# Properties

 Label 4600.2.e.v.4049.4 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.4 Root $$3.11721i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.v.4049.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.33689i q^{3} +3.16736i q^{7} +1.21273 q^{9} +O(q^{10})$$ $$q-1.33689i q^{3} +3.16736i q^{7} +1.21273 q^{9} -0.0955192 q^{11} +1.44343i q^{13} +2.29775i q^{17} -7.00371 q^{19} +4.23441 q^{21} -1.00000i q^{23} -5.63195i q^{27} -5.39076 q^{29} -0.584488 q^{31} +0.127699i q^{33} +9.29985i q^{37} +1.92970 q^{39} -2.86534 q^{41} +9.50208i q^{43} -7.09353i q^{47} -3.03218 q^{49} +3.07184 q^{51} -7.73922i q^{53} +9.36319i q^{57} -13.6426 q^{59} +0.234413 q^{61} +3.84114i q^{63} -7.49729i q^{67} -1.33689 q^{69} -5.18426 q^{71} -1.52384i q^{73} -0.302544i q^{77} +3.04068 q^{79} -3.89112 q^{81} -15.9909i q^{83} +7.20685i q^{87} -5.53325 q^{89} -4.57186 q^{91} +0.781396i q^{93} -2.58305i q^{97} -0.115839 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 8 q^{9} + O(q^{10})$$ $$10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.33689i − 0.771854i −0.922529 0.385927i $$-0.873882\pi$$
0.922529 0.385927i $$-0.126118\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.16736i 1.19715i 0.801067 + 0.598575i $$0.204266\pi$$
−0.801067 + 0.598575i $$0.795734\pi$$
$$8$$ 0 0
$$9$$ 1.21273 0.404242
$$10$$ 0 0
$$11$$ −0.0955192 −0.0288001 −0.0144001 0.999896i $$-0.504584\pi$$
−0.0144001 + 0.999896i $$0.504584\pi$$
$$12$$ 0 0
$$13$$ 1.44343i 0.400335i 0.979762 + 0.200167i $$0.0641486\pi$$
−0.979762 + 0.200167i $$0.935851\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.29775i 0.557287i 0.960395 + 0.278643i $$0.0898848\pi$$
−0.960395 + 0.278643i $$0.910115\pi$$
$$18$$ 0 0
$$19$$ −7.00371 −1.60676 −0.803381 0.595466i $$-0.796968\pi$$
−0.803381 + 0.595466i $$0.796968\pi$$
$$20$$ 0 0
$$21$$ 4.23441 0.924025
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.63195i − 1.08387i
$$28$$ 0 0
$$29$$ −5.39076 −1.00104 −0.500519 0.865725i $$-0.666858\pi$$
−0.500519 + 0.865725i $$0.666858\pi$$
$$30$$ 0 0
$$31$$ −0.584488 −0.104977 −0.0524886 0.998622i $$-0.516715\pi$$
−0.0524886 + 0.998622i $$0.516715\pi$$
$$32$$ 0 0
$$33$$ 0.127699i 0.0222295i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.29985i 1.52889i 0.644691 + 0.764443i $$0.276986\pi$$
−0.644691 + 0.764443i $$0.723014\pi$$
$$38$$ 0 0
$$39$$ 1.92970 0.309000
$$40$$ 0 0
$$41$$ −2.86534 −0.447492 −0.223746 0.974648i $$-0.571829\pi$$
−0.223746 + 0.974648i $$0.571829\pi$$
$$42$$ 0 0
$$43$$ 9.50208i 1.44905i 0.689246 + 0.724527i $$0.257942\pi$$
−0.689246 + 0.724527i $$0.742058\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 7.09353i − 1.03470i −0.855775 0.517349i $$-0.826919\pi$$
0.855775 0.517349i $$-0.173081\pi$$
$$48$$ 0 0
$$49$$ −3.03218 −0.433168
$$50$$ 0 0
$$51$$ 3.07184 0.430144
$$52$$ 0 0
$$53$$ − 7.73922i − 1.06306i −0.847038 0.531532i $$-0.821617\pi$$
0.847038 0.531532i $$-0.178383\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 9.36319i 1.24018i
$$58$$ 0 0
$$59$$ −13.6426 −1.77612 −0.888059 0.459730i $$-0.847946\pi$$
−0.888059 + 0.459730i $$0.847946\pi$$
$$60$$ 0 0
$$61$$ 0.234413 0.0300135 0.0150068 0.999887i $$-0.495223\pi$$
0.0150068 + 0.999887i $$0.495223\pi$$
$$62$$ 0 0
$$63$$ 3.84114i 0.483938i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 7.49729i − 0.915940i −0.888968 0.457970i $$-0.848577\pi$$
0.888968 0.457970i $$-0.151423\pi$$
$$68$$ 0 0
$$69$$ −1.33689 −0.160943
$$70$$ 0 0
$$71$$ −5.18426 −0.615258 −0.307629 0.951506i $$-0.599536\pi$$
−0.307629 + 0.951506i $$0.599536\pi$$
$$72$$ 0 0
$$73$$ − 1.52384i − 0.178352i −0.996016 0.0891760i $$-0.971577\pi$$
0.996016 0.0891760i $$-0.0284234\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 0.302544i − 0.0344781i
$$78$$ 0 0
$$79$$ 3.04068 0.342103 0.171052 0.985262i $$-0.445283\pi$$
0.171052 + 0.985262i $$0.445283\pi$$
$$80$$ 0 0
$$81$$ −3.89112 −0.432347
$$82$$ 0 0
$$83$$ − 15.9909i − 1.75523i −0.479369 0.877613i $$-0.659135\pi$$
0.479369 0.877613i $$-0.340865\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7.20685i 0.772655i
$$88$$ 0 0
