# Properties

 Label 9200.2.a.cw.1.3 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.791953.1 Defining polynomial: $$x^{5} - 2 x^{4} - 7 x^{3} + 7 x^{2} + 9 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.336890$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.33689 q^{3} +3.16736 q^{7} -1.21273 q^{9} +O(q^{10})$$ $$q+1.33689 q^{3} +3.16736 q^{7} -1.21273 q^{9} +0.0955192 q^{11} +1.44343 q^{13} -2.29775 q^{17} -7.00371 q^{19} +4.23441 q^{21} +1.00000 q^{23} -5.63195 q^{27} +5.39076 q^{29} +0.584488 q^{31} +0.127699 q^{33} -9.29985 q^{37} +1.92970 q^{39} -2.86534 q^{41} -9.50208 q^{43} -7.09353 q^{47} +3.03218 q^{49} -3.07184 q^{51} -7.73922 q^{53} -9.36319 q^{57} -13.6426 q^{59} +0.234413 q^{61} -3.84114 q^{63} -7.49729 q^{67} +1.33689 q^{69} +5.18426 q^{71} -1.52384 q^{73} +0.302544 q^{77} +3.04068 q^{79} -3.89112 q^{81} +15.9909 q^{83} +7.20685 q^{87} +5.53325 q^{89} +4.57186 q^{91} +0.781396 q^{93} +2.58305 q^{97} -0.115839 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} - q^{7} + 4 q^{9} + O(q^{10})$$ $$5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49} + 5 q^{51} - 18 q^{53} - 27 q^{57} - 23 q^{59} - 26 q^{61} + 5 q^{63} + 3 q^{67} + 3 q^{69} + 2 q^{71} - 4 q^{73} - 15 q^{77} - 43 q^{79} - 3 q^{81} + 30 q^{83} + 27 q^{87} + 15 q^{89} + 19 q^{91} + 15 q^{93} - 8 q^{97} - 37 q^{99} + O(q^{100})$$

## 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
$$3$$ 1.33689 0.771854 0.385927 0.922529i $$-0.373882\pi$$
0.385927 + 0.922529i $$0.373882\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.16736 1.19715 0.598575 0.801067i $$-0.295734\pi$$
0.598575 + 0.801067i $$0.295734\pi$$
$$8$$ 0 0
$$9$$ −1.21273 −0.404242
$$10$$ 0 0
$$11$$ 0.0955192 0.0288001 0.0144001 0.999896i $$-0.495416\pi$$
0.0144001 + 0.999896i $$0.495416\pi$$
$$12$$ 0 0
$$13$$ 1.44343 0.400335 0.200167 0.979762i $$-0.435851\pi$$
0.200167 + 0.979762i $$0.435851\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.29775 −0.557287 −0.278643 0.960395i $$-0.589885\pi$$
−0.278643 + 0.960395i $$0.589885\pi$$
$$18$$ 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
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.63195 −1.08387
$$28$$ 0 0
$$29$$ 5.39076 1.00104 0.500519 0.865725i $$-0.333142\pi$$
0.500519 + 0.865725i $$0.333142\pi$$
$$30$$ 0 0
$$31$$ 0.584488 0.104977 0.0524886 0.998622i $$-0.483285\pi$$
0.0524886 + 0.998622i $$0.483285\pi$$
$$32$$ 0 0
$$33$$ 0.127699 0.0222295
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.29985 −1.52889 −0.764443 0.644691i $$-0.776986\pi$$
−0.764443 + 0.644691i $$0.776986\pi$$
$$38$$ 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
$$43$$ −9.50208 −1.44905 −0.724527 0.689246i $$-0.757942\pi$$
−0.724527 + 0.689246i $$0.757942\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.09353 −1.03470 −0.517349 0.855775i $$-0.673081\pi$$
−0.517349 + 0.855775i $$0.673081\pi$$
$$48$$ 0 0
$$49$$ 3.03218 0.433168
$$50$$ 0 0
$$51$$ −3.07184 −0.430144
$$52$$ 0 0
$$53$$ −7.73922 −1.06306 −0.531532 0.847038i $$-0.678383\pi$$
−0.531532 + 0.847038i $$0.678383\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −9.36319 −1.24018
$$58$$ 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
$$63$$ −3.84114 −0.483938
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.49729 −0.915940 −0.457970 0.888968i $$-0.651423\pi$$
−0.457970 + 0.888968i $$0.651423\pi$$
$$68$$ 0 0
$$69$$ 1.33689 0.160943
$$70$$ 0 0
$$71$$ 5.18426 0.615258 0.307629 0.951506i $$-0.400464\pi$$
0.307629 + 0.951506i $$0.400464\pi$$
$$72$$ 0 0
$$73$$ −1.52384 −0.178352 −0.0891760 0.996016i $$-0.528423\pi$$
