# Properties

 Label 7350.2.a.dg.1.2 Level 7350 Weight 2 Character 7350.1 Self dual yes Analytic conductor 58.690 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.6900454856$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 7350.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.41421 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +5.82843 q^{17} -1.00000 q^{18} +1.41421 q^{19} -3.41421 q^{22} -0.414214 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} -2.41421 q^{31} -1.00000 q^{32} +3.41421 q^{33} -5.82843 q^{34} +1.00000 q^{36} +7.07107 q^{37} -1.41421 q^{38} +3.00000 q^{39} -5.82843 q^{41} +1.58579 q^{43} +3.41421 q^{44} +0.414214 q^{46} -1.65685 q^{47} +1.00000 q^{48} +5.82843 q^{51} +3.00000 q^{52} +12.6569 q^{53} -1.00000 q^{54} +1.41421 q^{57} -1.00000 q^{58} +7.24264 q^{59} -0.171573 q^{61} +2.41421 q^{62} +1.00000 q^{64} -3.41421 q^{66} -15.3137 q^{67} +5.82843 q^{68} -0.414214 q^{69} -9.31371 q^{71} -1.00000 q^{72} +15.0711 q^{73} -7.07107 q^{74} +1.41421 q^{76} -3.00000 q^{78} +1.07107 q^{79} +1.00000 q^{81} +5.82843 q^{82} -0.0710678 q^{83} -1.58579 q^{86} +1.00000 q^{87} -3.41421 q^{88} +4.00000 q^{89} -0.414214 q^{92} -2.41421 q^{93} +1.65685 q^{94} -1.00000 q^{96} -0.343146 q^{97} +3.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + 4q^{11} + 2q^{12} + 6q^{13} + 2q^{16} + 6q^{17} - 2q^{18} - 4q^{22} + 2q^{23} - 2q^{24} - 6q^{26} + 2q^{27} + 2q^{29} - 2q^{31} - 2q^{32} + 4q^{33} - 6q^{34} + 2q^{36} + 6q^{39} - 6q^{41} + 6q^{43} + 4q^{44} - 2q^{46} + 8q^{47} + 2q^{48} + 6q^{51} + 6q^{52} + 14q^{53} - 2q^{54} - 2q^{58} + 6q^{59} - 6q^{61} + 2q^{62} + 2q^{64} - 4q^{66} - 8q^{67} + 6q^{68} + 2q^{69} + 4q^{71} - 2q^{72} + 16q^{73} - 6q^{78} - 12q^{79} + 2q^{81} + 6q^{82} + 14q^{83} - 6q^{86} + 2q^{87} - 4q^{88} + 8q^{89} + 2q^{92} - 2q^{93} - 8q^{94} - 2q^{96} - 12q^{97} + 4q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.41421 1.02942 0.514712 0.857363i $$-0.327899\pi$$
0.514712 + 0.857363i $$0.327899\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.82843 1.41360 0.706801 0.707413i $$-0.250138\pi$$
0.706801 + 0.707413i $$0.250138\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.41421 0.324443 0.162221 0.986754i $$-0.448134\pi$$
0.162221 + 0.986754i $$0.448134\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.41421 −0.727913
$$23$$ −0.414214 −0.0863695 −0.0431847 0.999067i $$-0.513750\pi$$
−0.0431847 + 0.999067i $$0.513750\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −3.00000 −0.588348
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 0 0
$$31$$ −2.41421 −0.433606 −0.216803 0.976215i $$-0.569563\pi$$
−0.216803 + 0.976215i $$0.569563\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 3.41421 0.594338
$$34$$ −5.82843 −0.999567
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 7.07107 1.16248 0.581238 0.813733i $$-0.302568\pi$$
0.581238 + 0.813733i $$0.302568\pi$$
$$38$$ −1.41421 −0.229416
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ −5.82843 −0.910247 −0.455124 0.890428i $$-0.650405\pi$$
−0.455124 + 0.890428i $$0.650405\pi$$
$$42$$ 0 0
$$43$$ 1.58579 0.241830 0.120915 0.992663i $$-0.461417\pi$$
0.120915 + 0.992663i $$0.461417\pi$$
$$44$$ 3.41421 0.514712
$$45$$ 0 0
$$46$$ 0.414214 0.0610725
$$47$$ −1.65685 −0.241677 −0.120839 0.992672i $$-0.538558\pi$$
−0.120839 + 0.992672i $$0.538558\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 5.82843 0.816143
$$52$$ 3.00000 0.416025
$$53$$ 12.6569 1.73855 0.869276 0.494326i $$-0.164585\pi$$
0.869276 + 0.494326i $$0.164585\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.41421 0.187317
$$58$$ −1.00000 −0.131306
$$59$$ 7.24264 0.942912 0.471456 0.881890i $$-0.343729\pi$$
0.471456 + 0.881890i $$0.343729\pi$$
$$60$$ 0 0
$$61$$ −0.171573 −0.0219677 −0.0109838 0.999940i $$-0.503496\pi$$
−0.0109838 + 0.999940i $$0.503496\pi$$
$$62$$ 2.41421 0.306605
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −3.41421 −0.420261
$$67$$ −15.3137 −1.87087 −0.935434 0.353502i $$-0.884991\pi$$
−0.935434 + 0.353502i $$0.884991\pi$$
$$68$$ 5.82843 0.706801
$$69$$ −0.414214 −0.0498655
$$70$$ 0 0
$$71$$ −9.31371 −1.10533 −0.552667 0.833402i $$-0.686390\pi$$
−0.552667 + 0.833402i $$0.686390\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 15.0711 1.76394 0.881968 0.471310i $$-0.156219\pi$$
0.881968 + 0.471310i $$0.156219\pi$$
$$74$$ −7.07107 −0.821995
$$75$$ 0 0
$$76$$ 1.41421 0.162221
$$77$$ 0 0
$$78$$ −3.00000 −0.339683
$$79$$ 1.07107 0.120505 0.0602523 0.998183i $$-0.480809\pi$$
0.0602523 + 0.998183i $$0.480809\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 5.82843 0.643642
