Properties

 Label 3450.2.a.bo.1.2 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.42864 q^{11} +1.00000 q^{12} -1.37778 q^{13} +0.622216 q^{14} +1.00000 q^{16} -6.42864 q^{17} -1.00000 q^{18} -1.37778 q^{19} -0.622216 q^{21} -4.42864 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.37778 q^{26} +1.00000 q^{27} -0.622216 q^{28} -4.23506 q^{29} -4.00000 q^{31} -1.00000 q^{32} +4.42864 q^{33} +6.42864 q^{34} +1.00000 q^{36} -11.8064 q^{37} +1.37778 q^{38} -1.37778 q^{39} -10.8573 q^{41} +0.622216 q^{42} +1.05086 q^{43} +4.42864 q^{44} +1.00000 q^{46} +11.6128 q^{47} +1.00000 q^{48} -6.61285 q^{49} -6.42864 q^{51} -1.37778 q^{52} -3.37778 q^{53} -1.00000 q^{54} +0.622216 q^{56} -1.37778 q^{57} +4.23506 q^{58} +9.61285 q^{59} +8.66370 q^{61} +4.00000 q^{62} -0.622216 q^{63} +1.00000 q^{64} -4.42864 q^{66} -11.8064 q^{67} -6.42864 q^{68} -1.00000 q^{69} -6.99063 q^{71} -1.00000 q^{72} -2.75557 q^{73} +11.8064 q^{74} -1.37778 q^{76} -2.75557 q^{77} +1.37778 q^{78} +12.0415 q^{79} +1.00000 q^{81} +10.8573 q^{82} -2.19358 q^{83} -0.622216 q^{84} -1.05086 q^{86} -4.23506 q^{87} -4.42864 q^{88} +2.75557 q^{89} +0.857279 q^{91} -1.00000 q^{92} -4.00000 q^{93} -11.6128 q^{94} -1.00000 q^{96} +2.13335 q^{97} +6.61285 q^{98} +4.42864 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + 3q^{12} - 4q^{13} + 2q^{14} + 3q^{16} - 6q^{17} - 3q^{18} - 4q^{19} - 2q^{21} - 3q^{23} - 3q^{24} + 4q^{26} + 3q^{27} - 2q^{28} + 14q^{29} - 12q^{31} - 3q^{32} + 6q^{34} + 3q^{36} - 22q^{37} + 4q^{38} - 4q^{39} - 6q^{41} + 2q^{42} - 10q^{43} + 3q^{46} + 8q^{47} + 3q^{48} + 7q^{49} - 6q^{51} - 4q^{52} - 10q^{53} - 3q^{54} + 2q^{56} - 4q^{57} - 14q^{58} + 2q^{59} - 14q^{61} + 12q^{62} - 2q^{63} + 3q^{64} - 22q^{67} - 6q^{68} - 3q^{69} + 6q^{71} - 3q^{72} - 8q^{73} + 22q^{74} - 4q^{76} - 8q^{77} + 4q^{78} - 4q^{79} + 3q^{81} + 6q^{82} - 20q^{83} - 2q^{84} + 10q^{86} + 14q^{87} + 8q^{89} - 24q^{91} - 3q^{92} - 12q^{93} - 8q^{94} - 3q^{96} + 6q^{97} - 7q^{98} + O(q^{100})$$

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
$$7$$ −0.622216 −0.235175 −0.117588 0.993063i $$-0.537516\pi$$
−0.117588 + 0.993063i $$0.537516\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.42864 1.33529 0.667643 0.744482i $$-0.267303\pi$$
0.667643 + 0.744482i $$0.267303\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.37778 −0.382129 −0.191064 0.981578i $$-0.561194\pi$$
−0.191064 + 0.981578i $$0.561194\pi$$
$$14$$ 0.622216 0.166294
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.42864 −1.55917 −0.779587 0.626294i $$-0.784571\pi$$
−0.779587 + 0.626294i $$0.784571\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.37778 −0.316085 −0.158043 0.987432i $$-0.550518\pi$$
−0.158043 + 0.987432i $$0.550518\pi$$
$$20$$ 0 0
$$21$$ −0.622216 −0.135779
$$22$$ −4.42864 −0.944189
$$23$$ −1.00000 −0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 1.37778 0.270206
$$27$$ 1.00000 0.192450
$$28$$ −0.622216 −0.117588
$$29$$ −4.23506 −0.786432 −0.393216 0.919446i $$-0.628637\pi$$
−0.393216 + 0.919446i $$0.628637\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.42864 0.770927
$$34$$ 6.42864 1.10250
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −11.8064 −1.94096 −0.970482 0.241173i $$-0.922468\pi$$
−0.970482 + 0.241173i $$0.922468\pi$$
$$38$$ 1.37778 0.223506
$$39$$ −1.37778 −0.220622
$$40$$ 0 0
$$41$$ −10.8573 −1.69562 −0.847811 0.530298i $$-0.822080\pi$$
−0.847811 + 0.530298i $$0.822080\pi$$
$$42$$ 0.622216 0.0960100
$$43$$ 1.05086 0.160254 0.0801270 0.996785i $$-0.474467\pi$$
0.0801270 + 0.996785i $$0.474467\pi$$
$$44$$ 4.42864 0.667643
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 11.6128 1.69391 0.846954 0.531666i $$-0.178434\pi$$
0.846954 + 0.531666i $$0.178434\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.61285 −0.944693
$$50$$ 0 0
$$51$$ −6.42864 −0.900190
$$52$$ −1.37778 −0.191064
$$53$$ −3.37778 −0.463974 −0.231987 0.972719i $$-0.574523\pi$$
−0.231987 + 0.972719i $$0.574523\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0.622216 0.0831471
$$57$$ −1.37778 −0.182492
$$58$$ 4.23506 0.556091
$$59$$ 9.61285 1.25149 0.625743 0.780029i $$-0.284796\pi$$
0.625743 + 0.780029i $$0.284796\pi$$
$$60$$ 0 0
$$61$$ 8.66370 1.10927 0.554637 0.832093i $$-0.312857\pi$$
0.554637 + 0.832093i $$0.312857\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −0.622216 −0.0783918
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.42864 −0.545128
$$67$$ −11.8064 −1.44238 −0.721192 0.692735i $$-0.756406\pi$$
−0.721192 + 0.692735i $$0.756406\pi$$
$$68$$ −6.42864 −0.779587
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −6.99063 −0.829635 −0.414818 0.909905i $$-0.636155\pi$$
−0.414818 + 0.909905i $$0.636155\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −2.75557 −0.322515 −0.161257 0.986912i $$-0.551555\pi$$