−0.0891760 + 0.996016i $$0.528423\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0.302544 0.0344781
$$78$$ 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
$$83$$ 15.9909 1.75523 0.877613 0.479369i $$-0.159135\pi$$
0.877613 + 0.479369i $$0.159135\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7.20685 0.772655
$$88$$ 0 0
$$89$$ 5.53325 0.586523 0.293261 0.956032i $$-0.405259\pi$$
0.293261 + 0.956032i $$0.405259\pi$$
$$90$$ 0 0
$$91$$ 4.57186 0.479261
$$92$$ 0 0
$$93$$ 0.781396 0.0810270
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.58305 0.262269 0.131135 0.991365i $$-0.458138\pi$$
0.131135 + 0.991365i $$0.458138\pi$$
$$98$$ 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
$$103$$ 17.6668 1.74076 0.870378 0.492383i $$-0.163874\pi$$
0.870378 + 0.492383i $$0.163874\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 5.83635 0.564221 0.282111 0.959382i $$-0.408965\pi$$
0.282111 + 0.959382i $$0.408965\pi$$
$$108$$ 0 0
$$109$$ 7.02096 0.672486 0.336243 0.941775i $$-0.390844\pi$$
0.336243 + 0.941775i $$0.390844\pi$$
$$110$$ 0 0
$$111$$ −12.4329 −1.18008
$$112$$ 0 0
$$113$$ −17.1003 −1.60866 −0.804328 0.594185i $$-0.797475\pi$$
−0.804328 + 0.594185i $$0.797475\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.75048 −0.161832
$$118$$ 0 0
$$119$$ −7.27781 −0.667156
$$120$$ 0 0
$$121$$ −10.9909 −0.999171
$$122$$ 0 0
$$123$$ −3.83065 −0.345398
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.8050 0.958787 0.479394 0.877600i $$-0.340857\pi$$
0.479394 + 0.877600i $$0.340857\pi$$
$$128$$ 0 0
$$129$$ −12.7032 −1.11846
$$130$$ 0 0
$$131$$ 8.45211 0.738464 0.369232 0.929337i $$-0.379621\pi$$
0.369232 + 0.929337i $$0.379621\pi$$
$$132$$ 0 0
$$133$$ −22.1833 −1.92354
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.5322 −1.58331 −0.791655 0.610969i $$-0.790780\pi$$
−0.791655 + 0.610969i $$0.790780\pi$$
$$138$$ 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
$$143$$ 0.137875 0.0115297
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.05369 0.334343
$$148$$ 0 0
$$149$$ −4.53968 −0.371905 −0.185952 0.982559i $$-0.559537\pi$$
−0.185952 + 0.982559i $$0.559537\pi$$
$$150$$ 0 0
$$151$$ −0.00675366 −0.000549605 0 −0.000274803 1.00000i $$-0.500087\pi$$
−0.000274803 1.00000i $$0.500087\pi$$
$$152$$ 0 0
$$153$$ 2.78654 0.225279
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.1721 0.971439 0.485719 0.874115i $$-0.338558\pi$$
0.485719 + 0.874115i $$0.338558\pi$$
$$158$$ 0 0
$$159$$ −10.3465 −0.820529
$$160$$ 0 0
$$161$$ 3.16736 0.249623
$$162$$ 0 0
$$163$$ −20.2201 −1.58376 −0.791879 0.610677i $$-0.790897\pi$$
−0.791879 + 0.610677i $$0.790897\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.04709 −0.622702 −0.311351 0.950295i $$-0.600782\pi$$
−0.311351 + 0.950295i $$0.600782\pi$$
$$168$$ 0 0
$$169$$ −10.9165 −0.839732
$$170$$ 0 0
$$171$$ 8.49358 0.649520
$$172$$ 0 0
$$173$$ 16.8394 1.28028 0.640139 0.768259i $$-0.278877\pi$$
0.640139 + 0.768259i $$0.278877\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −18.2387 −1.37090
$$178$$ 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
$$183$$ 0.313385 0.0231661
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.219479 −0.0160499
$$188$$ 0 0
$$189$$ −17.8384 −1.29755
$$190$$ 0 0
$$191$$ −6.17586 −0.446870 −0.223435 0.974719i $$-0.571727\pi$$
−0.223435 + 0.974719i $$0.571727\pi$$
$$192$$ 0 0
$$193$$ 7.22267 0.519899 0.259949 0.965622i $$-0.416294\pi$$
0.259949 + 0.965622i $$0.416294\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −13.0019 −0.926349 −0.463174 0.886267i $$-0.653290\pi$$
