# Properties

 Label 2160.4.a.be.1.2 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.5637.1 Defining polynomial: $$x^{3} - x^{2} - 23x + 6$$ x^3 - x^2 - 23*x + 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 135) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.258712$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.00000 q^{5} -14.5174 q^{7} +O(q^{10})$$ $$q-5.00000 q^{5} -14.5174 q^{7} -49.2845 q^{11} +72.1800 q^{13} -118.017 q^{17} -123.389 q^{19} -91.4883 q^{23} +25.0000 q^{25} -174.400 q^{29} +46.2956 q^{31} +72.5871 q^{35} +154.977 q^{37} +364.203 q^{41} -125.714 q^{43} -221.523 q^{47} -132.244 q^{49} +13.6794 q^{53} +246.423 q^{55} +239.087 q^{59} -54.5457 q^{61} -360.900 q^{65} +76.0558 q^{67} +728.303 q^{71} -501.815 q^{73} +715.485 q^{77} -397.610 q^{79} -1369.46 q^{83} +590.084 q^{85} -1468.13 q^{89} -1047.87 q^{91} +616.945 q^{95} +335.023 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} - 44 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 - 44 * q^7 $$3 q - 15 q^{5} - 44 q^{7} + 38 q^{11} + 28 q^{13} + 19 q^{17} - 187 q^{19} - 81 q^{23} + 75 q^{25} - 160 q^{29} - 227 q^{31} + 220 q^{35} + 78 q^{37} + 338 q^{41} - 22 q^{43} - 472 q^{47} - 197 q^{49} - 521 q^{53} - 190 q^{55} + 140 q^{59} + 595 q^{61} - 140 q^{65} - 878 q^{67} - 602 q^{71} + 1294 q^{73} - 288 q^{77} - 629 q^{79} - 1287 q^{83} - 95 q^{85} - 2154 q^{89} + 440 q^{91} + 935 q^{95} + 1392 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 - 44 * q^7 + 38 * q^11 + 28 * q^13 + 19 * q^17 - 187 * q^19 - 81 * q^23 + 75 * q^25 - 160 * q^29 - 227 * q^31 + 220 * q^35 + 78 * q^37 + 338 * q^41 - 22 * q^43 - 472 * q^47 - 197 * q^49 - 521 * q^53 - 190 * q^55 + 140 * q^59 + 595 * q^61 - 140 * q^65 - 878 * q^67 - 602 * q^71 + 1294 * q^73 - 288 * q^77 - 629 * q^79 - 1287 * q^83 - 95 * q^85 - 2154 * q^89 + 440 * q^91 + 935 * q^95 + 1392 * 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
$$7$$ −14.5174 −0.783867 −0.391934 0.919993i $$-0.628194\pi$$
−0.391934 + 0.919993i $$0.628194\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −49.2845 −1.35090 −0.675448 0.737408i $$-0.736050\pi$$
−0.675448 + 0.737408i $$0.736050\pi$$
$$12$$ 0 0
$$13$$ 72.1800 1.53993 0.769967 0.638084i $$-0.220273\pi$$
0.769967 + 0.638084i $$0.220273\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −118.017 −1.68372 −0.841861 0.539694i $$-0.818540\pi$$
−0.841861 + 0.539694i $$0.818540\pi$$
$$18$$ 0 0
$$19$$ −123.389 −1.48986 −0.744932 0.667141i $$-0.767518\pi$$
−0.744932 + 0.667141i $$0.767518\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −91.4883 −0.829419 −0.414709 0.909954i $$-0.636117\pi$$
−0.414709 + 0.909954i $$0.636117\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −174.400 −1.11673 −0.558367 0.829594i $$-0.688572\pi$$
−0.558367 + 0.829594i $$0.688572\pi$$
$$30$$ 0 0
$$31$$ 46.2956 0.268224 0.134112 0.990966i $$-0.457182\pi$$
0.134112 + 0.990966i $$0.457182\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 72.5871 0.350556
$$36$$ 0 0
$$37$$ 154.977 0.688595 0.344297 0.938861i $$-0.388117\pi$$
0.344297 + 0.938861i $$0.388117\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 364.203 1.38729 0.693645 0.720317i $$-0.256004\pi$$
0.693645 + 0.720317i $$0.256004\pi$$
$$42$$ 0 0
$$43$$ −125.714 −0.445841 −0.222921 0.974837i $$-0.571559\pi$$
−0.222921 + 0.974837i $$0.571559\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −221.523 −0.687499 −0.343750 0.939061i $$-0.611697\pi$$
−0.343750 + 0.939061i $$0.611697\pi$$
$$48$$ 0 0
$$49$$ −132.244 −0.385552
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 13.6794 0.0354530 0.0177265 0.999843i $$-0.494357\pi$$
0.0177265 + 0.999843i $$0.494357\pi$$
$$54$$ 0 0
$$55$$ 246.423 0.604139
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 239.087 0.527567 0.263784 0.964582i $$-0.415030\pi$$
0.263784 + 0.964582i $$0.415030\pi$$
$$60$$ 0 0
$$61$$ −54.5457 −0.114490 −0.0572448 0.998360i $$-0.518232\pi$$
−0.0572448 + 0.998360i $$0.518232\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −360.900 −0.688679
$$66$$ 0 0
$$67$$ 76.0558 0.138682 0.0693410 0.997593i $$-0.477910\pi$$
