Properties

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

$q$-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 $$0.179509\pi$$
−0.845154 + 0.534522i $$0.820491\pi$$
$$8$$ 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 $$-0.967968\pi$$
0.994941 0.100462i $$-0.0320319\pi$$
$$18$$ 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 $$-0.654396\pi$$
0.466252 0.884652i $$-0.345604\pi$$
$$24$$ −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 $$0.407627\pi$$
−0.286141 + 0.958188i $$0.592373\pi$$
$$38$$ 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 $$0.263442\pi$$
−0.676625 + 0.736328i $$0.736558\pi$$
$$44$$ −5.65685 −0.852803
$$45$$ 0 0
$$46$$ −8.48528 −1.25109
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 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 $$0.367329\pi$$
−0.404836 + 0.914389i $$0.632671\pi$$
$$54$$ −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 $$-0.887693\pi$$
0.938401 0.345547i $$-0.112307\pi$$
$$68$$ 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 $$-0.677995\pi$$
0.530497 0.847687i $$-0.322005\pi$$
$$74$$ 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 $$0.113182\pi$$
−0.937448 + 0.348125i $$0.886818\pi$$
$$84$$ 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 $$0.0514759\pi$$
−0.986952 + 0.161012i $$0.948524\pi$$
$$98$$ 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 $$-0.973988\pi$$
0.996663 0.0816274i $$-0.0260117\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 13.3137 1.29314
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 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 $$0.0986705\pi$$
−0.952339 + 0.305042i $$0.901330\pi$$
$$114$$ −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 $$-0.859059\pi$$
0.903564 0.428454i $$-0.140941\pi$$
$$128$$ 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 $$0.264988\pi$$
−0.673041 + 0.739605i $$0.735012\pi$$
$$138$$ − 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 $$-0.931989\pi$$
0.977261 0.212040i $$-0.0680107\pi$$
$$158$$ 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 $$0.146114\pi$$
−0.896483 + 0.443079i $$0.853886\pi$$
$$164$$ 7.65685 0.597900
$$165$$ 0 0
$$166$$ 6.34315 0.492324
$$167$$ − 8.97056i − 0.694163i −0.937835 0.347081i $$-0.887173\pi$$
0.937835 0.347081i $$-0.112827\pi$$
$$168$$ − 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 $$-0.884803\pi$$
0.935225 0.354054i $$-0.115197\pi$$
$$174$$ 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 $$0.0285100\pi$$
−0.995992 + 0.0894472i $$0.971490\pi$$
$$194$$ 3.17157 0.227706
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 13.3137i 0.948562i 0.880373 + 0.474281i $$0.157292\pi$$
−0.880373 + 0.474281i $$0.842708\pi$$
$$198$$ 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 $$0.118098\pi$$
−0.931959 + 0.362563i $$0.881902\pi$$
$$224$$ 2.82843 0.188982
$$225$$ 0 0
$$226$$ 6.48528 0.431394
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 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 $$0.0578171\pi$$
−0.983549 + 0.180641i $$0.942183\pi$$
$$234$$ 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 $$0.0481188\pi$$
−0.988596 + 0.150595i $$0.951881\pi$$
$$258$$ 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 $$0.169712\pi$$
−0.861202 + 0.508263i $$0.830288\pi$$
$$264$$ −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 $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ 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 $$-0.824142\pi$$
0.851228 0.524796i $$-0.175858\pi$$
$$284$$ 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 $$-0.838568\pi$$
0.874130 0.485692i $$-0.161432\pi$$
$$294$$ −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 $$0.212061\pi$$
−0.786169 + 0.618011i $$0.787939\pi$$
$$308$$ − 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 $$-0.660677\pi$$
