# Properties

 Label 3450.2.a.bn.1.2 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +6.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} -6.47214 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +6.47214 q^{26} +1.00000 q^{27} +4.00000 q^{28} +8.47214 q^{29} +1.00000 q^{32} -6.47214 q^{34} +1.00000 q^{36} -8.47214 q^{37} +2.00000 q^{38} +6.47214 q^{39} -6.94427 q^{41} +4.00000 q^{42} -1.00000 q^{46} -12.9443 q^{47} +1.00000 q^{48} +9.00000 q^{49} -6.47214 q^{51} +6.47214 q^{52} +8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} +8.47214 q^{58} -6.00000 q^{59} +8.47214 q^{61} +4.00000 q^{63} +1.00000 q^{64} +12.9443 q^{67} -6.47214 q^{68} -1.00000 q^{69} -16.4721 q^{71} +1.00000 q^{72} -12.9443 q^{73} -8.47214 q^{74} +2.00000 q^{76} +6.47214 q^{78} +3.52786 q^{79} +1.00000 q^{81} -6.94427 q^{82} +10.4721 q^{83} +4.00000 q^{84} +8.47214 q^{87} -7.52786 q^{89} +25.8885 q^{91} -1.00000 q^{92} -12.9443 q^{94} +1.00000 q^{96} +13.4164 q^{97} +9.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + 2q^{12} + 4q^{13} + 8q^{14} + 2q^{16} - 4q^{17} + 2q^{18} + 4q^{19} + 8q^{21} - 2q^{23} + 2q^{24} + 4q^{26} + 2q^{27} + 8q^{28} + 8q^{29} + 2q^{32} - 4q^{34} + 2q^{36} - 8q^{37} + 4q^{38} + 4q^{39} + 4q^{41} + 8q^{42} - 2q^{46} - 8q^{47} + 2q^{48} + 18q^{49} - 4q^{51} + 4q^{52} + 2q^{54} + 8q^{56} + 4q^{57} + 8q^{58} - 12q^{59} + 8q^{61} + 8q^{63} + 2q^{64} + 8q^{67} - 4q^{68} - 2q^{69} - 24q^{71} + 2q^{72} - 8q^{73} - 8q^{74} + 4q^{76} + 4q^{78} + 16q^{79} + 2q^{81} + 4q^{82} + 12q^{83} + 8q^{84} + 8q^{87} - 24q^{89} + 16q^{91} - 2q^{92} - 8q^{94} + 2q^{96} + 18q^{98} + O(q^{100})$$

## 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
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 6.47214 1.79505 0.897524 0.440966i $$-0.145364\pi$$
0.897524 + 0.440966i $$0.145364\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.47214 −1.56972 −0.784862 0.619671i $$-0.787266\pi$$
−0.784862 + 0.619671i $$0.787266\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 6.47214 1.26929
$$27$$ 1.00000 0.192450
$$28$$ 4.00000 0.755929
$$29$$ 8.47214 1.57324 0.786618 0.617440i $$-0.211830\pi$$
0.786618 + 0.617440i $$0.211830\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −6.47214 −1.10996
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.47214 −1.39281 −0.696405 0.717649i $$-0.745218\pi$$
−0.696405 + 0.717649i $$0.745218\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 6.47214 1.03637
$$40$$ 0 0
$$41$$ −6.94427 −1.08451 −0.542257 0.840213i $$-0.682430\pi$$
−0.542257 + 0.840213i $$0.682430\pi$$
$$42$$ 4.00000 0.617213
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ −12.9443 −1.88812 −0.944058 0.329779i $$-0.893026\pi$$
−0.944058 + 0.329779i $$0.893026\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −6.47214 −0.906280
$$52$$ 6.47214 0.897524
$$53$$ 8.94427 1.22859 0.614295 0.789076i $$-0.289440\pi$$
0.614295 + 0.789076i $$0.289440\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 2.00000 0.264906
$$58$$ 8.47214 1.11245
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 8.47214 1.08475 0.542373 0.840138i $$-0.317526\pi$$
0.542373 + 0.840138i $$0.317526\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.9443 1.58139 0.790697 0.612207i $$-0.209718\pi$$
