# Properties

 Label 320.3.b.c.191.1 Level $320$ Weight $3$ Character 320.191 Analytic conductor $8.719$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 320.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.71936845953$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 191.1 Root $$-0.309017 - 0.951057i$$ of defining polynomial Character $$\chi$$ $$=$$ 320.191 Dual form 320.3.b.c.191.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.80423i q^{3} -2.23607 q^{5} -8.50651i q^{7} -5.47214 q^{9} +O(q^{10})$$ $$q-3.80423i q^{3} -2.23607 q^{5} -8.50651i q^{7} -5.47214 q^{9} -1.79611i q^{11} -0.472136 q^{13} +8.50651i q^{15} -23.8885 q^{17} +9.40456i q^{19} -32.3607 q^{21} -16.1150i q^{23} +5.00000 q^{25} -13.4208i q^{27} -6.94427 q^{29} +47.4468i q^{31} -6.83282 q^{33} +19.0211i q^{35} -26.3607 q^{37} +1.79611i q^{39} -41.4164 q^{41} +2.00811i q^{43} +12.2361 q^{45} -35.3481i q^{47} -23.3607 q^{49} +90.8774i q^{51} +21.6393 q^{53} +4.01623i q^{55} +35.7771 q^{57} -73.8644i q^{59} +26.1378 q^{61} +46.5488i q^{63} +1.05573 q^{65} -88.8693i q^{67} -61.3050 q^{69} -39.4144i q^{71} +137.554 q^{73} -19.0211i q^{75} -15.2786 q^{77} -113.703i q^{79} -100.305 q^{81} -21.2412i q^{83} +53.4164 q^{85} +26.4176i q^{87} +67.4427 q^{89} +4.01623i q^{91} +180.498 q^{93} -21.0292i q^{95} -39.1672 q^{97} +9.82857i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{9} + 16 q^{13} - 24 q^{17} - 40 q^{21} + 20 q^{25} + 8 q^{29} + 80 q^{33} - 16 q^{37} - 112 q^{41} + 40 q^{45} - 4 q^{49} + 176 q^{53} - 128 q^{61} + 40 q^{65} - 120 q^{69} + 264 q^{73} - 240 q^{77} - 276 q^{81} + 160 q^{85} - 88 q^{89} + 400 q^{93} - 264 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ − 3.80423i − 1.26808i −0.773302 0.634038i $$-0.781396\pi$$
0.773302 0.634038i $$-0.218604\pi$$
$$4$$ 0 0
$$5$$ −2.23607 −0.447214
$$6$$ 0 0
$$7$$ − 8.50651i − 1.21522i −0.794237 0.607608i $$-0.792129\pi$$
0.794237 0.607608i $$-0.207871\pi$$
$$8$$ 0 0
$$9$$ −5.47214 −0.608015
$$10$$ 0 0
$$11$$ − 1.79611i − 0.163283i −0.996662 0.0816415i $$-0.973984\pi$$
0.996662 0.0816415i $$-0.0260162\pi$$
$$12$$ 0 0
$$13$$ −0.472136 −0.0363182 −0.0181591 0.999835i $$-0.505781\pi$$
−0.0181591 + 0.999835i $$0.505781\pi$$
$$14$$ 0 0
$$15$$ 8.50651i 0.567101i
$$16$$ 0 0
$$17$$ −23.8885 −1.40521 −0.702604 0.711581i $$-0.747980\pi$$
−0.702604 + 0.711581i $$0.747980\pi$$
$$18$$ 0 0
$$19$$ 9.40456i 0.494977i 0.968891 + 0.247489i $$0.0796053\pi$$
−0.968891 + 0.247489i $$0.920395\pi$$
$$20$$ 0 0
$$21$$ −32.3607 −1.54098
$$22$$ 0 0
$$23$$ − 16.1150i − 0.700650i −0.936628 0.350325i $$-0.886071\pi$$
0.936628 0.350325i $$-0.113929\pi$$
$$24$$ 0 0
$$25$$ 5.00000 0.200000
$$26$$ 0 0
$$27$$ − 13.4208i − 0.497066i
$$28$$ 0 0
$$29$$ −6.94427 −0.239458 −0.119729 0.992807i $$-0.538203\pi$$
−0.119729 + 0.992807i $$0.538203\pi$$
$$30$$ 0 0
$$31$$ 47.4468i 1.53054i 0.643708 + 0.765271i $$0.277395\pi$$
−0.643708 + 0.765271i $$0.722605\pi$$
$$32$$ 0 0
$$33$$ −6.83282 −0.207055
$$34$$ 0 0
$$35$$ 19.0211i 0.543461i
$$36$$ 0 0
$$37$$ −26.3607 −0.712451 −0.356225 0.934400i $$-0.615936\pi$$
−0.356225 + 0.934400i $$0.615936\pi$$
$$38$$ 0 0
$$39$$ 1.79611i 0.0460542i
$$40$$ 0 0
$$41$$ −41.4164 −1.01016 −0.505078 0.863074i $$-0.668536\pi$$
−0.505078 + 0.863074i $$0.668536\pi$$
$$42$$ 0 0
$$43$$ 2.00811i 0.0467003i 0.999727 + 0.0233502i $$0.00743326\pi$$
−0.999727 + 0.0233502i $$0.992567\pi$$
$$44$$ 0 0
$$45$$ 12.2361 0.271913
$$46$$ 0 0
$$47$$ − 35.3481i − 0.752087i −0.926602 0.376044i $$-0.877284\pi$$
0.926602 0.376044i $$-0.122716\pi$$
$$48$$ 0 0
$$49$$ −23.3607 −0.476749
$$50$$ 0 0
$$51$$ 90.8774i 1.78191i
$$52$$ 0 0
$$53$$ 21.6393 0.408289 0.204145 0.978941i $$-0.434559\pi$$
0.204145 + 0.978941i $$0.434559\pi$$
$$54$$ 0 0
$$55$$ 4.01623i 0.0730223i
$$56$$ 0 0
$$57$$ 35.7771 0.627668
$$58$$ 0 0
$$59$$ − 73.8644i − 1.25194i −0.779848 0.625970i $$-0.784703\pi$$
0.779848 0.625970i $$-0.215297\pi$$
$$60$$ 0 0
$$61$$ 26.1378 0.428488 0.214244 0.976780i $$-0.431271\pi$$
0.214244 + 0.976780i $$0.431271\pi$$
$$62$$ 0 0
$$63$$ 46.5488i 0.738869i
$$64$$ 0 0
$$65$$ 1.05573 0.0162420
$$66$$ 0 0
$$67$$ − 88.8693i − 1.32641i −0.748439 0.663204i $$-0.769196\pi$$
0.748439 0.663204i $$-0.230804\pi$$
$$68$$ 0 0
$$69$$ −61.3050 −0.888478
$$70$$ 0 0
$$71$$ − 39.4144i − 0.555132i −0.960707 0.277566i $$-0.910472\pi$$
0.960707 0.277566i $$-0.0895277\pi$$
$$72$$ 0 0
$$73$$ 137.554 1.88430 0.942152 0.335186i $$-0.108799\pi$$
0.942152 + 0.335186i $$0.108799\pi$$
$$74$$ 0 0