$$89$$ −5.53325 −0.586523 −0.293261 0.956032i $$-0.594741\pi$$
−0.293261 + 0.956032i $$0.594741\pi$$
$$90$$ 0 0
$$91$$ −4.57186 −0.479261
$$92$$ 0 0
$$93$$ 0.781396i 0.0810270i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.58305i − 0.262269i −0.991365 0.131135i $$-0.958138\pi$$
0.991365 0.131135i $$-0.0418620\pi$$
$$98$$ 0 0
$$99$$ −0.115839 −0.0116422
$$100$$ 0 0
$$101$$ −6.65615 −0.662312 −0.331156 0.943576i $$-0.607439\pi$$
−0.331156 + 0.943576i $$0.607439\pi$$
$$102$$ 0 0
$$103$$ − 17.6668i − 1.74076i −0.492383 0.870378i $$-0.663874\pi$$
0.492383 0.870378i $$-0.336126\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 5.83635i 0.564221i 0.959382 + 0.282111i $$0.0910345\pi$$
−0.959382 + 0.282111i $$0.908965\pi$$
$$108$$ 0 0
$$109$$ −7.02096 −0.672486 −0.336243 0.941775i $$-0.609156\pi$$
−0.336243 + 0.941775i $$0.609156\pi$$
$$110$$ 0 0
$$111$$ 12.4329 1.18008
$$112$$ 0 0
$$113$$ − 17.1003i − 1.60866i −0.594185 0.804328i $$-0.702525\pi$$
0.594185 0.804328i $$-0.297475\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.75048i 0.161832i
$$118$$ 0 0
$$119$$ −7.27781 −0.667156
$$120$$ 0 0
$$121$$ −10.9909 −0.999171
$$122$$ 0 0
$$123$$ 3.83065i 0.345398i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.8050i 0.958787i 0.877600 + 0.479394i $$0.159143\pi$$
−0.877600 + 0.479394i $$0.840857\pi$$
$$128$$ 0 0
$$129$$ 12.7032 1.11846
$$130$$ 0 0
$$131$$ −8.45211 −0.738464 −0.369232 0.929337i $$-0.620379\pi$$
−0.369232 + 0.929337i $$0.620379\pi$$
$$132$$ 0 0
$$133$$ − 22.1833i − 1.92354i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.5322i 1.58331i 0.610969 + 0.791655i $$0.290780\pi$$
−0.610969 + 0.791655i $$0.709220\pi$$
$$138$$ 0 0
$$139$$ −7.42618 −0.629880 −0.314940 0.949112i $$-0.601984\pi$$
−0.314940 + 0.949112i $$0.601984\pi$$
$$140$$ 0 0
$$141$$ −9.48327 −0.798635
$$142$$ 0 0
$$143$$ − 0.137875i − 0.0115297i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.05369i 0.334343i
$$148$$ 0 0
$$149$$ 4.53968 0.371905 0.185952 0.982559i $$-0.440463\pi$$
0.185952 + 0.982559i $$0.440463\pi$$
$$150$$ 0 0
$$151$$ 0.00675366 0.000549605 0 0.000274803 1.00000i $$-0.499913\pi$$
0.000274803 1.00000i $$0.499913\pi$$
$$152$$ 0 0
$$153$$ 2.78654i 0.225279i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 12.1721i − 0.971439i −0.874115 0.485719i $$-0.838558\pi$$
0.874115 0.485719i $$-0.161442\pi$$
$$158$$ 0 0
$$159$$ −10.3465 −0.820529
$$160$$ 0 0
$$161$$ 3.16736 0.249623
$$162$$ 0 0
$$163$$ 20.2201i 1.58376i 0.610677 + 0.791879i $$0.290897\pi$$
−0.610677 + 0.791879i $$0.709103\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 8.04709i − 0.622702i −0.950295 0.311351i $$-0.899218\pi$$
0.950295 0.311351i $$-0.100782\pi$$
$$168$$ 0 0
$$169$$ 10.9165 0.839732
$$170$$ 0 0
$$171$$ −8.49358 −0.649520
$$172$$ 0 0
$$173$$ 16.8394i 1.28028i 0.768259 + 0.640139i $$0.221123\pi$$
−0.768259 + 0.640139i $$0.778877\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 18.2387i 1.37090i
$$178$$ 0 0
$$179$$ −23.7641 −1.77621 −0.888105 0.459641i $$-0.847978\pi$$
−0.888105 + 0.459641i $$0.847978\pi$$
$$180$$ 0 0
$$181$$ −3.13518 −0.233036 −0.116518 0.993189i $$-0.537173\pi$$
−0.116518 + 0.993189i $$0.537173\pi$$
$$182$$ 0 0
$$183$$ − 0.313385i − 0.0231661i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 0.219479i − 0.0160499i
$$188$$ 0 0
$$189$$ 17.8384 1.29755
$$190$$ 0 0
$$191$$ 6.17586 0.446870 0.223435 0.974719i $$-0.428273\pi$$
0.223435 + 0.974719i $$0.428273\pi$$
$$192$$ 0 0
$$193$$ 7.22267i 0.519899i 0.965622 + 0.259949i $$0.0837059\pi$$
−0.965622 + 0.259949i $$0.916294\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 13.0019i 0.926349i 0.886267 + 0.463174i $$0.153290\pi$$
−0.886267 + 0.463174i $$0.846710\pi$$
$$198$$ 0 0
$$199$$ 1.84114 0.130515 0.0652575 0.997868i $$-0.479213\pi$$
0.0652575 + 0.997868i $$0.479213\pi$$
$$200$$ 0 0
$$201$$ −10.0231 −0.706972
$$202$$ 0 0
$$203$$ − 17.0745i − 1.19839i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 1.21273i − 0.0842903i
$$208$$ 0 0
$$209$$ 0.668989 0.0462749
$$210$$ 0 0
$$211$$ −24.8791 −1.71275 −0.856375 0.516354i $$-0.827289\pi$$