0.790697 + 0.612207i $$0.209718\pi$$
$$68$$ −6.47214 −0.784862
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −16.4721 −1.95488 −0.977441 0.211207i $$-0.932261\pi$$
−0.977441 + 0.211207i $$0.932261\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −12.9443 −1.51501 −0.757506 0.652828i $$-0.773582\pi$$
−0.757506 + 0.652828i $$0.773582\pi$$
$$74$$ −8.47214 −0.984866
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 6.47214 0.732825
$$79$$ 3.52786 0.396916 0.198458 0.980109i $$-0.436407\pi$$
0.198458 + 0.980109i $$0.436407\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.94427 −0.766867
$$83$$ 10.4721 1.14947 0.574733 0.818341i $$-0.305106\pi$$
0.574733 + 0.818341i $$0.305106\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 8.47214 0.908308
$$88$$ 0 0
$$89$$ −7.52786 −0.797952 −0.398976 0.916961i $$-0.630634\pi$$
−0.398976 + 0.916961i $$0.630634\pi$$
$$90$$ 0 0
$$91$$ 25.8885 2.71386
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ −12.9443 −1.33510
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 13.4164 1.36223 0.681115 0.732177i $$-0.261495\pi$$
0.681115 + 0.732177i $$0.261495\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.47214 −0.843009 −0.421505 0.906826i $$-0.638498\pi$$
−0.421505 + 0.906826i $$0.638498\pi$$
$$102$$ −6.47214 −0.640837
$$103$$ −0.944272 −0.0930419 −0.0465209 0.998917i $$-0.514813\pi$$
−0.0465209 + 0.998917i $$0.514813\pi$$
$$104$$ 6.47214 0.634645
$$105$$ 0 0
$$106$$ 8.94427 0.868744
$$107$$ 6.47214 0.625685 0.312842 0.949805i $$-0.398719\pi$$
0.312842 + 0.949805i $$0.398719\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −8.47214 −0.811483 −0.405742 0.913988i $$-0.632987\pi$$
−0.405742 + 0.913988i $$0.632987\pi$$
$$110$$ 0 0
$$111$$ −8.47214 −0.804140
$$112$$ 4.00000 0.377964
$$113$$ −7.41641 −0.697677 −0.348838 0.937183i $$-0.613424\pi$$
−0.348838 + 0.937183i $$0.613424\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 8.47214 0.786618
$$117$$ 6.47214 0.598349
$$118$$ −6.00000 −0.552345
$$119$$ −25.8885 −2.37320
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 8.47214 0.767031
$$123$$ −6.94427 −0.626144
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ −15.4164 −1.36798 −0.683992 0.729489i $$-0.739758\pi$$
−0.683992 + 0.729489i $$0.739758\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.9443 0.956205 0.478103 0.878304i $$-0.341325\pi$$
0.478103 + 0.878304i $$0.341325\pi$$
$$132$$ 0 0
$$133$$ 8.00000 0.693688
$$134$$ 12.9443 1.11821
$$135$$ 0 0
$$136$$ −6.47214 −0.554981
$$137$$ 7.41641 0.633626 0.316813 0.948488i $$-0.397387\pi$$
0.316813 + 0.948488i $$0.397387\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −12.9443 −1.09010
$$142$$ −16.4721 −1.38231
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −12.9443 −1.07128
$$147$$ 9.00000 0.742307
$$148$$ −8.47214 −0.696405
$$149$$ 13.4164 1.09911 0.549557 0.835456i $$-0.314796\pi$$
0.549557 + 0.835456i $$0.314796\pi$$
$$150$$ 0 0
$$151$$ −0.944272 −0.0768438 −0.0384219 0.999262i $$-0.512233\pi$$
−0.0384219 + 0.999262i $$0.512233\pi$$
$$152$$ 2.00000 0.162221
$$153$$ −6.47214 −0.523241