0.483617 0.875280i $$-0.339323\pi$$
$$314$$ −5.31371 −0.299870
$$315$$ 0 0
$$316$$ 2.34315 0.131812
$$317$$ − 25.3137i − 1.42176i −0.703314 0.710880i $$-0.748297\pi$$
0.703314 0.710880i $$-0.251703\pi$$
$$318$$ 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 $$-0.903413\pi$$
0.954315 0.298802i $$-0.0965871\pi$$
$$338$$ 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 $$0.0834600\pi$$
−0.965823 + 0.259204i $$0.916540\pi$$
$$348$$ − 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 $$0.0451636\pi$$
−0.989951 + 0.141410i $$0.954836\pi$$
$$354$$ 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 $$-0.766449\pi$$
0.742687 0.669638i $$-0.233551\pi$$
$$368$$ − 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 $$-0.977845\pi$$
0.997579 0.0695455i $$-0.0221549\pi$$
$$374$$ −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 $$-0.726324\pi$$
0.652606 0.757697i $$-0.273676\pi$$
$$384$$ −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 $$-0.725218\pi$$
0.649968 0.759962i $$-0.274782\pi$$
$$398$$ 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 $$-0.813885\pi$$
0.833879 0.551947i $$-0.186115\pi$$
$$434$$ 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 $$0.256236\pi$$
−0.693119 + 0.720823i $$0.743764\pi$$
$$444$$ 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 $$-0.838034\pi$$
0.873315 0.487156i $$-0.161966\pi$$
$$458$$ 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 $$-0.971864\pi$$
0.996096 0.0882770i $$-0.0281361\pi$$
$$464$$ 8.82843 0.409849
$$465$$ 0 0
$$466$$ 5.51472 0.255464
$$467$$ − 7.31371i − 0.338438i −0.985578 0.169219i $$-0.945875\pi$$
0.985578 0.169219i $$-0.0541245\pi$$
$$468$$ − 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 $$-0.878154\pi$$
0.927626 0.373510i $$-0.121846\pi$$
$$488$$ 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 $$-0.824351\pi$$
0.851574 0.524235i $$-0.175649\pi$$
$$504$$ 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 $$-0.925358\pi$$
0.972632 0.232352i $$-0.0746422\pi$$
$$524$$ 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 $$-0.995330\pi$$
0.999892 0.0146719i $$-0.00467036\pi$$
$$548$$ − 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 $$-0.927306\pi$$
0.974035 0.226396i $$-0.0726945\pi$$
$$558$$ 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 $$-0.779181\pi$$
0.768870 0.639406i $$-0.220819\pi$$
$$564$$ 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 $$0.162361\pi$$
−0.872710 + 0.488240i $$0.837639\pi$$
$$578$$ − 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 $$-0.982732\pi$$
0.998529 0.0542226i $$-0.0172680\pi$$
$$588$$ 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 $$-0.997757\pi$$
0.999975 0.00704565i $$-0.00224272\pi$$
$$594$$ −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 $$-0.799939\pi$$
0.808905 0.587939i $$-0.200061\pi$$
$$608$$ 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 $$0.148586\pi$$
−0.893015 + 0.450027i $$0.851414\pi$$
$$614$$ 21.6569 0.874000
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ − 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 $$0.285186\pi$$
−0.624786 + 0.780796i $$0.714814\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 2.34315 0.0921898
$$647$$ − 8.48528i − 0.333591i −0.985992 0.166795i $$-0.946658\pi$$
0.985992 0.166795i $$-0.0533419\pi$$
$$648$$ 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 $$-0.689844\pi$$
0.561678 0.827356i $$-0.310156\pi$$
$$654$$ −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 $$0.216473\pi$$
−0.777529 + 0.628847i $$0.783527\pi$$
$$674$$ −10.9706 −0.422570
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ − 12.3431i − 0.474386i −0.971463 0.237193i $$-0.923773\pi$$
0.971463 0.237193i $$-0.0762273\pi$$
$$678$$ 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 $$0.222694\pi$$