0.0693410 + 0.997593i $$0.477910\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 728.303 1.21738 0.608688 0.793410i $$-0.291696\pi$$
0.608688 + 0.793410i $$0.291696\pi$$
$$72$$ 0 0
$$73$$ −501.815 −0.804562 −0.402281 0.915516i $$-0.631782\pi$$
−0.402281 + 0.915516i $$0.631782\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 715.485 1.05892
$$78$$ 0 0
$$79$$ −397.610 −0.566261 −0.283130 0.959081i $$-0.591373\pi$$
−0.283130 + 0.959081i $$0.591373\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1369.46 −1.81106 −0.905530 0.424283i $$-0.860526\pi$$
−0.905530 + 0.424283i $$0.860526\pi$$
$$84$$ 0 0
$$85$$ 590.084 0.752984
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1468.13 −1.74856 −0.874278 0.485425i $$-0.838665\pi$$
−0.874278 + 0.485425i $$0.838665\pi$$
$$90$$ 0 0
$$91$$ −1047.87 −1.20710
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 616.945 0.666287
$$96$$ 0 0
$$97$$ 335.023 0.350685 0.175343 0.984507i $$-0.443897\pi$$
0.175343 + 0.984507i $$0.443897\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1206.09 1.18822 0.594109 0.804384i $$-0.297505\pi$$
0.594109 + 0.804384i $$0.297505\pi$$
$$102$$ 0 0
$$103$$ −1061.11 −1.01509 −0.507545 0.861625i $$-0.669447\pi$$
−0.507545 + 0.861625i $$0.669447\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −475.578 −0.429681 −0.214841 0.976649i $$-0.568923\pi$$
−0.214841 + 0.976649i $$0.568923\pi$$
$$108$$ 0 0
$$109$$ 1320.42 1.16030 0.580152 0.814508i $$-0.302993\pi$$
0.580152 + 0.814508i $$0.302993\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 68.1750 0.0567555 0.0283777 0.999597i $$-0.490966\pi$$
0.0283777 + 0.999597i $$0.490966\pi$$
$$114$$ 0 0
$$115$$ 457.442 0.370927
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1713.30 1.31981
$$120$$ 0 0
$$121$$ 1097.97 0.824918
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 593.009 0.414339 0.207170 0.978305i $$-0.433575\pi$$
0.207170 + 0.978305i $$0.433575\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 338.937 0.226054 0.113027 0.993592i $$-0.463945\pi$$
0.113027 + 0.993592i $$0.463945\pi$$
$$132$$ 0 0
$$133$$ 1791.29 1.16785
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 811.442 0.506030 0.253015 0.967462i $$-0.418578\pi$$
0.253015 + 0.967462i $$0.418578\pi$$
$$138$$ 0 0
$$139$$ 3106.13 1.89538 0.947691 0.319189i $$-0.103410\pi$$
0.947691 + 0.319189i $$0.103410\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3557.36 −2.08029
$$144$$ 0 0
$$145$$ 872.001 0.499419
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2541.01 −1.39710 −0.698550 0.715561i $$-0.746171\pi$$
−0.698550 + 0.715561i $$0.746171\pi$$
$$150$$ 0 0
$$151$$ 1125.37 0.606499 0.303249 0.952911i $$-0.401928\pi$$
0.303249 + 0.952911i $$0.401928\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −231.478 −0.119953
$$156$$ 0 0
$$157$$ 3230.05 1.64195 0.820975 0.570963i $$-0.193430\pi$$
0.820975 + 0.570963i $$0.193430\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1328.17 0.650154
$$162$$ 0 0
$$163$$ 694.054 0.333512 0.166756 0.985998i $$-0.446671\pi$$
0.166756 + 0.985998i $$0.446671\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3216.04 1.49021 0.745103 0.666950i $$-0.232400\pi$$
0.745103 + 0.666950i $$0.232400\pi$$
$$168$$ 0 0
$$169$$ 3012.95 1.37139
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −297.546 −0.130763 −0.0653816 0.997860i $$-0.520826\pi$$
−0.0653816 + 0.997860i $$0.520826\pi$$
$$174$$ 0 0
$$175$$ −362.936 −0.156773
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −3450.12 −1.44064 −0.720320 0.693642i $$-0.756005\pi$$
−0.720320 + 0.693642i $$0.756005\pi$$
$$180$$ 0 0
$$181$$ −3089.75 −1.26883 −0.634417 0.772991i $$-0.718760\pi$$
−0.634417 + 0.772991i $$0.718760\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −774.883 −0.307949
$$186$$ 0 0
$$187$$ 5816.41 2.27453
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1532.11 0.580419 0.290209 0.956963i $$-0.406275\pi$$
0.290209 + 0.956963i $$0.406275\pi$$
$$192$$ 0 0
$$193$$ −5194.42 −1.93732 −0.968660 0.248389i $$-0.920099\pi$$
