.85 0.436746
$$216$$ 0 0
$$217$$ −203.323 −0.0636059
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 128.584 0.0391379
$$222$$ 0 0
$$223$$ −4026.25 −1.20905 −0.604524 0.796587i $$-0.706637\pi$$
−0.604524 + 0.796587i $$0.706637\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2288.33 0.669083 0.334542 0.942381i $$-0.391419\pi$$
0.334542 + 0.942381i $$0.391419\pi$$
$$228$$ 0 0
$$229$$ 2003.69 0.578200 0.289100 0.957299i $$-0.406644\pi$$
0.289100 + 0.957299i $$0.406644\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2837.72 −0.797877 −0.398939 0.916978i $$-0.630621\pi$$
−0.398939 + 0.916978i $$0.630621\pi$$
$$234$$ 0 0
$$235$$ −961.355 −0.266859
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1749.50 −0.473496 −0.236748 0.971571i $$-0.576082\pi$$
−0.236748 + 0.971571i $$0.576082\pi$$
$$240$$ 0 0
$$241$$ 3645.10 0.974279 0.487140 0.873324i $$-0.338040\pi$$
0.487140 + 0.873324i $$0.338040\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1640.55 −0.427800
$$246$$ 0 0
$$247$$ −144.022 −0.0371009
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1286.39 −0.323491 −0.161745 0.986833i $$-0.551712\pi$$
−0.161745 + 0.986833i $$0.551712\pi$$
$$252$$ 0 0
$$253$$ −8094.99 −2.01157
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2860.83 0.694372 0.347186 0.937796i $$-0.387137\pi$$
0.347186 + 0.937796i $$0.387137\pi$$
$$258$$ 0 0
$$259$$ −870.341 −0.208805
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4615.68 1.08219 0.541093 0.840963i $$-0.318011\pi$$
0.541093 + 0.840963i $$0.318011\pi$$
$$264$$ 0 0
$$265$$ −1376.02 −0.318974
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 3053.54 0.692111 0.346055 0.938214i $$-0.387521\pi$$
0.346055 + 0.938214i $$0.387521\pi$$
$$270$$ 0 0
$$271$$ 5867.51 1.31523 0.657613 0.753356i $$-0.271566\pi$$
0.657613 + 0.753356i $$0.271566\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1056.96 0.231771
$$276$$ 0 0
$$277$$ −7256.41 −1.57399 −0.786995 0.616959i $$-0.788364\pi$$
−0.786995 + 0.616959i $$0.788364\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1318.05 0.279816 0.139908 0.990165i $$-0.455319\pi$$
0.139908 + 0.990165i $$0.455319\pi$$
$$282$$ 0 0
$$283$$ −4810.74 −1.01049 −0.505245 0.862976i $$-0.668598\pi$$
−0.505245 + 0.862976i $$0.668598\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −285.511 −0.0587220
$$288$$ 0 0
$$289$$ −4242.83 −0.863593
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1512.59 −0.301593 −0.150796 0.988565i $$-0.548184\pi$$
−0.150796 + 0.988565i $$0.548184\pi$$
$$294$$ 0 0
$$295$$ 2485.42 0.490531
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −951.026 −0.183944
$$300$$ 0 0
$$301$$ 1062.58 0.203476
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 220.237 0.0413466
$$306$$ 0 0
$$307$$ −3092.67 −0.574945 −0.287472 0.957789i $$-0.592815\pi$$
−0.287472 + 0.957789i $$0.592815\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 21.7251 0.00396115 0.00198058 0.999998i $$-0.499370\pi$$
0.00198058 + 0.999998i $$0.499370\pi$$
$$312$$ 0 0
$$313$$ 758.733 0.137016 0.0685082 0.997651i $$-0.478176\pi$$
0.0685082 + 0.997651i $$0.478176\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1764.70 0.312668 0.156334 0.987704i $$-0.450032\pi$$
0.156334 + 0.987704i $$0.450032\pi$$
$$318$$ 0 0
$$319$$ −12159.0 −2.13409
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −750.632 −0.129307