−0.765091 + 0.643922i $$0.777306\pi$$
$$684$$ −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 $$0.324568\pi$$
−0.523654 + 0.851931i $$0.675432\pi$$
$$728$$ 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 $$-0.997983\pi$$
0.999980 0.00633719i $$-0.00201720\pi$$
$$734$$ −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 $$-0.768188\pi$$
0.746335 0.665570i $$-0.231812\pi$$
$$744$$ −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 $$-0.881962\pi$$
0.932028 0.362386i $$-0.118038\pi$$
$$758$$ − 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 $$-0.591711\pi$$
0.284148 0.958780i $$-0.408289\pi$$
$$774$$ −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 $$0.140695\pi$$
−0.903895 + 0.427754i $$0.859305\pi$$
$$788$$ − 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 $$-0.904851\pi$$
0.955656 0.294487i $$-0.0951487\pi$$
$$798$$ − 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 $$-0.625548\pi$$
0.384275 0.923219i $$-0.374452\pi$$
$$824$$ 1.65685 0.0577193
$$825$$ 0 0
$$826$$ 6.62742 0.230597
$$827$$ − 9.65685i − 0.335802i −0.985804 0.167901i $$-0.946301\pi$$
0.985804 0.167901i $$-0.0536989\pi$$
$$828$$ − 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 $$0.227506\pi$$
−0.755270 + 0.655414i $$0.772494\pi$$
$$854$$ 16.9706 0.580721
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 20.8284i 0.711486i 0.934584 + 0.355743i $$0.115772\pi$$
−0.934584 + 0.355743i $$0.884228\pi$$
$$858$$ − 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 $$-0.840127\pi$$
0.876499 0.481404i $$-0.159873\pi$$
$$864$$ −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 $$0.332921\pi$$
−0.501122 + 0.865376i $$0.667079\pi$$
$$878$$ 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 $$0.176644\pi$$
−0.849930 + 0.526895i $$0.823356\pi$$
$$884$$ 0.828427 0.0278630
$$885$$ 0 0
$$886$$ 30.3431 1.01940
$$887$$ 40.4853i 1.35936i 0.733508 + 0.679681i $$0.237882\pi$$
−0.733508 + 0.679681i $$0.762118\pi$$
$$888$$ − 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 $$-0.956081\pi$$
0.990497 0.137537i $$-0.0439187\pi$$
$$908$$ 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 $$-0.912448\pi$$
0.962411 0.271597i $$-0.0875518\pi$$
$$938$$ − 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 $$-0.835897\pi$$
0.870024 0.493010i $$-0.164103\pi$$
$$948$$ 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 $$-0.850995\pi$$
0.892422 0.451202i $$-0.149005\pi$$
$$954$$ −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 $$0.0385551\pi$$
−0.992673 + 0.120829i $$0.961445\pi$$
$$968$$ 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 $$0.0426089\pi$$
−0.991054 + 0.133460i $$0.957391\pi$$
$$978$$ 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 $$-0.988103\pi$$
0.999302 0.0373674i $$-0.0118972\pi$$
$$984$$ 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 $$-0.576962\pi$$
0.239435 0.970912i $$-0.423038\pi$$
$$998$$ 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.1249.2 4
3.2 odd 2 5850.2.e.bk.5149.4 4
5.2 odd 4 390.2.a.h.1.1 2
5.3 odd 4 1950.2.a.bd.1.2 2
5.4 even 2 inner 1950.2.e.o.1249.3 4
15.2 even 4 1170.2.a.o.1.1 2
15.8 even 4 5850.2.a.cl.1.2 2
15.14 odd 2 5850.2.e.bk.5149.1 4
20.7 even 4 3120.2.a.bc.1.2 2
60.47 odd 4 9360.2.a.ch.1.2 2
65.12 odd 4 5070.2.a.bc.1.2 2
65.47 even 4 5070.2.b.q.1351.4 4
65.57 even 4 5070.2.b.q.1351.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 5.2 odd 4
1170.2.a.o.1.1 2 15.2 even 4
1950.2.a.bd.1.2 2 5.3 odd 4
1950.2.e.o.1249.2 4 1.1 even 1 trivial
1950.2.e.o.1249.3 4 5.4 even 2 inner
3120.2.a.bc.1.2 2 20.7 even 4
5070.2.a.bc.1.2 2 65.12 odd 4
5070.2.b.q.1351.1 4 65.57 even 4
5070.2.b.q.1351.4 4 65.47 even 4
5850.2.a.cl.1.2 2 15.8 even 4
5850.2.e.bk.5149.1 4 15.14 odd 2
5850.2.e.bk.5149.4 4 3.2 odd 2
9360.2.a.ch.1.2 2 60.47 odd 4