−0.161257 + 0.986912i $$0.551555\pi$$
$$74$$ 11.8064 1.37247
$$75$$ 0 0
$$76$$ −1.37778 −0.158043
$$77$$ −2.75557 −0.314026
$$78$$ 1.37778 0.156003
$$79$$ 12.0415 1.35477 0.677387 0.735627i $$-0.263112\pi$$
0.677387 + 0.735627i $$0.263112\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.8573 1.19899
$$83$$ −2.19358 −0.240776 −0.120388 0.992727i $$-0.538414\pi$$
−0.120388 + 0.992727i $$0.538414\pi$$
$$84$$ −0.622216 −0.0678893
$$85$$ 0 0
$$86$$ −1.05086 −0.113317
$$87$$ −4.23506 −0.454046
$$88$$ −4.42864 −0.472095
$$89$$ 2.75557 0.292090 0.146045 0.989278i $$-0.453346\pi$$
0.146045 + 0.989278i $$0.453346\pi$$
$$90$$ 0 0
$$91$$ 0.857279 0.0898673
$$92$$ −1.00000 −0.104257
$$93$$ −4.00000 −0.414781
$$94$$ −11.6128 −1.19777
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 2.13335 0.216609 0.108305 0.994118i $$-0.465458\pi$$
0.108305 + 0.994118i $$0.465458\pi$$
$$98$$ 6.61285 0.667998
$$99$$ 4.42864 0.445095
$$100$$ 0 0
$$101$$ 16.2351 1.61545 0.807725 0.589560i $$-0.200699\pi$$
0.807725 + 0.589560i $$0.200699\pi$$
$$102$$ 6.42864 0.636530
$$103$$ −12.2351 −1.20556 −0.602778 0.797909i $$-0.705940\pi$$
−0.602778 + 0.797909i $$0.705940\pi$$
$$104$$ 1.37778 0.135103
$$105$$ 0 0
$$106$$ 3.37778 0.328079
$$107$$ 10.6637 1.03090 0.515450 0.856920i $$-0.327625\pi$$
0.515450 + 0.856920i $$0.327625\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −15.1526 −1.45135 −0.725676 0.688036i $$-0.758473\pi$$
−0.725676 + 0.688036i $$0.758473\pi$$
$$110$$ 0 0
$$111$$ −11.8064 −1.12062
$$112$$ −0.622216 −0.0587939
$$113$$ −14.4286 −1.35733 −0.678666 0.734447i $$-0.737442\pi$$
−0.678666 + 0.734447i $$0.737442\pi$$
$$114$$ 1.37778 0.129041
$$115$$ 0 0
$$116$$ −4.23506 −0.393216
$$117$$ −1.37778 −0.127376
$$118$$ −9.61285 −0.884934
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 8.61285 0.782986
$$122$$ −8.66370 −0.784375
$$123$$ −10.8573 −0.978968
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0.622216 0.0554314
$$127$$ 4.99063 0.442847 0.221423 0.975178i $$-0.428930\pi$$
0.221423 + 0.975178i $$0.428930\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.05086 0.0925226
$$130$$ 0 0
$$131$$ 0.755569 0.0660143 0.0330072 0.999455i $$-0.489492\pi$$
0.0330072 + 0.999455i $$0.489492\pi$$
$$132$$ 4.42864 0.385464
$$133$$ 0.857279 0.0743355
$$134$$ 11.8064 1.01992
$$135$$ 0 0
$$136$$ 6.42864 0.551251
$$137$$ −12.9175 −1.10362 −0.551808 0.833971i $$-0.686062\pi$$
−0.551808 + 0.833971i $$0.686062\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ −14.1017 −1.19609 −0.598046 0.801462i $$-0.704056\pi$$
−0.598046 + 0.801462i $$0.704056\pi$$
$$140$$ 0 0
$$141$$ 11.6128 0.977978
$$142$$ 6.99063 0.586641
$$143$$ −6.10171 −0.510251
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 2.75557 0.228052
$$147$$ −6.61285 −0.545418
$$148$$ −11.8064 −0.970482
$$149$$ 12.4286 1.01819 0.509097 0.860709i $$-0.329979\pi$$
0.509097 + 0.860709i $$0.329979\pi$$
$$150$$ 0 0
$$151$$ −4.85728 −0.395280 −0.197640 0.980275i $$-0.563328\pi$$
−0.197640 + 0.980275i $$0.563328\pi$$
$$152$$ 1.37778 0.111753
$$153$$ −6.42864 −0.519725
$$154$$ 2.75557 0.222050
$$155$$ 0 0
$$156$$ −1.37778 −0.110311
$$157$$ 9.90813 0.790755 0.395378 0.918519i $$-0.370614\pi$$
0.395378 + 0.918519i $$0.370614\pi$$
$$158$$ −12.0415 −0.957969
$$159$$ −3.37778 −0.267876
$$160$$ 0 0
$$161$$ 0.622216 0.0490375
$$162$$ −1.00000 −0.0785674
$$163$$ −10.1017 −0.791227 −0.395614 0.918417i $$-0.629468\pi$$
−0.395614 + 0.918417i $$0.629468\pi$$
$$164$$ −10.8573 −0.847811
$$165$$ 0 0
$$166$$ 2.19358 0.170255
$$167$$ −1.89829 −0.146894 −0.0734470 0.997299i $$-0.523400\pi$$
−0.0734470 + 0.997299i $$0.523400\pi$$
$$168$$ 0.622216 0.0480050
$$169$$ −11.1017 −0.853978
$$170$$ 0 0
$$171$$ −1.37778 −0.105362
$$172$$ 1.05086 0.0801270
$$173$$ −21.2257 −1.61376 −0.806880 0.590716i $$-0.798846\pi$$
−0.806880 + 0.590716i $$0.798846\pi$$
$$174$$ 4.23506 0.321059
$$175$$ 0 0
$$176$$ 4.42864 0.333821
$$177$$ 9.61285 0.722546
$$178$$ −2.75557 −0.206539
$$179$$ 0.488863 0.0365393 0.0182697 0.999833i $$-0.494184\pi$$
0.0182697 + 0.999833i $$0.494184\pi$$
$$180$$ 0 0
$$181$$ −20.9304 −1.55575 −0.777873 0.628422i $$-0.783701\pi$$
−0.777873 + 0.628422i $$0.783701\pi$$
$$182$$ −0.857279 −0.0635457
$$183$$ 8.66370 0.640439
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ −28.4701 −2.08194
$$188$$ 11.6128 0.846954
$$189$$ −0.622216 −0.0452595
$$190$$ 0 0
$$191$$ 18.9590 1.37182 0.685912 0.727684i $$-0.259403\pi$$
0.685912 + 0.727684i $$0.259403\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 1.24443 0.0895761 0.0447881 0.998997i $$-0.485739\pi$$
0.0447881 + 0.998997i $$0.485739\pi$$
$$194$$ −2.13335 −0.153166
$$195$$ 0 0
$$196$$ −6.61285 −0.472346
$$197$$ 15.2444 1.08612 0.543060 0.839694i $$-0.317265\pi$$