$$324$$ 0 0
$$325$$ 124.175 0.0211938
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −741.925 −0.124327
$$330$$ 0 0
$$331$$ −10098.1 −1.67687 −0.838433 0.545005i $$-0.816528\pi$$
−0.838433 + 0.545005i $$0.816528\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −3808.34 −0.621109
$$336$$ 0 0
$$337$$ −4485.28 −0.725012 −0.362506 0.931981i $$-0.618079\pi$$
−0.362506 + 0.931981i $$0.618079\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2227.71 −0.353775
$$342$$ 0 0
$$343$$ −2589.64 −0.407661
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −585.137 −0.0905239 −0.0452620 0.998975i $$-0.514412\pi$$
−0.0452620 + 0.998975i $$0.514412\pi$$
$$348$$ 0 0
$$349$$ 3461.97 0.530989 0.265494 0.964112i $$-0.414465\pi$$
0.265494 + 0.964112i $$0.414465\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6797.01 1.02484 0.512420 0.858735i $$-0.328749\pi$$
0.512420 + 0.858735i $$0.328749\pi$$
$$354$$ 0 0
$$355$$ 1321.42 0.197560
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4622.13 −0.679518 −0.339759 0.940513i $$-0.610345\pi$$
−0.339759 + 0.940513i $$0.610345\pi$$
$$360$$ 0 0
$$361$$ −6018.24 −0.877423
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3643.91 0.522550
$$366$$ 0 0
$$367$$ 147.843 0.0210281 0.0105141 0.999945i $$-0.496653\pi$$
0.0105141 + 0.999945i $$0.496653\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1061.94 −0.148607
$$372$$ 0 0
$$373$$ 8840.27 1.22716 0.613582 0.789631i $$-0.289728\pi$$
0.613582 + 0.789631i $$0.289728\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1428.48 −0.195148
$$378$$ 0 0
$$379$$ 6063.51 0.821799 0.410899 0.911681i $$-0.365215\pi$$
0.410899 + 0.911681i $$0.365215\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 496.589 0.0662520 0.0331260 0.999451i $$-0.489454\pi$$
0.0331260 + 0.999451i $$0.489454\pi$$
$$384$$ 0 0
$$385$$ 815.707 0.107980
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 3432.37 0.447373 0.223686 0.974661i $$-0.428191\pi$$
0.223686 + 0.974661i $$0.428191\pi$$
$$390$$ 0 0
$$391$$ −4956.66 −0.641098
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −3321.73 −0.423125
$$396$$ 0 0
$$397$$ −15311.8 −1.93571 −0.967856 0.251506i $$-0.919074\pi$$
−0.967856 + 0.251506i $$0.919074\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7088.63 −0.882766 −0.441383 0.897319i $$-0.645512\pi$$
−0.441383 + 0.897319i $$0.645512\pi$$
$$402$$ 0 0
$$403$$ −261.719 −0.0323502
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9535.89 −1.16137
$$408$$ 0 0
$$409$$ 9991.23 1.20791 0.603954 0.797019i $$-0.293591\pi$$
0.603954 + 0.797019i $$0.293591\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1918.12 0.228534
$$414$$ 0 0
$$415$$ 7456.65 0.882006
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −8172.83 −0.952909 −0.476455 0.879199i $$-0.658078\pi$$
−0.476455 + 0.879199i $$0.658078\pi$$
$$420$$ 0 0
$$421$$ −15475.1 −1.79147 −0.895734 0.444590i $$-0.853350\pi$$
−0.895734 + 0.444590i $$0.853350\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 647.189 0.0738666
$$426$$ 0 0
$$427$$ 169.968 0.0192630
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −3426.00 −0.382888 −0.191444 0.981504i $$-0.561317\pi$$
−0.191444 + 0.981504i $$0.561317\pi$$
$$432$$ 0 0
$$433$$ −1806.95 −0.200546 −0.100273 0.994960i $$-0.531972\pi$$