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.47214 0.518186
$$157$$ −20.4721 −1.63385 −0.816927 0.576741i $$-0.804324\pi$$
−0.816927 + 0.576741i $$0.804324\pi$$
$$158$$ 3.52786 0.280662
$$159$$ 8.94427 0.709327
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 1.00000 0.0785674
$$163$$ 12.9443 1.01387 0.506937 0.861983i $$-0.330778\pi$$
0.506937 + 0.861983i $$0.330778\pi$$
$$164$$ −6.94427 −0.542257
$$165$$ 0 0
$$166$$ 10.4721 0.812795
$$167$$ 20.9443 1.62072 0.810358 0.585935i $$-0.199273\pi$$
0.810358 + 0.585935i $$0.199273\pi$$
$$168$$ 4.00000 0.308607
$$169$$ 28.8885 2.22220
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ 2.94427 0.223849 0.111924 0.993717i $$-0.464299\pi$$
0.111924 + 0.993717i $$0.464299\pi$$
$$174$$ 8.47214 0.642271
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ −7.52786 −0.564237
$$179$$ −1.05573 −0.0789088 −0.0394544 0.999221i $$-0.512562\pi$$
−0.0394544 + 0.999221i $$0.512562\pi$$
$$180$$ 0 0
$$181$$ −0.472136 −0.0350936 −0.0175468 0.999846i $$-0.505586\pi$$
−0.0175468 + 0.999846i $$0.505586\pi$$
$$182$$ 25.8885 1.91899
$$183$$ 8.47214 0.626278
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −12.9443 −0.944058
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 9.88854 0.715510 0.357755 0.933816i $$-0.383542\pi$$
0.357755 + 0.933816i $$0.383542\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 20.0000 1.43963 0.719816 0.694165i $$-0.244226\pi$$
0.719816 + 0.694165i $$0.244226\pi$$
$$194$$ 13.4164 0.963242
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 1.05573 0.0752175 0.0376088 0.999293i $$-0.488026\pi$$
0.0376088 + 0.999293i $$0.488026\pi$$
$$198$$ 0 0
$$199$$ −9.41641 −0.667511 −0.333756 0.942660i $$-0.608316\pi$$
−0.333756 + 0.942660i $$0.608316\pi$$
$$200$$ 0 0
$$201$$ 12.9443 0.913019
$$202$$ −8.47214 −0.596097
$$203$$ 33.8885 2.37851
$$204$$ −6.47214 −0.453140
$$205$$ 0 0
$$206$$ −0.944272 −0.0657905
$$207$$ −1.00000 −0.0695048
$$208$$ 6.47214 0.448762
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 8.94427 0.614295
$$213$$ −16.4721 −1.12865
$$214$$ 6.47214 0.442426
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −8.47214 −0.573805
$$219$$ −12.9443 −0.874693
$$220$$ 0 0
$$221$$ −41.8885 −2.81773
$$222$$ −8.47214 −0.568613
$$223$$ −15.4164 −1.03236 −0.516180 0.856480i $$-0.672646\pi$$
−0.516180 + 0.856480i $$0.672646\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −7.41641 −0.493332
$$227$$ 9.52786 0.632387 0.316193 0.948695i $$-0.397595\pi$$
0.316193 + 0.948695i $$0.397595\pi$$
$$228$$ 2.00000 0.132453
$$229$$ −1.41641 −0.0935989 −0.0467994 0.998904i $$-0.514902\pi$$
−0.0467994 + 0.998904i $$0.514902\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 8.47214 0.556223
$$233$$ −2.94427 −0.192886 −0.0964428 0.995339i $$-0.530746\pi$$
−0.0964428 + 0.995339i $$0.530746\pi$$
$$234$$ 6.47214 0.423097
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 3.52786 0.229159
$$238$$ −25.8885 −1.67811
$$239$$ −3.52786 −0.228199 −0.114099 0.993469i $$-0.536398\pi$$
−0.114099 + 0.993469i $$0.536398\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ 8.47214 0.542373
$$245$$ 0 0
$$246$$ −6.94427 −0.442751
$$247$$ 12.9443 0.823624
$$248$$ 0 0
$$249$$ 10.4721 0.663645