0.543060 + 0.839694i $$0.317265\pi$$
$$198$$ −4.42864 −0.314730
$$199$$ −14.5303 −1.03003 −0.515015 0.857181i $$-0.672214\pi$$
−0.515015 + 0.857181i $$0.672214\pi$$
$$200$$ 0 0
$$201$$ −11.8064 −0.832761
$$202$$ −16.2351 −1.14230
$$203$$ 2.63512 0.184949
$$204$$ −6.42864 −0.450095
$$205$$ 0 0
$$206$$ 12.2351 0.852457
$$207$$ −1.00000 −0.0695048
$$208$$ −1.37778 −0.0955322
$$209$$ −6.10171 −0.422064
$$210$$ 0 0
$$211$$ 0.266706 0.0183608 0.00918041 0.999958i $$-0.497078\pi$$
0.00918041 + 0.999958i $$0.497078\pi$$
$$212$$ −3.37778 −0.231987
$$213$$ −6.99063 −0.478990
$$214$$ −10.6637 −0.728956
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 2.48886 0.168955
$$218$$ 15.1526 1.02626
$$219$$ −2.75557 −0.186204
$$220$$ 0 0
$$221$$ 8.85728 0.595805
$$222$$ 11.8064 0.792395
$$223$$ 0.990632 0.0663376 0.0331688 0.999450i $$-0.489440\pi$$
0.0331688 + 0.999450i $$0.489440\pi$$
$$224$$ 0.622216 0.0415735
$$225$$ 0 0
$$226$$ 14.4286 0.959779
$$227$$ 17.1526 1.13846 0.569228 0.822180i $$-0.307242\pi$$
0.569228 + 0.822180i $$0.307242\pi$$
$$228$$ −1.37778 −0.0912460
$$229$$ 7.15257 0.472655 0.236327 0.971673i $$-0.424056\pi$$
0.236327 + 0.971673i $$0.424056\pi$$
$$230$$ 0 0
$$231$$ −2.75557 −0.181303
$$232$$ 4.23506 0.278046
$$233$$ 3.71456 0.243349 0.121674 0.992570i $$-0.461174\pi$$
0.121674 + 0.992570i $$0.461174\pi$$
$$234$$ 1.37778 0.0900686
$$235$$ 0 0
$$236$$ 9.61285 0.625743
$$237$$ 12.0415 0.782179
$$238$$ −4.00000 −0.259281
$$239$$ 27.8479 1.80133 0.900666 0.434512i $$-0.143079\pi$$
0.900666 + 0.434512i $$0.143079\pi$$
$$240$$ 0 0
$$241$$ −7.12399 −0.458896 −0.229448 0.973321i $$-0.573692\pi$$
−0.229448 + 0.973321i $$0.573692\pi$$
$$242$$ −8.61285 −0.553655
$$243$$ 1.00000 0.0641500
$$244$$ 8.66370 0.554637
$$245$$ 0 0
$$246$$ 10.8573 0.692235
$$247$$ 1.89829 0.120785
$$248$$ 4.00000 0.254000
$$249$$ −2.19358 −0.139012
$$250$$ 0 0
$$251$$ −22.4099 −1.41450 −0.707250 0.706963i $$-0.750065\pi$$
−0.707250 + 0.706963i $$0.750065\pi$$
$$252$$ −0.622216 −0.0391959
$$253$$ −4.42864 −0.278426
$$254$$ −4.99063 −0.313140
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −11.7146 −0.730734 −0.365367 0.930864i $$-0.619057\pi$$
−0.365367 + 0.930864i $$0.619057\pi$$
$$258$$ −1.05086 −0.0654234
$$259$$ 7.34614 0.456467
$$260$$ 0 0
$$261$$ −4.23506 −0.262144
$$262$$ −0.755569 −0.0466792
$$263$$ −8.47013 −0.522290 −0.261145 0.965300i $$-0.584100\pi$$
−0.261145 + 0.965300i $$0.584100\pi$$
$$264$$ −4.42864 −0.272564
$$265$$ 0 0
$$266$$ −0.857279 −0.0525631
$$267$$ 2.75557 0.168638
$$268$$ −11.8064 −0.721192
$$269$$ 28.8256 1.75753 0.878765 0.477255i $$-0.158368\pi$$
0.878765 + 0.477255i $$0.158368\pi$$
$$270$$ 0 0
$$271$$ −13.8350 −0.840417 −0.420208 0.907428i $$-0.638043\pi$$
−0.420208 + 0.907428i $$0.638043\pi$$
$$272$$ −6.42864 −0.389794
$$273$$ 0.857279 0.0518849
$$274$$ 12.9175 0.780375
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ 2.88892 0.173578 0.0867892 0.996227i $$-0.472339\pi$$
0.0867892 + 0.996227i $$0.472339\pi$$
$$278$$ 14.1017 0.845764
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 5.51114 0.328767 0.164383 0.986397i $$-0.447437\pi$$
0.164383 + 0.986397i $$0.447437\pi$$
$$282$$ −11.6128 −0.691535
$$283$$ 8.19358 0.487058 0.243529 0.969894i $$-0.421695\pi$$
0.243529 + 0.969894i $$0.421695\pi$$
$$284$$ −6.99063 −0.414818
$$285$$ 0 0
$$286$$ 6.10171 0.360802
$$287$$ 6.75557 0.398769
$$288$$ −1.00000 −0.0589256
$$289$$ 24.3274 1.43102
$$290$$ 0 0
$$291$$ 2.13335 0.125059
$$292$$ −2.75557 −0.161257
$$293$$ 12.6222 0.737398 0.368699 0.929549i $$-0.379803\pi$$
0.368699 + 0.929549i $$0.379803\pi$$
$$294$$ 6.61285 0.385669
$$295$$ 0 0
$$296$$ 11.8064 0.686234
$$297$$ 4.42864 0.256976
$$298$$ −12.4286 −0.719972
$$299$$ 1.37778 0.0796793
$$300$$ 0 0
$$301$$ −0.653858 −0.0376878
$$302$$ 4.85728 0.279505
$$303$$ 16.2351 0.932680
$$304$$ −1.37778 −0.0790214
$$305$$ 0 0
$$306$$ 6.42864 0.367501
$$307$$ −0.653858 −0.0373177 −0.0186588 0.999826i $$-0.505940\pi$$
−0.0186588 + 0.999826i $$0.505940\pi$$
$$308$$ −2.75557 −0.157013
$$309$$ −12.2351 −0.696028
$$310$$ 0 0
$$311$$ −13.0923 −0.742399 −0.371199 0.928553i $$-0.621053\pi$$
−0.371199 + 0.928553i $$0.621053\pi$$
$$312$$ 1.37778 0.0780017
$$313$$ 11.2573 0.636302 0.318151 0.948040i $$-0.396938\pi$$
0.318151 + 0.948040i $$0.396938\pi$$
$$314$$ −9.90813 −0.559148
$$315$$ 0 0
$$316$$ 12.0415 0.677387
$$317$$ −22.4701 −1.26205 −0.631024 0.775763i $$-0.717365\pi$$
−0.631024 + 0.775763i $$0.717365\pi$$
$$318$$ 3.37778 0.189417
$$319$$ −18.7556 −1.05011
$$320$$ 0 0
$$321$$ 10.6637 0.595190
$$322$$ −0.622216 −0.0346747
$$323$$ 8.85728 0.492832
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 10.1017 0.559482
$$327$$ −15.1526 −0.837939
$$328$$ 10.8573 0.599493
$$329$$ −7.22570 −0.398365
$$330$$ 0 0