$$83$$ −0.0710678 −0.00780071 −0.00390035 0.999992i $$-0.501242\pi$$
−0.00390035 + 0.999992i $$0.501242\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.58579 −0.171000
$$87$$ 1.00000 0.107211
$$88$$ −3.41421 −0.363956
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −0.414214 −0.0431847
$$93$$ −2.41421 −0.250342
$$94$$ 1.65685 0.170891
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −0.343146 −0.0348412 −0.0174206 0.999848i $$-0.505545\pi$$
−0.0174206 + 0.999848i $$0.505545\pi$$
$$98$$ 0 0
$$99$$ 3.41421 0.343141
$$100$$ 0 0
$$101$$ 5.65685 0.562878 0.281439 0.959579i $$-0.409188\pi$$
0.281439 + 0.959579i $$0.409188\pi$$
$$102$$ −5.82843 −0.577100
$$103$$ −0.0710678 −0.00700252 −0.00350126 0.999994i $$-0.501114\pi$$
−0.00350126 + 0.999994i $$0.501114\pi$$
$$104$$ −3.00000 −0.294174
$$105$$ 0 0
$$106$$ −12.6569 −1.22934
$$107$$ 6.48528 0.626956 0.313478 0.949595i $$-0.398506\pi$$
0.313478 + 0.949595i $$0.398506\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −11.3137 −1.08366 −0.541828 0.840489i $$-0.682268\pi$$
−0.541828 + 0.840489i $$0.682268\pi$$
$$110$$ 0 0
$$111$$ 7.07107 0.671156
$$112$$ 0 0
$$113$$ 6.58579 0.619539 0.309769 0.950812i $$-0.399748\pi$$
0.309769 + 0.950812i $$0.399748\pi$$
$$114$$ −1.41421 −0.132453
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ 3.00000 0.277350
$$118$$ −7.24264 −0.666739
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0.656854 0.0597140
$$122$$ 0.171573 0.0155335
$$123$$ −5.82843 −0.525532
$$124$$ −2.41421 −0.216803
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −3.17157 −0.281432 −0.140716 0.990050i $$-0.544940\pi$$
−0.140716 + 0.990050i $$0.544940\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.58579 0.139621
$$130$$ 0 0
$$131$$ −18.8284 −1.64505 −0.822524 0.568731i $$-0.807435\pi$$
−0.822524 + 0.568731i $$0.807435\pi$$
$$132$$ 3.41421 0.297169
$$133$$ 0 0
$$134$$ 15.3137 1.32290
$$135$$ 0 0
$$136$$ −5.82843 −0.499784
$$137$$ −23.2132 −1.98324 −0.991619 0.129197i $$-0.958760\pi$$
−0.991619 + 0.129197i $$0.958760\pi$$
$$138$$ 0.414214 0.0352602
$$139$$ −15.5563 −1.31947 −0.659736 0.751497i $$-0.729332\pi$$
−0.659736 + 0.751497i $$0.729332\pi$$
$$140$$ 0 0
$$141$$ −1.65685 −0.139532
$$142$$ 9.31371 0.781589
$$143$$ 10.2426 0.856533
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −15.0711 −1.24729
$$147$$ 0 0
$$148$$ 7.07107 0.581238
$$149$$ 4.65685 0.381504 0.190752 0.981638i $$-0.438907\pi$$
0.190752 + 0.981638i $$0.438907\pi$$
$$150$$ 0 0
$$151$$ 7.89949 0.642852 0.321426 0.946935i $$-0.395838\pi$$
0.321426 + 0.946935i $$0.395838\pi$$
$$152$$ −1.41421 −0.114708
$$153$$ 5.82843 0.471200
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.00000 0.240192
$$157$$ −19.7990 −1.58013 −0.790066 0.613022i $$-0.789954\pi$$
−0.790066 + 0.613022i $$0.789954\pi$$
$$158$$ −1.07107 −0.0852096
$$159$$ 12.6569 1.00375
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 5.24264 0.410635 0.205318 0.978695i $$-0.434177\pi$$
0.205318 + 0.978695i $$0.434177\pi$$
$$164$$ −5.82843 −0.455124
$$165$$ 0 0
$$166$$ 0.0710678 0.00551593
$$167$$ −8.82843 −0.683164 −0.341582 0.939852i $$-0.610963\pi$$
−0.341582 + 0.939852i $$0.610963\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 1.41421 0.108148
$$172$$ 1.58579 0.120915
$$173$$ 14.2426 1.08285 0.541424 0.840750i $$-0.317885\pi$$
0.541424 + 0.840750i $$0.317885\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ 0 0
$$176$$ 3.41421 0.257356
$$177$$ 7.24264 0.544390
$$178$$ −4.00000 −0.299813
$$179$$ −5.07107 −0.379029 −0.189515 0.981878i $$-0.560691\pi$$
−0.189515 + 0.981878i $$0.560691\pi$$
$$180$$ 0 0
$$181$$ −23.3137 −1.73289 −0.866447 0.499269i $$-0.833602\pi$$
−0.866447 + 0.499269i $$0.833602\pi$$
$$182$$ 0 0
$$183$$ −0.171573 −0.0126830
$$184$$ 0.414214 0.0305362
$$185$$ 0 0
$$186$$ 2.41421 0.177019
$$187$$ 19.8995 1.45520
$$188$$ −1.65685 −0.120839
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0711 1.16286 0.581431 0.813596i $$-0.302493\pi$$
0.581431 + 0.813596i $$0.302493\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −6.82843 −0.491521 −0.245760 0.969331i $$-0.579038\pi$$
−0.245760 + 0.969331i $$0.579038\pi$$
$$194$$ 0.343146 0.0246364
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 23.1421 1.64881 0.824404 0.566001i $$-0.191510\pi$$
0.824404 + 0.566001i $$0.191510\pi$$
$$198$$ −3.41421 −0.242638
$$199$$ 19.6569 1.39344 0.696719 0.717344i $$-0.254643\pi$$
0.696719 + 0.717344i $$0.254643\pi$$
$$200$$ 0 0
$$201$$ −15.3137 −1.08015
$$202$$ −5.65685 −0.398015
$$203$$ 0 0
$$204$$ 5.82843 0.408072
$$205$$ 0 0
$$206$$ 0.0710678 0.00495153
