# Properties

 Label 1950.2.e.o.1249.3 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.3 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1249 Dual form 1950.2.e.o.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.82843i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.82843i q^{7} -1.00000i q^{8} -1.00000 q^{9} +5.65685 q^{11} +1.00000i q^{12} +1.00000i q^{13} +2.82843 q^{14} +1.00000 q^{16} +0.828427i q^{17} -1.00000i q^{18} -2.82843 q^{19} -2.82843 q^{21} +5.65685i q^{22} +8.48528i q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.82843i q^{28} +8.82843 q^{29} +4.00000 q^{31} +1.00000i q^{32} -5.65685i q^{33} -0.828427 q^{34} +1.00000 q^{36} -11.6569i q^{37} -2.82843i q^{38} +1.00000 q^{39} -7.65685 q^{41} -2.82843i q^{42} -9.65685i q^{43} -5.65685 q^{44} -8.48528 q^{46} -8.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +0.828427 q^{51} -1.00000i q^{52} -13.3137i q^{53} -1.00000 q^{54} -2.82843 q^{56} +2.82843i q^{57} +8.82843i q^{58} +2.34315 q^{59} +6.00000 q^{61} +4.00000i q^{62} +2.82843i q^{63} -1.00000 q^{64} +5.65685 q^{66} +5.65685i q^{67} -0.828427i q^{68} +8.48528 q^{69} -5.65685 q^{71} +1.00000i q^{72} +14.4853i q^{73} +11.6569 q^{74} +2.82843 q^{76} -16.0000i q^{77} +1.00000i q^{78} -2.34315 q^{79} +1.00000 q^{81} -7.65685i q^{82} -6.34315i q^{83} +2.82843 q^{84} +9.65685 q^{86} -8.82843i q^{87} -5.65685i q^{88} +15.6569 q^{89} +2.82843 q^{91} -8.48528i q^{92} -4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} -3.17157i q^{97} -1.00000i q^{98} -5.65685 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + 4q^{16} - 4q^{24} - 4q^{26} + 24q^{29} + 16q^{31} + 8q^{34} + 4q^{36} + 4q^{39} - 8q^{41} - 4q^{49} - 8q^{51} - 4q^{54} + 32q^{59} + 24q^{61} - 4q^{64} + 24q^{74} - 32q^{79} + 4q^{81} + 16q^{86} + 40q^{89} + 32q^{94} + 4q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 2.82843i − 1.06904i −0.845154 0.534522i $$-0.820491\pi$$
0.845154 0.534522i $$-0.179509\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 5.65685 1.70561 0.852803 0.522233i $$-0.174901\pi$$
0.852803 + 0.522233i $$0.174901\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 1.00000i 0.277350i
$$14$$ 2.82843 0.755929
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.828427i 0.200923i 0.994941 + 0.100462i $$0.0320319\pi$$
−0.994941 + 0.100462i $$0.967968\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ −2.82843 −0.617213
$$22$$ 5.65685i 1.20605i
$$23$$ 8.48528i 1.76930i 0.466252 + 0.884652i $$0.345604\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000i 0.192450i
$$28$$ 2.82843i 0.534522i
$$29$$ 8.82843 1.63940 0.819699 0.572795i $$-0.194141\pi$$
0.819699 + 0.572795i $$0.194141\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 5.65685i − 0.984732i
$$34$$ −0.828427 −0.142074
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 11.6569i − 1.91638i −0.286141 0.958188i $$-0.592373\pi$$
0.286141 0.958188i $$-0.407627\pi$$
$$38$$ − 2.82843i − 0.458831i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −7.65685 −1.19580 −0.597900 0.801571i $$-0.703998\pi$$
−0.597900 + 0.801571i $$0.703998\pi$$
$$42$$ − 2.82843i − 0.436436i
$$43$$ − 9.65685i − 1.47266i −0.676625 0.736328i $$-0.736558\pi$$
0.676625 0.736328i $$-0.263442\pi$$
$$44$$ −5.65685 −0.852803
$$45$$ 0 0
$$46$$ −8.48528 −1.25109
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0.828427 0.116003
$$52$$ − 1.00000i − 0.138675i
$$53$$ − 13.3137i − 1.82878i −0.404836 0.914389i $$-0.632671\pi$$
0.404836 0.914389i $$-0.367329\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.82843 −0.377964
$$57$$ 2.82843i 0.374634i
$$58$$ 8.82843i 1.15923i
$$59$$ 2.34315 0.305052 0.152526 0.988299i $$-0.451259\pi$$
0.152526 + 0.988299i $$0.451259\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 2.82843i 0.356348i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 5.65685 0.696311
$$67$$ 5.65685i 0.691095i 0.938401 + 0.345547i $$0.112307\pi$$
−0.938401 + 0.345547i $$0.887693\pi$$
$$68$$ − 0.828427i − 0.100462i
$$69$$ 8.48528 1.02151
$$70$$ 0 0
$$71$$ −5.65685 −0.671345 −0.335673 0.941979i $$-0.608964\pi$$
−0.335673 + 0.941979i $$0.608964\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 14.4853i 1.69537i 0.530497 + 0.847687i $$0.322005\pi$$
−0.530497 + 0.847687i $$0.677995\pi$$
$$74$$ 11.6569 1.35508
$$75$$ 0 0
$$76$$ 2.82843 0.324443
$$77$$ − 16.0000i − 1.82337i
$$78$$ 1.00000i 0.113228i
$$79$$ −2.34315 −0.263624 −0.131812 0.991275i $$-0.542080\pi$$
−0.131812 + 0.991275i $$0.542080\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 7.65685i − 0.845558i
$$83$$ − 6.34315i − 0.696251i −0.937448 0.348125i $$-0.886818\pi$$
0.937448 0.348125i $$-0.113182\pi$$
$$84$$ 2.82843 0.308607
$$85$$ 0 0
$$86$$ 9.65685 1.04133
$$87$$ − 8.82843i − 0.946507i
$$88$$ − 5.65685i − 0.603023i