−0.856375 + 0.516354i $$0.827289\pi$$
$$212$$ 0 0
$$213$$ 6.93078i 0.474889i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 1.85128i − 0.125673i
$$218$$ 0 0
$$219$$ −2.03721 −0.137662
$$220$$ 0 0
$$221$$ −3.31664 −0.223101
$$222$$ 0 0
$$223$$ − 4.04924i − 0.271157i −0.990767 0.135579i $$-0.956711\pi$$
0.990767 0.135579i $$-0.0432894\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.3916i 1.15432i 0.816630 + 0.577162i $$0.195840\pi$$
−0.816630 + 0.577162i $$0.804160\pi$$
$$228$$ 0 0
$$229$$ 3.51761 0.232450 0.116225 0.993223i $$-0.462921\pi$$
0.116225 + 0.993223i $$0.462921\pi$$
$$230$$ 0 0
$$231$$ −0.404468 −0.0266120
$$232$$ 0 0
$$233$$ 10.7910i 0.706941i 0.935446 + 0.353471i $$0.114999\pi$$
−0.935446 + 0.353471i $$0.885001\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 4.06506i − 0.264054i
$$238$$ 0 0
$$239$$ 2.86944 0.185608 0.0928042 0.995684i $$-0.470417\pi$$
0.0928042 + 0.995684i $$0.470417\pi$$
$$240$$ 0 0
$$241$$ 27.0486 1.74235 0.871176 0.490972i $$-0.163358\pi$$
0.871176 + 0.490972i $$0.163358\pi$$
$$242$$ 0 0
$$243$$ − 11.6939i − 0.750161i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 10.1093i − 0.643242i
$$248$$ 0 0
$$249$$ −21.3780 −1.35478
$$250$$ 0 0
$$251$$ −11.7794 −0.743510 −0.371755 0.928331i $$-0.621244\pi$$
−0.371755 + 0.928331i $$0.621244\pi$$
$$252$$ 0 0
$$253$$ 0.0955192i 0.00600524i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 8.47742i 0.528807i 0.964412 + 0.264404i $$0.0851751\pi$$
−0.964412 + 0.264404i $$0.914825\pi$$
$$258$$ 0 0
$$259$$ −29.4560 −1.83031
$$260$$ 0 0
$$261$$ −6.53751 −0.404662
$$262$$ 0 0
$$263$$ − 18.5118i − 1.14149i −0.821128 0.570745i $$-0.806655\pi$$
0.821128 0.570745i $$-0.193345\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 7.39734i 0.452710i
$$268$$ 0 0
$$269$$ 25.4583 1.55222 0.776109 0.630599i $$-0.217191\pi$$
0.776109 + 0.630599i $$0.217191\pi$$
$$270$$ 0 0
$$271$$ −14.1812 −0.861446 −0.430723 0.902484i $$-0.641741\pi$$
−0.430723 + 0.902484i $$0.641741\pi$$
$$272$$ 0 0
$$273$$ 6.11207i 0.369919i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 21.4281i 1.28749i 0.765241 + 0.643744i $$0.222620\pi$$
−0.765241 + 0.643744i $$0.777380\pi$$
$$278$$ 0 0
$$279$$ −0.708823 −0.0424361
$$280$$ 0 0
$$281$$ −8.90281 −0.531097 −0.265548 0.964098i $$-0.585553\pi$$
−0.265548 + 0.964098i $$0.585553\pi$$
$$282$$ 0 0
$$283$$ 13.0369i 0.774966i 0.921877 + 0.387483i $$0.126655\pi$$
−0.921877 + 0.387483i $$0.873345\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 9.07558i − 0.535715i
$$288$$ 0 0
$$289$$ 11.7203 0.689431
$$290$$ 0 0
$$291$$ −3.45325 −0.202433
$$292$$ 0 0
$$293$$ 6.74772i 0.394206i 0.980383 + 0.197103i $$0.0631533\pi$$
−0.980383 + 0.197103i $$0.936847\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0.537959i 0.0312156i
$$298$$ 0 0
$$299$$ 1.44343 0.0834755
$$300$$ 0 0
$$301$$ −30.0965 −1.73474
$$302$$ 0 0
$$303$$ 8.89854i 0.511208i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 15.6667i 0.894143i 0.894498 + 0.447071i $$0.147533\pi$$
−0.894498 + 0.447071i $$0.852467\pi$$
$$308$$ 0 0
$$309$$ −23.6185 −1.34361
$$310$$ 0 0
$$311$$ 8.27922 0.469472 0.234736 0.972059i $$-0.424577\pi$$
0.234736 + 0.972059i $$0.424577\pi$$
$$312$$ 0 0
$$313$$ − 15.4898i − 0.875534i −0.899088 0.437767i $$-0.855769\pi$$
0.899088 0.437767i $$-0.144231\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 10.5803i − 0.594251i −0.954838 0.297125i $$-0.903972\pi$$
0.954838 0.297125i $$-0.0960279\pi$$
$$318$$ 0 0
$$319$$ 0.514921 0.0288300
$$320$$ 0 0
$$321$$ 7.80256 0.435496
$$322$$ 0 0
$$323$$ − 16.0928i − 0.895427i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 9.38625i 0.519061i
$$328$$ 0 0
$$329$$ 22.4678 1.23869
$$330$$ 0 0
$$331$$ −1.32836 −0.0730133 −0.0365067 0.999333i $$-0.511623\pi$$
−0.0365067 + 0.999333i $$0.511623\pi$$
$$332$$ 0 0
$$333$$ 11.2782i 0.618040i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 33.1297i 1.80469i 0.431014 + 0.902345i $$0.358156\pi$$
−0.431014 + 0.902345i $$0.641844\pi$$
$$338$$ 0 0
$$339$$ −22.8611 −1.24165
$$340$$ 0 0
$$341$$ 0.0558298 0.00302335
$$342$$ 0 0