$$207$$ −0.414214 −0.0287898
$$208$$ 3.00000 0.208013
$$209$$ 4.82843 0.333989
$$210$$ 0 0
$$211$$ −26.5563 −1.82821 −0.914107 0.405473i $$-0.867107\pi$$
−0.914107 + 0.405473i $$0.867107\pi$$
$$212$$ 12.6569 0.869276
$$213$$ −9.31371 −0.638165
$$214$$ −6.48528 −0.443325
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 11.3137 0.766261
$$219$$ 15.0711 1.01841
$$220$$ 0 0
$$221$$ 17.4853 1.17619
$$222$$ −7.07107 −0.474579
$$223$$ 24.5563 1.64441 0.822207 0.569188i $$-0.192742\pi$$
0.822207 + 0.569188i $$0.192742\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.58579 −0.438080
$$227$$ 25.3848 1.68485 0.842423 0.538816i $$-0.181128\pi$$
0.842423 + 0.538816i $$0.181128\pi$$
$$228$$ 1.41421 0.0936586
$$229$$ 29.3137 1.93710 0.968552 0.248810i $$-0.0800396\pi$$
0.968552 + 0.248810i $$0.0800396\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1.00000 −0.0656532
$$233$$ 14.2426 0.933066 0.466533 0.884504i $$-0.345503\pi$$
0.466533 + 0.884504i $$0.345503\pi$$
$$234$$ −3.00000 −0.196116
$$235$$ 0 0
$$236$$ 7.24264 0.471456
$$237$$ 1.07107 0.0695733
$$238$$ 0 0
$$239$$ −11.3137 −0.731823 −0.365911 0.930650i $$-0.619243\pi$$
−0.365911 + 0.930650i $$0.619243\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ −0.656854 −0.0422242
$$243$$ 1.00000 0.0641500
$$244$$ −0.171573 −0.0109838
$$245$$ 0 0
$$246$$ 5.82843 0.371607
$$247$$ 4.24264 0.269953
$$248$$ 2.41421 0.153303
$$249$$ −0.0710678 −0.00450374
$$250$$ 0 0
$$251$$ −1.92893 −0.121753 −0.0608766 0.998145i $$-0.519390\pi$$
−0.0608766 + 0.998145i $$0.519390\pi$$
$$252$$ 0 0
$$253$$ −1.41421 −0.0889108
$$254$$ 3.17157 0.199002
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 13.3431 0.832323 0.416161 0.909291i $$-0.363375\pi$$
0.416161 + 0.909291i $$0.363375\pi$$
$$258$$ −1.58579 −0.0987268
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 18.8284 1.16322
$$263$$ −10.5563 −0.650932 −0.325466 0.945554i $$-0.605521\pi$$
−0.325466 + 0.945554i $$0.605521\pi$$
$$264$$ −3.41421 −0.210130
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 4.00000 0.244796
$$268$$ −15.3137 −0.935434
$$269$$ 0.443651 0.0270499 0.0135249 0.999909i $$-0.495695\pi$$
0.0135249 + 0.999909i $$0.495695\pi$$
$$270$$ 0 0
$$271$$ −11.5147 −0.699469 −0.349735 0.936849i $$-0.613728\pi$$
−0.349735 + 0.936849i $$0.613728\pi$$
$$272$$ 5.82843 0.353400
$$273$$ 0 0
$$274$$ 23.2132 1.40236
$$275$$ 0 0
$$276$$ −0.414214 −0.0249327
$$277$$ −24.0416 −1.44452 −0.722261 0.691621i $$-0.756897\pi$$
−0.722261 + 0.691621i $$0.756897\pi$$
$$278$$ 15.5563 0.933008
$$279$$ −2.41421 −0.144535
$$280$$ 0 0
$$281$$ 20.0000 1.19310 0.596550 0.802576i $$-0.296538\pi$$
0.596550 + 0.802576i $$0.296538\pi$$
$$282$$ 1.65685 0.0986642
$$283$$ −0.242641 −0.0144235 −0.00721175 0.999974i $$-0.502296\pi$$
−0.00721175 + 0.999974i $$0.502296\pi$$
$$284$$ −9.31371 −0.552667
$$285$$ 0 0
$$286$$ −10.2426 −0.605660
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 16.9706 0.998268
$$290$$ 0 0
$$291$$ −0.343146 −0.0201156
$$292$$ 15.0711 0.881968
$$293$$ −11.3137 −0.660954 −0.330477 0.943814i $$-0.607210\pi$$
−0.330477 + 0.943814i $$0.607210\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −7.07107 −0.410997
$$297$$ 3.41421 0.198113
$$298$$ −4.65685 −0.269764
$$299$$ −1.24264 −0.0718638
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −7.89949 −0.454565
$$303$$ 5.65685 0.324978
$$304$$ 1.41421 0.0811107
$$305$$ 0 0
$$306$$ −5.82843 −0.333189
$$307$$ 8.58579 0.490017 0.245008 0.969521i $$-0.421209\pi$$
0.245008 + 0.969521i $$0.421209\pi$$
$$308$$ 0 0
$$309$$ −0.0710678 −0.00404291
$$310$$ 0 0
$$311$$ −11.4142 −0.647241 −0.323620 0.946187i $$-0.604900\pi$$
−0.323620 + 0.946187i $$0.604900\pi$$
$$312$$ −3.00000 −0.169842
$$313$$ −16.0416 −0.906727 −0.453363 0.891326i $$-0.649776\pi$$
−0.453363 + 0.891326i $$0.649776\pi$$
$$314$$ 19.7990 1.11732
$$315$$ 0 0
$$316$$ 1.07107 0.0602523
$$317$$ 19.9706 1.12166 0.560829 0.827931i $$-0.310482\pi$$
0.560829 + 0.827931i $$0.310482\pi$$
$$318$$ −12.6569 −0.709761
$$319$$ 3.41421 0.191159
$$320$$ 0 0
$$321$$ 6.48528 0.361973
$$322$$ 0 0
$$323$$ 8.24264 0.458633
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −5.24264 −0.290363
$$327$$ −11.3137 −0.625650
$$328$$ 5.82843 0.321821
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −24.0711 −1.32307 −0.661533 0.749916i $$-0.730094\pi$$
−0.661533 + 0.749916i $$0.730094\pi$$
$$332$$ −0.0710678 −0.00390035
$$333$$ 7.07107 0.387492
$$334$$ 8.82843 0.483070
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4.79899 0.261418 0.130709 0.991421i $$-0.458275\pi$$
0.130709 + 0.991421i $$0.458275\pi$$