$$331$$ −29.4479 −1.61860 −0.809300 0.587395i $$-0.800153\pi$$
−0.809300 + 0.587395i $$0.800153\pi$$
$$332$$ −2.19358 −0.120388
$$333$$ −11.8064 −0.646988
$$334$$ 1.89829 0.103870
$$335$$ 0 0
$$336$$ −0.622216 −0.0339446
$$337$$ −24.6222 −1.34126 −0.670629 0.741793i $$-0.733976\pi$$
−0.670629 + 0.741793i $$0.733976\pi$$
$$338$$ 11.1017 0.603853
$$339$$ −14.4286 −0.783656
$$340$$ 0 0
$$341$$ −17.7146 −0.959297
$$342$$ 1.37778 0.0745020
$$343$$ 8.47013 0.457344
$$344$$ −1.05086 −0.0566583
$$345$$ 0 0
$$346$$ 21.2257 1.14110
$$347$$ 26.1017 1.40121 0.700607 0.713548i $$-0.252913\pi$$
0.700607 + 0.713548i $$0.252913\pi$$
$$348$$ −4.23506 −0.227023
$$349$$ −21.2257 −1.13619 −0.568093 0.822965i $$-0.692319\pi$$
−0.568093 + 0.822965i $$0.692319\pi$$
$$350$$ 0 0
$$351$$ −1.37778 −0.0735407
$$352$$ −4.42864 −0.236047
$$353$$ −4.28544 −0.228091 −0.114046 0.993476i $$-0.536381\pi$$
−0.114046 + 0.993476i $$0.536381\pi$$
$$354$$ −9.61285 −0.510917
$$355$$ 0 0
$$356$$ 2.75557 0.146045
$$357$$ 4.00000 0.211702
$$358$$ −0.488863 −0.0258372
$$359$$ −4.65386 −0.245621 −0.122811 0.992430i $$-0.539191\pi$$
−0.122811 + 0.992430i $$0.539191\pi$$
$$360$$ 0 0
$$361$$ −17.1017 −0.900090
$$362$$ 20.9304 1.10008
$$363$$ 8.61285 0.452057
$$364$$ 0.857279 0.0449336
$$365$$ 0 0
$$366$$ −8.66370 −0.452859
$$367$$ 4.88892 0.255200 0.127600 0.991826i $$-0.459273\pi$$
0.127600 + 0.991826i $$0.459273\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −10.8573 −0.565207
$$370$$ 0 0
$$371$$ 2.10171 0.109115
$$372$$ −4.00000 −0.207390
$$373$$ 23.4193 1.21260 0.606302 0.795235i $$-0.292652\pi$$
0.606302 + 0.795235i $$0.292652\pi$$
$$374$$ 28.4701 1.47216
$$375$$ 0 0
$$376$$ −11.6128 −0.598887
$$377$$ 5.83500 0.300518
$$378$$ 0.622216 0.0320033
$$379$$ −4.90766 −0.252089 −0.126045 0.992025i $$-0.540228\pi$$
−0.126045 + 0.992025i $$0.540228\pi$$
$$380$$ 0 0
$$381$$ 4.99063 0.255678
$$382$$ −18.9590 −0.970026
$$383$$ −19.8796 −1.01580 −0.507899 0.861417i $$-0.669578\pi$$
−0.507899 + 0.861417i $$0.669578\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −1.24443 −0.0633399
$$387$$ 1.05086 0.0534180
$$388$$ 2.13335 0.108305
$$389$$ −0.161933 −0.00821034 −0.00410517 0.999992i $$-0.501307\pi$$
−0.00410517 + 0.999992i $$0.501307\pi$$
$$390$$ 0 0
$$391$$ 6.42864 0.325110
$$392$$ 6.61285 0.333999
$$393$$ 0.755569 0.0381134
$$394$$ −15.2444 −0.768003
$$395$$ 0 0
$$396$$ 4.42864 0.222548
$$397$$ 1.76494 0.0885796 0.0442898 0.999019i $$-0.485898\pi$$
0.0442898 + 0.999019i $$0.485898\pi$$
$$398$$ 14.5303 0.728341
$$399$$ 0.857279 0.0429176
$$400$$ 0 0
$$401$$ −31.3461 −1.56535 −0.782676 0.622430i $$-0.786146\pi$$
−0.782676 + 0.622430i $$0.786146\pi$$
$$402$$ 11.8064 0.588851
$$403$$ 5.51114 0.274529
$$404$$ 16.2351 0.807725
$$405$$ 0 0
$$406$$ −2.63512 −0.130779
$$407$$ −52.2864 −2.59174
$$408$$ 6.42864 0.318265
$$409$$ −3.51114 −0.173615 −0.0868073 0.996225i $$-0.527666\pi$$
−0.0868073 + 0.996225i $$0.527666\pi$$
$$410$$ 0 0
$$411$$ −12.9175 −0.637173
$$412$$ −12.2351 −0.602778
$$413$$ −5.98126 −0.294319
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 1.37778 0.0675514
$$417$$ −14.1017 −0.690564
$$418$$ 6.10171 0.298444
$$419$$ −19.7748 −0.966061 −0.483031 0.875603i $$-0.660464\pi$$
−0.483031 + 0.875603i $$0.660464\pi$$
$$420$$ 0 0
$$421$$ 11.8064 0.575410 0.287705 0.957719i $$-0.407108\pi$$
0.287705 + 0.957719i $$0.407108\pi$$
$$422$$ −0.266706 −0.0129831
$$423$$ 11.6128 0.564636
$$424$$ 3.37778 0.164040
$$425$$ 0 0
$$426$$ 6.99063 0.338697
$$427$$ −5.39069 −0.260874
$$428$$ 10.6637 0.515450
$$429$$ −6.10171 −0.294593
$$430$$ 0 0
$$431$$ 24.9403 1.20133 0.600665 0.799501i $$-0.294903\pi$$
0.600665 + 0.799501i $$0.294903\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −32.0513 −1.54029 −0.770144 0.637870i $$-0.779816\pi$$
−0.770144 + 0.637870i $$0.779816\pi$$
$$434$$ −2.48886 −0.119469
$$435$$ 0 0
$$436$$ −15.1526 −0.725676
$$437$$ 1.37778 0.0659084
$$438$$ 2.75557 0.131666
$$439$$ 0.470127 0.0224379 0.0112190 0.999937i $$-0.496429\pi$$
0.0112190 + 0.999937i $$0.496429\pi$$
$$440$$ 0 0
$$441$$ −6.61285 −0.314898
$$442$$ −8.85728 −0.421298
$$443$$ −0.653858 −0.0310658 −0.0155329 0.999879i $$-0.504944\pi$$
−0.0155329 + 0.999879i $$0.504944\pi$$
$$444$$ −11.8064 −0.560308
$$445$$ 0 0
$$446$$ −0.990632 −0.0469078
$$447$$ 12.4286 0.587854
$$448$$ −0.622216 −0.0293969
$$449$$ 17.1427 0.809015 0.404508 0.914535i $$-0.367443\pi$$
0.404508 + 0.914535i $$0.367443\pi$$
$$450$$ 0 0
$$451$$ −48.0830 −2.26414
$$452$$ −14.4286 −0.678666
$$453$$ −4.85728 −0.228215
$$454$$ −17.1526 −0.805010
$$455$$ 0 0
$$456$$ 1.37778 0.0645207
$$457$$ 1.00937 0.0472162 0.0236081 0.999721i $$-0.492485\pi$$
0.0236081 + 0.999721i $$0.492485\pi$$
$$458$$ −7.15257 −0.334217
$$459$$ −6.42864 −0.300063
$$460$$ 0 0
$$461$$ 20.3180 0.946305 0.473153 0.880980i $$-0.343116\pi$$