$$89$$ 15.6569 1.65962 0.829812 0.558044i $$-0.188448\pi$$
0.829812 + 0.558044i $$0.188448\pi$$
$$90$$ 0 0
$$91$$ 2.82843 0.296500
$$92$$ − 8.48528i − 0.884652i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 3.17157i − 0.322024i −0.986952 0.161012i $$-0.948524\pi$$
0.986952 0.161012i $$-0.0514759\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ −5.65685 −0.568535
$$100$$ 0 0
$$101$$ 16.1421 1.60620 0.803101 0.595843i $$-0.203182\pi$$
0.803101 + 0.595843i $$0.203182\pi$$
$$102$$ 0.828427i 0.0820265i
$$103$$ 1.65685i 0.163255i 0.996663 + 0.0816274i $$0.0260117\pi$$
−0.996663 + 0.0816274i $$0.973988\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 13.3137 1.29314
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −8.82843 −0.845610 −0.422805 0.906221i $$-0.638954\pi$$
−0.422805 + 0.906221i $$0.638954\pi$$
$$110$$ 0 0
$$111$$ −11.6569 −1.10642
$$112$$ − 2.82843i − 0.267261i
$$113$$ − 6.48528i − 0.610084i −0.952339 0.305042i $$-0.901330\pi$$
0.952339 0.305042i $$-0.0986705\pi$$
$$114$$ −2.82843 −0.264906
$$115$$ 0 0
$$116$$ −8.82843 −0.819699
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 2.34315i 0.215704i
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 21.0000 1.90909
$$122$$ 6.00000i 0.543214i
$$123$$ 7.65685i 0.690395i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −2.82843 −0.251976
$$127$$ 9.65685i 0.856907i 0.903564 + 0.428454i $$0.140941\pi$$
−0.903564 + 0.428454i $$0.859059\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −9.65685 −0.850239
$$130$$ 0 0
$$131$$ −6.14214 −0.536641 −0.268320 0.963330i $$-0.586469\pi$$
−0.268320 + 0.963330i $$0.586469\pi$$
$$132$$ 5.65685i 0.492366i
$$133$$ 8.00000i 0.693688i
$$134$$ −5.65685 −0.488678
$$135$$ 0 0
$$136$$ 0.828427 0.0710370
$$137$$ − 17.3137i − 1.47921i −0.673041 0.739605i $$-0.735012\pi$$
0.673041 0.739605i $$-0.264988\pi$$
$$138$$ 8.48528i 0.722315i
$$139$$ −6.34315 −0.538019 −0.269009 0.963138i $$-0.586696\pi$$
−0.269009 + 0.963138i $$0.586696\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ − 5.65685i − 0.474713i
$$143$$ 5.65685i 0.473050i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −14.4853 −1.19881
$$147$$ 1.00000i 0.0824786i
$$148$$ 11.6569i 0.958188i
$$149$$ −3.65685 −0.299581 −0.149791 0.988718i $$-0.547860\pi$$
−0.149791 + 0.988718i $$0.547860\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 2.82843i 0.229416i
$$153$$ − 0.828427i − 0.0669744i
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 5.31371i 0.424080i 0.977261 + 0.212040i $$0.0680107\pi$$
−0.977261 + 0.212040i $$0.931989\pi$$
$$158$$ − 2.34315i − 0.186411i
$$159$$ −13.3137 −1.05585
$$160$$ 0 0
$$161$$ 24.0000 1.89146
$$162$$ 1.00000i 0.0785674i
$$163$$ − 11.3137i − 0.886158i −0.896483 0.443079i $$-0.853886\pi$$
0.896483 0.443079i $$-0.146114\pi$$
$$164$$ 7.65685 0.597900
$$165$$ 0 0
$$166$$ 6.34315 0.492324
$$167$$ 8.97056i 0.694163i 0.937835 + 0.347081i $$0.112827\pi$$
−0.937835 + 0.347081i $$0.887173\pi$$
$$168$$ 2.82843i 0.218218i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 2.82843 0.216295
$$172$$ 9.65685i 0.736328i
$$173$$ 9.31371i 0.708108i 0.935225 + 0.354054i $$0.115197\pi$$
−0.935225 + 0.354054i $$0.884803\pi$$
$$174$$ 8.82843 0.669281
$$175$$ 0 0
$$176$$ 5.65685 0.426401
$$177$$ − 2.34315i − 0.176122i
$$178$$ 15.6569i 1.17353i
$$179$$ −7.51472 −0.561676 −0.280838 0.959755i $$-0.590612\pi$$
−0.280838 + 0.959755i $$0.590612\pi$$
$$180$$ 0 0
$$181$$ −7.65685 −0.569129 −0.284565 0.958657i $$-0.591849\pi$$
−0.284565 + 0.958657i $$0.591849\pi$$
$$182$$ 2.82843i 0.209657i
$$183$$ − 6.00000i − 0.443533i
$$184$$ 8.48528 0.625543
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 4.68629i 0.342696i
$$188$$ 8.00000i 0.583460i
$$189$$ 2.82843 0.205738
$$190$$ 0 0
$$191$$ 11.3137 0.818631 0.409316 0.912393i $$-0.365768\pi$$
0.409316 + 0.912393i $$0.365768\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 2.48528i − 0.178894i −0.995992 0.0894472i $$-0.971490\pi$$
0.995992 0.0894472i $$-0.0285100\pi$$
$$194$$ 3.17157 0.227706
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 13.3137i − 0.948562i −0.880373 0.474281i $$-0.842708\pi$$
0.880373 0.474281i $$-0.157292\pi$$
$$198$$ − 5.65685i − 0.402015i
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ 5.65685 0.399004
$$202$$ 16.1421i 1.13576i
$$203$$ − 24.9706i − 1.75259i
$$204$$ −0.828427 −0.0580015
$$205$$ 0 0
$$206$$ −1.65685 −0.115439
$$207$$ − 8.48528i − 0.589768i
$$208$$ 1.00000i 0.0693375i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 0.686292 0.0472463 0.0236231 0.999721i $$-0.492480\pi$$
0.0236231 + 0.999721i $$0.492480\pi$$
$$212$$ 13.3137i 0.914389i
$$213$$ 5.65685i 0.387601i
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 11.3137i − 0.768025i
$$218$$ − 8.82843i − 0.597937i
$$219$$ 14.4853 0.978825