−0.463174 + 0.886267i $$0.653290\pi$$
$$198$$ 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
$$203$$ 17.0745 1.19839
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1.21273 −0.0842903
$$208$$ 0 0
$$209$$ −0.668989 −0.0462749
$$210$$ 0 0
$$211$$ 24.8791 1.71275 0.856375 0.516354i $$-0.172711\pi$$
0.856375 + 0.516354i $$0.172711\pi$$
$$212$$ 0 0
$$213$$ 6.93078 0.474889
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1.85128 0.125673
$$218$$ 0 0
$$219$$ −2.03721 −0.137662
$$220$$ 0 0
$$221$$ −3.31664 −0.223101
$$222$$ 0 0
$$223$$ 4.04924 0.271157 0.135579 0.990767i $$-0.456711\pi$$
0.135579 + 0.990767i $$0.456711\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.3916 1.15432 0.577162 0.816630i $$-0.304160\pi$$
0.577162 + 0.816630i $$0.304160\pi$$
$$228$$ 0 0
$$229$$ −3.51761 −0.232450 −0.116225 0.993223i $$-0.537079\pi$$
−0.116225 + 0.993223i $$0.537079\pi$$
$$230$$ 0 0
$$231$$ 0.404468 0.0266120
$$232$$ 0 0
$$233$$ 10.7910 0.706941 0.353471 0.935446i $$-0.385001\pi$$
0.353471 + 0.935446i $$0.385001\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4.06506 0.264054
$$238$$ 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
$$243$$ 11.6939 0.750161
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −10.1093 −0.643242
$$248$$ 0 0
$$249$$ 21.3780 1.35478
$$250$$ 0 0
$$251$$ 11.7794 0.743510 0.371755 0.928331i $$-0.378756\pi$$
0.371755 + 0.928331i $$0.378756\pi$$
$$252$$ 0 0
$$253$$ 0.0955192 0.00600524
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −8.47742 −0.528807 −0.264404 0.964412i $$-0.585175\pi$$
−0.264404 + 0.964412i $$0.585175\pi$$
$$258$$ 0 0
$$259$$ −29.4560 −1.83031
$$260$$ 0 0
$$261$$ −6.53751 −0.404662
$$262$$ 0 0
$$263$$ 18.5118 1.14149 0.570745 0.821128i $$-0.306655\pi$$
0.570745 + 0.821128i $$0.306655\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 7.39734 0.452710
$$268$$ 0 0
$$269$$ −25.4583 −1.55222 −0.776109 0.630599i $$-0.782809\pi$$
−0.776109 + 0.630599i $$0.782809\pi$$
$$270$$ 0 0
$$271$$ 14.1812 0.861446 0.430723 0.902484i $$-0.358259\pi$$
0.430723 + 0.902484i $$0.358259\pi$$
$$272$$ 0 0
$$273$$ 6.11207 0.369919
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −21.4281 −1.28749 −0.643744 0.765241i $$-0.722620\pi$$
−0.643744 + 0.765241i $$0.722620\pi$$
$$278$$ 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
$$283$$ −13.0369 −0.774966 −0.387483 0.921877i $$-0.626655\pi$$
−0.387483 + 0.921877i $$0.626655\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −9.07558 −0.535715
$$288$$ 0 0
$$289$$ −11.7203 −0.689431
$$290$$ 0 0
$$291$$ 3.45325 0.202433
$$292$$ 0 0
$$293$$ 6.74772 0.394206 0.197103 0.980383i $$-0.436847\pi$$
0.197103 + 0.980383i $$0.436847\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −0.537959 −0.0312156
$$298$$ 0 0
$$299$$ 1.44343 0.0834755
$$300$$ 0 0
$$301$$ −30.0965 −1.73474
$$302$$ 0 0
$$303$$ −8.89854 −0.511208
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 15.6667 0.894143 0.447071 0.894498i $$-0.352467\pi$$
0.447071 + 0.894498i $$0.352467\pi$$
$$308$$ 0 0
$$309$$ 23.6185 1.34361
$$310$$ 0 0
$$311$$ −8.27922 −0.469472 −0.234736 0.972059i $$-0.575423\pi$$
−0.234736 + 0.972059i $$0.575423\pi$$
$$312$$ 0 0
$$313$$ −15.4898 −0.875534 −0.437767 0.899088i $$-0.644231\pi$$
−0.437767 + 0.899088i $$0.644231\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10.5803 0.594251 0.297125 0.954838i $$-0.403972\pi$$
0.297125 + 0.954838i $$0.403972\pi$$
$$318$$ 0 0
$$319$$ 0.514921 0.0288300
$$320$$ 0 0
$$321$$ 7.80256 0.435496
$$322$$ 0 0
$$323$$ 16.0928 0.895427
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 9.38625 0.519061