$$338$$ 4.00000 0.217571
$$339$$ 6.58579 0.357691
$$340$$ 0 0
$$341$$ −8.24264 −0.446364
$$342$$ −1.41421 −0.0764719
$$343$$ 0 0
$$344$$ −1.58579 −0.0854999
$$345$$ 0 0
$$346$$ −14.2426 −0.765689
$$347$$ −9.79899 −0.526037 −0.263019 0.964791i $$-0.584718\pi$$
−0.263019 + 0.964791i $$0.584718\pi$$
$$348$$ 1.00000 0.0536056
$$349$$ 24.6569 1.31985 0.659926 0.751331i $$-0.270588\pi$$
0.659926 + 0.751331i $$0.270588\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ −3.41421 −0.181978
$$353$$ 20.4853 1.09032 0.545161 0.838332i $$-0.316469\pi$$
0.545161 + 0.838332i $$0.316469\pi$$
$$354$$ −7.24264 −0.384942
$$355$$ 0 0
$$356$$ 4.00000 0.212000
$$357$$ 0 0
$$358$$ 5.07107 0.268014
$$359$$ 33.8701 1.78759 0.893797 0.448472i $$-0.148032\pi$$
0.893797 + 0.448472i $$0.148032\pi$$
$$360$$ 0 0
$$361$$ −17.0000 −0.894737
$$362$$ 23.3137 1.22534
$$363$$ 0.656854 0.0344759
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0.171573 0.00896826
$$367$$ 0.757359 0.0395338 0.0197669 0.999805i $$-0.493708\pi$$
0.0197669 + 0.999805i $$0.493708\pi$$
$$368$$ −0.414214 −0.0215924
$$369$$ −5.82843 −0.303416
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −2.41421 −0.125171
$$373$$ 2.34315 0.121323 0.0606617 0.998158i $$-0.480679\pi$$
0.0606617 + 0.998158i $$0.480679\pi$$
$$374$$ −19.8995 −1.02898
$$375$$ 0 0
$$376$$ 1.65685 0.0854457
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ −34.2132 −1.75741 −0.878707 0.477361i $$-0.841593\pi$$
−0.878707 + 0.477361i $$0.841593\pi$$
$$380$$ 0 0
$$381$$ −3.17157 −0.162485
$$382$$ −16.0711 −0.822267
$$383$$ 27.5563 1.40806 0.704032 0.710168i $$-0.251381\pi$$
0.704032 + 0.710168i $$0.251381\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 6.82843 0.347558
$$387$$ 1.58579 0.0806101
$$388$$ −0.343146 −0.0174206
$$389$$ −10.3431 −0.524418 −0.262209 0.965011i $$-0.584451\pi$$
−0.262209 + 0.965011i $$0.584451\pi$$
$$390$$ 0 0
$$391$$ −2.41421 −0.122092
$$392$$ 0 0
$$393$$ −18.8284 −0.949769
$$394$$ −23.1421 −1.16588
$$395$$ 0 0
$$396$$ 3.41421 0.171571
$$397$$ 30.3137 1.52140 0.760701 0.649103i $$-0.224856\pi$$
0.760701 + 0.649103i $$0.224856\pi$$
$$398$$ −19.6569 −0.985309
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.1005 0.604270 0.302135 0.953265i $$-0.402301\pi$$
0.302135 + 0.953265i $$0.402301\pi$$
$$402$$ 15.3137 0.763778
$$403$$ −7.24264 −0.360782
$$404$$ 5.65685 0.281439
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.1421 1.19668
$$408$$ −5.82843 −0.288550
$$409$$ 22.5858 1.11680 0.558398 0.829573i $$-0.311416\pi$$
0.558398 + 0.829573i $$0.311416\pi$$
$$410$$ 0 0
$$411$$ −23.2132 −1.14502
$$412$$ −0.0710678 −0.00350126
$$413$$ 0 0
$$414$$ 0.414214 0.0203575
$$415$$ 0 0
$$416$$ −3.00000 −0.147087
$$417$$ −15.5563 −0.761798
$$418$$ −4.82843 −0.236166
$$419$$ 24.5563 1.19966 0.599828 0.800129i $$-0.295236\pi$$
0.599828 + 0.800129i $$0.295236\pi$$
$$420$$ 0 0
$$421$$ −35.6569 −1.73781 −0.868904 0.494980i $$-0.835175\pi$$
−0.868904 + 0.494980i $$0.835175\pi$$
$$422$$ 26.5563 1.29274
$$423$$ −1.65685 −0.0805590
$$424$$ −12.6569 −0.614671
$$425$$ 0 0
$$426$$ 9.31371 0.451251
$$427$$ 0 0
$$428$$ 6.48528 0.313478
$$429$$ 10.2426 0.494519
$$430$$ 0 0
$$431$$ 33.0416 1.59156 0.795780 0.605586i $$-0.207061\pi$$
0.795780 + 0.605586i $$0.207061\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 28.6274 1.37575 0.687873 0.725831i $$-0.258545\pi$$
0.687873 + 0.725831i $$0.258545\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −11.3137 −0.541828
$$437$$ −0.585786 −0.0280220
$$438$$ −15.0711 −0.720123
$$439$$ 1.58579 0.0756855 0.0378427 0.999284i $$-0.487951\pi$$
0.0378427 + 0.999284i $$0.487951\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −17.4853 −0.831690
$$443$$ −14.3431 −0.681463 −0.340732 0.940161i $$-0.610675\pi$$
−0.340732 + 0.940161i $$0.610675\pi$$
$$444$$ 7.07107 0.335578
$$445$$ 0 0
$$446$$ −24.5563 −1.16278
$$447$$ 4.65685 0.220262
$$448$$ 0 0
$$449$$ −25.0711 −1.18318 −0.591588 0.806240i $$-0.701499\pi$$
−0.591588 + 0.806240i $$0.701499\pi$$
$$450$$ 0 0
$$451$$ −19.8995 −0.937031
$$452$$ 6.58579 0.309769
$$453$$ 7.89949 0.371151
$$454$$ −25.3848 −1.19137
$$455$$ 0 0
$$456$$ −1.41421 −0.0662266
$$457$$ −3.82843 −0.179086 −0.0895431 0.995983i $$-0.528541\pi$$
−0.0895431 + 0.995983i $$0.528541\pi$$
$$458$$ −29.3137 −1.36974
$$459$$ 5.82843 0.272048
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 0 0
$$463$$ 29.8995 1.38955 0.694774 0.719228i $$-0.255505\pi$$
0.694774 + 0.719228i $$0.255505\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −14.2426 −0.659778
$$467$$ −1.24264 −0.0575026 −0.0287513 0.999587i $$-0.509153\pi$$