0.473153 + 0.880980i $$0.343116\pi$$
$$462$$ 2.75557 0.128201
$$463$$ 12.4572 0.578936 0.289468 0.957188i $$-0.406522\pi$$
0.289468 + 0.957188i $$0.406522\pi$$
$$464$$ −4.23506 −0.196608
$$465$$ 0 0
$$466$$ −3.71456 −0.172074
$$467$$ −15.7877 −0.730567 −0.365284 0.930896i $$-0.619028\pi$$
−0.365284 + 0.930896i $$0.619028\pi$$
$$468$$ −1.37778 −0.0636881
$$469$$ 7.34614 0.339213
$$470$$ 0 0
$$471$$ 9.90813 0.456543
$$472$$ −9.61285 −0.442467
$$473$$ 4.65386 0.213985
$$474$$ −12.0415 −0.553084
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ −3.37778 −0.154658
$$478$$ −27.8479 −1.27373
$$479$$ 5.12399 0.234121 0.117060 0.993125i $$-0.462653\pi$$
0.117060 + 0.993125i $$0.462653\pi$$
$$480$$ 0 0
$$481$$ 16.2667 0.741698
$$482$$ 7.12399 0.324489
$$483$$ 0.622216 0.0283118
$$484$$ 8.61285 0.391493
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −23.2128 −1.05187 −0.525936 0.850524i $$-0.676285\pi$$
−0.525936 + 0.850524i $$0.676285\pi$$
$$488$$ −8.66370 −0.392187
$$489$$ −10.1017 −0.456815
$$490$$ 0 0
$$491$$ −40.1847 −1.81351 −0.906755 0.421658i $$-0.861448\pi$$
−0.906755 + 0.421658i $$0.861448\pi$$
$$492$$ −10.8573 −0.489484
$$493$$ 27.2257 1.22618
$$494$$ −1.89829 −0.0854081
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 4.34968 0.195110
$$498$$ 2.19358 0.0982965
$$499$$ 30.5718 1.36858 0.684292 0.729208i $$-0.260112\pi$$
0.684292 + 0.729208i $$0.260112\pi$$
$$500$$ 0 0
$$501$$ −1.89829 −0.0848093
$$502$$ 22.4099 1.00020
$$503$$ −23.4291 −1.04465 −0.522326 0.852746i $$-0.674936\pi$$
−0.522326 + 0.852746i $$0.674936\pi$$
$$504$$ 0.622216 0.0277157
$$505$$ 0 0
$$506$$ 4.42864 0.196877
$$507$$ −11.1017 −0.493044
$$508$$ 4.99063 0.221423
$$509$$ −17.2128 −0.762943 −0.381472 0.924381i $$-0.624583\pi$$
−0.381472 + 0.924381i $$0.624583\pi$$
$$510$$ 0 0
$$511$$ 1.71456 0.0758476
$$512$$ −1.00000 −0.0441942
$$513$$ −1.37778 −0.0608307
$$514$$ 11.7146 0.516707
$$515$$ 0 0
$$516$$ 1.05086 0.0462613
$$517$$ 51.4291 2.26185
$$518$$ −7.34614 −0.322771
$$519$$ −21.2257 −0.931705
$$520$$ 0 0
$$521$$ 22.3684 0.979978 0.489989 0.871729i $$-0.337001\pi$$
0.489989 + 0.871729i $$0.337001\pi$$
$$522$$ 4.23506 0.185364
$$523$$ 30.2953 1.32472 0.662360 0.749186i $$-0.269555\pi$$
0.662360 + 0.749186i $$0.269555\pi$$
$$524$$ 0.755569 0.0330072
$$525$$ 0 0
$$526$$ 8.47013 0.369315
$$527$$ 25.7146 1.12014
$$528$$ 4.42864 0.192732
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 9.61285 0.417162
$$532$$ 0.857279 0.0371678
$$533$$ 14.9590 0.647946
$$534$$ −2.75557 −0.119245
$$535$$ 0 0
$$536$$ 11.8064 0.509960
$$537$$ 0.488863 0.0210960
$$538$$ −28.8256 −1.24276
$$539$$ −29.2859 −1.26143
$$540$$ 0 0
$$541$$ 31.4479 1.35205 0.676024 0.736879i $$-0.263701\pi$$
0.676024 + 0.736879i $$0.263701\pi$$
$$542$$ 13.8350 0.594264
$$543$$ −20.9304 −0.898210
$$544$$ 6.42864 0.275626
$$545$$ 0 0
$$546$$ −0.857279 −0.0366882
$$547$$ 20.8573 0.891793 0.445896 0.895085i $$-0.352885\pi$$
0.445896 + 0.895085i $$0.352885\pi$$
$$548$$ −12.9175 −0.551808
$$549$$ 8.66370 0.369758
$$550$$ 0 0
$$551$$ 5.83500 0.248580
$$552$$ 1.00000 0.0425628
$$553$$ −7.49240 −0.318609
$$554$$ −2.88892 −0.122739
$$555$$ 0 0
$$556$$ −14.1017 −0.598046
$$557$$ 36.3180 1.53884 0.769422 0.638740i $$-0.220544\pi$$
0.769422 + 0.638740i $$0.220544\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −1.44785 −0.0612376
$$560$$ 0 0
$$561$$ −28.4701 −1.20201
$$562$$ −5.51114 −0.232473
$$563$$ 2.58073 0.108765 0.0543824 0.998520i $$-0.482681\pi$$
0.0543824 + 0.998520i $$0.482681\pi$$
$$564$$ 11.6128 0.488989
$$565$$ 0 0
$$566$$ −8.19358 −0.344402
$$567$$ −0.622216 −0.0261306
$$568$$ 6.99063 0.293320
$$569$$ −36.3497 −1.52386 −0.761929 0.647661i $$-0.775747\pi$$
−0.761929 + 0.647661i $$0.775747\pi$$
$$570$$ 0 0
$$571$$ −45.9309 −1.92215 −0.961074 0.276292i $$-0.910894\pi$$
−0.961074 + 0.276292i $$0.910894\pi$$
$$572$$ −6.10171 −0.255125
$$573$$ 18.9590 0.792023
$$574$$ −6.75557 −0.281972
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 4.00000 0.166522 0.0832611 0.996528i $$-0.473466\pi$$
0.0832611 + 0.996528i $$0.473466\pi$$
$$578$$ −24.3274 −1.01189
$$579$$ 1.24443 0.0517168
$$580$$ 0 0
$$581$$ 1.36488 0.0566247
$$582$$ −2.13335 −0.0884303
$$583$$ −14.9590 −0.619538
$$584$$ 2.75557 0.114026
$$585$$ 0 0
$$586$$ −12.6222 −0.521419
$$587$$ 42.1847 1.74115 0.870574 0.492037i $$-0.163748\pi$$
0.870574 + 0.492037i $$0.163748\pi$$
$$588$$ −6.61285 −0.272709
$$589$$ 5.51114 0.227082
$$590$$ 0 0
$$591$$ 15.2444 0.627072
$$592$$ −11.8064 −0.485241
$$593$$ 18.7368 0.769430 0.384715 0.923036i $$-0.374300\pi$$
0.384715 + 0.923036i $$0.374300\pi$$
$$594$$ −4.42864 −0.181709
$$595$$ 0 0
$$596$$ 12.4286 0.509097
$$597$$ −14.5303 −0.594688
$$598$$ −1.37778 −0.0563418
$$599$$ −41.5625 −1.69820 −0.849098 0.528235i $$-0.822854\pi$$