$$75$$ − 19.0211i − 0.253615i
$$76$$ 0 0
$$77$$ −15.2786 −0.198424
$$78$$ 0 0
$$79$$ − 113.703i − 1.43928i −0.694350 0.719638i $$-0.744308\pi$$
0.694350 0.719638i $$-0.255692\pi$$
$$80$$ 0 0
$$81$$ −100.305 −1.23833
$$82$$ 0 0
$$83$$ − 21.2412i − 0.255919i −0.991779 0.127959i $$-0.959157\pi$$
0.991779 0.127959i $$-0.0408427\pi$$
$$84$$ 0 0
$$85$$ 53.4164 0.628428
$$86$$ 0 0
$$87$$ 26.4176i 0.303650i
$$88$$ 0 0
$$89$$ 67.4427 0.757783 0.378892 0.925441i $$-0.376305\pi$$
0.378892 + 0.925441i $$0.376305\pi$$
$$90$$ 0 0
$$91$$ 4.01623i 0.0441344i
$$92$$ 0 0
$$93$$ 180.498 1.94084
$$94$$ 0 0
$$95$$ − 21.0292i − 0.221360i
$$96$$ 0 0
$$97$$ −39.1672 −0.403785 −0.201893 0.979408i $$-0.564709\pi$$
−0.201893 + 0.979408i $$0.564709\pi$$
$$98$$ 0 0
$$99$$ 9.82857i 0.0992785i
$$100$$ 0 0
$$101$$ −99.8885 −0.988995 −0.494498 0.869179i $$-0.664648\pi$$
−0.494498 + 0.869179i $$0.664648\pi$$
$$102$$ 0 0
$$103$$ − 35.7721i − 0.347302i −0.984807 0.173651i $$-0.944444\pi$$
0.984807 0.173651i $$-0.0555565\pi$$
$$104$$ 0 0
$$105$$ 72.3607 0.689149
$$106$$ 0 0
$$107$$ 121.099i 1.13177i 0.824485 + 0.565884i $$0.191465\pi$$
−0.824485 + 0.565884i $$0.808535\pi$$
$$108$$ 0 0
$$109$$ 197.469 1.81164 0.905821 0.423660i $$-0.139255\pi$$
0.905821 + 0.423660i $$0.139255\pi$$
$$110$$ 0 0
$$111$$ 100.282i 0.903441i
$$112$$ 0 0
$$113$$ 81.2786 0.719280 0.359640 0.933091i $$-0.382900\pi$$
0.359640 + 0.933091i $$0.382900\pi$$
$$114$$ 0 0
$$115$$ 36.0341i 0.313340i
$$116$$ 0 0
$$117$$ 2.58359 0.0220820
$$118$$ 0 0
$$119$$ 203.208i 1.70763i
$$120$$ 0 0
$$121$$ 117.774 0.973339
$$122$$ 0 0
$$123$$ 157.557i 1.28095i
$$124$$ 0 0
$$125$$ −11.1803 −0.0894427
$$126$$ 0 0
$$127$$ − 1.84616i − 0.0145367i −0.999974 0.00726834i $$-0.997686\pi$$
0.999974 0.00726834i $$-0.00231361\pi$$
$$128$$ 0 0
$$129$$ 7.63932 0.0592195
$$130$$ 0 0
$$131$$ − 225.609i − 1.72221i −0.508428 0.861105i $$-0.669773\pi$$
0.508428 0.861105i $$-0.330227\pi$$
$$132$$ 0 0
$$133$$ 80.0000 0.601504
$$134$$ 0 0
$$135$$ 30.0098i 0.222295i
$$136$$ 0 0
$$137$$ −52.8328 −0.385641 −0.192820 0.981234i $$-0.561764\pi$$
−0.192820 + 0.981234i $$0.561764\pi$$
$$138$$ 0 0
$$139$$ 125.852i 0.905407i 0.891661 + 0.452703i $$0.149540\pi$$
−0.891661 + 0.452703i $$0.850460\pi$$
$$140$$ 0 0
$$141$$ −134.472 −0.953703
$$142$$ 0 0
$$143$$ 0.848009i 0.00593013i
$$144$$ 0 0
$$145$$ 15.5279 0.107089
$$146$$ 0 0
$$147$$ 88.8693i 0.604553i
$$148$$ 0 0
$$149$$ −132.971 −0.892420 −0.446210 0.894928i $$-0.647227\pi$$
−0.446210 + 0.894928i $$0.647227\pi$$
$$150$$ 0 0
$$151$$ − 151.221i − 1.00146i −0.865603 0.500732i $$-0.833064\pi$$
0.865603 0.500732i $$-0.166936\pi$$
$$152$$ 0 0
$$153$$ 130.721 0.854388
$$154$$ 0 0
$$155$$ − 106.094i − 0.684480i
$$156$$ 0 0
$$157$$ 36.7477 0.234062 0.117031 0.993128i $$-0.462662\pi$$
0.117031 + 0.993128i $$0.462662\pi$$
$$158$$ 0 0
$$159$$ − 82.3209i − 0.517741i
$$160$$ 0 0
$$161$$ −137.082 −0.851441
$$162$$ 0 0
$$163$$ − 302.854i − 1.85800i −0.370079 0.929000i $$-0.620669\pi$$
0.370079 0.929000i $$-0.379331\pi$$
$$164$$ 0 0
$$165$$ 15.2786 0.0925978
$$166$$ 0 0
$$167$$ 99.3839i 0.595113i 0.954704 + 0.297557i $$0.0961717\pi$$
−0.954704 + 0.297557i $$0.903828\pi$$
$$168$$ 0 0
$$169$$ −168.777 −0.998681
$$170$$ 0 0
$$171$$ − 51.4631i − 0.300954i
$$172$$ 0 0
$$173$$ 181.639 1.04994 0.524969 0.851121i $$-0.324077\pi$$
0.524969 + 0.851121i $$0.324077\pi$$
$$174$$ 0 0
$$175$$ − 42.5325i − 0.243043i
$$176$$ 0 0
$$177$$ −280.997 −1.58755
$$178$$ 0 0
$$179$$ 260.907i 1.45758i 0.684735 + 0.728792i $$0.259918\pi$$
−0.684735 + 0.728792i $$0.740082\pi$$
$$180$$ 0 0
$$181$$ −157.777 −0.871697 −0.435848 0.900020i $$-0.643552\pi$$
−0.435848 + 0.900020i $$0.643552\pi$$
$$182$$ 0 0
$$183$$ − 99.4340i − 0.543355i
$$184$$ 0 0
$$185$$ 58.9443 0.318618
$$186$$ 0 0
$$187$$ 42.9065i 0.229447i
$$188$$ 0 0
$$189$$ −114.164 −0.604043
$$190$$ 0 0
$$191$$ − 324.095i − 1.69683i −0.529328 0.848417i $$-0.677556\pi$$
0.529328 0.848417i $$-0.322444\pi$$
$$192$$ 0 0
$$193$$ 181.777 0.941850 0.470925 0.882173i $$-0.343920\pi$$
0.470925 + 0.882173i $$0.343920\pi$$
$$194$$ 0 0
$$195$$ − 4.01623i − 0.0205960i
$$196$$ 0 0
$$197$$ −140.525 −0.713324 −0.356662 0.934234i $$-0.616085\pi$$
−0.356662 + 0.934234i $$0.616085\pi$$
$$198$$ 0 0
$$199$$ 168.234i 0.845397i 0.906270 + 0.422698i $$0.138917\pi$$
−0.906270 + 0.422698i $$0.861083\pi$$
$$200$$ 0 0
$$201$$ −338.079 −1.68198
$$202$$ 0 0
$$203$$ 59.0715i 0.290993i
$$204$$ 0 0
$$205$$ 92.6099 0.451756
$$206$$ 0 0