−0.0287513 + 0.999587i $$0.509153\pi$$
$$468$$ 3.00000 0.138675
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −19.7990 −0.912289
$$472$$ −7.24264 −0.333370
$$473$$ 5.41421 0.248946
$$474$$ −1.07107 −0.0491958
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.6569 0.579518
$$478$$ 11.3137 0.517477
$$479$$ 20.2426 0.924910 0.462455 0.886643i $$-0.346969\pi$$
0.462455 + 0.886643i $$0.346969\pi$$
$$480$$ 0 0
$$481$$ 21.2132 0.967239
$$482$$ 6.00000 0.273293
$$483$$ 0 0
$$484$$ 0.656854 0.0298570
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 5.51472 0.249896 0.124948 0.992163i $$-0.460124\pi$$
0.124948 + 0.992163i $$0.460124\pi$$
$$488$$ 0.171573 0.00776674
$$489$$ 5.24264 0.237080
$$490$$ 0 0
$$491$$ −0.142136 −0.00641449 −0.00320725 0.999995i $$-0.501021\pi$$
−0.00320725 + 0.999995i $$0.501021\pi$$
$$492$$ −5.82843 −0.262766
$$493$$ 5.82843 0.262499
$$494$$ −4.24264 −0.190885
$$495$$ 0 0
$$496$$ −2.41421 −0.108401
$$497$$ 0 0
$$498$$ 0.0710678 0.00318462
$$499$$ −26.0711 −1.16710 −0.583551 0.812077i $$-0.698337\pi$$
−0.583551 + 0.812077i $$0.698337\pi$$
$$500$$ 0 0
$$501$$ −8.82843 −0.394425
$$502$$ 1.92893 0.0860925
$$503$$ −24.8701 −1.10890 −0.554451 0.832217i $$-0.687072\pi$$
−0.554451 + 0.832217i $$0.687072\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.41421 0.0628695
$$507$$ −4.00000 −0.177646
$$508$$ −3.17157 −0.140716
$$509$$ −6.04163 −0.267791 −0.133895 0.990995i $$-0.542749\pi$$
−0.133895 + 0.990995i $$0.542749\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 1.41421 0.0624391
$$514$$ −13.3431 −0.588541
$$515$$ 0 0
$$516$$ 1.58579 0.0698104
$$517$$ −5.65685 −0.248788
$$518$$ 0 0
$$519$$ 14.2426 0.625183
$$520$$ 0 0
$$521$$ −10.6569 −0.466885 −0.233443 0.972371i $$-0.574999\pi$$
−0.233443 + 0.972371i $$0.574999\pi$$
$$522$$ −1.00000 −0.0437688
$$523$$ −24.2843 −1.06188 −0.530939 0.847410i $$-0.678160\pi$$
−0.530939 + 0.847410i $$0.678160\pi$$
$$524$$ −18.8284 −0.822524
$$525$$ 0 0
$$526$$ 10.5563 0.460279
$$527$$ −14.0711 −0.612945
$$528$$ 3.41421 0.148585
$$529$$ −22.8284 −0.992540
$$530$$ 0 0
$$531$$ 7.24264 0.314304
$$532$$ 0 0
$$533$$ −17.4853 −0.757372
$$534$$ −4.00000 −0.173097
$$535$$ 0 0
$$536$$ 15.3137 0.661451
$$537$$ −5.07107 −0.218833
$$538$$ −0.443651 −0.0191271
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.2426 −1.47221 −0.736103 0.676869i $$-0.763336\pi$$
−0.736103 + 0.676869i $$0.763336\pi$$
$$542$$ 11.5147 0.494600
$$543$$ −23.3137 −1.00049
$$544$$ −5.82843 −0.249892
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 33.8701 1.44818 0.724090 0.689706i $$-0.242260\pi$$
0.724090 + 0.689706i $$0.242260\pi$$
$$548$$ −23.2132 −0.991619
$$549$$ −0.171573 −0.00732255
$$550$$ 0 0
$$551$$ 1.41421 0.0602475
$$552$$ 0.414214 0.0176301
$$553$$ 0 0
$$554$$ 24.0416 1.02143
$$555$$ 0 0
$$556$$ −15.5563 −0.659736
$$557$$ −19.3137 −0.818348 −0.409174 0.912456i $$-0.634183\pi$$
−0.409174 + 0.912456i $$0.634183\pi$$
$$558$$ 2.41421 0.102202
$$559$$ 4.75736 0.201215
$$560$$ 0 0
$$561$$ 19.8995 0.840157
$$562$$ −20.0000 −0.843649
$$563$$ 27.3848 1.15413 0.577065 0.816698i $$-0.304198\pi$$
0.577065 + 0.816698i $$0.304198\pi$$
$$564$$ −1.65685 −0.0697661
$$565$$ 0 0
$$566$$ 0.242641 0.0101989
$$567$$ 0 0
$$568$$ 9.31371 0.390795
$$569$$ −28.2426 −1.18399 −0.591997 0.805941i $$-0.701660\pi$$
−0.591997 + 0.805941i $$0.701660\pi$$
$$570$$ 0 0
$$571$$ 36.0122 1.50706 0.753532 0.657412i $$-0.228349\pi$$
0.753532 + 0.657412i $$0.228349\pi$$
$$572$$ 10.2426 0.428266
$$573$$ 16.0711 0.671378
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 19.6569 0.818326 0.409163 0.912461i $$-0.365821\pi$$
0.409163 + 0.912461i $$0.365821\pi$$
$$578$$ −16.9706 −0.705882
$$579$$ −6.82843 −0.283780
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0.343146 0.0142238
$$583$$ 43.2132 1.78971
$$584$$ −15.0711 −0.623645
$$585$$ 0 0
$$586$$ 11.3137 0.467365
$$587$$ −41.5269 −1.71400 −0.857000 0.515317i $$-0.827674\pi$$
−0.857000 + 0.515317i $$0.827674\pi$$
$$588$$ 0 0
$$589$$ −3.41421 −0.140680
$$590$$ 0 0
$$591$$ 23.1421 0.951940
$$592$$ 7.07107 0.290619
$$593$$ −26.0000 −1.06769 −0.533846 0.845582i $$-0.679254\pi$$
−0.533846 + 0.845582i $$0.679254\pi$$
$$594$$ −3.41421 −0.140087
$$595$$ 0 0
$$596$$ 4.65685 0.190752
$$597$$ 19.6569 0.804501
$$598$$ 1.24264 0.0508154
$$599$$ −30.4142 −1.24269 −0.621346 0.783537i $$-0.713414\pi$$
−0.621346 + 0.783537i $$0.713414\pi$$
$$600$$ 0 0
$$601$$ 1.27208 0.0518891 0.0259446 0.999663i $$-0.491741\pi$$
0.0259446 + 0.999663i $$0.491741\pi$$
$$602$$ 0 0
$$603$$ −15.3137 −0.623622
$$604$$ 7.89949 0.321426
$$605$$ 0 0
$$606$$ −5.65685 −0.229794