−0.968660 + 0.248389i $$0.920099\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2005.61 0.725349 0.362674 0.931916i $$-0.381864\pi$$
0.362674 + 0.931916i $$0.381864\pi$$
$$198$$ 0 0
$$199$$ 2874.68 1.02402 0.512011 0.858979i $$-0.328901\pi$$
0.512011 + 0.858979i $$0.328901\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2531.84 0.875372
$$204$$ 0 0
$$205$$ −1821.01 −0.620415
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 6081.18 2.01265
$$210$$ 0 0
$$211$$ 2749.94 0.897220 0.448610 0.893728i $$-0.351919\pi$$
0.448610 + 0.893728i $$0.351919\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 628.569 0.199386
$$216$$ 0 0
$$217$$ −672.093 −0.210252
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8518.45 −2.59282
$$222$$ 0 0
$$223$$ 783.727 0.235346 0.117673 0.993052i $$-0.462456\pi$$
0.117673 + 0.993052i $$0.462456\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −145.665 −0.0425909 −0.0212955 0.999773i $$-0.506779\pi$$
−0.0212955 + 0.999773i $$0.506779\pi$$
$$228$$ 0 0
$$229$$ −3411.82 −0.984539 −0.492270 0.870443i $$-0.663833\pi$$
−0.492270 + 0.870443i $$0.663833\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −134.977 −0.0379511 −0.0189756 0.999820i $$-0.506040\pi$$
−0.0189756 + 0.999820i $$0.506040\pi$$
$$234$$ 0 0
$$235$$ 1107.62 0.307459
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2245.32 −0.607690 −0.303845 0.952722i $$-0.598270\pi$$
−0.303845 + 0.952722i $$0.598270\pi$$
$$240$$ 0 0
$$241$$ 4158.54 1.11151 0.555757 0.831345i $$-0.312428\pi$$
0.555757 + 0.831345i $$0.312428\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 661.222 0.172424
$$246$$ 0 0
$$247$$ −8906.22 −2.29429
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 3946.14 0.992343 0.496171 0.868225i $$-0.334739\pi$$
0.496171 + 0.868225i $$0.334739\pi$$
$$252$$ 0 0
$$253$$ 4508.96 1.12046
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5695.84 1.38248 0.691239 0.722626i $$-0.257065\pi$$
0.691239 + 0.722626i $$0.257065\pi$$
$$258$$ 0 0
$$259$$ −2249.86 −0.539767
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2814.06 0.659781 0.329891 0.944019i $$-0.392988\pi$$
0.329891 + 0.944019i $$0.392988\pi$$
$$264$$ 0 0
$$265$$ −68.3970 −0.0158551
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −200.985 −0.0455548 −0.0227774 0.999741i $$-0.507251\pi$$
−0.0227774 + 0.999741i $$0.507251\pi$$
$$270$$ 0 0
$$271$$ 2406.05 0.539326 0.269663 0.962955i $$-0.413088\pi$$
0.269663 + 0.962955i $$0.413088\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1232.11 −0.270179
$$276$$ 0 0
$$277$$ −8429.33 −1.82841 −0.914205 0.405253i $$-0.867184\pi$$
−0.914205 + 0.405253i $$0.867184\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3974.26 −0.843717 −0.421859 0.906662i $$-0.638622\pi$$
−0.421859 + 0.906662i $$0.638622\pi$$
$$282$$ 0 0
$$283$$ 3072.41 0.645356 0.322678 0.946509i $$-0.395417\pi$$
0.322678 + 0.946509i $$0.395417\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −5287.28 −1.08745
$$288$$ 0 0
$$289$$ 9014.97 1.83492
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −3982.21 −0.794004 −0.397002 0.917818i $$-0.629949\pi$$
−0.397002 + 0.917818i $$0.629949\pi$$
$$294$$ 0 0
$$295$$ −1195.43 −0.235935
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −6603.63 −1.27725
$$300$$ 0 0
$$301$$ 1825.04 0.349480
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 272.729 0.0512013
$$306$$ 0 0
$$307$$ 2996.06 0.556984 0.278492 0.960439i $$-0.410165\pi$$
0.278492 + 0.960439i $$0.410165\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −3079.94 −0.561567 −0.280783 0.959771i $$-0.590594\pi$$
−0.280783 + 0.959771i $$0.590594\pi$$
$$312$$ 0 0
$$313$$ 7953.65 1.43632 0.718158 0.695880i $$-0.244986\pi$$
0.718158 + 0.695880i $$0.244986\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6832.98 1.21066 0.605328 0.795976i $$-0.293042\pi$$
0.605328 + 0.795976i $$0.293042\pi$$
$$318$$ 0 0
$$319$$ 8595.23 1.50859
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 14562.0 2.50852
$$324$$ 0 0
$$325$$ 1804.50 0.307987
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3215.95 0.538908
$$330$$ 0 0