−0.100273 + 0.994960i $$0.531972\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5551.80 0.607731
$$438$$ 0 0
$$439$$ −7974.34 −0.866957 −0.433479 0.901164i $$-0.642714\pi$$
−0.433479 + 0.901164i $$0.642714\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −3566.86 −0.382544 −0.191272 0.981537i $$-0.561261\pi$$
−0.191272 + 0.981537i $$0.561261\pi$$
$$444$$ 0 0
$$445$$ −533.916 −0.0568765
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −13687.2 −1.43862 −0.719308 0.694692i $$-0.755541\pi$$
−0.719308 + 0.694692i $$0.755541\pi$$
$$450$$ 0 0
$$451$$ −3128.21 −0.326611
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 95.8320 0.00987401
$$456$$ 0 0
$$457$$ 1691.89 0.173180 0.0865902 0.996244i $$-0.472403\pi$$
0.0865902 + 0.996244i $$0.472403\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 17881.6 1.80657 0.903285 0.429041i $$-0.141149\pi$$
0.903285 + 0.429041i $$0.141149\pi$$
$$462$$ 0 0
$$463$$ −11459.6 −1.15027 −0.575135 0.818059i $$-0.695050\pi$$
−0.575135 + 0.818059i $$0.695050\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18329.4 −1.81624 −0.908119 0.418711i $$-0.862482\pi$$
−0.908119 + 0.418711i $$0.862482\pi$$
$$468$$ 0 0
$$469$$ −2939.08 −0.289369
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 11642.2 1.13173
$$474$$ 0 0
$$475$$ −724.896 −0.0700221
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 15475.4 1.47618 0.738090 0.674703i $$-0.235728\pi$$
0.738090 + 0.674703i $$0.235728\pi$$
$$480$$ 0 0
$$481$$ −1120.31 −0.106199
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4622.80 −0.432805
$$486$$ 0 0
$$487$$ 16320.0 1.51854 0.759270 0.650776i $$-0.225556\pi$$
0.759270 + 0.650776i $$0.225556\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6138.33 −0.564193 −0.282097 0.959386i $$-0.591030\pi$$
−0.282097 + 0.959386i $$0.591030\pi$$
$$492$$ 0 0
$$493$$ −7445.13 −0.680146
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1019.81 0.0920413
$$498$$ 0 0
$$499$$ −3272.20 −0.293555 −0.146777 0.989170i $$-0.546890\pi$$
−0.146777 + 0.989170i $$0.546890\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −4696.96 −0.416356 −0.208178 0.978091i $$-0.566753\pi$$
−0.208178 + 0.978091i $$0.566753\pi$$
$$504$$ 0 0
$$505$$ −3139.87 −0.276678
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −10011.5 −0.871815 −0.435907 0.899992i $$-0.643572\pi$$
−0.435907 + 0.899992i $$0.643572\pi$$
$$510$$ 0 0
$$511$$ 2812.18 0.243451
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1124.39 0.0962067
$$516$$ 0 0
$$517$$ −8128.89 −0.691505
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 19483.2 1.63833 0.819167 0.573554i $$-0.194436\pi$$
0.819167 + 0.573554i $$0.194436\pi$$
$$522$$ 0 0
$$523$$ 5181.82 0.433241 0.216621 0.976256i $$-0.430497\pi$$
0.216621 + 0.976256i $$0.430497\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1364.06 −0.112750
$$528$$ 0 0
$$529$$ 24493.3 2.01310
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −367.512 −0.0298663
$$534$$ 0 0
$$535$$ −350.538 −0.0283273
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −13871.9 −1.10855
$$540$$ 0 0
$$541$$ 3178.24 0.252576 0.126288 0.991994i $$-0.459694\pi$$
0.126288 + 0.991994i $$0.459694\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1175.11 −0.0923599
$$546$$ 0 0
$$547$$ 13887.9 1.08556 0.542781 0.839874i $$-0.317371\pi$$