$$607$$ 39.9411 1.62116 0.810580 0.585628i $$-0.199152\pi$$
0.810580 + 0.585628i $$0.199152\pi$$
$$608$$ −1.41421 −0.0573539
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.97056 −0.201087
$$612$$ 5.82843 0.235600
$$613$$ −22.0000 −0.888572 −0.444286 0.895885i $$-0.646543\pi$$
−0.444286 + 0.895885i $$0.646543\pi$$
$$614$$ −8.58579 −0.346494
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.3848 0.901177 0.450589 0.892732i $$-0.351214\pi$$
0.450589 + 0.892732i $$0.351214\pi$$
$$618$$ 0.0710678 0.00285877
$$619$$ −12.1421 −0.488034 −0.244017 0.969771i $$-0.578465\pi$$
−0.244017 + 0.969771i $$0.578465\pi$$
$$620$$ 0 0
$$621$$ −0.414214 −0.0166218
$$622$$ 11.4142 0.457668
$$623$$ 0 0
$$624$$ 3.00000 0.120096
$$625$$ 0 0
$$626$$ 16.0416 0.641153
$$627$$ 4.82843 0.192829
$$628$$ −19.7990 −0.790066
$$629$$ 41.2132 1.64328
$$630$$ 0 0
$$631$$ −40.8284 −1.62535 −0.812677 0.582714i $$-0.801991\pi$$
−0.812677 + 0.582714i $$0.801991\pi$$
$$632$$ −1.07107 −0.0426048
$$633$$ −26.5563 −1.05552
$$634$$ −19.9706 −0.793132
$$635$$ 0 0
$$636$$ 12.6569 0.501877
$$637$$ 0 0
$$638$$ −3.41421 −0.135170
$$639$$ −9.31371 −0.368445
$$640$$ 0 0
$$641$$ 31.7574 1.25434 0.627170 0.778882i $$-0.284213\pi$$
0.627170 + 0.778882i $$0.284213\pi$$
$$642$$ −6.48528 −0.255954
$$643$$ −13.7574 −0.542537 −0.271269 0.962504i $$-0.587443\pi$$
−0.271269 + 0.962504i $$0.587443\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.24264 −0.324302
$$647$$ 4.87006 0.191462 0.0957309 0.995407i $$-0.469481\pi$$
0.0957309 + 0.995407i $$0.469481\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 24.7279 0.970656
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5.24264 0.205318
$$653$$ −12.6863 −0.496453 −0.248226 0.968702i $$-0.579848\pi$$
−0.248226 + 0.968702i $$0.579848\pi$$
$$654$$ 11.3137 0.442401
$$655$$ 0 0
$$656$$ −5.82843 −0.227562
$$657$$ 15.0711 0.587978
$$658$$ 0 0
$$659$$ 3.17157 0.123547 0.0617735 0.998090i $$-0.480324\pi$$
0.0617735 + 0.998090i $$0.480324\pi$$
$$660$$ 0 0
$$661$$ 34.3431 1.33579 0.667897 0.744254i $$-0.267195\pi$$
0.667897 + 0.744254i $$0.267195\pi$$
$$662$$ 24.0711 0.935549
$$663$$ 17.4853 0.679072
$$664$$ 0.0710678 0.00275797
$$665$$ 0 0
$$666$$ −7.07107 −0.273998
$$667$$ −0.414214 −0.0160384
$$668$$ −8.82843 −0.341582
$$669$$ 24.5563 0.949403
$$670$$ 0 0
$$671$$ −0.585786 −0.0226140
$$672$$ 0 0
$$673$$ −0.798990 −0.0307988 −0.0153994 0.999881i $$-0.504902\pi$$
−0.0153994 + 0.999881i $$0.504902\pi$$
$$674$$ −4.79899 −0.184850
$$675$$ 0 0
$$676$$ −4.00000 −0.153846
$$677$$ −4.38478 −0.168521 −0.0842603 0.996444i $$-0.526853\pi$$
−0.0842603 + 0.996444i $$0.526853\pi$$
$$678$$ −6.58579 −0.252926
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 25.3848 0.972747
$$682$$ 8.24264 0.315627
$$683$$ 25.0711 0.959318 0.479659 0.877455i $$-0.340760\pi$$
0.479659 + 0.877455i $$0.340760\pi$$
$$684$$ 1.41421 0.0540738
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 29.3137 1.11839
$$688$$ 1.58579 0.0604575
$$689$$ 37.9706 1.44656
$$690$$ 0 0
$$691$$ 33.1127 1.25967 0.629833 0.776730i $$-0.283123\pi$$
0.629833 + 0.776730i $$0.283123\pi$$
$$692$$ 14.2426 0.541424
$$693$$ 0 0
$$694$$ 9.79899 0.371965
$$695$$ 0 0
$$696$$ −1.00000 −0.0379049
$$697$$ −33.9706 −1.28673
$$698$$ −24.6569 −0.933276
$$699$$ 14.2426 0.538706
$$700$$ 0 0
$$701$$ 36.3137 1.37155 0.685775 0.727814i $$-0.259463\pi$$
0.685775 + 0.727814i $$0.259463\pi$$
$$702$$ −3.00000 −0.113228
$$703$$ 10.0000 0.377157
$$704$$ 3.41421 0.128678
$$705$$ 0 0
$$706$$ −20.4853 −0.770974
$$707$$ 0 0
$$708$$ 7.24264 0.272195
$$709$$ 19.8995 0.747341 0.373671 0.927561i $$-0.378099\pi$$
0.373671 + 0.927561i $$0.378099\pi$$
$$710$$ 0 0
$$711$$ 1.07107 0.0401682
$$712$$ −4.00000 −0.149906
$$713$$ 1.00000 0.0374503
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −5.07107 −0.189515
$$717$$ −11.3137 −0.422518
$$718$$ −33.8701 −1.26402
$$719$$ 18.9289 0.705930 0.352965 0.935637i $$-0.385173\pi$$
0.352965 + 0.935637i $$0.385173\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 17.0000 0.632674
$$723$$ −6.00000 −0.223142
$$724$$ −23.3137 −0.866447
$$725$$ 0 0
$$726$$ −0.656854 −0.0243781
$$727$$ −37.8701 −1.40452 −0.702261 0.711919i $$-0.747826\pi$$
−0.702261 + 0.711919i $$0.747826\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 9.24264 0.341851
$$732$$ −0.171573 −0.00634152
$$733$$ −0.514719 −0.0190116 −0.00950578 0.999955i $$-0.503026\pi$$
−0.00950578 + 0.999955i $$0.503026\pi$$
$$734$$ −0.757359 −0.0279546
$$735$$ 0 0
$$736$$ 0.414214 0.0152681
$$737$$ −52.2843 −1.92592
$$738$$ 5.82843 0.214547
$$739$$ 53.8701 1.98164 0.990821 0.135180i $$-0.0431613\pi$$