−0.849098 + 0.528235i $$0.822854\pi$$
$$600$$ 0 0
$$601$$ 23.7146 0.967337 0.483668 0.875251i $$-0.339304\pi$$
0.483668 + 0.875251i $$0.339304\pi$$
$$602$$ 0.653858 0.0266493
$$603$$ −11.8064 −0.480795
$$604$$ −4.85728 −0.197640
$$605$$ 0 0
$$606$$ −16.2351 −0.659504
$$607$$ 2.74266 0.111321 0.0556606 0.998450i $$-0.482274\pi$$
0.0556606 + 0.998450i $$0.482274\pi$$
$$608$$ 1.37778 0.0558765
$$609$$ 2.63512 0.106781
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ −6.42864 −0.259862
$$613$$ −24.0731 −0.972305 −0.486152 0.873874i $$-0.661600\pi$$
−0.486152 + 0.873874i $$0.661600\pi$$
$$614$$ 0.653858 0.0263876
$$615$$ 0 0
$$616$$ 2.75557 0.111025
$$617$$ −39.4893 −1.58978 −0.794890 0.606753i $$-0.792472\pi$$
−0.794890 + 0.606753i $$0.792472\pi$$
$$618$$ 12.2351 0.492166
$$619$$ −22.6222 −0.909264 −0.454632 0.890679i $$-0.650229\pi$$
−0.454632 + 0.890679i $$0.650229\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 13.0923 0.524955
$$623$$ −1.71456 −0.0686923
$$624$$ −1.37778 −0.0551555
$$625$$ 0 0
$$626$$ −11.2573 −0.449934
$$627$$ −6.10171 −0.243679
$$628$$ 9.90813 0.395378
$$629$$ 75.8992 3.02630
$$630$$ 0 0
$$631$$ 5.93978 0.236459 0.118229 0.992986i $$-0.462278\pi$$
0.118229 + 0.992986i $$0.462278\pi$$
$$632$$ −12.0415 −0.478985
$$633$$ 0.266706 0.0106006
$$634$$ 22.4701 0.892403
$$635$$ 0 0
$$636$$ −3.37778 −0.133938
$$637$$ 9.11108 0.360994
$$638$$ 18.7556 0.742540
$$639$$ −6.99063 −0.276545
$$640$$ 0 0
$$641$$ 19.8163 0.782696 0.391348 0.920243i $$-0.372009\pi$$
0.391348 + 0.920243i $$0.372009\pi$$
$$642$$ −10.6637 −0.420863
$$643$$ 44.7467 1.76464 0.882318 0.470653i $$-0.155982\pi$$
0.882318 + 0.470653i $$0.155982\pi$$
$$644$$ 0.622216 0.0245187
$$645$$ 0 0
$$646$$ −8.85728 −0.348485
$$647$$ 32.9403 1.29501 0.647507 0.762059i $$-0.275811\pi$$
0.647507 + 0.762059i $$0.275811\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 42.5718 1.67109
$$650$$ 0 0
$$651$$ 2.48886 0.0975462
$$652$$ −10.1017 −0.395614
$$653$$ 37.2257 1.45675 0.728377 0.685177i $$-0.240275\pi$$
0.728377 + 0.685177i $$0.240275\pi$$
$$654$$ 15.1526 0.592512
$$655$$ 0 0
$$656$$ −10.8573 −0.423906
$$657$$ −2.75557 −0.107505
$$658$$ 7.22570 0.281687
$$659$$ 24.6953 0.961994 0.480997 0.876722i $$-0.340275\pi$$
0.480997 + 0.876722i $$0.340275\pi$$
$$660$$ 0 0
$$661$$ −14.8287 −0.576770 −0.288385 0.957515i $$-0.593118\pi$$
−0.288385 + 0.957515i $$0.593118\pi$$
$$662$$ 29.4479 1.14452
$$663$$ 8.85728 0.343988
$$664$$ 2.19358 0.0851273
$$665$$ 0 0
$$666$$ 11.8064 0.457490
$$667$$ 4.23506 0.163982
$$668$$ −1.89829 −0.0734470
$$669$$ 0.990632 0.0383000
$$670$$ 0 0
$$671$$ 38.3684 1.48120
$$672$$ 0.622216 0.0240025
$$673$$ 8.77430 0.338225 0.169112 0.985597i $$-0.445910\pi$$
0.169112 + 0.985597i $$0.445910\pi$$
$$674$$ 24.6222 0.948412
$$675$$ 0 0
$$676$$ −11.1017 −0.426989
$$677$$ −12.8256 −0.492929 −0.246465 0.969152i $$-0.579269\pi$$
−0.246465 + 0.969152i $$0.579269\pi$$
$$678$$ 14.4286 0.554129
$$679$$ −1.32741 −0.0509412
$$680$$ 0 0
$$681$$ 17.1526 0.657288
$$682$$ 17.7146 0.678325
$$683$$ 8.47013 0.324100 0.162050 0.986783i $$-0.448189\pi$$
0.162050 + 0.986783i $$0.448189\pi$$
$$684$$ −1.37778 −0.0526809
$$685$$ 0 0
$$686$$ −8.47013 −0.323391
$$687$$ 7.15257 0.272887
$$688$$ 1.05086 0.0400635
$$689$$ 4.65386 0.177298
$$690$$ 0 0
$$691$$ 25.7975 0.981384 0.490692 0.871333i $$-0.336744\pi$$
0.490692 + 0.871333i $$0.336744\pi$$
$$692$$ −21.2257 −0.806880
$$693$$ −2.75557 −0.104675
$$694$$ −26.1017 −0.990807
$$695$$ 0 0
$$696$$ 4.23506 0.160530
$$697$$ 69.7975 2.64377
$$698$$ 21.2257 0.803404
$$699$$ 3.71456 0.140497
$$700$$ 0 0
$$701$$ 7.45091 0.281417 0.140709 0.990051i $$-0.455062\pi$$
0.140709 + 0.990051i $$0.455062\pi$$
$$702$$ 1.37778 0.0520011
$$703$$ 16.2667 0.613510
$$704$$ 4.42864 0.166911
$$705$$ 0 0
$$706$$ 4.28544 0.161285
$$707$$ −10.1017 −0.379914
$$708$$ 9.61285 0.361273
$$709$$ 13.2543 0.497775 0.248887 0.968532i $$-0.419935\pi$$
0.248887 + 0.968532i $$0.419935\pi$$
$$710$$ 0 0
$$711$$ 12.0415 0.451591
$$712$$ −2.75557 −0.103269
$$713$$ 4.00000 0.149801
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 0.488863 0.0182697
$$717$$ 27.8479 1.04000
$$718$$ 4.65386 0.173680
$$719$$ −2.33677 −0.0871469 −0.0435735 0.999050i $$-0.513874\pi$$
−0.0435735 + 0.999050i $$0.513874\pi$$
$$720$$ 0 0
$$721$$ 7.61285 0.283517
$$722$$ 17.1017 0.636460
$$723$$ −7.12399 −0.264944
$$724$$ −20.9304 −0.777873
$$725$$ 0 0
$$726$$ −8.61285 −0.319653
$$727$$ −3.37778 −0.125275 −0.0626375 0.998036i $$-0.519951\pi$$
−0.0626375 + 0.998036i $$0.519951\pi$$
$$728$$ −0.857279 −0.0317729
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −6.75557 −0.249864
$$732$$ 8.66370 0.320220
$$733$$ 21.0509 0.777531 0.388766 0.921337i $$-0.372902\pi$$
0.388766 + 0.921337i $$0.372902\pi$$