$$328$$ 0 0
$$329$$ −22.4678 −1.23869
$$330$$ 0 0
$$331$$ 1.32836 0.0730133 0.0365067 0.999333i $$-0.488377\pi$$
0.0365067 + 0.999333i $$0.488377\pi$$
$$332$$ 0 0
$$333$$ 11.2782 0.618040
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −33.1297 −1.80469 −0.902345 0.431014i $$-0.858156\pi$$
−0.902345 + 0.431014i $$0.858156\pi$$
$$338$$ 0 0
$$339$$ −22.8611 −1.24165
$$340$$ 0 0
$$341$$ 0.0558298 0.00302335
$$342$$ 0 0
$$343$$ −12.5675 −0.678582
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 15.7579 0.845926 0.422963 0.906147i $$-0.360990\pi$$
0.422963 + 0.906147i $$0.360990\pi$$
$$348$$ 0 0
$$349$$ −14.5904 −0.781004 −0.390502 0.920602i $$-0.627699\pi$$
−0.390502 + 0.920602i $$0.627699\pi$$
$$350$$ 0 0
$$351$$ −8.12931 −0.433910
$$352$$ 0 0
$$353$$ 28.0097 1.49080 0.745402 0.666615i $$-0.232258\pi$$
0.745402 + 0.666615i $$0.232258\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −9.72964 −0.514947
$$358$$ 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
$$363$$ −14.6936 −0.771213
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 21.0197 1.09722 0.548610 0.836079i $$-0.315157\pi$$
0.548610 + 0.836079i $$0.315157\pi$$
$$368$$ 0 0
$$369$$ 3.47488 0.180895
$$370$$ 0 0
$$371$$ −24.5129 −1.27265
$$372$$ 0 0
$$373$$ 10.2558 0.531023 0.265511 0.964108i $$-0.414459\pi$$
0.265511 + 0.964108i $$0.414459\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.78116 0.400750
$$378$$ 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
$$383$$ 18.7419 0.957667 0.478833 0.877906i $$-0.341060\pi$$
0.478833 + 0.877906i $$0.341060\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 11.5234 0.585769
$$388$$ 0 0
$$389$$ −3.07719 −0.156020 −0.0780100 0.996953i $$-0.524857\pi$$
−0.0780100 + 0.996953i $$0.524857\pi$$
$$390$$ 0 0
$$391$$ −2.29775 −0.116202
$$392$$ 0 0
$$393$$ 11.2995 0.569986
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −11.4670 −0.575513 −0.287756 0.957704i $$-0.592909\pi$$
−0.287756 + 0.957704i $$0.592909\pi$$
$$398$$ 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
$$403$$ 0.843666 0.0420260
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −0.888314 −0.0440321
$$408$$ 0 0
$$409$$ −0.851429 −0.0421005 −0.0210502 0.999778i $$-0.506701\pi$$
−0.0210502 + 0.999778i $$0.506701\pi$$
$$410$$ 0 0
$$411$$ −24.7755 −1.22208
$$412$$ 0 0
$$413$$ −43.2111 −2.12628
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −9.92799 −0.486176
$$418$$ 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
$$423$$ 8.60251 0.418268
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.742472 0.0359307
$$428$$ 0 0
$$429$$ 0.184324 0.00889923
$$430$$ 0 0
$$431$$ −15.2881 −0.736402 −0.368201 0.929746i $$-0.620026\pi$$
−0.368201 + 0.929746i $$0.620026\pi$$
$$432$$ 0 0
$$433$$ 13.2632 0.637390 0.318695 0.947857i $$-0.396755\pi$$
0.318695 + 0.947857i $$0.396755\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −7.00371 −0.335033
$$438$$ 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
$$443$$ 5.49270 0.260966 0.130483 0.991451i $$-0.458347\pi$$
0.130483 + 0.991451i $$0.458347\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −6.06905 −0.287056
$$448$$ 0 0
$$449$$ 13.1705 0.621553 0.310777 0.950483i $$-0.399411\pi$$
0.310777 + 0.950483i $$0.399411\pi$$
$$450$$ 0 0
$$451$$ −0.273695 −0.0128878
$$452$$ 0 0
$$453$$ −0.00902890 −0.000424215 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 20.9676 0.980821 0.490410 0.871492i $$-0.336847\pi$$
0.490410 + 0.871492i $$0.336847\pi$$
$$458$$ 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