0.990821 + 0.135180i $$0.0431613\pi$$
$$740$$ 0 0
$$741$$ 4.24264 0.155857
$$742$$ 0 0
$$743$$ −36.5563 −1.34112 −0.670561 0.741854i $$-0.733947\pi$$
−0.670561 + 0.741854i $$0.733947\pi$$
$$744$$ 2.41421 0.0885094
$$745$$ 0 0
$$746$$ −2.34315 −0.0857887
$$747$$ −0.0710678 −0.00260024
$$748$$ 19.8995 0.727598
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −12.9706 −0.473303 −0.236651 0.971595i $$-0.576050\pi$$
−0.236651 + 0.971595i $$0.576050\pi$$
$$752$$ −1.65685 −0.0604193
$$753$$ −1.92893 −0.0702942
$$754$$ −3.00000 −0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −44.2426 −1.60803 −0.804013 0.594612i $$-0.797306\pi$$
−0.804013 + 0.594612i $$0.797306\pi$$
$$758$$ 34.2132 1.24268
$$759$$ −1.41421 −0.0513327
$$760$$ 0 0
$$761$$ 15.0294 0.544817 0.272408 0.962182i $$-0.412180\pi$$
0.272408 + 0.962182i $$0.412180\pi$$
$$762$$ 3.17157 0.114894
$$763$$ 0 0
$$764$$ 16.0711 0.581431
$$765$$ 0 0
$$766$$ −27.5563 −0.995651
$$767$$ 21.7279 0.784550
$$768$$ 1.00000 0.0360844
$$769$$ −15.3137 −0.552226 −0.276113 0.961125i $$-0.589046\pi$$
−0.276113 + 0.961125i $$0.589046\pi$$
$$770$$ 0 0
$$771$$ 13.3431 0.480542
$$772$$ −6.82843 −0.245760
$$773$$ −19.3137 −0.694666 −0.347333 0.937742i $$-0.612913\pi$$
−0.347333 + 0.937742i $$0.612913\pi$$
$$774$$ −1.58579 −0.0569999
$$775$$ 0 0
$$776$$ 0.343146 0.0123182
$$777$$ 0 0
$$778$$ 10.3431 0.370820
$$779$$ −8.24264 −0.295323
$$780$$ 0 0
$$781$$ −31.7990 −1.13786
$$782$$ 2.41421 0.0863321
$$783$$ 1.00000 0.0357371
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 18.8284 0.671588
$$787$$ −43.4558 −1.54903 −0.774517 0.632553i $$-0.782007\pi$$
−0.774517 + 0.632553i $$0.782007\pi$$
$$788$$ 23.1421 0.824404
$$789$$ −10.5563 −0.375816
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −3.41421 −0.121319
$$793$$ −0.514719 −0.0182782
$$794$$ −30.3137 −1.07579
$$795$$ 0 0
$$796$$ 19.6569 0.696719
$$797$$ 16.2426 0.575344 0.287672 0.957729i $$-0.407119\pi$$
0.287672 + 0.957729i $$0.407119\pi$$
$$798$$ 0 0
$$799$$ −9.65685 −0.341635
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ −12.1005 −0.427284
$$803$$ 51.4558 1.81584
$$804$$ −15.3137 −0.540073
$$805$$ 0 0
$$806$$ 7.24264 0.255111
$$807$$ 0.443651 0.0156172
$$808$$ −5.65685 −0.199007
$$809$$ −34.6274 −1.21744 −0.608718 0.793387i $$-0.708316\pi$$
−0.608718 + 0.793387i $$0.708316\pi$$
$$810$$ 0 0
$$811$$ 33.7574 1.18538 0.592691 0.805430i $$-0.298066\pi$$
0.592691 + 0.805430i $$0.298066\pi$$
$$812$$ 0 0
$$813$$ −11.5147 −0.403839
$$814$$ −24.1421 −0.846181
$$815$$ 0 0
$$816$$ 5.82843 0.204036
$$817$$ 2.24264 0.0784601
$$818$$ −22.5858 −0.789694
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 13.1716 0.459691 0.229846 0.973227i $$-0.426178\pi$$
0.229846 + 0.973227i $$0.426178\pi$$
$$822$$ 23.2132 0.809653
$$823$$ 28.5858 0.996438 0.498219 0.867051i $$-0.333988\pi$$
0.498219 + 0.867051i $$0.333988\pi$$
$$824$$ 0.0710678 0.00247576
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −11.8995 −0.413786 −0.206893 0.978364i $$-0.566335\pi$$
−0.206893 + 0.978364i $$0.566335\pi$$
$$828$$ −0.414214 −0.0143949
$$829$$ −0.798990 −0.0277501 −0.0138750 0.999904i $$-0.504417\pi$$
−0.0138750 + 0.999904i $$0.504417\pi$$
$$830$$ 0 0
$$831$$ −24.0416 −0.833995
$$832$$ 3.00000 0.104006
$$833$$ 0 0
$$834$$ 15.5563 0.538672
$$835$$ 0 0
$$836$$ 4.82843 0.166995
$$837$$ −2.41421 −0.0834474
$$838$$ −24.5563 −0.848285
$$839$$ 17.4142 0.601205 0.300603 0.953749i $$-0.402812\pi$$
0.300603 + 0.953749i $$0.402812\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 35.6569 1.22882
$$843$$ 20.0000 0.688837
$$844$$ −26.5563 −0.914107
$$845$$ 0 0
$$846$$ 1.65685 0.0569638
$$847$$ 0 0
$$848$$ 12.6569 0.434638
$$849$$ −0.242641 −0.00832741
$$850$$ 0 0
$$851$$ −2.92893 −0.100403
$$852$$ −9.31371 −0.319082
$$853$$ 14.3137 0.490092 0.245046 0.969511i $$-0.421197\pi$$
0.245046 + 0.969511i $$0.421197\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −6.48528 −0.221662
$$857$$ 29.4558 1.00619 0.503096 0.864230i $$-0.332194\pi$$
0.503096 + 0.864230i $$0.332194\pi$$
$$858$$ −10.2426 −0.349678
$$859$$ 21.9411 0.748622 0.374311 0.927303i $$-0.377879\pi$$
0.374311 + 0.927303i $$0.377879\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −33.0416 −1.12540
$$863$$ 24.9706 0.850008 0.425004 0.905192i $$-0.360273\pi$$
0.425004 + 0.905192i $$0.360273\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −28.6274 −0.972799
$$867$$ 16.9706 0.576351
$$868$$ 0 0
$$869$$ 3.65685 0.124050
$$870$$ 0 0
$$871$$ −45.9411 −1.55666
$$872$$ 11.3137 0.383131
$$873$$ −0.343146 −0.0116137
$$874$$ 0.585786 0.0198145