$$343$$ 12.5675i 0.678582i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 15.7579i 0.845926i 0.906147 + 0.422963i $$0.139010\pi$$
−0.906147 + 0.422963i $$0.860990\pi$$
$$348$$ 0 0
$$349$$ 14.5904 0.781004 0.390502 0.920602i $$-0.372301\pi$$
0.390502 + 0.920602i $$0.372301\pi$$
$$350$$ 0 0
$$351$$ 8.12931 0.433910
$$352$$ 0 0
$$353$$ 28.0097i 1.49080i 0.666615 + 0.745402i $$0.267742\pi$$
−0.666615 + 0.745402i $$0.732258\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 9.72964i 0.514947i
$$358$$ 0 0
$$359$$ −0.214689 −0.0113308 −0.00566542 0.999984i $$-0.501803\pi$$
−0.00566542 + 0.999984i $$0.501803\pi$$
$$360$$ 0 0
$$361$$ 30.0520 1.58168
$$362$$ 0 0
$$363$$ 14.6936i 0.771213i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 21.0197i 1.09722i 0.836079 + 0.548610i $$0.184843\pi$$
−0.836079 + 0.548610i $$0.815157\pi$$
$$368$$ 0 0
$$369$$ −3.47488 −0.180895
$$370$$ 0 0
$$371$$ 24.5129 1.27265
$$372$$ 0 0
$$373$$ 10.2558i 0.531023i 0.964108 + 0.265511i $$0.0855408\pi$$
−0.964108 + 0.265511i $$0.914459\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 7.78116i − 0.400750i
$$378$$ 0 0
$$379$$ 20.6274 1.05956 0.529780 0.848135i $$-0.322274\pi$$
0.529780 + 0.848135i $$0.322274\pi$$
$$380$$ 0 0
$$381$$ 14.4451 0.740044
$$382$$ 0 0
$$383$$ − 18.7419i − 0.957667i −0.877906 0.478833i $$-0.841060\pi$$
0.877906 0.478833i $$-0.158940\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 11.5234i 0.585769i
$$388$$ 0 0
$$389$$ 3.07719 0.156020 0.0780100 0.996953i $$-0.475143\pi$$
0.0780100 + 0.996953i $$0.475143\pi$$
$$390$$ 0 0
$$391$$ 2.29775 0.116202
$$392$$ 0 0
$$393$$ 11.2995i 0.569986i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.4670i 0.575513i 0.957704 + 0.287756i $$0.0929093\pi$$
−0.957704 + 0.287756i $$0.907091\pi$$
$$398$$ 0 0
$$399$$ −29.6566 −1.48469
$$400$$ 0 0
$$401$$ 14.1227 0.705252 0.352626 0.935764i $$-0.385289\pi$$
0.352626 + 0.935764i $$0.385289\pi$$
$$402$$ 0 0
$$403$$ − 0.843666i − 0.0420260i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 0.888314i − 0.0440321i
$$408$$ 0 0
$$409$$ 0.851429 0.0421005 0.0210502 0.999778i $$-0.493299\pi$$
0.0210502 + 0.999778i $$0.493299\pi$$
$$410$$ 0 0
$$411$$ 24.7755 1.22208
$$412$$ 0 0
$$413$$ − 43.2111i − 2.12628i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 9.92799i 0.486176i
$$418$$ 0 0
$$419$$ −0.680728 −0.0332557 −0.0166279 0.999862i $$-0.505293\pi$$
−0.0166279 + 0.999862i $$0.505293\pi$$
$$420$$ 0 0
$$421$$ −35.9671 −1.75293 −0.876466 0.481465i $$-0.840105\pi$$
−0.876466 + 0.481465i $$0.840105\pi$$
$$422$$ 0 0
$$423$$ − 8.60251i − 0.418268i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.742472i 0.0359307i
$$428$$ 0 0
$$429$$ −0.184324 −0.00889923
$$430$$ 0 0
$$431$$ 15.2881 0.736402 0.368201 0.929746i $$-0.379974\pi$$
0.368201 + 0.929746i $$0.379974\pi$$
$$432$$ 0 0
$$433$$ 13.2632i 0.637390i 0.947857 + 0.318695i $$0.103245\pi$$
−0.947857 + 0.318695i $$0.896755\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.00371i 0.335033i
$$438$$ 0 0
$$439$$ −18.7740 −0.896035 −0.448018 0.894025i $$-0.647870\pi$$
−0.448018 + 0.894025i $$0.647870\pi$$
$$440$$ 0 0
$$441$$ −3.67720 −0.175105
$$442$$ 0 0
$$443$$ − 5.49270i − 0.260966i −0.991451 0.130483i $$-0.958347\pi$$
0.991451 0.130483i $$-0.0416528\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 6.06905i − 0.287056i
$$448$$ 0 0
$$449$$ −13.1705 −0.621553 −0.310777 0.950483i $$-0.600589\pi$$
−0.310777 + 0.950483i $$0.600589\pi$$
$$450$$ 0 0
$$451$$ 0.273695 0.0128878
$$452$$ 0 0
$$453$$ − 0.00902890i 0 0.000424215i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 20.9676i − 0.980821i −0.871492 0.490410i $$-0.836847\pi$$
0.871492 0.490410i $$-0.163153\pi$$
$$458$$ 0 0
$$459$$ 12.9408 0.604026
$$460$$ 0 0
$$461$$ 6.27335 0.292179 0.146089 0.989271i $$-0.453331\pi$$
0.146089 + 0.989271i $$0.453331\pi$$
$$462$$ 0 0
$$463$$ 9.49565i 0.441300i 0.975353 + 0.220650i $$0.0708179\pi$$
−0.975353 + 0.220650i $$0.929182\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 13.9066i − 0.643518i −0.946822 0.321759i $$-0.895726\pi$$
0.946822 0.321759i $$-0.104274\pi$$