$$463$$ −9.49565 −0.441300 −0.220650 0.975353i $$-0.570818\pi$$
−0.220650 + 0.975353i $$0.570818\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −13.9066 −0.643518 −0.321759 0.946822i $$-0.604274\pi$$
−0.321759 + 0.946822i $$0.604274\pi$$
$$468$$ 0 0
$$469$$ −23.7466 −1.09652
$$470$$ 0 0
$$471$$ 16.2727 0.749809
$$472$$ 0 0
$$473$$ −0.907631 −0.0417329
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 9.38555 0.429735
$$478$$ 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
$$483$$ 4.23441 0.192672
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −28.9654 −1.31255 −0.656274 0.754523i $$-0.727868\pi$$
−0.656274 + 0.754523i $$0.727868\pi$$
$$488$$ 0 0
$$489$$ −27.0320 −1.22243
$$490$$ 0 0
$$491$$ 37.8991 1.71036 0.855182 0.518328i $$-0.173446\pi$$
0.855182 + 0.518328i $$0.173446\pi$$
$$492$$ 0 0
$$493$$ −12.3866 −0.557866
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 16.4204 0.736557
$$498$$ 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
$$503$$ 18.2182 0.812309 0.406154 0.913804i $$-0.366870\pi$$
0.406154 + 0.913804i $$0.366870\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −14.5942 −0.648150
$$508$$ 0 0
$$509$$ −31.0138 −1.37466 −0.687332 0.726344i $$-0.741218\pi$$
−0.687332 + 0.726344i $$0.741218\pi$$
$$510$$ 0 0
$$511$$ −4.82655 −0.213514
$$512$$ 0 0
$$513$$ 39.4446 1.74152
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −0.677568 −0.0297994
$$518$$ 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
$$523$$ −10.3111 −0.450873 −0.225437 0.974258i $$-0.572381\pi$$
−0.225437 + 0.974258i $$0.572381\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.34301 −0.0585024
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 16.5448 0.717981
$$532$$ 0 0
$$533$$ −4.13592 −0.179146
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −31.7699 −1.37097
$$538$$ 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
$$543$$ −4.19139 −0.179870
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2.47986 −0.106031 −0.0530156 0.998594i $$-0.516883\pi$$
−0.0530156 + 0.998594i $$0.516883\pi$$
$$548$$ 0 0
$$549$$ −0.284279 −0.0121327
$$550$$ 0 0
$$551$$ −37.7553 −1.60843
$$552$$ 0 0
$$553$$ 9.63094 0.409549
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.9814 0.973755 0.486877 0.873470i $$-0.338136\pi$$
0.486877 + 0.873470i $$0.338136\pi$$
$$558$$ 0 0
$$559$$ −13.7156 −0.580107
$$560$$ 0 0
$$561$$ −0.293420 −0.0123882
$$562$$ 0 0
$$563$$ 18.4939 0.779427 0.389713 0.920936i $$-0.372574\pi$$
0.389713 + 0.920936i $$0.372574\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −12.3246 −0.517584
$$568$$ 0 0
$$569$$ 35.6081 1.49277 0.746385 0.665514i $$-0.231788\pi$$
0.746385 + 0.665514i $$0.231788\pi$$
$$570$$ 0 0
$$571$$ −20.5827 −0.861359 −0.430680 0.902505i $$-0.641726\pi$$
−0.430680 + 0.902505i $$0.641726\pi$$
$$572$$ 0 0
$$573$$ −8.25645 −0.344918
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −19.0378 −0.792554 −0.396277 0.918131i $$-0.629698\pi$$
−0.396277 + 0.918131i $$0.629698\pi$$
$$578$$ 0 0
$$579$$ 9.65591 0.401286
$$580$$ 0 0
$$581$$ 50.6489 2.10127
$$582$$ 0 0
$$583$$ −0.739244 −0.0306163
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −7.29062 −0.300916 −0.150458 0.988616i $$-0.548075\pi$$
−0.150458 + 0.988616i $$0.548075\pi$$
$$588$$ 0 0
$$589$$ −4.09358 −0.168673
$$590$$ 0 0
$$591$$ −17.3821 −0.715006
$$592$$ 0 0
$$593$$ 10.5832 0.434598 0.217299 0.976105i $$-0.430275\pi$$
0.217299 + 0.976105i $$0.430275\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.46140 0.100739
$$598$$ 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
$$603$$ 9.09216 0.370261