$$875$$ 0 0
$$876$$ 15.0711 0.509204
$$877$$ −26.3431 −0.889545 −0.444772 0.895644i $$-0.646715\pi$$
−0.444772 + 0.895644i $$0.646715\pi$$
$$878$$ −1.58579 −0.0535177
$$879$$ −11.3137 −0.381602
$$880$$ 0 0
$$881$$ 41.6274 1.40246 0.701232 0.712933i $$-0.252634\pi$$
0.701232 + 0.712933i $$0.252634\pi$$
$$882$$ 0 0
$$883$$ −31.5269 −1.06097 −0.530483 0.847696i $$-0.677989\pi$$
−0.530483 + 0.847696i $$0.677989\pi$$
$$884$$ 17.4853 0.588094
$$885$$ 0 0
$$886$$ 14.3431 0.481867
$$887$$ −25.7574 −0.864847 −0.432424 0.901671i $$-0.642342\pi$$
−0.432424 + 0.901671i $$0.642342\pi$$
$$888$$ −7.07107 −0.237289
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.41421 0.114380
$$892$$ 24.5563 0.822207
$$893$$ −2.34315 −0.0784104
$$894$$ −4.65685 −0.155749
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.24264 −0.0414906
$$898$$ 25.0711 0.836632
$$899$$ −2.41421 −0.0805185
$$900$$ 0 0
$$901$$ 73.7696 2.45762
$$902$$ 19.8995 0.662581
$$903$$ 0 0
$$904$$ −6.58579 −0.219040
$$905$$ 0 0
$$906$$ −7.89949 −0.262443
$$907$$ 40.6985 1.35137 0.675686 0.737190i $$-0.263848\pi$$
0.675686 + 0.737190i $$0.263848\pi$$
$$908$$ 25.3848 0.842423
$$909$$ 5.65685 0.187626
$$910$$ 0 0
$$911$$ 22.0711 0.731247 0.365624 0.930763i $$-0.380856\pi$$
0.365624 + 0.930763i $$0.380856\pi$$
$$912$$ 1.41421 0.0468293
$$913$$ −0.242641 −0.00803023
$$914$$ 3.82843 0.126633
$$915$$ 0 0
$$916$$ 29.3137 0.968552
$$917$$ 0 0
$$918$$ −5.82843 −0.192367
$$919$$ 4.44365 0.146583 0.0732913 0.997311i $$-0.476650\pi$$
0.0732913 + 0.997311i $$0.476650\pi$$
$$920$$ 0 0
$$921$$ 8.58579 0.282911
$$922$$ −20.0000 −0.658665
$$923$$ −27.9411 −0.919693
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −29.8995 −0.982558
$$927$$ −0.0710678 −0.00233417
$$928$$ −1.00000 −0.0328266
$$929$$ −8.65685 −0.284022 −0.142011 0.989865i $$-0.545357\pi$$
−0.142011 + 0.989865i $$0.545357\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 14.2426 0.466533
$$933$$ −11.4142 −0.373685
$$934$$ 1.24264 0.0406604
$$935$$ 0 0
$$936$$ −3.00000 −0.0980581
$$937$$ 0.242641 0.00792673 0.00396336 0.999992i $$-0.498738\pi$$
0.00396336 + 0.999992i $$0.498738\pi$$
$$938$$ 0 0
$$939$$ −16.0416 −0.523499
$$940$$ 0 0
$$941$$ 51.3137 1.67278 0.836390 0.548136i $$-0.184662\pi$$
0.836390 + 0.548136i $$0.184662\pi$$
$$942$$ 19.7990 0.645086
$$943$$ 2.41421 0.0786176
$$944$$ 7.24264 0.235728
$$945$$ 0 0
$$946$$ −5.41421 −0.176031
$$947$$ −25.2132 −0.819319 −0.409660 0.912239i $$-0.634352\pi$$
−0.409660 + 0.912239i $$0.634352\pi$$
$$948$$ 1.07107 0.0347867
$$949$$ 45.2132 1.46768
$$950$$ 0 0
$$951$$ 19.9706 0.647590
$$952$$ 0 0
$$953$$ −31.5980 −1.02356 −0.511779 0.859117i $$-0.671014\pi$$
−0.511779 + 0.859117i $$0.671014\pi$$
$$954$$ −12.6569 −0.409781
$$955$$ 0 0
$$956$$ −11.3137 −0.365911
$$957$$ 3.41421 0.110366
$$958$$ −20.2426 −0.654010
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25.1716 −0.811986
$$962$$ −21.2132 −0.683941
$$963$$ 6.48528 0.208985
$$964$$ −6.00000 −0.193247
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −44.0416 −1.41628 −0.708142 0.706070i $$-0.750466\pi$$
−0.708142 + 0.706070i $$0.750466\pi$$
$$968$$ −0.656854 −0.0211121
$$969$$ 8.24264 0.264792
$$970$$ 0 0
$$971$$ −44.9706 −1.44317 −0.721587 0.692324i $$-0.756587\pi$$
−0.721587 + 0.692324i $$0.756587\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −5.51472 −0.176703
$$975$$ 0 0
$$976$$ −0.171573 −0.00549191
$$977$$ −8.44365 −0.270136 −0.135068 0.990836i $$-0.543125\pi$$
−0.135068 + 0.990836i $$0.543125\pi$$
$$978$$ −5.24264 −0.167641
$$979$$ 13.6569 0.436475
$$980$$ 0 0
$$981$$ −11.3137 −0.361219
$$982$$ 0.142136 0.00453573
$$983$$ 28.6863 0.914951 0.457475 0.889222i $$-0.348754\pi$$
0.457475 + 0.889222i $$0.348754\pi$$
$$984$$ 5.82843 0.185803
$$985$$ 0 0
$$986$$ −5.82843 −0.185615
$$987$$ 0 0
$$988$$ 4.24264 0.134976
$$989$$ −0.656854 −0.0208867
$$990$$ 0 0
$$991$$ −18.3431 −0.582689 −0.291345 0.956618i $$-0.594103\pi$$
−0.291345 + 0.956618i $$0.594103\pi$$
$$992$$ 2.41421 0.0766514
$$993$$ −24.0711 −0.763872
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −0.0710678 −0.00225187
$$997$$ −25.3137 −0.801693 −0.400847 0.916145i $$-0.631284\pi$$
−0.400847 + 0.916145i $$0.631284\pi$$
$$998$$ 26.0711 0.825265
$$999$$ 7.07107 0.223719
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 7350.2.a.dg.1.2 yes 2
5.4 even 2 7350.2.a.dj.1.2 yes 2
7.6 odd 2 7350.2.a.dc.1.2 2
35.34 odd 2 7350.2.a.dm.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
7350.2.a.dc.1.2 2 7.6 odd 2
7350.2.a.dg.1.2 yes 2 1.1 even 1 trivial
7350.2.a.dj.1.2 yes 2 5.4 even 2
7350.2.a.dm.1.2 yes 2 35.34 odd 2