$$220$$ 0 0
$$221$$ −0.828427 −0.0557260
$$222$$ − 11.6569i − 0.782357i
$$223$$ − 10.8284i − 0.725125i −0.931959 0.362563i $$-0.881902\pi$$
0.931959 0.362563i $$-0.118098\pi$$
$$224$$ 2.82843 0.188982
$$225$$ 0 0
$$226$$ 6.48528 0.431394
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ − 2.82843i − 0.187317i
$$229$$ −4.14214 −0.273720 −0.136860 0.990590i $$-0.543701\pi$$
−0.136860 + 0.990590i $$0.543701\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ − 8.82843i − 0.579615i
$$233$$ − 5.51472i − 0.361281i −0.983549 0.180641i $$-0.942183\pi$$
0.983549 0.180641i $$-0.0578171\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ −2.34315 −0.152526
$$237$$ 2.34315i 0.152204i
$$238$$ 2.34315i 0.151884i
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 5.31371 0.342286 0.171143 0.985246i $$-0.445254\pi$$
0.171143 + 0.985246i $$0.445254\pi$$
$$242$$ 21.0000i 1.34993i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ −7.65685 −0.488183
$$247$$ − 2.82843i − 0.179969i
$$248$$ − 4.00000i − 0.254000i
$$249$$ −6.34315 −0.401981
$$250$$ 0 0
$$251$$ 10.8284 0.683484 0.341742 0.939794i $$-0.388983\pi$$
0.341742 + 0.939794i $$0.388983\pi$$
$$252$$ − 2.82843i − 0.178174i
$$253$$ 48.0000i 3.01773i
$$254$$ −9.65685 −0.605925
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 4.82843i − 0.301189i −0.988596 0.150595i $$-0.951881\pi$$
0.988596 0.150595i $$-0.0481188\pi$$
$$258$$ − 9.65685i − 0.601209i
$$259$$ −32.9706 −2.04869
$$260$$ 0 0
$$261$$ −8.82843 −0.546466
$$262$$ − 6.14214i − 0.379462i
$$263$$ − 16.4853i − 1.01653i −0.861202 0.508263i $$-0.830288\pi$$
0.861202 0.508263i $$-0.169712\pi$$
$$264$$ −5.65685 −0.348155
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ − 15.6569i − 0.958184i
$$268$$ − 5.65685i − 0.345547i
$$269$$ 14.4853 0.883183 0.441592 0.897216i $$-0.354414\pi$$
0.441592 + 0.897216i $$0.354414\pi$$
$$270$$ 0 0
$$271$$ 7.31371 0.444276 0.222138 0.975015i $$-0.428696\pi$$
0.222138 + 0.975015i $$0.428696\pi$$
$$272$$ 0.828427i 0.0502308i
$$273$$ − 2.82843i − 0.171184i
$$274$$ 17.3137 1.04596
$$275$$ 0 0
$$276$$ −8.48528 −0.510754
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ − 6.34315i − 0.380437i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 8.34315 0.497710 0.248855 0.968541i $$-0.419946\pi$$
0.248855 + 0.968541i $$0.419946\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ 17.6569i 1.04959i 0.851228 + 0.524796i $$0.175858\pi$$
−0.851228 + 0.524796i $$0.824142\pi$$
$$284$$ 5.65685 0.335673
$$285$$ 0 0
$$286$$ −5.65685 −0.334497
$$287$$ 21.6569i 1.27836i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 16.3137 0.959630
$$290$$ 0 0
$$291$$ −3.17157 −0.185921
$$292$$ − 14.4853i − 0.847687i
$$293$$ 16.6274i 0.971384i 0.874130 + 0.485692i $$0.161432\pi$$
−0.874130 + 0.485692i $$0.838568\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ −11.6569 −0.677541
$$297$$ 5.65685i 0.328244i
$$298$$ − 3.65685i − 0.211836i
$$299$$ −8.48528 −0.490716
$$300$$ 0 0
$$301$$ −27.3137 −1.57434
$$302$$ 12.0000i 0.690522i
$$303$$ − 16.1421i − 0.927341i
$$304$$ −2.82843 −0.162221
$$305$$ 0 0
$$306$$ 0.828427 0.0473580
$$307$$ − 21.6569i − 1.23602i −0.786169 0.618011i $$-0.787939\pi$$
0.786169 0.618011i $$-0.212061\pi$$
$$308$$ 16.0000i 0.911685i
$$309$$ 1.65685 0.0942551
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 30.9706i 1.75056i 0.483617 + 0.875280i $$0.339323\pi$$
−0.483617 + 0.875280i $$0.660677\pi$$
$$314$$ −5.31371 −0.299870
$$315$$ 0 0
$$316$$ 2.34315 0.131812
$$317$$ 25.3137i 1.42176i 0.703314 + 0.710880i $$0.251703\pi$$
−0.703314 + 0.710880i $$0.748297\pi$$
$$318$$ − 13.3137i − 0.746596i
$$319$$ 49.9411 2.79617
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 24.0000i 1.33747i
$$323$$ − 2.34315i − 0.130376i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 11.3137 0.626608
$$327$$ 8.82843i 0.488213i
$$328$$ 7.65685i 0.422779i
$$329$$ −22.6274 −1.24749
$$330$$ 0 0
$$331$$ −8.48528 −0.466393 −0.233197 0.972430i $$-0.574919\pi$$
−0.233197 + 0.972430i $$0.574919\pi$$
$$332$$ 6.34315i 0.348125i
$$333$$ 11.6569i 0.638792i
$$334$$ −8.97056 −0.490847
$$335$$ 0 0
$$336$$ −2.82843 −0.154303
$$337$$ 10.9706i 0.597605i 0.954315 + 0.298802i $$0.0965871\pi$$
−0.954315 + 0.298802i $$0.903413\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ −6.48528 −0.352232
$$340$$ 0 0
$$341$$ 22.6274 1.22534
$$342$$ 2.82843i 0.152944i
$$343$$ − 16.9706i − 0.916324i
$$344$$ −9.65685 −0.520663
$$345$$ 0 0
$$346$$ −9.31371 −0.500708
$$347$$ − 9.65685i − 0.518407i −0.965823 0.259204i $$-0.916540\pi$$
0.965823 0.259204i $$-0.0834600\pi$$
$$348$$ 8.82843i 0.473253i
$$349$$ −12.1421 −0.649954 −0.324977 0.945722i $$-0.605356\pi$$
−0.324977 + 0.945722i $$0.605356\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 5.65685i 0.301511i