$$207$$ 88.1833i 0.426006i
$$208$$ 0 0
$$209$$ 16.8916 0.0808213
$$210$$ 0 0
$$211$$ − 93.9455i − 0.445240i −0.974905 0.222620i $$-0.928539\pi$$
0.974905 0.222620i $$-0.0714608\pi$$
$$212$$ 0 0
$$213$$ −149.941 −0.703949
$$214$$ 0 0
$$215$$ − 4.49028i − 0.0208850i
$$216$$ 0 0
$$217$$ 403.607 1.85994
$$218$$ 0 0
$$219$$ − 523.287i − 2.38944i
$$220$$ 0 0
$$221$$ 11.2786 0.0510346
$$222$$ 0 0
$$223$$ 214.035i 0.959797i 0.877324 + 0.479899i $$0.159327\pi$$
−0.877324 + 0.479899i $$0.840673\pi$$
$$224$$ 0 0
$$225$$ −27.3607 −0.121603
$$226$$ 0 0
$$227$$ 41.4225i 0.182478i 0.995829 + 0.0912389i $$0.0290827\pi$$
−0.995829 + 0.0912389i $$0.970917\pi$$
$$228$$ 0 0
$$229$$ −73.2786 −0.319994 −0.159997 0.987117i $$-0.551148\pi$$
−0.159997 + 0.987117i $$0.551148\pi$$
$$230$$ 0 0
$$231$$ 58.1234i 0.251616i
$$232$$ 0 0
$$233$$ −307.050 −1.31781 −0.658905 0.752227i $$-0.728980\pi$$
−0.658905 + 0.752227i $$0.728980\pi$$
$$234$$ 0 0
$$235$$ 79.0407i 0.336344i
$$236$$ 0 0
$$237$$ −432.551 −1.82511
$$238$$ 0 0
$$239$$ − 42.9065i − 0.179525i −0.995963 0.0897625i $$-0.971389\pi$$
0.995963 0.0897625i $$-0.0286108\pi$$
$$240$$ 0 0
$$241$$ −135.082 −0.560506 −0.280253 0.959926i $$-0.590418\pi$$
−0.280253 + 0.959926i $$0.590418\pi$$
$$242$$ 0 0
$$243$$ 260.796i 1.07323i
$$244$$ 0 0
$$245$$ 52.2361 0.213208
$$246$$ 0 0
$$247$$ − 4.44023i − 0.0179767i
$$248$$ 0 0
$$249$$ −80.8065 −0.324524
$$250$$ 0 0
$$251$$ 221.169i 0.881152i 0.897715 + 0.440576i $$0.145226\pi$$
−0.897715 + 0.440576i $$0.854774\pi$$
$$252$$ 0 0
$$253$$ −28.9443 −0.114404
$$254$$ 0 0
$$255$$ − 203.208i − 0.796894i
$$256$$ 0 0
$$257$$ −257.056 −1.00022 −0.500108 0.865963i $$-0.666707\pi$$
−0.500108 + 0.865963i $$0.666707\pi$$
$$258$$ 0 0
$$259$$ 224.237i 0.865781i
$$260$$ 0 0
$$261$$ 38.0000 0.145594
$$262$$ 0 0
$$263$$ − 164.168i − 0.624212i −0.950047 0.312106i $$-0.898966\pi$$
0.950047 0.312106i $$-0.101034\pi$$
$$264$$ 0 0
$$265$$ −48.3870 −0.182592
$$266$$ 0 0
$$267$$ − 256.567i − 0.960926i
$$268$$ 0 0
$$269$$ 35.4752 0.131878 0.0659391 0.997824i $$-0.478996\pi$$
0.0659391 + 0.997824i $$0.478996\pi$$
$$270$$ 0 0
$$271$$ 298.950i 1.10314i 0.834130 + 0.551568i $$0.185970\pi$$
−0.834130 + 0.551568i $$0.814030\pi$$
$$272$$ 0 0
$$273$$ 15.2786 0.0559657
$$274$$ 0 0
$$275$$ − 8.98056i − 0.0326566i
$$276$$ 0 0
$$277$$ 457.246 1.65071 0.825354 0.564616i $$-0.190976\pi$$
0.825354 + 0.564616i $$0.190976\pi$$
$$278$$ 0 0
$$279$$ − 259.635i − 0.930593i
$$280$$ 0 0
$$281$$ −5.63932 −0.0200688 −0.0100344 0.999950i $$-0.503194\pi$$
−0.0100344 + 0.999950i $$0.503194\pi$$
$$282$$ 0 0
$$283$$ 169.918i 0.600418i 0.953874 + 0.300209i $$0.0970563\pi$$
−0.953874 + 0.300209i $$0.902944\pi$$
$$284$$ 0 0
$$285$$ −80.0000 −0.280702
$$286$$ 0 0
$$287$$ 352.309i 1.22756i
$$288$$ 0 0
$$289$$ 281.663 0.974611
$$290$$ 0 0
$$291$$ 149.001i 0.512030i
$$292$$ 0 0
$$293$$ 26.8591 0.0916694 0.0458347 0.998949i $$-0.485405\pi$$
0.0458347 + 0.998949i $$0.485405\pi$$
$$294$$ 0 0
$$295$$ 165.166i 0.559884i
$$296$$ 0 0
$$297$$ −24.1052 −0.0811624
$$298$$ 0 0
$$299$$ 7.60845i 0.0254463i
$$300$$ 0 0
$$301$$ 17.0820 0.0567510
$$302$$ 0 0
$$303$$ 379.999i 1.25412i
$$304$$ 0 0
$$305$$ −58.4458 −0.191626
$$306$$ 0 0
$$307$$ 118.031i 0.384466i 0.981349 + 0.192233i $$0.0615730\pi$$
−0.981349 + 0.192233i $$0.938427\pi$$
$$308$$ 0 0
$$309$$ −136.085 −0.440405
$$310$$ 0 0
$$311$$ 121.835i 0.391753i 0.980629 + 0.195877i $$0.0627552\pi$$
−0.980629 + 0.195877i $$0.937245\pi$$
$$312$$ 0 0
$$313$$ 219.548 0.701431 0.350716 0.936482i $$-0.385938\pi$$
0.350716 + 0.936482i $$0.385938\pi$$
$$314$$ 0 0
$$315$$ − 104.086i − 0.330432i
$$316$$ 0 0
$$317$$ −366.859 −1.15728 −0.578642 0.815582i $$-0.696417\pi$$
−0.578642 + 0.815582i $$0.696417\pi$$
$$318$$ 0 0
$$319$$ 12.4727i 0.0390993i
$$320$$ 0 0
$$321$$ 460.689 1.43517
$$322$$ 0 0
$$323$$ − 224.661i − 0.695546i
$$324$$ 0 0
$$325$$ −2.36068 −0.00726363
$$326$$ 0 0
$$327$$ − 751.217i − 2.29730i
$$328$$ 0 0
$$329$$ −300.689 −0.913948
$$330$$ 0 0
$$331$$ 162.846i 0.491981i 0.969272 + 0.245990i $$0.0791132\pi$$
−0.969272 + 0.245990i $$0.920887\pi$$
$$332$$ 0 0
$$333$$ 144.249 0.433181
$$334$$ 0 0
$$335$$ 198.718i 0.593187i
$$336$$ 0 0
$$337$$ 17.1084 0.0507666 0.0253833 0.999678i $$-0.491919\pi$$
0.0253833 + 0.999678i $$0.491919\pi$$
$$338$$ 0 0
$$339$$ − 309.202i − 0.912101i
$$340$$ 0 0
$$341$$ 85.2198 0.249911
$$342$$ 0 0
$$343$$ − 218.101i − 0.635863i
$$344$$ 0 0
$$345$$ 137.082 0.397339
$$346$$ 0 0