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 4.00000 0.251976
$$253$$ 0 0
$$254$$ −15.4164 −0.967311
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −33.8885 −2.10573
$$260$$ 0 0
$$261$$ 8.47214 0.524412
$$262$$ 10.9443 0.676139
$$263$$ 3.05573 0.188424 0.0942121 0.995552i $$-0.469967\pi$$
0.0942121 + 0.995552i $$0.469967\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ −7.52786 −0.460698
$$268$$ 12.9443 0.790697
$$269$$ −8.47214 −0.516555 −0.258278 0.966071i $$-0.583155\pi$$
−0.258278 + 0.966071i $$0.583155\pi$$
$$270$$ 0 0
$$271$$ −16.9443 −1.02929 −0.514646 0.857403i $$-0.672076\pi$$
−0.514646 + 0.857403i $$0.672076\pi$$
$$272$$ −6.47214 −0.392431
$$273$$ 25.8885 1.56685
$$274$$ 7.41641 0.448042
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ −0.583592 −0.0350647 −0.0175323 0.999846i $$-0.505581\pi$$
−0.0175323 + 0.999846i $$0.505581\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.4721 0.744025 0.372013 0.928228i $$-0.378668\pi$$
0.372013 + 0.928228i $$0.378668\pi$$
$$282$$ −12.9443 −0.770820
$$283$$ 24.0000 1.42665 0.713326 0.700832i $$-0.247188\pi$$
0.713326 + 0.700832i $$0.247188\pi$$
$$284$$ −16.4721 −0.977441
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −27.7771 −1.63963
$$288$$ 1.00000 0.0589256
$$289$$ 24.8885 1.46403
$$290$$ 0 0
$$291$$ 13.4164 0.786484
$$292$$ −12.9443 −0.757506
$$293$$ 16.9443 0.989895 0.494947 0.868923i $$-0.335187\pi$$
0.494947 + 0.868923i $$0.335187\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −8.47214 −0.492433
$$297$$ 0 0
$$298$$ 13.4164 0.777192
$$299$$ −6.47214 −0.374293
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −0.944272 −0.0543367
$$303$$ −8.47214 −0.486711
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ −6.47214 −0.369987
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ −0.944272 −0.0537178
$$310$$ 0 0
$$311$$ −15.5279 −0.880504 −0.440252 0.897874i $$-0.645111\pi$$
−0.440252 + 0.897874i $$0.645111\pi$$
$$312$$ 6.47214 0.366413
$$313$$ 1.41641 0.0800601 0.0400301 0.999198i $$-0.487255\pi$$
0.0400301 + 0.999198i $$0.487255\pi$$
$$314$$ −20.4721 −1.15531
$$315$$ 0 0
$$316$$ 3.52786 0.198458
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 8.94427 0.501570
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 6.47214 0.361239
$$322$$ −4.00000 −0.222911
$$323$$ −12.9443 −0.720239
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 12.9443 0.716917
$$327$$ −8.47214 −0.468510
$$328$$ −6.94427 −0.383433
$$329$$ −51.7771 −2.85456
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 10.4721 0.574733
$$333$$ −8.47214 −0.464270
$$334$$ 20.9443 1.14602
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ −1.41641 −0.0771567 −0.0385783 0.999256i $$-0.512283\pi$$
−0.0385783 + 0.999256i $$0.512283\pi$$
$$338$$ 28.8885 1.57133
$$339$$ −7.41641 −0.402804
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.00000 0.108148
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 2.94427 0.158285
$$347$$ −8.94427 −0.480154 −0.240077 0.970754i $$-0.577173\pi$$
−0.240077 + 0.970754i $$0.577173\pi$$
$$348$$ 8.47214 0.454154
$$349$$ 7.88854 0.422264 0.211132 0.977458i $$-0.432285\pi$$
0.211132 + 0.977458i $$0.432285\pi$$
$$350$$ 0 0