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.0996 −0.896996 −0.448498 0.893784i $$-0.648041\pi$$
−0.448498 + 0.893784i $$0.648041\pi$$
$$608$$ 0 0
$$609$$ 22.8267 0.924984
$$610$$ 0 0
$$611$$ −10.2390 −0.414225
$$612$$ 0 0
$$613$$ 5.08532 0.205394 0.102697 0.994713i $$-0.467253\pi$$
0.102697 + 0.994713i $$0.467253\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.63073 0.307202 0.153601 0.988133i $$-0.450913\pi$$
0.153601 + 0.988133i $$0.450913\pi$$
$$618$$ 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
$$623$$ 17.5258 0.702156
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −0.894364 −0.0357175
$$628$$ 0 0
$$629$$ 21.3688 0.852028
$$630$$ 0 0
$$631$$ 1.86751 0.0743444 0.0371722 0.999309i $$-0.488165\pi$$
0.0371722 + 0.999309i $$0.488165\pi$$
$$632$$ 0 0
$$633$$ 33.2607 1.32199
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.37673 0.173412
$$638$$ 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
$$643$$ −5.06670 −0.199811 −0.0999055 0.994997i $$-0.531854\pi$$
−0.0999055 + 0.994997i $$0.531854\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −37.0429 −1.45631 −0.728153 0.685414i $$-0.759621\pi$$
−0.728153 + 0.685414i $$0.759621\pi$$
$$648$$ 0 0
$$649$$ −1.30313 −0.0511524
$$650$$ 0 0
$$651$$ 2.47496 0.0970014
$$652$$ 0 0
$$653$$ −11.9884 −0.469143 −0.234572 0.972099i $$-0.575369\pi$$
−0.234572 + 0.972099i $$0.575369\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.84800 0.0720974
$$658$$ 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
$$663$$ −4.43398 −0.172202
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.39076 0.208731
$$668$$ 0 0
$$669$$ 5.41339 0.209294
$$670$$ 0 0
$$671$$ 0.0223910 0.000864393 0
$$672$$ 0 0
$$673$$ −19.0620 −0.734784 −0.367392 0.930066i $$-0.619749\pi$$
−0.367392 + 0.930066i $$0.619749\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −2.54346 −0.0977533 −0.0488766 0.998805i $$-0.515564\pi$$
−0.0488766 + 0.998805i $$0.515564\pi$$
$$678$$ 0 0
$$679$$ 8.18146 0.313975
$$680$$ 0 0
$$681$$ 23.2507 0.890969
$$682$$ 0 0
$$683$$ −0.377109 −0.0144297 −0.00721483 0.999974i $$-0.502297\pi$$
−0.00721483 + 0.999974i $$0.502297\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −4.70266 −0.179418
$$688$$ 0 0
$$689$$ −11.1710 −0.425581
$$690$$ 0 0
$$691$$ −7.24743 −0.275705 −0.137853 0.990453i $$-0.544020\pi$$
−0.137853 + 0.990453i $$0.544020\pi$$
$$692$$ 0 0
$$693$$ −0.366903 −0.0139375
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.58385 0.249381
$$698$$ 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
$$703$$ 65.1335 2.45656
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −21.0824 −0.792887
$$708$$ 0 0
$$709$$ −45.9062 −1.72404 −0.862022 0.506870i $$-0.830802\pi$$
−0.862022 + 0.506870i $$0.830802\pi$$
$$710$$ 0 0
$$711$$ −3.68751 −0.138293
$$712$$ 0 0
$$713$$ 0.584488 0.0218892
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3.83612 0.143263
$$718$$ 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
$$723$$ 36.1609 1.34484
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −19.0273 −0.705685 −0.352842 0.935683i $$-0.614785\pi$$
−0.352842 + 0.935683i $$0.614785\pi$$
$$728$$ 0 0
$$729$$ 27.3067 1.01136
$$730$$ 0 0
$$731$$ 21.8334 0.807539
$$732$$ 0 0
$$733$$ 42.1027 1.55510 0.777549 0.628822i $$-0.216463\pi$$
0.777549 + 0.628822i $$0.216463\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −0.716135 −0.0263792
$$738$$ 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
$$743$$ 21.3492 0.783225 0.391613 0.920130i $$-0.371917\pi$$
0.391613 + 0.920130i $$0.371917\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −19.3925 −0.709536