$$347$$ − 167.498i − 0.482703i −0.970438 0.241351i $$-0.922409\pi$$
0.970438 0.241351i $$-0.0775906\pi$$
$$348$$ 0 0
$$349$$ 483.495 1.38537 0.692687 0.721239i $$-0.256427\pi$$
0.692687 + 0.721239i $$0.256427\pi$$
$$350$$ 0 0
$$351$$ 6.33644i 0.0180525i
$$352$$ 0 0
$$353$$ 307.994 0.872504 0.436252 0.899825i $$-0.356306\pi$$
0.436252 + 0.899825i $$0.356306\pi$$
$$354$$ 0 0
$$355$$ 88.1332i 0.248263i
$$356$$ 0 0
$$357$$ 773.050 2.16540
$$358$$ 0 0
$$359$$ − 23.2494i − 0.0647615i −0.999476 0.0323807i $$-0.989691\pi$$
0.999476 0.0323807i $$-0.0103089\pi$$
$$360$$ 0 0
$$361$$ 272.554 0.754998
$$362$$ 0 0
$$363$$ − 448.039i − 1.23427i
$$364$$ 0 0
$$365$$ −307.580 −0.842686
$$366$$ 0 0
$$367$$ − 517.325i − 1.40960i −0.709404 0.704802i $$-0.751036\pi$$
0.709404 0.704802i $$-0.248964\pi$$
$$368$$ 0 0
$$369$$ 226.636 0.614190
$$370$$ 0 0
$$371$$ − 184.075i − 0.496159i
$$372$$ 0 0
$$373$$ 88.3545 0.236875 0.118438 0.992961i $$-0.462211\pi$$
0.118438 + 0.992961i $$0.462211\pi$$
$$374$$ 0 0
$$375$$ 42.5325i 0.113420i
$$376$$ 0 0
$$377$$ 3.27864 0.00869666
$$378$$ 0 0
$$379$$ − 19.3332i − 0.0510112i −0.999675 0.0255056i $$-0.991880\pi$$
0.999675 0.0255056i $$-0.00811956\pi$$
$$380$$ 0 0
$$381$$ −7.02321 −0.0184336
$$382$$ 0 0
$$383$$ − 431.612i − 1.12692i −0.826142 0.563462i $$-0.809469\pi$$
0.826142 0.563462i $$-0.190531\pi$$
$$384$$ 0 0
$$385$$ 34.1641 0.0887379
$$386$$ 0 0
$$387$$ − 10.9887i − 0.0283945i
$$388$$ 0 0
$$389$$ 296.354 0.761837 0.380918 0.924609i $$-0.375608\pi$$
0.380918 + 0.924609i $$0.375608\pi$$
$$390$$ 0 0
$$391$$ 384.963i 0.984560i
$$392$$ 0 0
$$393$$ −858.269 −2.18389
$$394$$ 0 0
$$395$$ 254.247i 0.643664i
$$396$$ 0 0
$$397$$ 86.1904 0.217104 0.108552 0.994091i $$-0.465379\pi$$
0.108552 + 0.994091i $$0.465379\pi$$
$$398$$ 0 0
$$399$$ − 304.338i − 0.762752i
$$400$$ 0 0
$$401$$ 442.997 1.10473 0.552365 0.833602i $$-0.313725\pi$$
0.552365 + 0.833602i $$0.313725\pi$$
$$402$$ 0 0
$$403$$ − 22.4014i − 0.0555865i
$$404$$ 0 0
$$405$$ 224.289 0.553799
$$406$$ 0 0
$$407$$ 47.3467i 0.116331i
$$408$$ 0 0
$$409$$ 63.4102 0.155037 0.0775186 0.996991i $$-0.475300\pi$$
0.0775186 + 0.996991i $$0.475300\pi$$
$$410$$ 0 0
$$411$$ 200.988i 0.489022i
$$412$$ 0 0
$$413$$ −628.328 −1.52138
$$414$$ 0 0
$$415$$ 47.4969i 0.114450i
$$416$$ 0 0
$$417$$ 478.768 1.14812
$$418$$ 0 0
$$419$$ 435.678i 1.03980i 0.854226 + 0.519902i $$0.174032\pi$$
−0.854226 + 0.519902i $$0.825968\pi$$
$$420$$ 0 0
$$421$$ 582.912 1.38459 0.692294 0.721615i $$-0.256600\pi$$
0.692294 + 0.721615i $$0.256600\pi$$
$$422$$ 0 0
$$423$$ 193.430i 0.457280i
$$424$$ 0 0
$$425$$ −119.443 −0.281042
$$426$$ 0 0
$$427$$ − 222.341i − 0.520705i
$$428$$ 0 0
$$429$$ 3.22602 0.00751986
$$430$$ 0 0
$$431$$ 375.882i 0.872117i 0.899918 + 0.436058i $$0.143626\pi$$
−0.899918 + 0.436058i $$0.856374\pi$$
$$432$$ 0 0
$$433$$ −368.164 −0.850263 −0.425132 0.905131i $$-0.639772\pi$$
−0.425132 + 0.905131i $$0.639772\pi$$
$$434$$ 0 0
$$435$$ − 59.0715i − 0.135797i
$$436$$ 0 0
$$437$$ 151.554 0.346806
$$438$$ 0 0
$$439$$ 483.549i 1.10148i 0.834677 + 0.550739i $$0.185654\pi$$
−0.834677 + 0.550739i $$0.814346\pi$$
$$440$$ 0 0
$$441$$ 127.833 0.289870
$$442$$ 0 0
$$443$$ 279.181i 0.630205i 0.949058 + 0.315102i $$0.102039\pi$$
−0.949058 + 0.315102i $$0.897961\pi$$
$$444$$ 0 0
$$445$$ −150.807 −0.338891
$$446$$ 0 0
$$447$$ 505.850i 1.13166i
$$448$$ 0 0
$$449$$ 756.079 1.68392 0.841959 0.539542i $$-0.181403\pi$$
0.841959 + 0.539542i $$0.181403\pi$$
$$450$$ 0 0
$$451$$ 74.3885i 0.164941i
$$452$$ 0 0
$$453$$ −575.279 −1.26993
$$454$$ 0 0
$$455$$ − 8.98056i − 0.0197375i
$$456$$ 0 0
$$457$$ 285.672 0.625103 0.312551 0.949901i $$-0.398816\pi$$
0.312551 + 0.949901i $$0.398816\pi$$
$$458$$ 0 0
$$459$$ 320.603i 0.698482i
$$460$$ 0 0
$$461$$ 99.1146 0.214999 0.107500 0.994205i $$-0.465716\pi$$
0.107500 + 0.994205i $$0.465716\pi$$
$$462$$ 0 0
$$463$$ − 630.603i − 1.36199i −0.732286 0.680997i $$-0.761547\pi$$
0.732286 0.680997i $$-0.238453\pi$$
$$464$$ 0 0
$$465$$ −403.607 −0.867972
$$466$$ 0 0
$$467$$ − 496.010i − 1.06212i −0.847334 0.531060i $$-0.821794\pi$$
0.847334 0.531060i $$-0.178206\pi$$
$$468$$ 0 0
$$469$$ −755.967 −1.61187
$$470$$ 0 0
$$471$$ − 139.796i − 0.296808i
$$472$$ 0 0
$$473$$ 3.60680 0.00762537
$$474$$ 0 0
$$475$$ 47.0228i 0.0989954i
$$476$$ 0 0
$$477$$ −118.413 −0.248246
$$478$$ 0 0
$$479$$ 579.090i 1.20896i 0.796621 + 0.604478i $$0.206618\pi$$
−0.796621 + 0.604478i $$0.793382\pi$$
$$480$$ 0 0
$$481$$ 12.4458 0.0258749
$$482$$ 0 0
$$483$$ 521.491i 1.07969i