$$468$$ 0 0
$$469$$ 23.7466 1.09652
$$470$$ 0 0
$$471$$ −16.2727 −0.749809
$$472$$ 0 0
$$473$$ − 0.907631i − 0.0417329i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 9.38555i − 0.429735i
$$478$$ 0 0
$$479$$ −3.66220 −0.167330 −0.0836652 0.996494i $$-0.526663\pi$$
−0.0836652 + 0.996494i $$0.526663\pi$$
$$480$$ 0 0
$$481$$ −13.4237 −0.612066
$$482$$ 0 0
$$483$$ − 4.23441i − 0.192672i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 28.9654i − 1.31255i −0.754523 0.656274i $$-0.772132\pi$$
0.754523 0.656274i $$-0.227868\pi$$
$$488$$ 0 0
$$489$$ 27.0320 1.22243
$$490$$ 0 0
$$491$$ −37.8991 −1.71036 −0.855182 0.518328i $$-0.826554\pi$$
−0.855182 + 0.518328i $$0.826554\pi$$
$$492$$ 0 0
$$493$$ − 12.3866i − 0.557866i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 16.4204i − 0.736557i
$$498$$ 0 0
$$499$$ 26.6914 1.19487 0.597436 0.801917i $$-0.296186\pi$$
0.597436 + 0.801917i $$0.296186\pi$$
$$500$$ 0 0
$$501$$ −10.7581 −0.480635
$$502$$ 0 0
$$503$$ − 18.2182i − 0.812309i −0.913804 0.406154i $$-0.866870\pi$$
0.913804 0.406154i $$-0.133130\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 14.5942i − 0.648150i
$$508$$ 0 0
$$509$$ 31.0138 1.37466 0.687332 0.726344i $$-0.258782\pi$$
0.687332 + 0.726344i $$0.258782\pi$$
$$510$$ 0 0
$$511$$ 4.82655 0.213514
$$512$$ 0 0
$$513$$ 39.4446i 1.74152i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0.677568i 0.0297994i
$$518$$ 0 0
$$519$$ 22.5125 0.988188
$$520$$ 0 0
$$521$$ −39.9824 −1.75166 −0.875831 0.482618i $$-0.839686\pi$$
−0.875831 + 0.482618i $$0.839686\pi$$
$$522$$ 0 0
$$523$$ 10.3111i 0.450873i 0.974258 + 0.225437i $$0.0723808\pi$$
−0.974258 + 0.225437i $$0.927619\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1.34301i − 0.0585024i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −16.5448 −0.717981
$$532$$ 0 0
$$533$$ − 4.13592i − 0.179146i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 31.7699i 1.37097i
$$538$$ 0 0
$$539$$ 0.289631 0.0124753
$$540$$ 0 0
$$541$$ −42.5472 −1.82925 −0.914623 0.404307i $$-0.867513\pi$$
−0.914623 + 0.404307i $$0.867513\pi$$
$$542$$ 0 0
$$543$$ 4.19139i 0.179870i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 2.47986i − 0.106031i −0.998594 0.0530156i $$-0.983117\pi$$
0.998594 0.0530156i $$-0.0168833\pi$$
$$548$$ 0 0
$$549$$ 0.284279 0.0121327
$$550$$ 0 0
$$551$$ 37.7553 1.60843
$$552$$ 0 0
$$553$$ 9.63094i 0.409549i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 22.9814i − 0.973755i −0.873470 0.486877i $$-0.838136\pi$$
0.873470 0.486877i $$-0.161864\pi$$
$$558$$ 0 0
$$559$$ −13.7156 −0.580107
$$560$$ 0 0
$$561$$ −0.293420 −0.0123882
$$562$$ 0 0
$$563$$ − 18.4939i − 0.779427i −0.920936 0.389713i $$-0.872574\pi$$
0.920936 0.389713i $$-0.127426\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 12.3246i − 0.517584i
$$568$$ 0 0
$$569$$ −35.6081 −1.49277 −0.746385 0.665514i $$-0.768212\pi$$
−0.746385 + 0.665514i $$0.768212\pi$$
$$570$$ 0 0
$$571$$ 20.5827 0.861359 0.430680 0.902505i $$-0.358274\pi$$
0.430680 + 0.902505i $$0.358274\pi$$
$$572$$ 0 0
$$573$$ − 8.25645i − 0.344918i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 19.0378i 0.792554i 0.918131 + 0.396277i $$0.129698\pi$$
−0.918131 + 0.396277i $$0.870302\pi$$
$$578$$ 0 0
$$579$$ 9.65591 0.401286
$$580$$ 0 0
$$581$$ 50.6489 2.10127
$$582$$ 0 0
$$583$$ 0.739244i 0.0306163i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 7.29062i − 0.300916i −0.988616 0.150458i $$-0.951925\pi$$
0.988616 0.150458i $$-0.0480748\pi$$
$$588$$ 0 0
$$589$$ 4.09358 0.168673
$$590$$ 0 0
$$591$$ 17.3821 0.715006
$$592$$ 0 0
$$593$$ 10.5832i 0.434598i 0.976105 + 0.217299i $$0.0697247\pi$$
−0.976105 + 0.217299i $$0.930275\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 2.46140i − 0.100739i
$$598$$ 0 0
$$599$$ 5.63520 0.230248 0.115124 0.993351i $$-0.463273\pi$$
0.115124 + 0.993351i $$0.463273\pi$$
$$600$$ 0 0
$$601$$ −17.7093 −0.722377 −0.361188 0.932493i $$-0.617629\pi$$
−0.361188 + 0.932493i $$0.617629\pi$$
$$602$$ 0 0
$$603$$ − 9.09216i − 0.370261i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 22.0996i − 0.896996i −0.893784 0.448498i $$-0.851959\pi$$