$$734$$ −4.88892 −0.180453
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ −52.2864 −1.92599
$$738$$ 10.8573 0.399662
$$739$$ −22.6351 −0.832646 −0.416323 0.909217i $$-0.636682\pi$$
−0.416323 + 0.909217i $$0.636682\pi$$
$$740$$ 0 0
$$741$$ 1.89829 0.0697354
$$742$$ −2.10171 −0.0771562
$$743$$ −3.87955 −0.142327 −0.0711635 0.997465i $$-0.522671\pi$$
−0.0711635 + 0.997465i $$0.522671\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −23.4193 −0.857440
$$747$$ −2.19358 −0.0802588
$$748$$ −28.4701 −1.04097
$$749$$ −6.63512 −0.242442
$$750$$ 0 0
$$751$$ 19.7748 0.721592 0.360796 0.932645i $$-0.382505\pi$$
0.360796 + 0.932645i $$0.382505\pi$$
$$752$$ 11.6128 0.423477
$$753$$ −22.4099 −0.816662
$$754$$ −5.83500 −0.212498
$$755$$ 0 0
$$756$$ −0.622216 −0.0226298
$$757$$ 8.19358 0.297801 0.148900 0.988852i $$-0.452427\pi$$
0.148900 + 0.988852i $$0.452427\pi$$
$$758$$ 4.90766 0.178254
$$759$$ −4.42864 −0.160749
$$760$$ 0 0
$$761$$ −45.1624 −1.63714 −0.818568 0.574410i $$-0.805232\pi$$
−0.818568 + 0.574410i $$0.805232\pi$$
$$762$$ −4.99063 −0.180792
$$763$$ 9.42816 0.341322
$$764$$ 18.9590 0.685912
$$765$$ 0 0
$$766$$ 19.8796 0.718277
$$767$$ −13.2444 −0.478229
$$768$$ 1.00000 0.0360844
$$769$$ 29.4924 1.06352 0.531762 0.846894i $$-0.321530\pi$$
0.531762 + 0.846894i $$0.321530\pi$$
$$770$$ 0 0
$$771$$ −11.7146 −0.421889
$$772$$ 1.24443 0.0447881
$$773$$ −41.0923 −1.47799 −0.738994 0.673712i $$-0.764699\pi$$
−0.738994 + 0.673712i $$0.764699\pi$$
$$774$$ −1.05086 −0.0377722
$$775$$ 0 0
$$776$$ −2.13335 −0.0765829
$$777$$ 7.34614 0.263541
$$778$$ 0.161933 0.00580559
$$779$$ 14.9590 0.535961
$$780$$ 0 0
$$781$$ −30.9590 −1.10780
$$782$$ −6.42864 −0.229888
$$783$$ −4.23506 −0.151349
$$784$$ −6.61285 −0.236173
$$785$$ 0 0
$$786$$ −0.755569 −0.0269502
$$787$$ 10.0919 0.359736 0.179868 0.983691i $$-0.442433\pi$$
0.179868 + 0.983691i $$0.442433\pi$$
$$788$$ 15.2444 0.543060
$$789$$ −8.47013 −0.301544
$$790$$ 0 0
$$791$$ 8.97773 0.319211
$$792$$ −4.42864 −0.157365
$$793$$ −11.9367 −0.423885
$$794$$ −1.76494 −0.0626353
$$795$$ 0 0
$$796$$ −14.5303 −0.515015
$$797$$ −25.2958 −0.896022 −0.448011 0.894028i $$-0.647867\pi$$
−0.448011 + 0.894028i $$0.647867\pi$$
$$798$$ −0.857279 −0.0303473
$$799$$ −74.6548 −2.64110
$$800$$ 0 0
$$801$$ 2.75557 0.0973632
$$802$$ 31.3461 1.10687
$$803$$ −12.2034 −0.430649
$$804$$ −11.8064 −0.416380
$$805$$ 0 0
$$806$$ −5.51114 −0.194122
$$807$$ 28.8256 1.01471
$$808$$ −16.2351 −0.571148
$$809$$ 43.4479 1.52755 0.763773 0.645485i $$-0.223345\pi$$
0.763773 + 0.645485i $$0.223345\pi$$
$$810$$ 0 0
$$811$$ −8.65386 −0.303878 −0.151939 0.988390i $$-0.548552\pi$$
−0.151939 + 0.988390i $$0.548552\pi$$
$$812$$ 2.63512 0.0924747
$$813$$ −13.8350 −0.485215
$$814$$ 52.2864 1.83264
$$815$$ 0 0
$$816$$ −6.42864 −0.225047
$$817$$ −1.44785 −0.0506539
$$818$$ 3.51114 0.122764
$$819$$ 0.857279 0.0299558
$$820$$ 0 0
$$821$$ 2.78721 0.0972744 0.0486372 0.998817i $$-0.484512\pi$$
0.0486372 + 0.998817i $$0.484512\pi$$
$$822$$ 12.9175 0.450550
$$823$$ 29.9684 1.04463 0.522316 0.852752i $$-0.325068\pi$$
0.522316 + 0.852752i $$0.325068\pi$$
$$824$$ 12.2351 0.426229
$$825$$ 0 0
$$826$$ 5.98126 0.208115
$$827$$ 47.2543 1.64319 0.821596 0.570070i $$-0.193084\pi$$
0.821596 + 0.570070i $$0.193084\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ −53.1624 −1.84641 −0.923203 0.384312i $$-0.874439\pi$$
−0.923203 + 0.384312i $$0.874439\pi$$
$$830$$ 0 0
$$831$$ 2.88892 0.100216
$$832$$ −1.37778 −0.0477661
$$833$$ 42.5116 1.47294
$$834$$ 14.1017 0.488302
$$835$$ 0 0
$$836$$ −6.10171 −0.211032
$$837$$ −4.00000 −0.138260
$$838$$ 19.7748 0.683108
$$839$$ −17.3274 −0.598208 −0.299104 0.954220i $$-0.596688\pi$$
−0.299104 + 0.954220i $$0.596688\pi$$
$$840$$ 0 0
$$841$$ −11.0642 −0.381525
$$842$$ −11.8064 −0.406876
$$843$$ 5.51114 0.189814
$$844$$ 0.266706 0.00918041
$$845$$ 0 0
$$846$$ −11.6128 −0.399258
$$847$$ −5.35905 −0.184139
$$848$$ −3.37778 −0.115994
$$849$$ 8.19358 0.281203
$$850$$ 0 0
$$851$$ 11.8064 0.404719
$$852$$ −6.99063 −0.239495
$$853$$ 17.7649 0.608260 0.304130 0.952631i $$-0.401634\pi$$
0.304130 + 0.952631i $$0.401634\pi$$
$$854$$ 5.39069 0.184466
$$855$$ 0 0
$$856$$ −10.6637 −0.364478
$$857$$ 10.0830 0.344428 0.172214 0.985060i $$-0.444908\pi$$
0.172214 + 0.985060i $$0.444908\pi$$
$$858$$ 6.10171 0.208309
$$859$$ 5.12399 0.174828 0.0874141 0.996172i $$-0.472140\pi$$
0.0874141 + 0.996172i $$0.472140\pi$$
$$860$$ 0 0
$$861$$ 6.75557 0.230229
$$862$$ −24.9403 −0.849468
$$863$$ 49.2070 1.67502 0.837512 0.546419i $$-0.184009\pi$$
0.837512 + 0.546419i $$0.184009\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 32.0513 1.08915
$$867$$ 24.3274 0.826202
$$868$$ 2.48886 0.0844775
$$869$$ 53.3274 1.80901
$$870$$ 0 0
$$871$$ 16.2667 0.551176
$$872$$ 15.1526 0.513131
$$873$$ 2.13335 0.0722031