$$484$$ 0 0
$$485$$ 87.5805 0.180578
$$486$$ 0 0
$$487$$ 626.363i 1.28617i 0.765796 + 0.643084i $$0.222345\pi$$
−0.765796 + 0.643084i $$0.777655\pi$$
$$488$$ 0 0
$$489$$ −1152.13 −2.35608
$$490$$ 0 0
$$491$$ − 22.3013i − 0.0454201i −0.999742 0.0227100i $$-0.992771\pi$$
0.999742 0.0227100i $$-0.00722945\pi$$
$$492$$ 0 0
$$493$$ 165.889 0.336488
$$494$$ 0 0
$$495$$ − 21.9773i − 0.0443987i
$$496$$ 0 0
$$497$$ −335.279 −0.674605
$$498$$ 0 0
$$499$$ − 627.362i − 1.25724i −0.777714 0.628619i $$-0.783621\pi$$
0.777714 0.628619i $$-0.216379\pi$$
$$500$$ 0 0
$$501$$ 378.079 0.754649
$$502$$ 0 0
$$503$$ 780.853i 1.55239i 0.630492 + 0.776196i $$0.282853\pi$$
−0.630492 + 0.776196i $$0.717147\pi$$
$$504$$ 0 0
$$505$$ 223.358 0.442292
$$506$$ 0 0
$$507$$ 642.066i 1.26640i
$$508$$ 0 0
$$509$$ 288.950 0.567683 0.283841 0.958871i $$-0.408391\pi$$
0.283841 + 0.958871i $$0.408391\pi$$
$$510$$ 0 0
$$511$$ − 1170.11i − 2.28984i
$$512$$ 0 0
$$513$$ 126.217 0.246036
$$514$$ 0 0
$$515$$ 79.9888i 0.155318i
$$516$$ 0 0
$$517$$ −63.4891 −0.122803
$$518$$ 0 0
$$519$$ − 690.997i − 1.33140i
$$520$$ 0 0
$$521$$ −602.984 −1.15736 −0.578680 0.815555i $$-0.696432\pi$$
−0.578680 + 0.815555i $$0.696432\pi$$
$$522$$ 0 0
$$523$$ − 367.962i − 0.703560i −0.936083 0.351780i $$-0.885577\pi$$
0.936083 0.351780i $$-0.114423\pi$$
$$524$$ 0 0
$$525$$ −161.803 −0.308197
$$526$$ 0 0
$$527$$ − 1133.44i − 2.15073i
$$528$$ 0 0
$$529$$ 269.308 0.509089
$$530$$ 0 0
$$531$$ 404.196i 0.761198i
$$532$$ 0 0
$$533$$ 19.5542 0.0366870
$$534$$ 0 0
$$535$$ − 270.786i − 0.506142i
$$536$$ 0 0
$$537$$ 992.551 1.84833
$$538$$ 0 0
$$539$$ 41.9584i 0.0778449i
$$540$$ 0 0
$$541$$ 616.885 1.14027 0.570134 0.821551i $$-0.306891\pi$$
0.570134 + 0.821551i $$0.306891\pi$$
$$542$$ 0 0
$$543$$ 600.220i 1.10538i
$$544$$ 0 0
$$545$$ −441.554 −0.810191
$$546$$ 0 0
$$547$$ 97.8499i 0.178885i 0.995992 + 0.0894423i $$0.0285085\pi$$
−0.995992 + 0.0894423i $$0.971492\pi$$
$$548$$ 0 0
$$549$$ −143.029 −0.260527
$$550$$ 0 0
$$551$$ − 65.3078i − 0.118526i
$$552$$ 0 0
$$553$$ −967.214 −1.74903
$$554$$ 0 0
$$555$$ − 224.237i − 0.404031i
$$556$$ 0 0
$$557$$ −896.302 −1.60916 −0.804580 0.593845i $$-0.797609\pi$$
−0.804580 + 0.593845i $$0.797609\pi$$
$$558$$ 0 0
$$559$$ − 0.948103i − 0.00169607i
$$560$$ 0 0
$$561$$ 163.226 0.290955
$$562$$ 0 0
$$563$$ − 771.186i − 1.36978i −0.728647 0.684890i $$-0.759850\pi$$
0.728647 0.684890i $$-0.240150\pi$$
$$564$$ 0 0
$$565$$ −181.745 −0.321672
$$566$$ 0 0
$$567$$ 853.245i 1.50484i
$$568$$ 0 0
$$569$$ 8.74767 0.0153738 0.00768688 0.999970i $$-0.497553\pi$$
0.00768688 + 0.999970i $$0.497553\pi$$
$$570$$ 0 0
$$571$$ − 511.138i − 0.895164i −0.894243 0.447582i $$-0.852285\pi$$
0.894243 0.447582i $$-0.147715\pi$$
$$572$$ 0 0
$$573$$ −1232.93 −2.15171
$$574$$ 0 0
$$575$$ − 80.5748i − 0.140130i
$$576$$ 0 0
$$577$$ −713.712 −1.23694 −0.618468 0.785810i $$-0.712246\pi$$
−0.618468 + 0.785810i $$0.712246\pi$$
$$578$$ 0 0
$$579$$ − 691.521i − 1.19434i
$$580$$ 0 0
$$581$$ −180.689 −0.310996
$$582$$ 0 0
$$583$$ − 38.8666i − 0.0666666i
$$584$$ 0 0
$$585$$ −5.77709 −0.00987536
$$586$$ 0 0
$$587$$ 422.169i 0.719198i 0.933107 + 0.359599i $$0.117086\pi$$
−0.933107 + 0.359599i $$0.882914\pi$$
$$588$$ 0 0
$$589$$ −446.217 −0.757584
$$590$$ 0 0
$$591$$ 534.588i 0.904548i
$$592$$ 0 0
$$593$$ −308.663 −0.520510 −0.260255 0.965540i $$-0.583807\pi$$
−0.260255 + 0.965540i $$0.583807\pi$$
$$594$$ 0 0
$$595$$ − 454.387i − 0.763676i
$$596$$ 0 0
$$597$$ 640.000 1.07203
$$598$$ 0 0
$$599$$ − 462.196i − 0.771612i −0.922580 0.385806i $$-0.873923\pi$$
0.922580 0.385806i $$-0.126077\pi$$
$$600$$ 0 0
$$601$$ 355.358 0.591277 0.295639 0.955300i $$-0.404468\pi$$
0.295639 + 0.955300i $$0.404468\pi$$
$$602$$ 0 0
$$603$$ 486.305i 0.806476i
$$604$$ 0 0
$$605$$ −263.351 −0.435290
$$606$$ 0 0
$$607$$ 630.403i 1.03856i 0.854605 + 0.519278i $$0.173799\pi$$
−0.854605 + 0.519278i $$0.826201\pi$$
$$608$$ 0 0
$$609$$ 224.721 0.369001
$$610$$ 0 0
$$611$$ 16.6891i 0.0273144i
$$612$$ 0 0
$$613$$ −812.525 −1.32549 −0.662745 0.748846i $$-0.730608\pi$$
−0.662745 + 0.748846i $$0.730608\pi$$
$$614$$ 0 0
$$615$$ − 352.309i − 0.572860i
$$616$$ 0 0
$$617$$ −437.935 −0.709781 −0.354891 0.934908i $$-0.615482\pi$$
−0.354891 + 0.934908i $$0.615482\pi$$
$$618$$ 0 0
$$619$$ 770.250i 1.24435i 0.782880 + 0.622173i $$0.213750\pi$$
−0.782880 + 0.622173i $$0.786250\pi$$
$$620$$ 0 0
$$621$$ −216.276 −0.348270
$$622$$ 0 0
$$623$$ − 573.702i − 0.920870i
$$624$$ 0 0
$$625$$ 25.0000 0.0400000