$$353$$ − 5.31371i − 0.282820i −0.989951 0.141410i $$-0.954836\pi$$
0.989951 0.141410i $$-0.0451636\pi$$
$$354$$ 2.34315 0.124537
$$355$$ 0 0
$$356$$ −15.6569 −0.829812
$$357$$ − 2.34315i − 0.124012i
$$358$$ − 7.51472i − 0.397165i
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ − 7.65685i − 0.402435i
$$363$$ − 21.0000i − 1.10221i
$$364$$ −2.82843 −0.148250
$$365$$ 0 0
$$366$$ 6.00000 0.313625
$$367$$ 25.6569i 1.33928i 0.742687 + 0.669638i $$0.233551\pi$$
−0.742687 + 0.669638i $$0.766449\pi$$
$$368$$ 8.48528i 0.442326i
$$369$$ 7.65685 0.398600
$$370$$ 0 0
$$371$$ −37.6569 −1.95505
$$372$$ 4.00000i 0.207390i
$$373$$ 2.68629i 0.139091i 0.997579 + 0.0695455i $$0.0221549\pi$$
−0.997579 + 0.0695455i $$0.977845\pi$$
$$374$$ −4.68629 −0.242322
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 8.82843i 0.454687i
$$378$$ 2.82843i 0.145479i
$$379$$ 7.51472 0.386005 0.193003 0.981198i $$-0.438177\pi$$
0.193003 + 0.981198i $$0.438177\pi$$
$$380$$ 0 0
$$381$$ 9.65685 0.494736
$$382$$ 11.3137i 0.578860i
$$383$$ 29.6569i 1.51539i 0.652606 + 0.757697i $$0.273676\pi$$
−0.652606 + 0.757697i $$0.726324\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.48528 0.126497
$$387$$ 9.65685i 0.490885i
$$388$$ 3.17157i 0.161012i
$$389$$ 6.48528 0.328817 0.164408 0.986392i $$-0.447429\pi$$
0.164408 + 0.986392i $$0.447429\pi$$
$$390$$ 0 0
$$391$$ −7.02944 −0.355494
$$392$$ 1.00000i 0.0505076i
$$393$$ 6.14214i 0.309830i
$$394$$ 13.3137 0.670735
$$395$$ 0 0
$$396$$ 5.65685 0.284268
$$397$$ 30.2843i 1.51992i 0.649968 + 0.759962i $$0.274782\pi$$
−0.649968 + 0.759962i $$0.725218\pi$$
$$398$$ − 10.3431i − 0.518455i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −26.9706 −1.34685 −0.673423 0.739258i $$-0.735177\pi$$
−0.673423 + 0.739258i $$0.735177\pi$$
$$402$$ 5.65685i 0.282138i
$$403$$ 4.00000i 0.199254i
$$404$$ −16.1421 −0.803101
$$405$$ 0 0
$$406$$ 24.9706 1.23927
$$407$$ − 65.9411i − 3.26858i
$$408$$ − 0.828427i − 0.0410133i
$$409$$ 3.65685 0.180820 0.0904099 0.995905i $$-0.471182\pi$$
0.0904099 + 0.995905i $$0.471182\pi$$
$$410$$ 0 0
$$411$$ −17.3137 −0.854022
$$412$$ − 1.65685i − 0.0816274i
$$413$$ − 6.62742i − 0.326114i
$$414$$ 8.48528 0.417029
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 6.34315i 0.310625i
$$418$$ − 16.0000i − 0.782586i
$$419$$ 10.8284 0.529003 0.264502 0.964385i $$-0.414793\pi$$
0.264502 + 0.964385i $$0.414793\pi$$
$$420$$ 0 0
$$421$$ −24.1421 −1.17662 −0.588308 0.808637i $$-0.700206\pi$$
−0.588308 + 0.808637i $$0.700206\pi$$
$$422$$ 0.686292i 0.0334081i
$$423$$ 8.00000i 0.388973i
$$424$$ −13.3137 −0.646571
$$425$$ 0 0
$$426$$ −5.65685 −0.274075
$$427$$ − 16.9706i − 0.821263i
$$428$$ 4.00000i 0.193347i
$$429$$ 5.65685 0.273115
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 22.9706i 1.10389i 0.833879 + 0.551947i $$0.186115\pi$$
−0.833879 + 0.551947i $$0.813885\pi$$
$$434$$ 11.3137 0.543075
$$435$$ 0 0
$$436$$ 8.82843 0.422805
$$437$$ − 24.0000i − 1.14808i
$$438$$ 14.4853i 0.692134i
$$439$$ −22.6274 −1.07995 −0.539974 0.841682i $$-0.681566\pi$$
−0.539974 + 0.841682i $$0.681566\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ − 0.828427i − 0.0394043i
$$443$$ − 30.3431i − 1.44165i −0.693119 0.720823i $$-0.743764\pi$$
0.693119 0.720823i $$-0.256236\pi$$
$$444$$ 11.6569 0.553210
$$445$$ 0 0
$$446$$ 10.8284 0.512741
$$447$$ 3.65685i 0.172963i
$$448$$ 2.82843i 0.133631i
$$449$$ −26.2843 −1.24043 −0.620216 0.784431i $$-0.712955\pi$$
−0.620216 + 0.784431i $$0.712955\pi$$
$$450$$ 0 0
$$451$$ −43.3137 −2.03956
$$452$$ 6.48528i 0.305042i
$$453$$ − 12.0000i − 0.563809i
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 2.82843 0.132453
$$457$$ 20.8284i 0.974313i 0.873315 + 0.487156i $$0.161966\pi$$
−0.873315 + 0.487156i $$0.838034\pi$$
$$458$$ − 4.14214i − 0.193549i
$$459$$ −0.828427 −0.0386677
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ − 16.0000i − 0.744387i
$$463$$ 3.79899i 0.176554i 0.996096 + 0.0882770i $$0.0281361\pi$$
−0.996096 + 0.0882770i $$0.971864\pi$$
$$464$$ 8.82843 0.409849
$$465$$ 0 0
$$466$$ 5.51472 0.255464
$$467$$ 7.31371i 0.338438i 0.985578 + 0.169219i $$0.0541245\pi$$
−0.985578 + 0.169219i $$0.945875\pi$$
$$468$$ 1.00000i 0.0462250i
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 5.31371 0.244843
$$472$$ − 2.34315i − 0.107852i
$$473$$ − 54.6274i − 2.51177i
$$474$$ −2.34315 −0.107624
$$475$$ 0 0
$$476$$ −2.34315 −0.107398
$$477$$ 13.3137i 0.609593i
$$478$$ − 16.0000i − 0.731823i
$$479$$ 11.3137 0.516937 0.258468 0.966020i $$-0.416782\pi$$
0.258468 + 0.966020i $$0.416782\pi$$
$$480$$ 0 0
$$481$$ 11.6569 0.531507
$$482$$ 5.31371i 0.242033i
$$483$$ − 24.0000i − 1.09204i
$$484$$ −21.0000 −0.954545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 16.4853i 0.747019i 0.927626 + 0.373510i $$0.121846\pi$$