$$331$$ 2296.57 0.381363 0.190682 0.981652i $$-0.438930\pi$$
0.190682 + 0.981652i $$0.438930\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −380.279 −0.0620205
$$336$$ 0 0
$$337$$ 7261.48 1.17376 0.586881 0.809673i $$-0.300355\pi$$
0.586881 + 0.809673i $$0.300355\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2281.66 −0.362342
$$342$$ 0 0
$$343$$ 6899.32 1.08609
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7425.22 1.14872 0.574361 0.818602i $$-0.305251\pi$$
0.574361 + 0.818602i $$0.305251\pi$$
$$348$$ 0 0
$$349$$ −478.160 −0.0733390 −0.0366695 0.999327i $$-0.511675\pi$$
−0.0366695 + 0.999327i $$0.511675\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4993.09 0.752847 0.376424 0.926448i $$-0.377154\pi$$
0.376424 + 0.926448i $$0.377154\pi$$
$$354$$ 0 0
$$355$$ −3641.52 −0.544427
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6873.09 −1.01044 −0.505219 0.862991i $$-0.668588\pi$$
−0.505219 + 0.862991i $$0.668588\pi$$
$$360$$ 0 0
$$361$$ 8365.87 1.21969
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2509.08 0.359811
$$366$$ 0 0
$$367$$ 8688.72 1.23582 0.617912 0.786247i $$-0.287979\pi$$
0.617912 + 0.786247i $$0.287979\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −198.590 −0.0277904
$$372$$ 0 0
$$373$$ 3494.54 0.485095 0.242548 0.970140i $$-0.422017\pi$$
0.242548 + 0.970140i $$0.422017\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12588.2 −1.71970
$$378$$ 0 0
$$379$$ 5802.83 0.786468 0.393234 0.919438i $$-0.371356\pi$$
0.393234 + 0.919438i $$0.371356\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −3358.56 −0.448080 −0.224040 0.974580i $$-0.571925\pi$$
−0.224040 + 0.974580i $$0.571925\pi$$
$$384$$ 0 0
$$385$$ −3577.42 −0.473565
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −19.1370 −0.00249430 −0.00124715 0.999999i $$-0.500397\pi$$
−0.00124715 + 0.999999i $$0.500397\pi$$
$$390$$ 0 0
$$391$$ 10797.2 1.39651
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1988.05 0.253239
$$396$$ 0 0
$$397$$ −4348.59 −0.549747 −0.274873 0.961480i $$-0.588636\pi$$
−0.274873 + 0.961480i $$0.588636\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8501.61 −1.05873 −0.529364 0.848395i $$-0.677570\pi$$
−0.529364 + 0.848395i $$0.677570\pi$$
$$402$$ 0 0
$$403$$ 3341.62 0.413047
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7637.95 −0.930219
$$408$$ 0 0
$$409$$ −2810.67 −0.339801 −0.169900 0.985461i $$-0.554345\pi$$
−0.169900 + 0.985461i $$0.554345\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −3470.93 −0.413543
$$414$$ 0 0
$$415$$ 6847.31 0.809930
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −16355.4 −1.90696 −0.953478 0.301461i $$-0.902526\pi$$
−0.953478 + 0.301461i $$0.902526\pi$$
$$420$$ 0 0
$$421$$ −4510.90 −0.522204 −0.261102 0.965311i $$-0.584086\pi$$
−0.261102 + 0.965311i $$0.584086\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2950.42 −0.336745
$$426$$ 0 0
$$427$$ 791.864 0.0897447
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5850.47 0.653845 0.326923 0.945051i $$-0.393988\pi$$
0.326923 + 0.945051i $$0.393988\pi$$
$$432$$ 0 0
$$433$$ −3836.82 −0.425833 −0.212916 0.977070i $$-0.568296\pi$$
−0.212916 + 0.977070i $$0.568296\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 11288.7 1.23572
$$438$$ 0 0
$$439$$ −16227.3 −1.76421 −0.882106 0.471052i $$-0.843875\pi$$
−0.882106 + 0.471052i $$0.843875\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 6705.13 0.719120 0.359560 0.933122i $$-0.382927\pi$$
0.359560 + 0.933122i $$0.382927\pi$$
$$444$$ 0 0
$$445$$ 7340.65 0.781978
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −213.100 −0.0223982 −0.0111991 0.999937i $$-0.503565\pi$$
−0.0111991 + 0.999937i $$0.503565\pi$$
$$450$$ 0 0
$$451$$ −17949.6 −1.87408
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 5239.34 0.539833
$$456$$ 0 0
$$457$$ 16462.1 1.68504 0.842520 0.538665i $$-0.181071\pi$$
0.842520 + 0.538665i $$0.181071\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1562.06 −0.157814 −0.0789071 0.996882i $$-0.525143\pi$$
−0.0789071 + 0.996882i $$0.525143\pi$$
$$462$$ 0 0
$$463$$ 5924.27 0.594653 0.297326 0.954776i $$-0.403905\pi$$