$$351$$ 6.47214 0.345457
$$352$$ 0 0
$$353$$ −14.9443 −0.795403 −0.397702 0.917515i $$-0.630192\pi$$
−0.397702 + 0.917515i $$0.630192\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ 0 0
$$356$$ −7.52786 −0.398976
$$357$$ −25.8885 −1.37017
$$358$$ −1.05573 −0.0557970
$$359$$ −15.0557 −0.794611 −0.397305 0.917686i $$-0.630055\pi$$
−0.397305 + 0.917686i $$0.630055\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −0.472136 −0.0248149
$$363$$ −11.0000 −0.577350
$$364$$ 25.8885 1.35693
$$365$$ 0 0
$$366$$ 8.47214 0.442846
$$367$$ −12.0000 −0.626395 −0.313197 0.949688i $$-0.601400\pi$$
−0.313197 + 0.949688i $$0.601400\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −6.94427 −0.361504
$$370$$ 0 0
$$371$$ 35.7771 1.85745
$$372$$ 0 0
$$373$$ 5.41641 0.280451 0.140225 0.990120i $$-0.455217\pi$$
0.140225 + 0.990120i $$0.455217\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.9443 −0.667550
$$377$$ 54.8328 2.82403
$$378$$ 4.00000 0.205738
$$379$$ −36.8328 −1.89197 −0.945987 0.324204i $$-0.894904\pi$$
−0.945987 + 0.324204i $$0.894904\pi$$
$$380$$ 0 0
$$381$$ −15.4164 −0.789807
$$382$$ 9.88854 0.505942
$$383$$ 21.8885 1.11845 0.559226 0.829015i $$-0.311098\pi$$
0.559226 + 0.829015i $$0.311098\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 20.0000 1.01797
$$387$$ 0 0
$$388$$ 13.4164 0.681115
$$389$$ −4.47214 −0.226746 −0.113373 0.993552i $$-0.536166\pi$$
−0.113373 + 0.993552i $$0.536166\pi$$
$$390$$ 0 0
$$391$$ 6.47214 0.327310
$$392$$ 9.00000 0.454569
$$393$$ 10.9443 0.552065
$$394$$ 1.05573 0.0531868
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −33.3050 −1.67153 −0.835764 0.549089i $$-0.814975\pi$$
−0.835764 + 0.549089i $$0.814975\pi$$
$$398$$ −9.41641 −0.472002
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −1.41641 −0.0707320 −0.0353660 0.999374i $$-0.511260\pi$$
−0.0353660 + 0.999374i $$0.511260\pi$$
$$402$$ 12.9443 0.645602
$$403$$ 0 0
$$404$$ −8.47214 −0.421505
$$405$$ 0 0
$$406$$ 33.8885 1.68186
$$407$$ 0 0
$$408$$ −6.47214 −0.320418
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ 7.41641 0.365824
$$412$$ −0.944272 −0.0465209
$$413$$ −24.0000 −1.18096
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 6.47214 0.317323
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −2.11146 −0.103151 −0.0515757 0.998669i $$-0.516424\pi$$
−0.0515757 + 0.998669i $$0.516424\pi$$
$$420$$ 0 0
$$421$$ −10.5836 −0.515813 −0.257906 0.966170i $$-0.583033\pi$$
−0.257906 + 0.966170i $$0.583033\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ −12.9443 −0.629372
$$424$$ 8.94427 0.434372
$$425$$ 0 0
$$426$$ −16.4721 −0.798078
$$427$$ 33.8885 1.63998
$$428$$ 6.47214 0.312842
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 5.41641 0.260296 0.130148 0.991495i $$-0.458455\pi$$
0.130148 + 0.991495i $$0.458455\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −8.47214 −0.405742
$$437$$ −2.00000 −0.0956730
$$438$$ −12.9443 −0.618501
$$439$$ 26.8328 1.28066 0.640330 0.768100i $$-0.278798\pi$$
0.640330 + 0.768100i $$0.278798\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ −41.8885 −1.99243
$$443$$ 8.94427 0.424955 0.212478 0.977166i $$-0.431847\pi$$
0.212478 + 0.977166i $$0.431847\pi$$