$$748$$ 0 0
$$749$$ 18.4858 0.675458
$$750$$ 0 0
$$751$$ 27.8218 1.01523 0.507617 0.861583i $$-0.330527\pi$$
0.507617 + 0.861583i $$0.330527\pi$$
$$752$$ 0 0
$$753$$ 15.7478 0.573881
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −8.74990 −0.318020 −0.159010 0.987277i $$-0.550830\pi$$
−0.159010 + 0.987277i $$0.550830\pi$$
$$758$$ 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
$$763$$ 22.2379 0.805066
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −19.6921 −0.711041
$$768$$ 0 0
$$769$$ −31.3229 −1.12953 −0.564766 0.825251i $$-0.691034\pi$$
−0.564766 + 0.825251i $$0.691034\pi$$
$$770$$ 0 0
$$771$$ −11.3334 −0.408162
$$772$$ 0 0
$$773$$ −32.3308 −1.16286 −0.581429 0.813597i $$-0.697506\pi$$
−0.581429 + 0.813597i $$0.697506\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −39.3794 −1.41273
$$778$$ 0 0
$$779$$ 20.0680 0.719012
$$780$$ 0 0
$$781$$ 0.495196 0.0177195
$$782$$ 0 0
$$783$$ −30.3605 −1.08499
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −18.5712 −0.661990 −0.330995 0.943632i $$-0.607384\pi$$
−0.330995 + 0.943632i $$0.607384\pi$$
$$788$$ 0 0
$$789$$ 24.7483 0.881063
$$790$$ 0 0
$$791$$ −54.1627 −1.92580
$$792$$ 0 0
$$793$$ 0.338358 0.0120155
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 24.1985 0.857154 0.428577 0.903505i $$-0.359015\pi$$
0.428577 + 0.903505i $$0.359015\pi$$
$$798$$ 0 0
$$799$$ 16.2992 0.576624
$$800$$ 0 0
$$801$$ −6.71031 −0.237097
$$802$$ 0 0
$$803$$ −0.145556 −0.00513656
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −34.0349 −1.19809
$$808$$ 0 0
$$809$$ −30.4574 −1.07082 −0.535412 0.844591i $$-0.679844\pi$$
−0.535412 + 0.844591i $$0.679844\pi$$
$$810$$ 0 0
$$811$$ −4.05626 −0.142435 −0.0712174 0.997461i $$-0.522688\pi$$
−0.0712174 + 0.997461i $$0.522688\pi$$
$$812$$ 0 0
$$813$$ 18.9587 0.664910
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 66.5499 2.32829
$$818$$ 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
$$823$$ −42.9810 −1.49822 −0.749111 0.662444i $$-0.769519\pi$$
−0.749111 + 0.662444i $$0.769519\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 51.6777 1.79701 0.898505 0.438963i $$-0.144654\pi$$
0.898505 + 0.438963i $$0.144654\pi$$
$$828$$ 0 0
$$829$$ 25.6228 0.889915 0.444958 0.895552i $$-0.353219\pi$$
0.444958 + 0.895552i $$0.353219\pi$$
$$830$$ 0 0
$$831$$ −28.6470 −0.993753
$$832$$ 0 0
$$833$$ −6.96720 −0.241399
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −3.29181 −0.113781
$$838$$ 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
$$843$$ −11.9021 −0.409929
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −34.8121 −1.19616
$$848$$ 0 0
$$849$$ −17.4290 −0.598160
$$850$$ 0 0
$$851$$ −9.29985 −0.318795
$$852$$ 0 0
$$853$$ −13.8668 −0.474791 −0.237395 0.971413i $$-0.576294\pi$$
−0.237395 + 0.971413i $$0.576294\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −16.3518 −0.558567 −0.279283 0.960209i $$-0.590097\pi$$
−0.279283 + 0.960209i $$0.590097\pi$$
$$858$$ 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
$$863$$ −15.2130 −0.517856 −0.258928 0.965897i $$-0.583369\pi$$
−0.258928 + 0.965897i $$0.583369\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −15.6688 −0.532140
$$868$$ 0 0
$$869$$ 0.290443 0.00985262
$$870$$ 0 0
$$871$$ −10.8218 −0.366683
$$872$$ 0 0
$$873$$ −3.13253 −0.106020
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 53.6934 1.81310 0.906549 0.422101i $$-0.138707\pi$$
0.906549 + 0.422101i $$0.138707\pi$$
$$878$$ 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
$$883$$ −19.9787 −0.672337 −0.336169 0.941802i $$-0.609131\pi$$