0.297326 + 0.954776i $$0.403905\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −17905.1 −1.77420 −0.887098 0.461582i $$-0.847282\pi$$
−0.887098 + 0.461582i $$0.847282\pi$$
$$468$$ 0 0
$$469$$ −1104.13 −0.108708
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6195.75 0.602285
$$474$$ 0 0
$$475$$ −3084.73 −0.297973
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9915.44 0.945820 0.472910 0.881111i $$-0.343204\pi$$
0.472910 + 0.881111i $$0.343204\pi$$
$$480$$ 0 0
$$481$$ 11186.2 1.06039
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1675.12 −0.156831
$$486$$ 0 0
$$487$$ −11910.8 −1.10828 −0.554138 0.832425i $$-0.686952\pi$$
−0.554138 + 0.832425i $$0.686952\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 11063.8 1.01691 0.508453 0.861090i $$-0.330218\pi$$
0.508453 + 0.861090i $$0.330218\pi$$
$$492$$ 0 0
$$493$$ 20582.2 1.88027
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10573.1 −0.954261
$$498$$ 0 0
$$499$$ −9347.25 −0.838557 −0.419279 0.907858i $$-0.637717\pi$$
−0.419279 + 0.907858i $$0.637717\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −19474.2 −1.72627 −0.863135 0.504973i $$-0.831502\pi$$
−0.863135 + 0.504973i $$0.831502\pi$$
$$504$$ 0 0
$$505$$ −6030.43 −0.531387
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −22164.1 −1.93007 −0.965035 0.262121i $$-0.915578\pi$$
−0.965035 + 0.262121i $$0.915578\pi$$
$$510$$ 0 0
$$511$$ 7285.06 0.630670
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 5305.55 0.453962
$$516$$ 0 0
$$517$$ 10917.7 0.928740
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 254.564 0.0214062 0.0107031 0.999943i $$-0.496593\pi$$
0.0107031 + 0.999943i $$0.496593\pi$$
$$522$$ 0 0
$$523$$ −4049.92 −0.338606 −0.169303 0.985564i $$-0.554152\pi$$
−0.169303 + 0.985564i $$0.554152\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5463.66 −0.451614
$$528$$ 0 0
$$529$$ −3796.89 −0.312064
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 26288.1 2.13633
$$534$$ 0 0
$$535$$ 2377.89 0.192159
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6517.60 0.520841
$$540$$ 0 0
$$541$$ −4085.88 −0.324705 −0.162353 0.986733i $$-0.551908\pi$$
−0.162353 + 0.986733i $$0.551908\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6602.09 −0.518904
$$546$$ 0 0
$$547$$ 15392.2 1.20315 0.601575 0.798816i $$-0.294540\pi$$
0.601575 + 0.798816i $$0.294540\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 21519.1 1.66378
$$552$$ 0 0
$$553$$ 5772.27 0.443873
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10897.6 0.828987 0.414493 0.910052i $$-0.363959\pi$$
0.414493 + 0.910052i $$0.363959\pi$$
$$558$$ 0 0
$$559$$ −9074.02 −0.686566
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1551.69 −0.116156 −0.0580781 0.998312i $$-0.518497\pi$$
−0.0580781 + 0.998312i $$0.518497\pi$$
$$564$$ 0 0
$$565$$ −340.875 −0.0253818
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1246.95 0.0918715 0.0459357 0.998944i $$-0.485373\pi$$
0.0459357 + 0.998944i $$0.485373\pi$$
$$570$$ 0 0
$$571$$ −4196.58 −0.307568 −0.153784 0.988104i $$-0.549146\pi$$
−0.153784 + 0.988104i $$0.549146\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2287.21 −0.165884
$$576$$ 0 0
$$577$$ 20585.1 1.48521 0.742607 0.669728i $$-0.233589\pi$$
0.742607 + 0.669728i $$0.233589\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 19881.1 1.41963
$$582$$ 0 0
$$583$$ −674.183 −0.0478933
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −4855.78 −0.341430 −0.170715 0.985320i $$-0.554608\pi$$
−0.170715 + 0.985320i $$0.554608\pi$$
$$588$$ 0 0
$$589$$ −5712.37 −0.399617
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 23965.6 1.65961 0.829804 0.558055i $$-0.188452\pi$$
0.829804 + 0.558055i $$0.188452\pi$$
$$594$$ 0 0
$$595$$ −8566.50 −0.590239
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 14229.1 0.970595 0.485297 0.874349i $$-0.338711\pi$$
0.485297 + 0.874349i $$0.338711\pi$$
$$600$$ 0 0
$$601$$ −8877.97 −0.602562 −0.301281 0.953535i $$-0.597414\pi$$
−0.301281 + 0.953535i $$0.597414\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −5489.83 −0.368915