−0.927626 + 0.373510i $$0.878154\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ −11.3137 −0.511624
$$490$$ 0 0
$$491$$ −38.1421 −1.72133 −0.860665 0.509171i $$-0.829952\pi$$
−0.860665 + 0.509171i $$0.829952\pi$$
$$492$$ − 7.65685i − 0.345198i
$$493$$ 7.31371i 0.329393i
$$494$$ 2.82843 0.127257
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 16.0000i 0.717698i
$$498$$ − 6.34315i − 0.284243i
$$499$$ −0.485281 −0.0217242 −0.0108621 0.999941i $$-0.503458\pi$$
−0.0108621 + 0.999941i $$0.503458\pi$$
$$500$$ 0 0
$$501$$ 8.97056 0.400775
$$502$$ 10.8284i 0.483296i
$$503$$ 23.5147i 1.04847i 0.851574 + 0.524235i $$0.175649\pi$$
−0.851574 + 0.524235i $$0.824351\pi$$
$$504$$ 2.82843 0.125988
$$505$$ 0 0
$$506$$ −48.0000 −2.13386
$$507$$ 1.00000i 0.0444116i
$$508$$ − 9.65685i − 0.428454i
$$509$$ 37.3137 1.65390 0.826951 0.562275i $$-0.190074\pi$$
0.826951 + 0.562275i $$0.190074\pi$$
$$510$$ 0 0
$$511$$ 40.9706 1.81243
$$512$$ 1.00000i 0.0441942i
$$513$$ − 2.82843i − 0.124878i
$$514$$ 4.82843 0.212973
$$515$$ 0 0
$$516$$ 9.65685 0.425119
$$517$$ − 45.2548i − 1.99031i
$$518$$ − 32.9706i − 1.44864i
$$519$$ 9.31371 0.408826
$$520$$ 0 0
$$521$$ 26.9706 1.18160 0.590801 0.806817i $$-0.298812\pi$$
0.590801 + 0.806817i $$0.298812\pi$$
$$522$$ − 8.82843i − 0.386410i
$$523$$ 10.6274i 0.464704i 0.972632 + 0.232352i $$0.0746422\pi$$
−0.972632 + 0.232352i $$0.925358\pi$$
$$524$$ 6.14214 0.268320
$$525$$ 0 0
$$526$$ 16.4853 0.718792
$$527$$ 3.31371i 0.144347i
$$528$$ − 5.65685i − 0.246183i
$$529$$ −49.0000 −2.13043
$$530$$ 0 0
$$531$$ −2.34315 −0.101684
$$532$$ − 8.00000i − 0.346844i
$$533$$ − 7.65685i − 0.331655i
$$534$$ 15.6569 0.677538
$$535$$ 0 0
$$536$$ 5.65685 0.244339
$$537$$ 7.51472i 0.324284i
$$538$$ 14.4853i 0.624505i
$$539$$ −5.65685 −0.243658
$$540$$ 0 0
$$541$$ 14.4853 0.622771 0.311385 0.950284i $$-0.399207\pi$$
0.311385 + 0.950284i $$0.399207\pi$$
$$542$$ 7.31371i 0.314151i
$$543$$ 7.65685i 0.328587i
$$544$$ −0.828427 −0.0355185
$$545$$ 0 0
$$546$$ 2.82843 0.121046
$$547$$ 0.686292i 0.0293437i 0.999892 + 0.0146719i $$0.00467036\pi$$
−0.999892 + 0.0146719i $$0.995330\pi$$
$$548$$ 17.3137i 0.739605i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −24.9706 −1.06378
$$552$$ − 8.48528i − 0.361158i
$$553$$ 6.62742i 0.281826i
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ 6.34315 0.269009
$$557$$ 10.6863i 0.452793i 0.974035 + 0.226396i $$0.0726945\pi$$
−0.974035 + 0.226396i $$0.927306\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ 9.65685 0.408441
$$560$$ 0 0
$$561$$ 4.68629 0.197855
$$562$$ 8.34315i 0.351934i
$$563$$ 30.3431i 1.27881i 0.768870 + 0.639406i $$0.220819\pi$$
−0.768870 + 0.639406i $$0.779181\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ −17.6569 −0.742173
$$567$$ − 2.82843i − 0.118783i
$$568$$ 5.65685i 0.237356i
$$569$$ −31.6569 −1.32712 −0.663562 0.748121i $$-0.730956\pi$$
−0.663562 + 0.748121i $$0.730956\pi$$
$$570$$ 0 0
$$571$$ 20.9706 0.877591 0.438795 0.898587i $$-0.355405\pi$$
0.438795 + 0.898587i $$0.355405\pi$$
$$572$$ − 5.65685i − 0.236525i
$$573$$ − 11.3137i − 0.472637i
$$574$$ −21.6569 −0.903940
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 23.4558i − 0.976480i −0.872710 0.488240i $$-0.837639\pi$$
0.872710 0.488240i $$-0.162361\pi$$
$$578$$ 16.3137i 0.678561i
$$579$$ −2.48528 −0.103285
$$580$$ 0 0
$$581$$ −17.9411 −0.744323
$$582$$ − 3.17157i − 0.131466i
$$583$$ − 75.3137i − 3.11918i
$$584$$ 14.4853 0.599405
$$585$$ 0 0
$$586$$ −16.6274 −0.686872
$$587$$ 2.62742i 0.108445i 0.998529 + 0.0542226i $$0.0172680\pi$$
−0.998529 + 0.0542226i $$0.982732\pi$$
$$588$$ − 1.00000i − 0.0412393i
$$589$$ −11.3137 −0.466173
$$590$$ 0 0
$$591$$ −13.3137 −0.547653
$$592$$ − 11.6569i − 0.479094i
$$593$$ 0.343146i 0.0140913i 0.999975 + 0.00704565i $$0.00224272\pi$$
−0.999975 + 0.00704565i $$0.997757\pi$$
$$594$$ −5.65685 −0.232104
$$595$$ 0 0
$$596$$ 3.65685 0.149791
$$597$$ 10.3431i 0.423317i
$$598$$ − 8.48528i − 0.346989i
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 29.3137 1.19573 0.597866 0.801596i $$-0.296016\pi$$
0.597866 + 0.801596i $$0.296016\pi$$
$$602$$ − 27.3137i − 1.11322i
$$603$$ − 5.65685i − 0.230365i
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ 16.1421 0.655729
$$607$$ 28.9706i 1.17588i 0.808905 + 0.587939i $$0.200061\pi$$
−0.808905 + 0.587939i $$0.799939\pi$$
$$608$$ − 2.82843i − 0.114708i
$$609$$ −24.9706 −1.01186
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0.828427i 0.0334872i
$$613$$ − 22.2843i − 0.900053i −0.893015 0.450027i $$-0.851414\pi$$
0.893015 0.450027i $$-0.148586\pi$$
$$614$$ 21.6569 0.874000
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ 2.00000i 0.0805170i 0.999189 + 0.0402585i $$0.0128181\pi$$
−0.999189 + 0.0402585i $$0.987182\pi$$
$$618$$ 1.65685i 0.0666485i
$$619$$ 34.8284 1.39987 0.699936 0.714205i $$-0.253212\pi$$