0.542781 + 0.839874i $$0.317371\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8339.05 0.644747
$$552$$ 0 0
$$553$$ −2563.55 −0.197130
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10574.5 0.804406 0.402203 0.915551i $$-0.368245\pi$$
0.402203 + 0.915551i $$0.368245\pi$$
$$558$$ 0 0
$$559$$ 1367.76 0.103489
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 7815.91 0.585083 0.292541 0.956253i $$-0.405499\pi$$
0.292541 + 0.956253i $$0.405499\pi$$
$$564$$ 0 0
$$565$$ −8238.48 −0.613443
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 20024.0 1.47531 0.737655 0.675178i $$-0.235933\pi$$
0.737655 + 0.675178i $$0.235933\pi$$
$$570$$ 0 0
$$571$$ −23886.5 −1.75065 −0.875323 0.483538i $$-0.839351\pi$$
−0.875323 + 0.483538i $$0.839351\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −4786.72 −0.347165
$$576$$ 0 0
$$577$$ −18777.2 −1.35478 −0.677389 0.735625i $$-0.736888\pi$$
−0.677389 + 0.735625i $$0.736888\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5754.67 0.410919
$$582$$ 0 0
$$583$$ −11635.2 −0.826551
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 23209.8 1.63198 0.815988 0.578069i $$-0.196194\pi$$
0.815988 + 0.578069i $$0.196194\pi$$
$$588$$ 0 0
$$589$$ 1527.83 0.106882
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4204.44 0.291157 0.145578 0.989347i $$-0.453496\pi$$
0.145578 + 0.989347i $$0.453496\pi$$
$$594$$ 0 0
$$595$$ 499.468 0.0344138
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −26962.9 −1.83919 −0.919593 0.392872i $$-0.871482\pi$$
−0.919593 + 0.392872i $$0.871482\pi$$
$$600$$ 0 0
$$601$$ 13595.2 0.922727 0.461363 0.887211i $$-0.347360\pi$$
0.461363 + 0.887211i $$0.347360\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 2282.29 0.153369
$$606$$ 0 0
$$607$$ −14337.5 −0.958720 −0.479360 0.877618i $$-0.659131\pi$$
−0.479360 + 0.877618i $$0.659131\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −955.010 −0.0632333
$$612$$ 0 0
$$613$$ 7963.62 0.524710 0.262355 0.964971i $$-0.415501\pi$$
0.262355 + 0.964971i $$0.415501\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 23594.2 1.53949 0.769745 0.638352i $$-0.220383\pi$$
0.769745 + 0.638352i $$0.220383\pi$$
$$618$$ 0 0
$$619$$ −12132.6 −0.787803 −0.393901 0.919153i $$-0.628875\pi$$
−0.393901 + 0.919153i $$0.628875\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −412.049 −0.0264982
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −5838.95 −0.370134
$$630$$ 0 0
$$631$$ 614.270 0.0387539 0.0193769 0.999812i $$-0.493832\pi$$
0.0193769 + 0.999812i $$0.493832\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 6413.21 0.400788
$$636$$ 0 0
$$637$$ −1629.72 −0.101369
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2057.06 −0.126754 −0.0633769 0.997990i $$-0.520187\pi$$
−0.0633769 + 0.997990i $$0.520187\pi$$
$$642$$ 0 0
$$643$$ −17827.2 −1.09337 −0.546684 0.837339i $$-0.684110\pi$$
−0.546684 + 0.837339i $$0.684110\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17499.4 1.06333 0.531664 0.846955i $$-0.321567\pi$$
0.531664 + 0.846955i $$0.321567\pi$$
$$648$$ 0 0
$$649$$ 21015.9 1.27110
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6049.52 0.362536 0.181268 0.983434i $$-0.441980\pi$$
0.181268 + 0.983434i $$0.441980\pi$$
$$654$$ 0 0
$$655$$ 5353.22 0.319340
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1436.32 −0.0849027 −0.0424514 0.999099i $$-0.513517\pi$$