0.893784 0.448498i $$-0.148041\pi$$
$$608$$ 0 0
$$609$$ −22.8267 −0.924984
$$610$$ 0 0
$$611$$ 10.2390 0.414225
$$612$$ 0 0
$$613$$ 5.08532i 0.205394i 0.994713 + 0.102697i $$0.0327472\pi$$
−0.994713 + 0.102697i $$0.967253\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 7.63073i − 0.307202i −0.988133 0.153601i $$-0.950913\pi$$
0.988133 0.153601i $$-0.0490870\pi$$
$$618$$ 0 0
$$619$$ −38.5591 −1.54982 −0.774910 0.632072i $$-0.782205\pi$$
−0.774910 + 0.632072i $$0.782205\pi$$
$$620$$ 0 0
$$621$$ −5.63195 −0.226002
$$622$$ 0 0
$$623$$ − 17.5258i − 0.702156i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 0.894364i − 0.0357175i
$$628$$ 0 0
$$629$$ −21.3688 −0.852028
$$630$$ 0 0
$$631$$ −1.86751 −0.0743444 −0.0371722 0.999309i $$-0.511835\pi$$
−0.0371722 + 0.999309i $$0.511835\pi$$
$$632$$ 0 0
$$633$$ 33.2607i 1.32199i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 4.37673i − 0.173412i
$$638$$ 0 0
$$639$$ −6.28708 −0.248713
$$640$$ 0 0
$$641$$ −1.91236 −0.0755338 −0.0377669 0.999287i $$-0.512024\pi$$
−0.0377669 + 0.999287i $$0.512024\pi$$
$$642$$ 0 0
$$643$$ 5.06670i 0.199811i 0.994997 + 0.0999055i $$0.0318541\pi$$
−0.994997 + 0.0999055i $$0.968146\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 37.0429i − 1.45631i −0.685414 0.728153i $$-0.740379\pi$$
0.685414 0.728153i $$-0.259621\pi$$
$$648$$ 0 0
$$649$$ 1.30313 0.0511524
$$650$$ 0 0
$$651$$ −2.47496 −0.0970014
$$652$$ 0 0
$$653$$ − 11.9884i − 0.469143i −0.972099 0.234572i $$-0.924631\pi$$
0.972099 0.234572i $$-0.0753687\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 1.84800i − 0.0720974i
$$658$$ 0 0
$$659$$ 2.82513 0.110051 0.0550257 0.998485i $$-0.482476\pi$$
0.0550257 + 0.998485i $$0.482476\pi$$
$$660$$ 0 0
$$661$$ −26.5004 −1.03075 −0.515374 0.856965i $$-0.672347\pi$$
−0.515374 + 0.856965i $$0.672347\pi$$
$$662$$ 0 0
$$663$$ 4.43398i 0.172202i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.39076i 0.208731i
$$668$$ 0 0
$$669$$ −5.41339 −0.209294
$$670$$ 0 0
$$671$$ −0.0223910 −0.000864393 0
$$672$$ 0 0
$$673$$ − 19.0620i − 0.734784i −0.930066 0.367392i $$-0.880251\pi$$
0.930066 0.367392i $$-0.119749\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.54346i 0.0977533i 0.998805 + 0.0488766i $$0.0155641\pi$$
−0.998805 + 0.0488766i $$0.984436\pi$$
$$678$$ 0 0
$$679$$ 8.18146 0.313975
$$680$$ 0 0
$$681$$ 23.2507 0.890969
$$682$$ 0 0
$$683$$ 0.377109i 0.0144297i 0.999974 + 0.00721483i $$0.00229657\pi$$
−0.999974 + 0.00721483i $$0.997703\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 4.70266i − 0.179418i
$$688$$ 0 0
$$689$$ 11.1710 0.425581
$$690$$ 0 0
$$691$$ 7.24743 0.275705 0.137853 0.990453i $$-0.455980\pi$$
0.137853 + 0.990453i $$0.455980\pi$$
$$692$$ 0 0
$$693$$ − 0.366903i − 0.0139375i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 6.58385i − 0.249381i
$$698$$ 0 0
$$699$$ 14.4264 0.545655
$$700$$ 0 0
$$701$$ −15.7381 −0.594420 −0.297210 0.954812i $$-0.596056\pi$$
−0.297210 + 0.954812i $$0.596056\pi$$
$$702$$ 0 0
$$703$$ − 65.1335i − 2.45656i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 21.0824i − 0.792887i
$$708$$ 0 0
$$709$$ 45.9062 1.72404 0.862022 0.506870i $$-0.169198\pi$$
0.862022 + 0.506870i $$0.169198\pi$$
$$710$$ 0 0
$$711$$ 3.68751 0.138293
$$712$$ 0 0
$$713$$ 0.584488i 0.0218892i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 3.83612i − 0.143263i
$$718$$ 0 0
$$719$$ 50.4975 1.88324 0.941619 0.336682i $$-0.109305\pi$$
0.941619 + 0.336682i $$0.109305\pi$$
$$720$$ 0 0
$$721$$ 55.9570 2.08395
$$722$$ 0 0
$$723$$ − 36.1609i − 1.34484i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 19.0273i − 0.705685i −0.935683 0.352842i $$-0.885215\pi$$
0.935683 0.352842i $$-0.114785\pi$$
$$728$$ 0 0
$$729$$ −27.3067 −1.01136
$$730$$ 0 0
$$731$$ −21.8334 −0.807539
$$732$$ 0 0
$$733$$ 42.1027i 1.55510i 0.628822 + 0.777549i $$0.283537\pi$$
−0.628822 + 0.777549i $$0.716463\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0.716135i 0.0263792i
$$738$$ 0 0
$$739$$ −47.9212 −1.76281 −0.881405 0.472361i $$-0.843402\pi$$
−0.881405 + 0.472361i $$0.843402\pi$$