0.699936 + 0.714205i $$0.253212\pi$$
$$620$$ 0 0
$$621$$ −8.48528 −0.340503
$$622$$ − 24.0000i − 0.962312i
$$623$$ − 44.2843i − 1.77421i
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −30.9706 −1.23783
$$627$$ 16.0000i 0.638978i
$$628$$ − 5.31371i − 0.212040i
$$629$$ 9.65685 0.385044
$$630$$ 0 0
$$631$$ −33.6569 −1.33986 −0.669929 0.742425i $$-0.733676\pi$$
−0.669929 + 0.742425i $$0.733676\pi$$
$$632$$ 2.34315i 0.0932053i
$$633$$ − 0.686292i − 0.0272776i
$$634$$ −25.3137 −1.00534
$$635$$ 0 0
$$636$$ 13.3137 0.527923
$$637$$ − 1.00000i − 0.0396214i
$$638$$ 49.9411i 1.97719i
$$639$$ 5.65685 0.223782
$$640$$ 0 0
$$641$$ −4.62742 −0.182772 −0.0913860 0.995816i $$-0.529130\pi$$
−0.0913860 + 0.995816i $$0.529130\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ − 39.5980i − 1.56159i −0.624786 0.780796i $$-0.714814\pi$$
0.624786 0.780796i $$-0.285186\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 2.34315 0.0921898
$$647$$ 8.48528i 0.333591i 0.985992 + 0.166795i $$0.0533419\pi$$
−0.985992 + 0.166795i $$0.946658\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 13.2548 0.520298
$$650$$ 0 0
$$651$$ −11.3137 −0.443419
$$652$$ 11.3137i 0.443079i
$$653$$ 42.2843i 1.65471i 0.561678 + 0.827356i $$0.310156\pi$$
−0.561678 + 0.827356i $$0.689844\pi$$
$$654$$ −8.82843 −0.345219
$$655$$ 0 0
$$656$$ −7.65685 −0.298950
$$657$$ − 14.4853i − 0.565125i
$$658$$ − 22.6274i − 0.882109i
$$659$$ 7.51472 0.292732 0.146366 0.989231i $$-0.453242\pi$$
0.146366 + 0.989231i $$0.453242\pi$$
$$660$$ 0 0
$$661$$ −8.14214 −0.316692 −0.158346 0.987384i $$-0.550616\pi$$
−0.158346 + 0.987384i $$0.550616\pi$$
$$662$$ − 8.48528i − 0.329790i
$$663$$ 0.828427i 0.0321734i
$$664$$ −6.34315 −0.246162
$$665$$ 0 0
$$666$$ −11.6569 −0.451694
$$667$$ 74.9117i 2.90059i
$$668$$ − 8.97056i − 0.347081i
$$669$$ −10.8284 −0.418651
$$670$$ 0 0
$$671$$ 33.9411 1.31028
$$672$$ − 2.82843i − 0.109109i
$$673$$ − 32.6274i − 1.25769i −0.777529 0.628847i $$-0.783527\pi$$
0.777529 0.628847i $$-0.216473\pi$$
$$674$$ −10.9706 −0.422570
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 12.3431i 0.474386i 0.971463 + 0.237193i $$0.0762273\pi$$
−0.971463 + 0.237193i $$0.923773\pi$$
$$678$$ − 6.48528i − 0.249066i
$$679$$ −8.97056 −0.344259
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 22.6274i 0.866449i
$$683$$ − 33.6569i − 1.28784i −0.765091 0.643922i $$-0.777306\pi$$
0.765091 0.643922i $$-0.222694\pi$$
$$684$$ −2.82843 −0.108148
$$685$$ 0 0
$$686$$ 16.9706 0.647939
$$687$$ 4.14214i 0.158032i
$$688$$ − 9.65685i − 0.368164i
$$689$$ 13.3137 0.507212
$$690$$ 0 0
$$691$$ −27.7990 −1.05752 −0.528762 0.848770i $$-0.677343\pi$$
−0.528762 + 0.848770i $$0.677343\pi$$
$$692$$ − 9.31371i − 0.354054i
$$693$$ 16.0000i 0.607790i
$$694$$ 9.65685 0.366569
$$695$$ 0 0
$$696$$ −8.82843 −0.334641
$$697$$ − 6.34315i − 0.240264i
$$698$$ − 12.1421i − 0.459587i
$$699$$ −5.51472 −0.208586
$$700$$ 0 0
$$701$$ 0.142136 0.00536839 0.00268419 0.999996i $$-0.499146\pi$$
0.00268419 + 0.999996i $$0.499146\pi$$
$$702$$ − 1.00000i − 0.0377426i
$$703$$ 32.9706i 1.24351i
$$704$$ −5.65685 −0.213201
$$705$$ 0 0
$$706$$ 5.31371 0.199984
$$707$$ − 45.6569i − 1.71710i
$$708$$ 2.34315i 0.0880608i
$$709$$ 7.17157 0.269334 0.134667 0.990891i $$-0.457004\pi$$
0.134667 + 0.990891i $$0.457004\pi$$
$$710$$ 0 0
$$711$$ 2.34315 0.0878748
$$712$$ − 15.6569i − 0.586765i
$$713$$ 33.9411i 1.27111i
$$714$$ 2.34315 0.0876900
$$715$$ 0 0
$$716$$ 7.51472 0.280838
$$717$$ 16.0000i 0.597531i
$$718$$ 28.2843i 1.05556i
$$719$$ 29.6569 1.10601 0.553007 0.833177i $$-0.313480\pi$$
0.553007 + 0.833177i $$0.313480\pi$$
$$720$$ 0 0
$$721$$ 4.68629 0.174527
$$722$$ − 11.0000i − 0.409378i
$$723$$ − 5.31371i − 0.197619i
$$724$$ 7.65685 0.284565
$$725$$ 0 0
$$726$$ 21.0000 0.779383
$$727$$ − 45.9411i − 1.70386i −0.523654 0.851931i $$-0.675432\pi$$
0.523654 0.851931i $$-0.324568\pi$$
$$728$$ − 2.82843i − 0.104828i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 6.00000i 0.221766i
$$733$$ 0.343146i 0.0126744i 0.999980 + 0.00633719i $$0.00201720\pi$$
−0.999980 + 0.00633719i $$0.997983\pi$$
$$734$$ −25.6569 −0.947012
$$735$$ 0 0
$$736$$ −8.48528 −0.312772
$$737$$ 32.0000i 1.17874i
$$738$$ 7.65685i 0.281853i
$$739$$ 14.1421 0.520227 0.260113 0.965578i $$-0.416240\pi$$
0.260113 + 0.965578i $$0.416240\pi$$
$$740$$ 0 0
$$741$$ −2.82843 −0.103905
$$742$$ − 37.6569i − 1.38243i
$$743$$ 36.2843i 1.33114i 0.746335 + 0.665570i $$0.231812\pi$$
−0.746335 + 0.665570i $$0.768188\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −2.68629 −0.0983521
$$747$$ 6.34315i 0.232084i
$$748$$ − 4.68629i − 0.171348i
$$749$$ −11.3137 −0.413394
$$750$$ 0 0
$$751$$ 11.3137 0.412843 0.206422 0.978463i $$-0.433818\pi$$
0.206422 + 0.978463i $$0.433818\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 10.8284i − 0.394610i