−0.336169 + 0.941802i $$0.609131\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 51.9251 1.74347 0.871737 0.489974i $$-0.162994\pi$$
0.871737 + 0.489974i $$0.162994\pi$$
$$888$$ 0 0
$$889$$ 34.2233 1.14781
$$890$$ 0 0
$$891$$ −0.371676 −0.0124516
$$892$$ 0 0
$$893$$ 49.6810 1.66251
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.92970 0.0644309
$$898$$ 0 0
$$899$$ 3.15083 0.105086
$$900$$ 0 0
$$901$$ 17.7828 0.592431
$$902$$ 0 0
$$903$$ −40.2358 −1.33896
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −6.64335 −0.220589 −0.110294 0.993899i $$-0.535179\pi$$
−0.110294 + 0.993899i $$0.535179\pi$$
$$908$$ 0 0
$$909$$ 8.07209 0.267734
$$910$$ 0 0
$$911$$ −15.2064 −0.503811 −0.251905 0.967752i $$-0.581057\pi$$
−0.251905 + 0.967752i $$0.581057\pi$$
$$912$$ 0 0
$$913$$ 1.52744 0.0505507
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 26.7709 0.884052
$$918$$ 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
$$923$$ 7.48310 0.246309
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −21.4249 −0.703687
$$928$$ 0 0
$$929$$ −4.86715 −0.159686 −0.0798431 0.996807i $$-0.525442\pi$$
−0.0798431 + 0.996807i $$0.525442\pi$$
$$930$$ 0 0
$$931$$ −21.2365 −0.695999
$$932$$ 0 0
$$933$$ −11.0684 −0.362364
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.44469 0.112533 0.0562666 0.998416i $$-0.482080\pi$$
0.0562666 + 0.998416i $$0.482080\pi$$
$$938$$ 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
$$943$$ −2.86534 −0.0933084
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −5.17298 −0.168099 −0.0840496 0.996462i $$-0.526785\pi$$
−0.0840496 + 0.996462i $$0.526785\pi$$
$$948$$ 0 0
$$949$$ −2.19955 −0.0714005
$$950$$ 0 0
$$951$$ 14.1447 0.458675
$$952$$ 0 0
$$953$$ −3.41090 −0.110490 −0.0552449 0.998473i $$-0.517594\pi$$
−0.0552449 + 0.998473i $$0.517594\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0.688392 0.0222526
$$958$$ 0 0
$$959$$ −58.6981 −1.89546
$$960$$ 0 0
$$961$$ −30.6584 −0.988980
$$962$$ 0 0
$$963$$ −7.07789 −0.228082
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 17.5047 0.562914 0.281457 0.959574i $$-0.409182\pi$$
0.281457 + 0.959574i $$0.409182\pi$$
$$968$$ 0 0
$$969$$ 21.5143 0.691139
$$970$$ 0 0
$$971$$ −35.9954 −1.15515 −0.577574 0.816338i $$-0.696001\pi$$
−0.577574 + 0.816338i $$0.696001\pi$$
$$972$$ 0 0
$$973$$ −23.5214 −0.754062
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 31.3634 1.00340 0.501702 0.865040i $$-0.332707\pi$$
0.501702 + 0.865040i $$0.332707\pi$$
$$978$$ 0 0
$$979$$ 0.528531 0.0168919
$$980$$ 0 0
$$981$$ −8.51450 −0.271847
$$982$$ 0 0
$$983$$ −36.4262 −1.16182 −0.580908 0.813969i $$-0.697302\pi$$
−0.580908 + 0.813969i $$0.697302\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −30.0369 −0.956086
$$988$$ 0 0
$$989$$ −9.50208 −0.302149
$$990$$ 0 0
$$991$$ −6.22316 −0.197685 −0.0988426 0.995103i $$-0.531514\pi$$
−0.0988426 + 0.995103i $$0.531514\pi$$
$$992$$ 0 0
$$993$$ 1.77587 0.0563556
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 24.0896 0.762925 0.381463 0.924384i $$-0.375420\pi$$
0.381463 + 0.924384i $$0.375420\pi$$
$$998$$ 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 9200.2.a.cw.1.3 5
4.3 odd 2 4600.2.a.bc.1.3 5
5.4 even 2 9200.2.a.cs.1.3 5
20.3 even 4 4600.2.e.v.4049.4 10
20.7 even 4 4600.2.e.v.4049.7 10
20.19 odd 2 4600.2.a.bg.1.3 yes 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.3 5 4.3 odd 2
4600.2.a.bg.1.3 yes 5 20.19 odd 2
4600.2.e.v.4049.4 10 20.3 even 4
4600.2.e.v.4049.7 10 20.7 even 4
9200.2.a.cs.1.3 5 5.4 even 2
9200.2.a.cw.1.3 5 1.1 even 1 trivial