$$606$$ 0 0
$$607$$ 10876.7 0.727302 0.363651 0.931535i $$-0.381530\pi$$
0.363651 + 0.931535i $$0.381530\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −15989.5 −1.05870
$$612$$ 0 0
$$613$$ −19544.8 −1.28778 −0.643890 0.765118i $$-0.722680\pi$$
−0.643890 + 0.765118i $$0.722680\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −5041.75 −0.328968 −0.164484 0.986380i $$-0.552596\pi$$
−0.164484 + 0.986380i $$0.552596\pi$$
$$618$$ 0 0
$$619$$ 5208.05 0.338173 0.169087 0.985601i $$-0.445918\pi$$
0.169087 + 0.985601i $$0.445918\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 21313.5 1.37064
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −18289.9 −1.15940
$$630$$ 0 0
$$631$$ 20284.6 1.27974 0.639872 0.768482i $$-0.278987\pi$$
0.639872 + 0.768482i $$0.278987\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2965.05 −0.185298
$$636$$ 0 0
$$637$$ −9545.40 −0.593724
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 20852.4 1.28490 0.642449 0.766329i $$-0.277919\pi$$
0.642449 + 0.766329i $$0.277919\pi$$
$$642$$ 0 0
$$643$$ 2187.22 0.134146 0.0670729 0.997748i $$-0.478634\pi$$
0.0670729 + 0.997748i $$0.478634\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17044.1 1.03566 0.517831 0.855483i $$-0.326740\pi$$
0.517831 + 0.855483i $$0.326740\pi$$
$$648$$ 0 0
$$649$$ −11783.3 −0.712688
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −8474.26 −0.507846 −0.253923 0.967224i $$-0.581721\pi$$
−0.253923 + 0.967224i $$0.581721\pi$$
$$654$$ 0 0
$$655$$ −1694.69 −0.101094
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 25560.2 1.51090 0.755450 0.655207i $$-0.227419\pi$$
0.755450 + 0.655207i $$0.227419\pi$$
$$660$$ 0 0
$$661$$ 1209.59 0.0711766 0.0355883 0.999367i $$-0.488670\pi$$
0.0355883 + 0.999367i $$0.488670\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8956.46 −0.522281
$$666$$ 0 0
$$667$$ 15955.6 0.926241
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2688.26 0.154664
$$672$$ 0 0
$$673$$ −8698.21 −0.498204 −0.249102 0.968477i $$-0.580135\pi$$
−0.249102 + 0.968477i $$0.580135\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8424.49 0.478256 0.239128 0.970988i $$-0.423138\pi$$
0.239128 + 0.970988i $$0.423138\pi$$
$$678$$ 0 0
$$679$$ −4863.68 −0.274891
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −17828.6 −0.998817 −0.499408 0.866367i $$-0.666449\pi$$
−0.499408 + 0.866367i $$0.666449\pi$$
$$684$$ 0 0
$$685$$ −4057.21 −0.226304
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 987.378 0.0545952
$$690$$ 0 0
$$691$$ −14525.1 −0.799652 −0.399826 0.916591i $$-0.630930\pi$$
−0.399826 + 0.916591i $$0.630930\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −15530.6 −0.847641
$$696$$ 0 0
$$697$$ −42982.0 −2.33581
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18815.5 −1.01377 −0.506883 0.862015i $$-0.669202\pi$$
−0.506883 + 0.862015i $$0.669202\pi$$
$$702$$ 0 0
$$703$$ −19122.4 −1.02591
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −17509.3 −0.931405
$$708$$ 0 0
$$709$$ −12934.4 −0.685137 −0.342569 0.939493i $$-0.611297\pi$$
−0.342569 + 0.939493i $$0.611297\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −4235.51 −0.222470
$$714$$ 0 0
$$715$$ 17786.8 0.930333
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −8471.10 −0.439386 −0.219693 0.975569i $$-0.570506\pi$$
−0.219693 + 0.975569i $$0.570506\pi$$
$$720$$ 0 0
$$721$$ 15404.6 0.795695
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −4360.00 −0.223347
$$726$$ 0 0
$$727$$ −24369.5 −1.24321 −0.621605 0.783331i $$-0.713519\pi$$
−0.621605 + 0.783331i $$0.713519\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 14836.3 0.750673
$$732$$ 0 0
$$733$$ −35411.8 −1.78440 −0.892199 0.451642i $$-0.850838\pi$$
−0.892199 + 0.451642i $$0.850838\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3748.37 −0.187345
$$738$$ 0 0
$$739$$ 24447.0 1.21691 0.608456 0.793588i $$-0.291789\pi$$
0.608456 + 0.793588i $$0.291789\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24125.9 −1.19125 −0.595623 0.803264i $$-0.703095\pi$$
−0.595623 + 0.803264i $$0.703095\pi$$
$$744$$ 0 0