$$754$$ −8.82843 −0.321512
$$755$$ 0 0
$$756$$ −2.82843 −0.102869
$$757$$ 19.9411i 0.724773i 0.932028 + 0.362386i $$0.118038\pi$$
−0.932028 + 0.362386i $$0.881962\pi$$
$$758$$ 7.51472i 0.272947i
$$759$$ 48.0000 1.74229
$$760$$ 0 0
$$761$$ 27.6569 1.00256 0.501280 0.865285i $$-0.332863\pi$$
0.501280 + 0.865285i $$0.332863\pi$$
$$762$$ 9.65685i 0.349831i
$$763$$ 24.9706i 0.903995i
$$764$$ −11.3137 −0.409316
$$765$$ 0 0
$$766$$ −29.6569 −1.07155
$$767$$ 2.34315i 0.0846061i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −4.82843 −0.173892
$$772$$ 2.48528i 0.0894472i
$$773$$ 53.3137i 1.91756i 0.284148 + 0.958780i $$0.408289\pi$$
−0.284148 + 0.958780i $$0.591711\pi$$
$$774$$ −9.65685 −0.347108
$$775$$ 0 0
$$776$$ −3.17157 −0.113853
$$777$$ 32.9706i 1.18281i
$$778$$ 6.48528i 0.232509i
$$779$$ 21.6569 0.775937
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ − 7.02944i − 0.251372i
$$783$$ 8.82843i 0.315502i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ −6.14214 −0.219083
$$787$$ − 24.0000i − 0.855508i −0.903895 0.427754i $$-0.859305\pi$$
0.903895 0.427754i $$-0.140695\pi$$
$$788$$ 13.3137i 0.474281i
$$789$$ −16.4853 −0.586892
$$790$$ 0 0
$$791$$ −18.3431 −0.652207
$$792$$ 5.65685i 0.201008i
$$793$$ 6.00000i 0.213066i
$$794$$ −30.2843 −1.07475
$$795$$ 0 0
$$796$$ 10.3431 0.366603
$$797$$ 16.6274i 0.588973i 0.955656 + 0.294487i $$0.0951487\pi$$
−0.955656 + 0.294487i $$0.904851\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ 6.62742 0.234461
$$800$$ 0 0
$$801$$ −15.6569 −0.553208
$$802$$ − 26.9706i − 0.952364i
$$803$$ 81.9411i 2.89164i
$$804$$ −5.65685 −0.199502
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ − 14.4853i − 0.509906i
$$808$$ − 16.1421i − 0.567878i
$$809$$ −13.3137 −0.468085 −0.234043 0.972226i $$-0.575196\pi$$
−0.234043 + 0.972226i $$0.575196\pi$$
$$810$$ 0 0
$$811$$ −1.85786 −0.0652384 −0.0326192 0.999468i $$-0.510385\pi$$
−0.0326192 + 0.999468i $$0.510385\pi$$
$$812$$ 24.9706i 0.876295i
$$813$$ − 7.31371i − 0.256503i
$$814$$ 65.9411 2.31124
$$815$$ 0 0
$$816$$ 0.828427 0.0290008
$$817$$ 27.3137i 0.955586i
$$818$$ 3.65685i 0.127859i
$$819$$ −2.82843 −0.0988332
$$820$$ 0 0
$$821$$ 34.2843 1.19653 0.598265 0.801299i $$-0.295857\pi$$
0.598265 + 0.801299i $$0.295857\pi$$
$$822$$ − 17.3137i − 0.603885i
$$823$$ 52.9706i 1.84644i 0.384275 + 0.923219i $$0.374452\pi$$
−0.384275 + 0.923219i $$0.625548\pi$$
$$824$$ 1.65685 0.0577193
$$825$$ 0 0
$$826$$ 6.62742 0.230597
$$827$$ 9.65685i 0.335802i 0.985804 + 0.167901i $$0.0536989\pi$$
−0.985804 + 0.167901i $$0.946301\pi$$
$$828$$ 8.48528i 0.294884i
$$829$$ 53.3137 1.85166 0.925831 0.377938i $$-0.123367\pi$$
0.925831 + 0.377938i $$0.123367\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ − 1.00000i − 0.0346688i
$$833$$ − 0.828427i − 0.0287033i
$$834$$ −6.34315 −0.219645
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 4.00000i 0.138260i
$$838$$ 10.8284i 0.374062i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 48.9411 1.68763
$$842$$ − 24.1421i − 0.831993i
$$843$$ − 8.34315i − 0.287353i
$$844$$ −0.686292 −0.0236231
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ − 59.3970i − 2.04090i
$$848$$ − 13.3137i − 0.457195i
$$849$$ 17.6569 0.605982
$$850$$ 0 0
$$851$$ 98.9117 3.39065
$$852$$ − 5.65685i − 0.193801i
$$853$$ − 38.2843i − 1.31083i −0.755270 0.655414i $$-0.772494\pi$$
0.755270 0.655414i $$-0.227506\pi$$
$$854$$ 16.9706 0.580721
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 20.8284i − 0.711486i −0.934584 0.355743i $$-0.884228\pi$$
0.934584 0.355743i $$-0.115772\pi$$
$$858$$ 5.65685i 0.193122i
$$859$$ −37.9411 −1.29453 −0.647267 0.762263i $$-0.724088\pi$$
−0.647267 + 0.762263i $$0.724088\pi$$
$$860$$ 0 0
$$861$$ 21.6569 0.738064
$$862$$ − 16.0000i − 0.544962i
$$863$$ 28.2843i 0.962808i 0.876499 + 0.481404i $$0.159873\pi$$
−0.876499 + 0.481404i $$0.840127\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −22.9706 −0.780571
$$867$$ − 16.3137i − 0.554043i
$$868$$ 11.3137i 0.384012i
$$869$$ −13.2548 −0.449639
$$870$$ 0 0
$$871$$ −5.65685 −0.191675
$$872$$ 8.82843i 0.298968i
$$873$$ 3.17157i 0.107341i
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ −14.4853 −0.489412
$$877$$ − 51.2548i − 1.73075i −0.501122 0.865376i $$-0.667079\pi$$
0.501122 0.865376i $$-0.332921\pi$$
$$878$$ − 22.6274i − 0.763638i
$$879$$ 16.6274 0.560829
$$880$$ 0 0
$$881$$ −10.2843 −0.346486 −0.173243 0.984879i $$-0.555425\pi$$
−0.173243 + 0.984879i $$0.555425\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ − 31.3137i − 1.05379i −0.849930 0.526895i $$-0.823356\pi$$
0.849930 0.526895i $$-0.176644\pi$$
$$884$$ 0.828427 0.0278630
$$885$$ 0 0
$$886$$ 30.3431 1.01940
$$887$$ − 40.4853i − 1.35936i −0.733508 0.679681i $$-0.762118\pi$$