$$874$$ −1.37778 −0.0466043
$$875$$ 0 0
$$876$$ −2.75557 −0.0931020
$$877$$ −9.11108 −0.307659 −0.153830 0.988097i $$-0.549161\pi$$
−0.153830 + 0.988097i $$0.549161\pi$$
$$878$$ −0.470127 −0.0158660
$$879$$ 12.6222 0.425737
$$880$$ 0 0
$$881$$ −9.71456 −0.327292 −0.163646 0.986519i $$-0.552325\pi$$
−0.163646 + 0.986519i $$0.552325\pi$$
$$882$$ 6.61285 0.222666
$$883$$ 33.3274 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$884$$ 8.85728 0.297903
$$885$$ 0 0
$$886$$ 0.653858 0.0219668
$$887$$ 53.5941 1.79951 0.899757 0.436391i $$-0.143744\pi$$
0.899757 + 0.436391i $$0.143744\pi$$
$$888$$ 11.8064 0.396198
$$889$$ −3.10525 −0.104147
$$890$$ 0 0
$$891$$ 4.42864 0.148365
$$892$$ 0.990632 0.0331688
$$893$$ −16.0000 −0.535420
$$894$$ −12.4286 −0.415676
$$895$$ 0 0
$$896$$ 0.622216 0.0207868
$$897$$ 1.37778 0.0460029
$$898$$ −17.1427 −0.572060
$$899$$ 16.9403 0.564989
$$900$$ 0 0
$$901$$ 21.7146 0.723417
$$902$$ 48.0830 1.60099
$$903$$ −0.653858 −0.0217590
$$904$$ 14.4286 0.479889
$$905$$ 0 0
$$906$$ 4.85728 0.161372
$$907$$ −2.56199 −0.0850696 −0.0425348 0.999095i $$-0.513543\pi$$
−0.0425348 + 0.999095i $$0.513543\pi$$
$$908$$ 17.1526 0.569228
$$909$$ 16.2351 0.538483
$$910$$ 0 0
$$911$$ 8.47013 0.280628 0.140314 0.990107i $$-0.455189\pi$$
0.140314 + 0.990107i $$0.455189\pi$$
$$912$$ −1.37778 −0.0456230
$$913$$ −9.71456 −0.321505
$$914$$ −1.00937 −0.0333869
$$915$$ 0 0
$$916$$ 7.15257 0.236327
$$917$$ −0.470127 −0.0155250
$$918$$ 6.42864 0.212177
$$919$$ −36.8988 −1.21718 −0.608589 0.793486i $$-0.708264\pi$$
−0.608589 + 0.793486i $$0.708264\pi$$
$$920$$ 0 0
$$921$$ −0.653858 −0.0215454
$$922$$ −20.3180 −0.669139
$$923$$ 9.63158 0.317027
$$924$$ −2.75557 −0.0906516
$$925$$ 0 0
$$926$$ −12.4572 −0.409370
$$927$$ −12.2351 −0.401852
$$928$$ 4.23506 0.139023
$$929$$ −0.164996 −0.00541334 −0.00270667 0.999996i $$-0.500862\pi$$
−0.00270667 + 0.999996i $$0.500862\pi$$
$$930$$ 0 0
$$931$$ 9.11108 0.298604
$$932$$ 3.71456 0.121674
$$933$$ −13.0923 −0.428624
$$934$$ 15.7877 0.516589
$$935$$ 0 0
$$936$$ 1.37778 0.0450343
$$937$$ 40.3180 1.31713 0.658566 0.752523i $$-0.271163\pi$$
0.658566 + 0.752523i $$0.271163\pi$$
$$938$$ −7.34614 −0.239860
$$939$$ 11.2573 0.367369
$$940$$ 0 0
$$941$$ 53.3689 1.73978 0.869888 0.493249i $$-0.164191\pi$$
0.869888 + 0.493249i $$0.164191\pi$$
$$942$$ −9.90813 −0.322824
$$943$$ 10.8573 0.353562
$$944$$ 9.61285 0.312872
$$945$$ 0 0
$$946$$ −4.65386 −0.151310
$$947$$ −44.6735 −1.45170 −0.725848 0.687856i $$-0.758552\pi$$
−0.725848 + 0.687856i $$0.758552\pi$$
$$948$$ 12.0415 0.391089
$$949$$ 3.79658 0.123242
$$950$$ 0 0
$$951$$ −22.4701 −0.728644
$$952$$ −4.00000 −0.129641
$$953$$ 21.2672 0.688912 0.344456 0.938803i $$-0.388063\pi$$
0.344456 + 0.938803i $$0.388063\pi$$
$$954$$ 3.37778 0.109360
$$955$$ 0 0
$$956$$ 27.8479 0.900666
$$957$$ −18.7556 −0.606281
$$958$$ −5.12399 −0.165548
$$959$$ 8.03747 0.259543
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −16.2667 −0.524460
$$963$$ 10.6637 0.343633
$$964$$ −7.12399 −0.229448
$$965$$ 0 0
$$966$$ −0.622216 −0.0200195
$$967$$ −10.9461 −0.352002 −0.176001 0.984390i $$-0.556316\pi$$
−0.176001 + 0.984390i $$0.556316\pi$$
$$968$$ −8.61285 −0.276827
$$969$$ 8.85728 0.284537
$$970$$ 0 0
$$971$$ −16.8988 −0.542307 −0.271154 0.962536i $$-0.587405\pi$$
−0.271154 + 0.962536i $$0.587405\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 8.77430 0.281291
$$974$$ 23.2128 0.743786
$$975$$ 0 0
$$976$$ 8.66370 0.277318
$$977$$ −59.2484 −1.89553 −0.947763 0.318976i $$-0.896661\pi$$
−0.947763 + 0.318976i $$0.896661\pi$$
$$978$$ 10.1017 0.323017
$$979$$ 12.2034 0.390023
$$980$$ 0 0
$$981$$ −15.1526 −0.483784
$$982$$ 40.1847 1.28234
$$983$$ 23.8163 0.759621 0.379810 0.925064i $$-0.375989\pi$$
0.379810 + 0.925064i $$0.375989\pi$$
$$984$$ 10.8573 0.346117
$$985$$ 0 0
$$986$$ −27.2257 −0.867043
$$987$$ −7.22570 −0.229996
$$988$$ 1.89829 0.0603926
$$989$$ −1.05086 −0.0334152
$$990$$ 0 0
$$991$$ 29.7146 0.943914 0.471957 0.881622i $$-0.343548\pi$$
0.471957 + 0.881622i $$0.343548\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −29.4479 −0.934499
$$994$$ −4.34968 −0.137963
$$995$$ 0 0
$$996$$ −2.19358 −0.0695061
$$997$$ −22.7052 −0.719081 −0.359540 0.933130i $$-0.617066\pi$$
−0.359540 + 0.933130i $$0.617066\pi$$
$$998$$ −30.5718 −0.967735
$$999$$ −11.8064 −0.373539
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 3450.2.a.bo.1.2 3
5.2 odd 4 690.2.d.c.139.3 6
5.3 odd 4 690.2.d.c.139.6 yes 6
5.4 even 2 3450.2.a.bt.1.2 3
15.2 even 4 2070.2.d.e.829.4 6
15.8 even 4 2070.2.d.e.829.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.c.139.3 6 5.2 odd 4
690.2.d.c.139.6 yes 6 5.3 odd 4
2070.2.d.e.829.1 6 15.8 even 4
2070.2.d.e.829.4 6 15.2 even 4
3450.2.a.bo.1.2 3 1.1 even 1 trivial
3450.2.a.bt.1.2 3 5.4 even 2