$$745$$ 12705.1 0.624802
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 6904.17 0.336813
$$750$$ 0 0
$$751$$ −11882.4 −0.577356 −0.288678 0.957426i $$-0.593216\pi$$
−0.288678 + 0.957426i $$0.593216\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −5626.85 −0.271234
$$756$$ 0 0
$$757$$ −14601.3 −0.701049 −0.350525 0.936554i $$-0.613997\pi$$
−0.350525 + 0.936554i $$0.613997\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 20296.3 0.966809 0.483404 0.875397i $$-0.339400\pi$$
0.483404 + 0.875397i $$0.339400\pi$$
$$762$$ 0 0
$$763$$ −19169.1 −0.909524
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 17257.3 0.812418
$$768$$ 0 0
$$769$$ 36322.0 1.70326 0.851629 0.524146i $$-0.175615\pi$$
0.851629 + 0.524146i $$0.175615\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −28930.9 −1.34615 −0.673073 0.739576i $$-0.735026\pi$$
−0.673073 + 0.739576i $$0.735026\pi$$
$$774$$ 0 0
$$775$$ 1157.39 0.0536448
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −44938.6 −2.06687
$$780$$ 0 0
$$781$$ −35894.1 −1.64455
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16150.3 −0.734303
$$786$$ 0 0
$$787$$ −21128.3 −0.956978 −0.478489 0.878094i $$-0.658815\pi$$
−0.478489 + 0.878094i $$0.658815\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −989.726 −0.0444887
$$792$$ 0 0
$$793$$ −3937.11 −0.176306
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2765.87 0.122926 0.0614632 0.998109i $$-0.480423\pi$$
0.0614632 + 0.998109i $$0.480423\pi$$
$$798$$ 0 0
$$799$$ 26143.5 1.15756
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 24731.7 1.08688
$$804$$ 0 0
$$805$$ −6640.87 −0.290758
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −16756.6 −0.728220 −0.364110 0.931356i $$-0.618627\pi$$
−0.364110 + 0.931356i $$0.618627\pi$$
$$810$$ 0 0
$$811$$ −17829.6 −0.771987 −0.385993 0.922502i $$-0.626141\pi$$
−0.385993 + 0.922502i $$0.626141\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −3470.27 −0.149151
$$816$$ 0 0
$$817$$ 15511.7 0.664243
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6757.48 0.287256 0.143628 0.989632i $$-0.454123\pi$$
0.143628 + 0.989632i $$0.454123\pi$$
$$822$$ 0 0
$$823$$ −7121.28 −0.301619 −0.150809 0.988563i $$-0.548188\pi$$
−0.150809 + 0.988563i $$0.548188\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1171.74 −0.0492688 −0.0246344 0.999697i $$-0.507842\pi$$
−0.0246344 + 0.999697i $$0.507842\pi$$
$$828$$ 0 0
$$829$$ 23617.8 0.989483 0.494742 0.869040i $$-0.335263\pi$$
0.494742 + 0.869040i $$0.335263\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 15607.1 0.649163
$$834$$ 0 0
$$835$$ −16080.2 −0.666440
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −35054.3 −1.44244 −0.721222 0.692704i $$-0.756419\pi$$
−0.721222 + 0.692704i $$0.756419\pi$$
$$840$$ 0 0
$$841$$ 6026.42 0.247096
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −15064.8 −0.613306
$$846$$ 0 0
$$847$$ −15939.6 −0.646627
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −14178.6 −0.571133
$$852$$ 0 0
$$853$$ −32772.3 −1.31548 −0.657740 0.753245i $$-0.728487\pi$$
−0.657740 + 0.753245i $$0.728487\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3503.93 0.139664 0.0698319 0.997559i $$-0.477754\pi$$
0.0698319 + 0.997559i $$0.477754\pi$$
$$858$$ 0 0
$$859$$ −31044.1 −1.23307 −0.616537 0.787326i $$-0.711465\pi$$
−0.616537 + 0.787326i $$0.711465\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 26333.6 1.03871 0.519354 0.854559i $$-0.326173\pi$$
0.519354 + 0.854559i $$0.326173\pi$$
$$864$$ 0 0
$$865$$ 1487.73 0.0584790
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 19596.0 0.764959
$$870$$ 0 0
$$871$$ 5489.71 0.213561
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 1814.68 0.0701112
$$876$$ 0 0
$$877$$ −40977.3 −1.57777 −0.788886 0.614540i $$-0.789342\pi$$
−0.788886 + 0.614540i $$0.789342\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −37022.4 −1.41579 −0.707897 0.706315i $$-0.750356\pi$$
−0.707897 + 0.706315i $$0.750356\pi$$
$$882$$ 0 0
$$883$$ 36037.9 1.37347 0.686734 0.726909i $$-0.259044\pi$$