0.733508 0.679681i $$-0.237882\pi$$
$$888$$ 11.6569i 0.391178i
$$889$$ 27.3137 0.916072
$$890$$ 0 0
$$891$$ 5.65685 0.189512
$$892$$ 10.8284i 0.362563i
$$893$$ 22.6274i 0.757198i
$$894$$ −3.65685 −0.122304
$$895$$ 0 0
$$896$$ −2.82843 −0.0944911
$$897$$ 8.48528i 0.283315i
$$898$$ − 26.2843i − 0.877117i
$$899$$ 35.3137 1.17778
$$900$$ 0 0
$$901$$ 11.0294 0.367444
$$902$$ − 43.3137i − 1.44219i
$$903$$ 27.3137i 0.908943i
$$904$$ −6.48528 −0.215697
$$905$$ 0 0
$$906$$ 12.0000 0.398673
$$907$$ 8.28427i 0.275075i 0.990497 + 0.137537i $$0.0439187\pi$$
−0.990497 + 0.137537i $$0.956081\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ −16.1421 −0.535401
$$910$$ 0 0
$$911$$ 24.9706 0.827312 0.413656 0.910433i $$-0.364252\pi$$
0.413656 + 0.910433i $$0.364252\pi$$
$$912$$ 2.82843i 0.0936586i
$$913$$ − 35.8823i − 1.18753i
$$914$$ −20.8284 −0.688943
$$915$$ 0 0
$$916$$ 4.14214 0.136860
$$917$$ 17.3726i 0.573693i
$$918$$ − 0.828427i − 0.0273422i
$$919$$ −41.9411 −1.38351 −0.691755 0.722132i $$-0.743162\pi$$
−0.691755 + 0.722132i $$0.743162\pi$$
$$920$$ 0 0
$$921$$ −21.6569 −0.713618
$$922$$ 14.0000i 0.461065i
$$923$$ − 5.65685i − 0.186198i
$$924$$ 16.0000 0.526361
$$925$$ 0 0
$$926$$ −3.79899 −0.124843
$$927$$ − 1.65685i − 0.0544182i
$$928$$ 8.82843i 0.289807i
$$929$$ 33.5980 1.10231 0.551157 0.834402i $$-0.314187\pi$$
0.551157 + 0.834402i $$0.314187\pi$$
$$930$$ 0 0
$$931$$ 2.82843 0.0926980
$$932$$ 5.51472i 0.180641i
$$933$$ 24.0000i 0.785725i
$$934$$ −7.31371 −0.239312
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 16.6274i 0.543194i 0.962411 + 0.271597i $$0.0875518\pi$$
−0.962411 + 0.271597i $$0.912448\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 30.9706 1.01069
$$940$$ 0 0
$$941$$ 54.9706 1.79199 0.895995 0.444065i $$-0.146464\pi$$
0.895995 + 0.444065i $$0.146464\pi$$
$$942$$ 5.31371i 0.173130i
$$943$$ − 64.9706i − 2.11573i
$$944$$ 2.34315 0.0762629
$$945$$ 0 0
$$946$$ 54.6274 1.77609
$$947$$ 30.3431i 0.986020i 0.870024 + 0.493010i $$0.164103\pi$$
−0.870024 + 0.493010i $$0.835897\pi$$
$$948$$ − 2.34315i − 0.0761018i
$$949$$ −14.4853 −0.470212
$$950$$ 0 0
$$951$$ 25.3137 0.820853
$$952$$ − 2.34315i − 0.0759418i
$$953$$ 27.8579i 0.902405i 0.892422 + 0.451202i $$0.149005\pi$$
−0.892422 + 0.451202i $$0.850995\pi$$
$$954$$ −13.3137 −0.431047
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ − 49.9411i − 1.61437i
$$958$$ 11.3137i 0.365529i
$$959$$ −48.9706 −1.58134
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 11.6569i 0.375832i
$$963$$ 4.00000i 0.128898i
$$964$$ −5.31371 −0.171143
$$965$$ 0 0
$$966$$ 24.0000 0.772187
$$967$$ − 7.51472i − 0.241657i −0.992673 0.120829i $$-0.961445\pi$$
0.992673 0.120829i $$-0.0385551\pi$$
$$968$$ − 21.0000i − 0.674966i
$$969$$ −2.34315 −0.0752727
$$970$$ 0 0
$$971$$ 15.5147 0.497891 0.248946 0.968517i $$-0.419916\pi$$
0.248946 + 0.968517i $$0.419916\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 17.9411i 0.575166i
$$974$$ −16.4853 −0.528222
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ − 8.34315i − 0.266921i −0.991054 0.133460i $$-0.957391\pi$$
0.991054 0.133460i $$-0.0426089\pi$$
$$978$$ − 11.3137i − 0.361773i
$$979$$ 88.5685 2.83066
$$980$$ 0 0
$$981$$ 8.82843 0.281870
$$982$$ − 38.1421i − 1.21716i
$$983$$ 2.34315i 0.0747347i 0.999302 + 0.0373674i $$0.0118972\pi$$
−0.999302 + 0.0373674i $$0.988103\pi$$
$$984$$ 7.65685 0.244092
$$985$$ 0 0
$$986$$ −7.31371 −0.232916
$$987$$ 22.6274i 0.720239i
$$988$$ 2.82843i 0.0899843i
$$989$$ 81.9411 2.60558
$$990$$ 0 0
$$991$$ −42.9117 −1.36313 −0.681567 0.731755i $$-0.738701\pi$$
−0.681567 + 0.731755i $$0.738701\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 8.48528i 0.269272i
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 6.34315 0.200990
$$997$$ 61.3137i 1.94182i 0.239435 + 0.970912i $$0.423038\pi$$
−0.239435 + 0.970912i $$0.576962\pi$$
$$998$$ − 0.485281i − 0.0153613i
$$999$$ 11.6569 0.368807
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 1950.2.e.o.1249.3 4
3.2 odd 2 5850.2.e.bk.5149.1 4
5.2 odd 4 1950.2.a.bd.1.2 2
5.3 odd 4 390.2.a.h.1.1 2
5.4 even 2 inner 1950.2.e.o.1249.2 4
15.2 even 4 5850.2.a.cl.1.2 2
15.8 even 4 1170.2.a.o.1.1 2
15.14 odd 2 5850.2.e.bk.5149.4 4
20.3 even 4 3120.2.a.bc.1.2 2
60.23 odd 4 9360.2.a.ch.1.2 2
65.8 even 4 5070.2.b.q.1351.4 4
65.18 even 4 5070.2.b.q.1351.1 4
65.38 odd 4 5070.2.a.bc.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 5.3 odd 4
1170.2.a.o.1.1 2 15.8 even 4
1950.2.a.bd.1.2 2 5.2 odd 4
1950.2.e.o.1249.2 4 5.4 even 2 inner
1950.2.e.o.1249.3 4 1.1 even 1 trivial
3120.2.a.bc.1.2 2 20.3 even 4
5070.2.a.bc.1.2 2 65.38 odd 4
5070.2.b.q.1351.1 4 65.18 even 4
5070.2.b.q.1351.4 4 65.8 even 4
5850.2.a.cl.1.2 2 15.2 even 4
5850.2.e.bk.5149.1 4 3.2 odd 2
5850.2.e.bk.5149.4 4 15.14 odd 2
9360.2.a.ch.1.2 2 60.23 odd 4