$$740$$ 0 0
$$741$$ −13.5151 −0.496489
$$742$$ 0 0
$$743$$ − 21.3492i − 0.783225i −0.920130 0.391613i $$-0.871917\pi$$
0.920130 0.391613i $$-0.128083\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 19.3925i − 0.709536i
$$748$$ 0 0
$$749$$ −18.4858 −0.675458
$$750$$ 0 0
$$751$$ −27.8218 −1.01523 −0.507617 0.861583i $$-0.669473\pi$$
−0.507617 + 0.861583i $$0.669473\pi$$
$$752$$ 0 0
$$753$$ 15.7478i 0.573881i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8.74990i 0.318020i 0.987277 + 0.159010i $$0.0508303\pi$$
−0.987277 + 0.159010i $$0.949170\pi$$
$$758$$ 0 0
$$759$$ 0.127699 0.00463517
$$760$$ 0 0
$$761$$ 21.9923 0.797221 0.398610 0.917120i $$-0.369493\pi$$
0.398610 + 0.917120i $$0.369493\pi$$
$$762$$ 0 0
$$763$$ − 22.2379i − 0.805066i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 19.6921i − 0.711041i
$$768$$ 0 0
$$769$$ 31.3229 1.12953 0.564766 0.825251i $$-0.308966\pi$$
0.564766 + 0.825251i $$0.308966\pi$$
$$770$$ 0 0
$$771$$ 11.3334 0.408162
$$772$$ 0 0
$$773$$ − 32.3308i − 1.16286i −0.813597 0.581429i $$-0.802494\pi$$
0.813597 0.581429i $$-0.197506\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 39.3794i 1.41273i
$$778$$ 0 0
$$779$$ 20.0680 0.719012
$$780$$ 0 0
$$781$$ 0.495196 0.0177195
$$782$$ 0 0
$$783$$ 30.3605i 1.08499i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 18.5712i − 0.661990i −0.943632 0.330995i $$-0.892616\pi$$
0.943632 0.330995i $$-0.107384\pi$$
$$788$$ 0 0
$$789$$ −24.7483 −0.881063
$$790$$ 0 0
$$791$$ 54.1627 1.92580
$$792$$ 0 0
$$793$$ 0.338358i 0.0120155i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 24.1985i − 0.857154i −0.903505 0.428577i $$-0.859015\pi$$
0.903505 0.428577i $$-0.140985\pi$$
$$798$$ 0 0
$$799$$ 16.2992 0.576624
$$800$$ 0 0
$$801$$ −6.71031 −0.237097
$$802$$ 0 0
$$803$$ 0.145556i 0.00513656i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 34.0349i − 1.19809i
$$808$$ 0 0
$$809$$ 30.4574 1.07082 0.535412 0.844591i $$-0.320156\pi$$
0.535412 + 0.844591i $$0.320156\pi$$
$$810$$ 0 0
$$811$$ 4.05626 0.142435 0.0712174 0.997461i $$-0.477312\pi$$
0.0712174 + 0.997461i $$0.477312\pi$$
$$812$$ 0 0
$$813$$ 18.9587i 0.664910i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 66.5499i − 2.32829i
$$818$$ 0 0
$$819$$ −5.54441 −0.193737
$$820$$ 0 0
$$821$$ 19.0615 0.665252 0.332626 0.943059i $$-0.392065\pi$$
0.332626 + 0.943059i $$0.392065\pi$$
$$822$$ 0 0
$$823$$ 42.9810i 1.49822i 0.662444 + 0.749111i $$0.269519\pi$$
−0.662444 + 0.749111i $$0.730481\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 51.6777i 1.79701i 0.438963 + 0.898505i $$0.355346\pi$$
−0.438963 + 0.898505i $$0.644654\pi$$
$$828$$ 0 0
$$829$$ −25.6228 −0.889915 −0.444958 0.895552i $$-0.646781\pi$$
−0.444958 + 0.895552i $$0.646781\pi$$
$$830$$ 0 0
$$831$$ 28.6470 0.993753
$$832$$ 0 0
$$833$$ − 6.96720i − 0.241399i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.29181i 0.113781i
$$838$$ 0 0
$$839$$ 53.0313 1.83084 0.915422 0.402495i $$-0.131857\pi$$
0.915422 + 0.402495i $$0.131857\pi$$
$$840$$ 0 0
$$841$$ 0.0602571 0.00207783
$$842$$ 0 0
$$843$$ 11.9021i 0.409929i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 34.8121i − 1.19616i
$$848$$ 0 0
$$849$$ 17.4290 0.598160
$$850$$ 0 0
$$851$$ 9.29985 0.318795
$$852$$ 0 0
$$853$$ − 13.8668i − 0.474791i −0.971413 0.237395i $$-0.923706\pi$$
0.971413 0.237395i $$-0.0762937\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.3518i 0.558567i 0.960209 + 0.279283i $$0.0900969\pi$$
−0.960209 + 0.279283i $$0.909903\pi$$
$$858$$ 0 0
$$859$$ −42.1443 −1.43795 −0.718973 0.695038i $$-0.755388\pi$$
−0.718973 + 0.695038i $$0.755388\pi$$
$$860$$ 0 0
$$861$$ −12.1331 −0.413493
$$862$$ 0 0
$$863$$ 15.2130i 0.517856i 0.965897 + 0.258928i $$0.0833692\pi$$
−0.965897 + 0.258928i $$0.916631\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 15.6688i − 0.532140i
$$868$$ 0 0
$$869$$ −0.290443 −0.00985262
$$870$$ 0 0
$$871$$ 10.8218 0.366683
$$872$$ 0 0
$$873$$ − 3.13253i − 0.106020i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 53.6934i − 1.81310i −0.422101 0.906549i $$-0.638707\pi$$