$$626$$ 0 0
$$627$$ − 64.2597i − 0.102487i
$$628$$ 0 0
$$629$$ 629.718 1.00114
$$630$$ 0 0
$$631$$ − 875.496i − 1.38747i −0.720228 0.693737i $$-0.755963\pi$$
0.720228 0.693737i $$-0.244037\pi$$
$$632$$ 0 0
$$633$$ −357.390 −0.564597
$$634$$ 0 0
$$635$$ 4.12814i 0.00650100i
$$636$$ 0 0
$$637$$ 11.0294 0.0173146
$$638$$ 0 0
$$639$$ 215.681i 0.337529i
$$640$$ 0 0
$$641$$ 842.571 1.31446 0.657232 0.753689i $$-0.271727\pi$$
0.657232 + 0.753689i $$0.271727\pi$$
$$642$$ 0 0
$$643$$ 1153.20i 1.79348i 0.442563 + 0.896738i $$0.354069\pi$$
−0.442563 + 0.896738i $$0.645931\pi$$
$$644$$ 0 0
$$645$$ −17.0820 −0.0264838
$$646$$ 0 0
$$647$$ − 355.751i − 0.549847i −0.961466 0.274924i $$-0.911347\pi$$
0.961466 0.274924i $$-0.0886526\pi$$
$$648$$ 0 0
$$649$$ −132.669 −0.204420
$$650$$ 0 0
$$651$$ − 1535.41i − 2.35854i
$$652$$ 0 0
$$653$$ 557.915 0.854387 0.427194 0.904160i $$-0.359502\pi$$
0.427194 + 0.904160i $$0.359502\pi$$
$$654$$ 0 0
$$655$$ 504.478i 0.770195i
$$656$$ 0 0
$$657$$ −752.715 −1.14569
$$658$$ 0 0
$$659$$ − 284.157i − 0.431194i −0.976482 0.215597i $$-0.930830\pi$$
0.976482 0.215597i $$-0.0691697\pi$$
$$660$$ 0 0
$$661$$ −716.735 −1.08432 −0.542160 0.840275i $$-0.682393\pi$$
−0.542160 + 0.840275i $$0.682393\pi$$
$$662$$ 0 0
$$663$$ − 42.9065i − 0.0647157i
$$664$$ 0 0
$$665$$ −178.885 −0.269001
$$666$$ 0 0
$$667$$ 111.907i 0.167776i
$$668$$ 0 0
$$669$$ 814.237 1.21710
$$670$$ 0 0
$$671$$ − 46.9464i − 0.0699648i
$$672$$ 0 0
$$673$$ −695.378 −1.03325 −0.516625 0.856212i $$-0.672812\pi$$
−0.516625 + 0.856212i $$0.672812\pi$$
$$674$$ 0 0
$$675$$ − 67.1040i − 0.0994133i
$$676$$ 0 0
$$677$$ 820.237 1.21158 0.605788 0.795626i $$-0.292858\pi$$
0.605788 + 0.795626i $$0.292858\pi$$
$$678$$ 0 0
$$679$$ 333.176i 0.490686i
$$680$$ 0 0
$$681$$ 157.580 0.231396
$$682$$ 0 0
$$683$$ − 335.508i − 0.491227i −0.969368 0.245613i $$-0.921011\pi$$
0.969368 0.245613i $$-0.0789894\pi$$
$$684$$ 0 0
$$685$$ 118.138 0.172464
$$686$$ 0 0
$$687$$ 278.769i 0.405777i
$$688$$ 0 0
$$689$$ −10.2167 −0.0148283
$$690$$ 0 0
$$691$$ − 336.568i − 0.487074i −0.969892 0.243537i $$-0.921692\pi$$
0.969892 0.243537i $$-0.0783077\pi$$
$$692$$ 0 0
$$693$$ 83.6068 0.120645
$$694$$ 0 0
$$695$$ − 281.413i − 0.404910i
$$696$$ 0 0
$$697$$ 989.378 1.41948
$$698$$ 0 0
$$699$$ 1168.09i 1.67108i
$$700$$ 0 0
$$701$$ 429.364 0.612502 0.306251 0.951951i $$-0.400925\pi$$
0.306251 + 0.951951i $$0.400925\pi$$
$$702$$ 0 0
$$703$$ − 247.911i − 0.352647i
$$704$$ 0 0
$$705$$ 300.689 0.426509
$$706$$ 0 0
$$707$$ 849.703i 1.20184i
$$708$$ 0 0
$$709$$ −1224.60 −1.72722 −0.863609 0.504162i $$-0.831801\pi$$
−0.863609 + 0.504162i $$0.831801\pi$$
$$710$$ 0 0
$$711$$ 622.197i 0.875101i
$$712$$ 0 0
$$713$$ 764.604 1.07238
$$714$$ 0 0
$$715$$ − 1.89621i − 0.00265204i
$$716$$ 0 0
$$717$$ −163.226 −0.227651
$$718$$ 0 0
$$719$$ 496.022i 0.689877i 0.938625 + 0.344939i $$0.112100\pi$$
−0.938625 + 0.344939i $$0.887900\pi$$
$$720$$ 0 0
$$721$$ −304.296 −0.422047
$$722$$ 0 0
$$723$$ 513.883i 0.710764i
$$724$$ 0 0
$$725$$ −34.7214 −0.0478915
$$726$$ 0 0
$$727$$ − 152.843i − 0.210238i −0.994460 0.105119i $$-0.966478\pi$$
0.994460 0.105119i $$-0.0335224\pi$$
$$728$$ 0 0
$$729$$ 89.3808 0.122607
$$730$$ 0 0
$$731$$ − 47.9709i − 0.0656237i
$$732$$ 0 0
$$733$$ 761.286 1.03859 0.519295 0.854595i $$-0.326195\pi$$
0.519295 + 0.854595i $$0.326195\pi$$
$$734$$ 0 0
$$735$$ − 198.718i − 0.270364i
$$736$$ 0 0
$$737$$ −159.619 −0.216580
$$738$$ 0 0
$$739$$ 183.975i 0.248951i 0.992223 + 0.124476i $$0.0397249\pi$$
−0.992223 + 0.124476i $$0.960275\pi$$
$$740$$ 0 0
$$741$$ −16.8916 −0.0227957
$$742$$ 0 0
$$743$$ 495.247i 0.666551i 0.942830 + 0.333275i $$0.108154\pi$$
−0.942830 + 0.333275i $$0.891846\pi$$
$$744$$ 0 0
$$745$$ 297.331 0.399102
$$746$$ 0 0
$$747$$ 116.235i 0.155602i
$$748$$ 0 0
$$749$$ 1030.13 1.37534
$$750$$ 0 0
$$751$$ 800.059i 1.06533i 0.846328 + 0.532663i $$0.178809\pi$$
−0.846328 + 0.532663i $$0.821191\pi$$
$$752$$ 0 0
$$753$$ 841.378 1.11737
$$754$$ 0 0
$$755$$ 338.140i 0.447868i
$$756$$ 0 0
$$757$$ 276.367 0.365082 0.182541 0.983198i $$-0.441568\pi$$
0.182541 + 0.983198i $$0.441568\pi$$
$$758$$ 0 0
$$759$$ 110.111i 0.145073i
$$760$$ 0 0
$$761$$ 891.207 1.17110 0.585550 0.810636i $$-0.300879\pi$$
0.585550 + 0.810636i $$0.300879\pi$$
$$762$$ 0 0
$$763$$ − 1679.77i − 2.20154i
$$764$$ 0 0
$$765$$ −292.302 −0.382094
$$766$$ 0 0
$$767$$ 34.8740i 0.0454681i
$$768$$ 0 0
$$769$$ −835.430 −1.08639 −0.543193 0.839608i $$-0.682785\pi$$