−0.0424514 + 0.999099i $$0.513517\pi$$
$$660$$ 0 0
$$661$$ 9287.18 0.546489 0.273245 0.961945i $$-0.411903\pi$$
0.273245 + 0.961945i $$0.411903\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −559.438 −0.0326226
$$666$$ 0 0
$$667$$ 55065.4 3.19662
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1862.25 0.107141
$$672$$ 0 0
$$673$$ 15014.9 0.860000 0.430000 0.902829i $$-0.358514\pi$$
0.430000 + 0.902829i $$0.358514\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −9147.19 −0.519284 −0.259642 0.965705i $$-0.583604\pi$$
−0.259642 + 0.965705i $$0.583604\pi$$
$$678$$ 0 0
$$679$$ −3567.64 −0.201640
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 10388.4 0.581992 0.290996 0.956724i $$-0.406013\pi$$
0.290996 + 0.956724i $$0.406013\pi$$
$$684$$ 0 0
$$685$$ −10228.5 −0.570527
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1366.94 −0.0755823
$$690$$ 0 0
$$691$$ 25955.9 1.42895 0.714477 0.699659i $$-0.246665\pi$$
0.714477 + 0.699659i $$0.246665\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3921.28 −0.214018
$$696$$ 0 0
$$697$$ −1915.44 −0.104093
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −31771.2 −1.71181 −0.855906 0.517131i $$-0.827000\pi$$
−0.855906 + 0.517131i $$0.827000\pi$$
$$702$$ 0 0
$$703$$ 6540.02 0.350870
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2423.19 −0.128902
$$708$$ 0 0
$$709$$ −7002.47 −0.370921 −0.185461 0.982652i $$-0.559378\pi$$
−0.185461 + 0.982652i $$0.559378\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 10088.8 0.529913
$$714$$ 0 0
$$715$$ 1049.98 0.0549191
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −34634.3 −1.79644 −0.898221 0.439545i $$-0.855140\pi$$
−0.898221 + 0.439545i $$0.855140\pi$$
$$720$$ 0 0
$$721$$ 867.745 0.0448218
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7189.87 −0.368310
$$726$$ 0 0
$$727$$ −10821.6 −0.552064 −0.276032 0.961148i $$-0.589020\pi$$
−0.276032 + 0.961148i $$0.589020\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7128.66 0.360688
$$732$$ 0 0
$$733$$ −5576.04 −0.280977 −0.140488 0.990082i $$-0.544867\pi$$
−0.140488 + 0.990082i $$0.544867\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −32202.0 −1.60947
$$738$$ 0 0
$$739$$ −29317.7 −1.45936 −0.729681 0.683788i $$-0.760331\pi$$
−0.729681 + 0.683788i $$0.760331\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −7521.08 −0.371361 −0.185681 0.982610i $$-0.559449\pi$$
−0.185681 + 0.982610i $$0.559449\pi$$
$$744$$ 0 0
$$745$$ −6503.40 −0.319820
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −270.528 −0.0131974
$$750$$ 0 0
$$751$$ 37300.7 1.81241 0.906206 0.422837i $$-0.138966\pi$$
0.906206 + 0.422837i $$0.138966\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −7558.62 −0.364353
$$756$$ 0 0
$$757$$ −26432.1 −1.26908 −0.634538 0.772891i $$-0.718810\pi$$
−0.634538 + 0.772891i $$0.718810\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −23440.4 −1.11658 −0.558288 0.829647i $$-0.688542\pi$$
−0.558288 + 0.829647i $$0.688542\pi$$
$$762$$ 0 0
$$763$$ −906.889 −0.0430296
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2469.02 0.116233
$$768$$ 0 0
$$769$$ 34075.2 1.59790 0.798948 0.601401i $$-0.205390\pi$$
0.798948 + 0.601401i $$0.205390\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14623.5 0.680427 0.340214 0.940348i $$-0.389501\pi$$
0.340214 + 0.940348i $$0.389501\pi$$