0.686734 + 0.726909i $$0.259044\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1465.05 0.0554584 0.0277292 0.999615i $$-0.491172\pi$$
0.0277292 + 0.999615i $$0.491172\pi$$
$$888$$ 0 0
$$889$$ −8608.97 −0.324787
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 27333.5 1.02428
$$894$$ 0 0
$$895$$ 17250.6 0.644273
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −8073.96 −0.299535
$$900$$ 0 0
$$901$$ −1614.40 −0.0596930
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 15448.7 0.567440
$$906$$ 0 0
$$907$$ −33660.8 −1.23229 −0.616146 0.787632i $$-0.711307\pi$$
−0.616146 + 0.787632i $$0.711307\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 25992.7 0.945311 0.472655 0.881247i $$-0.343296\pi$$
0.472655 + 0.881247i $$0.343296\pi$$
$$912$$ 0 0
$$913$$ 67493.3 2.44655
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −4920.50 −0.177196
$$918$$ 0 0
$$919$$ −1149.54 −0.0412620 −0.0206310 0.999787i $$-0.506568\pi$$
−0.0206310 + 0.999787i $$0.506568\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 52568.9 1.87468
$$924$$ 0 0
$$925$$ 3874.42 0.137719
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −1923.20 −0.0679204 −0.0339602 0.999423i $$-0.510812\pi$$
−0.0339602 + 0.999423i $$0.510812\pi$$
$$930$$ 0 0
$$931$$ 16317.5 0.574420
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −29082.0 −1.01720
$$936$$ 0 0
$$937$$ 3511.90 0.122443 0.0612213 0.998124i $$-0.480500\pi$$
0.0612213 + 0.998124i $$0.480500\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6848.16 0.237241 0.118620 0.992940i $$-0.462153\pi$$
0.118620 + 0.992940i $$0.462153\pi$$
$$942$$ 0 0
$$943$$ −33320.3 −1.15064
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 48357.3 1.65935 0.829673 0.558250i $$-0.188527\pi$$
0.829673 + 0.558250i $$0.188527\pi$$
$$948$$ 0 0
$$949$$ −36221.0 −1.23897
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −38701.1 −1.31548 −0.657740 0.753245i $$-0.728488\pi$$
−0.657740 + 0.753245i $$0.728488\pi$$
$$954$$ 0 0
$$955$$ −7660.57 −0.259571
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −11780.0 −0.396660
$$960$$ 0 0
$$961$$ −27647.7 −0.928056
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 25972.1 0.866396
$$966$$ 0 0
$$967$$ 24312.7 0.808526 0.404263 0.914643i $$-0.367528\pi$$
0.404263 + 0.914643i $$0.367528\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −37464.3 −1.23820 −0.619098 0.785314i $$-0.712501\pi$$
−0.619098 + 0.785314i $$0.712501\pi$$
$$972$$ 0 0
$$973$$ −45092.9 −1.48573
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −3186.09 −0.104332 −0.0521659 0.998638i $$-0.516612\pi$$
−0.0521659 + 0.998638i $$0.516612\pi$$
$$978$$ 0 0
$$979$$ 72356.1 2.36212
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 30345.6 0.984614 0.492307 0.870422i $$-0.336154\pi$$
0.492307 + 0.870422i $$0.336154\pi$$
$$984$$ 0 0
$$985$$ −10028.0 −0.324386
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11501.3 0.369789
$$990$$ 0 0
$$991$$ −3443.75 −0.110388 −0.0551940 0.998476i $$-0.517578\pi$$
−0.0551940 + 0.998476i $$0.517578\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −14373.4 −0.457957
$$996$$ 0 0
$$997$$ −4567.89 −0.145102 −0.0725510 0.997365i $$-0.523114\pi$$
−0.0725510 + 0.997365i $$0.523114\pi$$
$$998$$ 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.be.1.2 3
3.2 odd 2 2160.4.a.bm.1.2 3
4.3 odd 2 135.4.a.g.1.2 yes 3
12.11 even 2 135.4.a.f.1.2 3
20.3 even 4 675.4.b.k.649.3 6
20.7 even 4 675.4.b.k.649.4 6
20.19 odd 2 675.4.a.q.1.2 3
36.7 odd 6 405.4.e.r.271.2 6
36.11 even 6 405.4.e.t.271.2 6
36.23 even 6 405.4.e.t.136.2 6
36.31 odd 6 405.4.e.r.136.2 6
60.23 odd 4 675.4.b.l.649.4 6
60.47 odd 4 675.4.b.l.649.3 6
60.59 even 2 675.4.a.r.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.2 3 12.11 even 2
135.4.a.g.1.2 yes 3 4.3 odd 2
405.4.e.r.136.2 6 36.31 odd 6
405.4.e.r.271.2 6 36.7 odd 6
405.4.e.t.136.2 6 36.23 even 6
405.4.e.t.271.2 6 36.11 even 6
675.4.a.q.1.2 3 20.19 odd 2
675.4.a.r.1.2 3 60.59 even 2
675.4.b.k.649.3 6 20.3 even 4
675.4.b.k.649.4 6 20.7 even 4
675.4.b.l.649.3 6 60.47 odd 4
675.4.b.l.649.4 6 60.23 odd 4
2160.4.a.be.1.2 3 1.1 even 1 trivial
2160.4.a.bm.1.2 3 3.2 odd 2