−0.543193 + 0.839608i $$0.682785\pi$$
$$770$$ 0 0
$$771$$ 977.898i 1.26835i
$$772$$ 0 0
$$773$$ −213.522 −0.276225 −0.138112 0.990417i $$-0.544103\pi$$
−0.138112 + 0.990417i $$0.544103\pi$$
$$774$$ 0 0
$$775$$ 237.234i 0.306109i
$$776$$ 0 0
$$777$$ 853.050 1.09788
$$778$$ 0 0
$$779$$ − 389.503i − 0.500004i
$$780$$ 0 0
$$781$$ −70.7926 −0.0906436
$$782$$ 0 0
$$783$$ 93.1976i 0.119026i
$$784$$ 0 0
$$785$$ −82.1703 −0.104676
$$786$$ 0 0
$$787$$ 370.182i 0.470371i 0.971951 + 0.235185i $$0.0755697\pi$$
−0.971951 + 0.235185i $$0.924430\pi$$
$$788$$ 0 0
$$789$$ −624.531 −0.791547
$$790$$ 0 0
$$791$$ − 691.397i − 0.874080i
$$792$$ 0 0
$$793$$ −12.3406 −0.0155619
$$794$$ 0 0
$$795$$ 184.075i 0.231541i
$$796$$ 0 0
$$797$$ −274.426 −0.344323 −0.172162 0.985069i $$-0.555075\pi$$
−0.172162 + 0.985069i $$0.555075\pi$$
$$798$$ 0 0
$$799$$ 844.414i 1.05684i
$$800$$ 0 0
$$801$$ −369.056 −0.460744
$$802$$ 0 0
$$803$$ − 247.063i − 0.307675i
$$804$$ 0 0
$$805$$ 306.525 0.380776
$$806$$ 0 0
$$807$$ − 134.956i − 0.167232i
$$808$$ 0 0
$$809$$ 665.214 0.822266 0.411133 0.911575i $$-0.365133\pi$$
0.411133 + 0.911575i $$0.365133\pi$$
$$810$$ 0 0
$$811$$ 360.665i 0.444717i 0.974965 + 0.222358i $$0.0713755\pi$$
−0.974965 + 0.222358i $$0.928624\pi$$
$$812$$ 0 0
$$813$$ 1137.27 1.39886
$$814$$ 0 0
$$815$$ 677.202i 0.830923i
$$816$$ 0 0
$$817$$ −18.8854 −0.0231156
$$818$$ 0 0
$$819$$ − 21.9773i − 0.0268344i
$$820$$ 0 0
$$821$$ 666.899 0.812301 0.406151 0.913806i $$-0.366871\pi$$
0.406151 + 0.913806i $$0.366871\pi$$
$$822$$ 0 0
$$823$$ − 122.433i − 0.148764i −0.997230 0.0743822i $$-0.976302\pi$$
0.997230 0.0743822i $$-0.0236985\pi$$
$$824$$ 0 0
$$825$$ −34.1641 −0.0414110
$$826$$ 0 0
$$827$$ − 1532.98i − 1.85366i −0.375477 0.926832i $$-0.622521\pi$$
0.375477 0.926832i $$-0.377479\pi$$
$$828$$ 0 0
$$829$$ 195.475 0.235796 0.117898 0.993026i $$-0.462384\pi$$
0.117898 + 0.993026i $$0.462384\pi$$
$$830$$ 0 0
$$831$$ − 1739.47i − 2.09322i
$$832$$ 0 0
$$833$$ 558.053 0.669931
$$834$$ 0 0
$$835$$ − 222.229i − 0.266143i
$$836$$ 0 0
$$837$$ 636.774 0.760781
$$838$$ 0 0
$$839$$ − 1325.97i − 1.58041i −0.612840 0.790207i $$-0.709973\pi$$
0.612840 0.790207i $$-0.290027\pi$$
$$840$$ 0 0
$$841$$ −792.777 −0.942660
$$842$$ 0 0
$$843$$ 21.4532i 0.0254487i
$$844$$ 0 0
$$845$$ 377.397 0.446624
$$846$$ 0 0
$$847$$ − 1001.85i − 1.18282i
$$848$$ 0 0
$$849$$ 646.407 0.761375
$$850$$ 0 0
$$851$$ 424.801i 0.499179i
$$852$$ 0 0
$$853$$ −1055.28 −1.23714 −0.618570 0.785730i $$-0.712288\pi$$
−0.618570 + 0.785730i $$0.712288\pi$$
$$854$$ 0 0
$$855$$ 115.075i 0.134591i
$$856$$ 0 0
$$857$$ 155.378 0.181304 0.0906521 0.995883i $$-0.471105\pi$$
0.0906521 + 0.995883i $$0.471105\pi$$
$$858$$ 0 0
$$859$$ − 226.033i − 0.263136i −0.991307 0.131568i $$-0.957999\pi$$
0.991307 0.131568i $$-0.0420011\pi$$
$$860$$ 0 0
$$861$$ 1340.26 1.55664
$$862$$ 0 0
$$863$$ − 930.702i − 1.07845i −0.842162 0.539225i $$-0.818717\pi$$
0.842162 0.539225i $$-0.181283\pi$$
$$864$$ 0 0
$$865$$ −406.158 −0.469547
$$866$$ 0 0
$$867$$ − 1071.51i − 1.23588i
$$868$$ 0 0
$$869$$ −204.223 −0.235009
$$870$$ 0 0
$$871$$ 41.9584i 0.0481727i
$$872$$ 0 0
$$873$$ 214.328 0.245508
$$874$$ 0 0
$$875$$ 95.1057i 0.108692i
$$876$$ 0 0
$$877$$ 33.5217 0.0382231 0.0191115 0.999817i $$-0.493916\pi$$
0.0191115 + 0.999817i $$0.493916\pi$$
$$878$$ 0 0
$$879$$ − 102.178i − 0.116244i
$$880$$ 0 0
$$881$$ −933.850 −1.05999 −0.529994 0.848001i $$-0.677806\pi$$
−0.529994 + 0.848001i $$0.677806\pi$$
$$882$$ 0 0
$$883$$ 542.308i 0.614166i 0.951683 + 0.307083i $$0.0993529\pi$$
−0.951683 + 0.307083i $$0.900647\pi$$
$$884$$ 0 0
$$885$$ 628.328 0.709975
$$886$$ 0 0
$$887$$ 714.720i 0.805773i 0.915250 + 0.402886i $$0.131993\pi$$
−0.915250 + 0.402886i $$0.868007\pi$$
$$888$$ 0 0
$$889$$ −15.7044 −0.0176652
$$890$$ 0 0
$$891$$ 180.159i 0.202199i
$$892$$ 0 0
$$893$$ 332.433 0.372266
$$894$$ 0 0
$$895$$ − 583.407i − 0.651851i
$$896$$ 0 0
$$897$$ 28.9443 0.0322679
$$898$$ 0 0
$$899$$ − 329.484i − 0.366500i
$$900$$ 0 0
$$901$$ −516.932 −0.573731
$$902$$ 0 0
$$903$$ − 64.9839i − 0.0719645i
$$904$$ 0 0
$$905$$ 352.800 0.389835
$$906$$ 0 0
$$907$$ − 347.233i − 0.382837i −0.981509 0.191418i $$-0.938691\pi$$
0.981509 0.191418i $$-0.0613087\pi$$
$$908$$ 0 0
$$909$$ 546.604 0.601324
$$910$$ 0 0
$$911$$ 1427.54i 1.56701i 0.621386 + 0.783504i $$0.286570\pi$$
−0.621386 + 0.783504i $$0.713430\pi$$
$$912$$ 0 0
$$913$$ −38.1517 −0.0417871
$$914$$ 0 0
$$915$$ 222.341i 0.242996i
$$916$$ 0 0
$$917$$ −1919.15 −2.09286