$$774$$ 0 0
$$775$$ −1317.29 −0.0610560
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2145.42 0.0986749
$$780$$ 0 0
$$781$$ 11173.5 0.511932
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 839.837 0.0381848
$$786$$ 0 0
$$787$$ 33523.1 1.51839 0.759193 0.650865i $$-0.225594\pi$$
0.759193 + 0.650865i $$0.225594\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6358.04 −0.285797
$$792$$ 0 0
$$793$$ 218.783 0.00979725
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −29447.5 −1.30876 −0.654381 0.756165i $$-0.727071\pi$$
−0.654381 + 0.756165i $$0.727071\pi$$
$$798$$ 0 0
$$799$$ −4977.43 −0.220386
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 30811.6 1.35407
$$804$$ 0 0
$$805$$ −3694.15 −0.161741
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −26891.1 −1.16866 −0.584328 0.811518i $$-0.698642\pi$$
−0.584328 + 0.811518i $$0.698642\pi$$
$$810$$ 0 0
$$811$$ −14912.6 −0.645687 −0.322844 0.946452i $$-0.604639\pi$$
−0.322844 + 0.946452i $$0.604639\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 335.745 0.0144302
$$816$$ 0 0
$$817$$ −7984.58 −0.341916
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 42644.2 1.81278 0.906391 0.422439i $$-0.138826\pi$$
0.906391 + 0.422439i $$0.138826\pi$$
$$822$$ 0 0
$$823$$ 25121.7 1.06402 0.532010 0.846738i $$-0.321437\pi$$
0.532010 + 0.846738i $$0.321437\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −4918.93 −0.206830 −0.103415 0.994638i $$-0.532977\pi$$
−0.103415 + 0.994638i $$0.532977\pi$$
$$828$$ 0 0
$$829$$ −22807.6 −0.955538 −0.477769 0.878486i $$-0.658554\pi$$
−0.477769 + 0.878486i $$0.658554\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −8493.97 −0.353300
$$834$$ 0 0
$$835$$ 12841.8 0.532225
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 25772.5 1.06051 0.530255 0.847839i $$-0.322096\pi$$
0.530255 + 0.847839i $$0.322096\pi$$
$$840$$ 0 0
$$841$$ 58321.7 2.39131
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −10861.6 −0.442192
$$846$$ 0 0
$$847$$ 1761.35 0.0714532
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 43185.9 1.73959
$$852$$ 0 0
$$853$$ 41636.0 1.67126 0.835632 0.549289i $$-0.185101\pi$$
0.835632 + 0.549289i $$0.185101\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 17941.6 0.715137 0.357569 0.933887i $$-0.383606\pi$$
0.357569 + 0.933887i $$0.383606\pi$$
$$858$$ 0 0
$$859$$ −23917.0 −0.949985 −0.474993 0.879990i $$-0.657549\pi$$
−0.474993 + 0.879990i $$0.657549\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 9786.53 0.386022 0.193011 0.981197i $$-0.438175\pi$$
0.193011 + 0.981197i $$0.438175\pi$$
$$864$$ 0 0
$$865$$ 15062.3 0.592061
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −28087.5 −1.09644
$$870$$ 0 0
$$871$$ −3783.20 −0.147174
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 482.343 0.0186356
$$876$$ 0 0
$$877$$ −29186.8 −1.12379 −0.561897 0.827207i $$-0.689928\pi$$
−0.561897 + 0.827207i $$0.689928\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 18426.6 0.704664 0.352332 0.935875i $$-0.385389\pi$$
0.352332 + 0.935875i $$0.385389\pi$$
$$882$$ 0 0
$$883$$ −3942.63 −0.150261 −0.0751303 0.997174i $$-0.523937\pi$$
−0.0751303 + 0.997174i $$0.523937\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −436.876 −0.0165376 −0.00826881 0.999966i $$-0.502632\pi$$
−0.00826881 + 0.999966i $$0.502632\pi$$
$$888$$ 0 0