0.422101 0.906549i $$-0.361293\pi$$
$$878$$ 0 0
$$879$$ 9.02096 0.304269
$$880$$ 0 0
$$881$$ 46.5787 1.56928 0.784638 0.619954i $$-0.212849\pi$$
0.784638 + 0.619954i $$0.212849\pi$$
$$882$$ 0 0
$$883$$ 19.9787i 0.672337i 0.941802 + 0.336169i $$0.109131\pi$$
−0.941802 + 0.336169i $$0.890869\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 51.9251i 1.74347i 0.489974 + 0.871737i $$0.337006\pi$$
−0.489974 + 0.871737i $$0.662994\pi$$
$$888$$ 0 0
$$889$$ −34.2233 −1.14781
$$890$$ 0 0
$$891$$ 0.371676 0.0124516
$$892$$ 0 0
$$893$$ 49.6810i 1.66251i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 1.92970i − 0.0644309i
$$898$$ 0 0
$$899$$ 3.15083 0.105086
$$900$$ 0 0
$$901$$ 17.7828 0.592431
$$902$$ 0 0
$$903$$ 40.2358i 1.33896i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 6.64335i − 0.220589i −0.993899 0.110294i $$-0.964821\pi$$
0.993899 0.110294i $$-0.0351794\pi$$
$$908$$ 0 0
$$909$$ −8.07209 −0.267734
$$910$$ 0 0
$$911$$ 15.2064 0.503811 0.251905 0.967752i $$-0.418943\pi$$
0.251905 + 0.967752i $$0.418943\pi$$
$$912$$ 0 0
$$913$$ 1.52744i 0.0505507i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 26.7709i − 0.884052i
$$918$$ 0 0
$$919$$ −41.2068 −1.35929 −0.679644 0.733542i $$-0.737866\pi$$
−0.679644 + 0.733542i $$0.737866\pi$$
$$920$$ 0 0
$$921$$ 20.9446 0.690148
$$922$$ 0 0
$$923$$ − 7.48310i − 0.246309i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 21.4249i − 0.703687i
$$928$$ 0 0
$$929$$ 4.86715 0.159686 0.0798431 0.996807i $$-0.474558\pi$$
0.0798431 + 0.996807i $$0.474558\pi$$
$$930$$ 0 0
$$931$$ 21.2365 0.695999
$$932$$ 0 0
$$933$$ − 11.0684i − 0.362364i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 3.44469i − 0.112533i −0.998416 0.0562666i $$-0.982080\pi$$
0.998416 0.0562666i $$-0.0179197\pi$$
$$938$$ 0 0
$$939$$ −20.7081 −0.675784
$$940$$ 0 0
$$941$$ −34.8591 −1.13637 −0.568187 0.822900i $$-0.692355\pi$$
−0.568187 + 0.822900i $$0.692355\pi$$
$$942$$ 0 0
$$943$$ 2.86534i 0.0933084i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 5.17298i − 0.168099i −0.996462 0.0840496i $$-0.973215\pi$$
0.996462 0.0840496i $$-0.0267854\pi$$
$$948$$ 0 0
$$949$$ 2.19955 0.0714005
$$950$$ 0 0
$$951$$ −14.1447 −0.458675
$$952$$ 0 0
$$953$$ − 3.41090i − 0.110490i −0.998473 0.0552449i $$-0.982406\pi$$
0.998473 0.0552449i $$-0.0175940\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 0.688392i − 0.0222526i
$$958$$ 0 0
$$959$$ −58.6981 −1.89546
$$960$$ 0 0
$$961$$ −30.6584 −0.988980
$$962$$ 0 0
$$963$$ 7.07789i 0.228082i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 17.5047i 0.562914i 0.959574 + 0.281457i $$0.0908176\pi$$
−0.959574 + 0.281457i $$0.909182\pi$$
$$968$$ 0 0
$$969$$ −21.5143 −0.691139
$$970$$ 0 0
$$971$$ 35.9954 1.15515 0.577574 0.816338i $$-0.303999\pi$$
0.577574 + 0.816338i $$0.303999\pi$$
$$972$$ 0 0
$$973$$ − 23.5214i − 0.754062i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 31.3634i − 1.00340i −0.865040 0.501702i $$-0.832707\pi$$
0.865040 0.501702i $$-0.167293\pi$$
$$978$$ 0 0
$$979$$ 0.528531 0.0168919
$$980$$ 0 0
$$981$$ −8.51450 −0.271847
$$982$$ 0 0
$$983$$ 36.4262i 1.16182i 0.813969 + 0.580908i $$0.197302\pi$$
−0.813969 + 0.580908i $$0.802698\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 30.0369i − 0.956086i
$$988$$ 0 0
$$989$$ 9.50208 0.302149
$$990$$ 0 0
$$991$$ 6.22316 0.197685 0.0988426 0.995103i $$-0.468486\pi$$
0.0988426 + 0.995103i $$0.468486\pi$$
$$992$$ 0 0
$$993$$ 1.77587i 0.0563556i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 24.0896i − 0.762925i −0.924384 0.381463i $$-0.875420\pi$$
0.924384 0.381463i $$-0.124580\pi$$
$$998$$ 0 0
$$999$$ 52.3763 1.65711
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 4600.2.e.v.4049.4 10
5.2 odd 4 4600.2.a.bc.1.3 5
5.3 odd 4 4600.2.a.bg.1.3 yes 5
5.4 even 2 inner 4600.2.e.v.4049.7 10
20.3 even 4 9200.2.a.cs.1.3 5
20.7 even 4 9200.2.a.cw.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.3 5 5.2 odd 4
4600.2.a.bg.1.3 yes 5 5.3 odd 4
4600.2.e.v.4049.4 10 1.1 even 1 trivial
4600.2.e.v.4049.7 10 5.4 even 2 inner
9200.2.a.cs.1.3 5 20.3 even 4
9200.2.a.cw.1.3 5 20.7 even 4