$$918$$ 0 0
$$919$$ − 569.162i − 0.619327i −0.950846 0.309664i $$-0.899784\pi$$
0.950846 0.309664i $$-0.100216\pi$$
$$920$$ 0 0
$$921$$ 449.017 0.487532
$$922$$ 0 0
$$923$$ 18.6089i 0.0201614i
$$924$$ 0 0
$$925$$ −131.803 −0.142490
$$926$$ 0 0
$$927$$ 195.750i 0.211165i
$$928$$ 0 0
$$929$$ 1535.96 1.65335 0.826675 0.562680i $$-0.190230\pi$$
0.826675 + 0.562680i $$0.190230\pi$$
$$930$$ 0 0
$$931$$ − 219.697i − 0.235980i
$$932$$ 0 0
$$933$$ 463.489 0.496773
$$934$$ 0 0
$$935$$ − 95.9418i − 0.102612i
$$936$$ 0 0
$$937$$ 338.721 0.361496 0.180748 0.983529i $$-0.442148\pi$$
0.180748 + 0.983529i $$0.442148\pi$$
$$938$$ 0 0
$$939$$ − 835.210i − 0.889468i
$$940$$ 0 0
$$941$$ −1439.77 −1.53004 −0.765022 0.644004i $$-0.777272\pi$$
−0.765022 + 0.644004i $$0.777272\pi$$
$$942$$ 0 0
$$943$$ 667.424i 0.707766i
$$944$$ 0 0
$$945$$ 255.279 0.270136
$$946$$ 0 0
$$947$$ − 656.135i − 0.692856i −0.938077 0.346428i $$-0.887394\pi$$
0.938077 0.346428i $$-0.112606\pi$$
$$948$$ 0 0
$$949$$ −64.9443 −0.0684344
$$950$$ 0 0
$$951$$ 1395.62i 1.46752i
$$952$$ 0 0
$$953$$ 436.675 0.458211 0.229105 0.973402i $$-0.426420\pi$$
0.229105 + 0.973402i $$0.426420\pi$$
$$954$$ 0 0
$$955$$ 724.699i 0.758847i
$$956$$ 0 0
$$957$$ 47.4489 0.0495809
$$958$$ 0 0
$$959$$ 449.423i 0.468637i
$$960$$ 0 0
$$961$$ −1290.20 −1.34256
$$962$$ 0 0
$$963$$ − 662.671i − 0.688132i
$$964$$ 0 0
$$965$$ −406.466 −0.421208
$$966$$ 0 0
$$967$$ − 903.436i − 0.934267i −0.884187 0.467133i $$-0.845287\pi$$
0.884187 0.467133i $$-0.154713\pi$$
$$968$$ 0 0
$$969$$ −854.663 −0.882005
$$970$$ 0 0
$$971$$ 1866.89i 1.92265i 0.275420 + 0.961324i $$0.411183\pi$$
−0.275420 + 0.961324i $$0.588817\pi$$
$$972$$ 0 0
$$973$$ 1070.56 1.10026
$$974$$ 0 0
$$975$$ 8.98056i 0.00921083i
$$976$$ 0 0
$$977$$ −1073.95 −1.09923 −0.549615 0.835418i $$-0.685226\pi$$
−0.549615 + 0.835418i $$0.685226\pi$$
$$978$$ 0 0
$$979$$ − 121.135i − 0.123733i
$$980$$ 0 0
$$981$$ −1080.58 −1.10151
$$982$$ 0 0
$$983$$ 534.114i 0.543351i 0.962389 + 0.271675i $$0.0875777\pi$$
−0.962389 + 0.271675i $$0.912422\pi$$
$$984$$ 0 0
$$985$$ 314.223 0.319008
$$986$$ 0 0
$$987$$ 1143.89i 1.15895i
$$988$$ 0 0
$$989$$ 32.3607 0.0327206
$$990$$ 0 0
$$991$$ 520.419i 0.525146i 0.964912 + 0.262573i $$0.0845710\pi$$
−0.964912 + 0.262573i $$0.915429\pi$$
$$992$$ 0 0
$$993$$ 619.502 0.623869
$$994$$ 0 0
$$995$$ − 376.183i − 0.378073i
$$996$$ 0 0
$$997$$ −457.680 −0.459057 −0.229528 0.973302i $$-0.573718\pi$$
−0.229528 + 0.973302i $$0.573718\pi$$
$$998$$ 0 0
$$999$$ 353.781i 0.354135i
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 320.3.b.c.191.1 4
3.2 odd 2 2880.3.e.e.2431.3 4
4.3 odd 2 inner 320.3.b.c.191.4 4
5.2 odd 4 1600.3.h.n.1599.1 8
5.3 odd 4 1600.3.h.n.1599.7 8
5.4 even 2 1600.3.b.s.1151.4 4
8.3 odd 2 20.3.b.a.11.1 4
8.5 even 2 20.3.b.a.11.2 yes 4
12.11 even 2 2880.3.e.e.2431.4 4
16.3 odd 4 1280.3.g.e.1151.2 8
16.5 even 4 1280.3.g.e.1151.1 8
16.11 odd 4 1280.3.g.e.1151.7 8
16.13 even 4 1280.3.g.e.1151.8 8
20.3 even 4 1600.3.h.n.1599.2 8
20.7 even 4 1600.3.h.n.1599.8 8
20.19 odd 2 1600.3.b.s.1151.1 4
24.5 odd 2 180.3.c.a.91.3 4
24.11 even 2 180.3.c.a.91.4 4
40.3 even 4 100.3.d.b.99.4 8
40.13 odd 4 100.3.d.b.99.6 8
40.19 odd 2 100.3.b.f.51.4 4
40.27 even 4 100.3.d.b.99.5 8
40.29 even 2 100.3.b.f.51.3 4
40.37 odd 4 100.3.d.b.99.3 8
120.29 odd 2 900.3.c.k.451.2 4
120.53 even 4 900.3.f.e.199.3 8
120.59 even 2 900.3.c.k.451.1 4
120.77 even 4 900.3.f.e.199.6 8
120.83 odd 4 900.3.f.e.199.5 8
120.107 odd 4 900.3.f.e.199.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.1 4 8.3 odd 2
20.3.b.a.11.2 yes 4 8.5 even 2
100.3.b.f.51.3 4 40.29 even 2
100.3.b.f.51.4 4 40.19 odd 2
100.3.d.b.99.3 8 40.37 odd 4
100.3.d.b.99.4 8 40.3 even 4
100.3.d.b.99.5 8 40.27 even 4
100.3.d.b.99.6 8 40.13 odd 4
180.3.c.a.91.3 4 24.5 odd 2
180.3.c.a.91.4 4 24.11 even 2
320.3.b.c.191.1 4 1.1 even 1 trivial
320.3.b.c.191.4 4 4.3 odd 2 inner
900.3.c.k.451.1 4 120.59 even 2
900.3.c.k.451.2 4 120.29 odd 2
900.3.f.e.199.3 8 120.53 even 4
900.3.f.e.199.4 8 120.107 odd 4
900.3.f.e.199.5 8 120.83 odd 4
900.3.f.e.199.6 8 120.77 even 4
1280.3.g.e.1151.1 8 16.5 even 4
1280.3.g.e.1151.2 8 16.3 odd 4
1280.3.g.e.1151.7 8 16.11 odd 4
1280.3.g.e.1151.8 8 16.13 even 4
1600.3.b.s.1151.1 4 20.19 odd 2
1600.3.b.s.1151.4 4 5.4 even 2
1600.3.h.n.1599.1 8 5.2 odd 4
1600.3.h.n.1599.2 8 20.3 even 4
1600.3.h.n.1599.7 8 5.3 odd 4
1600.3.h.n.1599.8 8 20.7 even 4
2880.3.e.e.2431.3 4 3.2 odd 2
2880.3.e.e.2431.4 4 12.11 even 2