$$889$$ 4949.39 0.186723
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 5575.05 0.208916
$$894$$ 0 0
$$895$$ 12314.0 0.459901
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 15153.8 0.562189
$$900$$ 0 0
$$901$$ −7124.36 −0.263426
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −18455.5 −0.677879
$$906$$ 0 0
$$907$$ 43243.2 1.58309 0.791547 0.611108i $$-0.209276\pi$$
0.791547 + 0.611108i $$0.209276\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −14141.5 −0.514301 −0.257150 0.966371i $$-0.582784\pi$$
−0.257150 + 0.966371i $$0.582784\pi$$
$$912$$ 0 0
$$913$$ 63051.0 2.28552
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4131.34 0.148778
$$918$$ 0 0
$$919$$ −1506.36 −0.0540697 −0.0270349 0.999634i $$-0.508607\pi$$
−0.0270349 + 0.999634i $$0.508607\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1312.70 0.0468126
$$924$$ 0 0
$$925$$ −5638.76 −0.200434
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −32173.3 −1.13624 −0.568122 0.822944i $$-0.692330\pi$$
−0.568122 + 0.822944i $$0.692330\pi$$
$$930$$ 0 0
$$931$$ 9513.82 0.334912
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 5472.42 0.191409
$$936$$ 0 0
$$937$$ 34833.5 1.21447 0.607236 0.794522i $$-0.292278\pi$$
0.607236 + 0.794522i $$0.292278\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 3709.66 0.128514 0.0642569 0.997933i $$-0.479532\pi$$
0.0642569 + 0.997933i $$0.479532\pi$$
$$942$$ 0 0
$$943$$ 14166.9 0.489224
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 32085.0 1.10097 0.550487 0.834844i $$-0.314442\pi$$
0.550487 + 0.834844i $$0.314442\pi$$
$$948$$ 0 0
$$949$$ 3619.86 0.123820
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −35654.2 −1.21191 −0.605956 0.795498i $$-0.707209\pi$$
−0.605956 + 0.795498i $$0.707209\pi$$
$$954$$ 0 0
$$955$$ 9845.08 0.333591
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −7893.84 −0.265803
$$960$$ 0 0
$$961$$ −27014.6 −0.906804
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 2595.60 0.0865857
$$966$$ 0 0
$$967$$ 32345.4 1.07565 0.537827 0.843055i $$-0.319245\pi$$
0.537827 + 0.843055i $$0.319245\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20137.4 −0.665542 −0.332771 0.943008i $$-0.607984\pi$$
−0.332771 + 0.943008i $$0.607984\pi$$
$$972$$ 0 0
$$973$$ −3026.25 −0.0997091
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 30087.1 0.985231 0.492615 0.870247i $$-0.336041\pi$$
0.492615 + 0.870247i $$0.336041\pi$$
$$978$$ 0 0
$$979$$ −4514.62 −0.147383
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 59417.5 1.92790 0.963950 0.266084i $$-0.0857299\pi$$
0.963950 + 0.266084i $$0.0857299\pi$$
$$984$$ 0 0
$$985$$ −16431.6 −0.531529
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −52724.8 −1.69520
$$990$$ 0 0
$$991$$ 20153.5 0.646011 0.323006 0.946397i $$-0.395307\pi$$
0.323006 + 0.946397i $$0.395307\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −18132.4 −0.577723
$$996$$ 0 0
$$997$$ −20772.3 −0.659846 −0.329923 0.944008i $$-0.607023\pi$$
−0.329923 + 0.944008i $$0.607023\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 2160.4.a.bs.1.2 3
3.2 odd 2 2160.4.a.bk.1.2 3
4.3 odd 2 1080.4.a.j.1.2 yes 3
12.11 even 2 1080.4.a.d.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.d.1.2 3 12.11 even 2
1080.4.a.j.1.2 yes 3 4.3 odd 2
2160.4.a.bk.1.2 3 3.2 odd 2
2160.4.a.bs.1.2 3 1.1 even 1 trivial