$$444$$ −8.47214 −0.402070
$$445$$ 0 0
$$446$$ −15.4164 −0.729988
$$447$$ 13.4164 0.634574
$$448$$ 4.00000 0.188982
$$449$$ −17.0557 −0.804910 −0.402455 0.915440i $$-0.631843\pi$$
−0.402455 + 0.915440i $$0.631843\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −7.41641 −0.348838
$$453$$ −0.944272 −0.0443658
$$454$$ 9.52786 0.447165
$$455$$ 0 0
$$456$$ 2.00000 0.0936586
$$457$$ −1.41641 −0.0662568 −0.0331284 0.999451i $$-0.510547\pi$$
−0.0331284 + 0.999451i $$0.510547\pi$$
$$458$$ −1.41641 −0.0661844
$$459$$ −6.47214 −0.302093
$$460$$ 0 0
$$461$$ 30.3607 1.41404 0.707019 0.707195i $$-0.250040\pi$$
0.707019 + 0.707195i $$0.250040\pi$$
$$462$$ 0 0
$$463$$ 11.4164 0.530565 0.265283 0.964171i $$-0.414535\pi$$
0.265283 + 0.964171i $$0.414535\pi$$
$$464$$ 8.47214 0.393309
$$465$$ 0 0
$$466$$ −2.94427 −0.136391
$$467$$ 17.5279 0.811093 0.405546 0.914074i $$-0.367081\pi$$
0.405546 + 0.914074i $$0.367081\pi$$
$$468$$ 6.47214 0.299175
$$469$$ 51.7771 2.39084
$$470$$ 0 0
$$471$$ −20.4721 −0.943306
$$472$$ −6.00000 −0.276172
$$473$$ 0 0
$$474$$ 3.52786 0.162040
$$475$$ 0 0
$$476$$ −25.8885 −1.18660
$$477$$ 8.94427 0.409530
$$478$$ −3.52786 −0.161361
$$479$$ 0.944272 0.0431449 0.0215724 0.999767i $$-0.493133\pi$$
0.0215724 + 0.999767i $$0.493133\pi$$
$$480$$ 0 0
$$481$$ −54.8328 −2.50016
$$482$$ −2.00000 −0.0910975
$$483$$ −4.00000 −0.182006
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −4.58359 −0.207702 −0.103851 0.994593i $$-0.533117\pi$$
−0.103851 + 0.994593i $$0.533117\pi$$
$$488$$ 8.47214 0.383516
$$489$$ 12.9443 0.585360
$$490$$ 0 0
$$491$$ −7.88854 −0.356005 −0.178002 0.984030i $$-0.556964\pi$$
−0.178002 + 0.984030i $$0.556964\pi$$
$$492$$ −6.94427 −0.313072
$$493$$ −54.8328 −2.46955
$$494$$ 12.9443 0.582390
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −65.8885 −2.95551
$$498$$ 10.4721 0.469268
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 0 0
$$501$$ 20.9443 0.935721
$$502$$ −12.0000 −0.535586
$$503$$ −21.8885 −0.975962 −0.487981 0.872854i $$-0.662266\pi$$
−0.487981 + 0.872854i $$0.662266\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 28.8885 1.28299
$$508$$ −15.4164 −0.683992
$$509$$ −25.4164 −1.12656 −0.563281 0.826265i $$-0.690461\pi$$
−0.563281 + 0.826265i $$0.690461\pi$$
$$510$$ 0 0
$$511$$ −51.7771 −2.29048
$$512$$ 1.00000 0.0441942
$$513$$ 2.00000 0.0883022
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −33.8885 −1.48898
$$519$$ 2.94427 0.129239
$$520$$ 0 0
$$521$$ −28.4721 −1.24739 −0.623693 0.781669i $$-0.714369\pi$$
−0.623693 + 0.781669i $$0.714369\pi$$
$$522$$ 8.47214 0.370815
$$523$$ −12.9443 −0.566013 −0.283007 0.959118i $$-0.591332\pi$$
−0.283007 + 0.959118i $$0.591332\pi$$
$$524$$ 10.9443 0.478103
$$525$$ 0 0
$$526$$ 3.05573 0.133236
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 8.00000 0.346844
$$533$$ −44.9443 −1.94675
$$534$$ −7.52786 −0.325763
$$535$$ 0 0
$$536$$ 12.9443 0.559107
$$537$$ −1.05573 −0.0455580
$$538$$ −8.47214 −0.365260
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −12.8328 −0.551726 −0.275863 0.961197i $$-0.588964\pi$$
−0.275863 + 0.961197i $$0.588964\pi$$
$$542$$ −16.9443 −0.727819
$$543$$ −0.472136 −0.0202613
$$544$$ −6.47214 −0.277491
$$545$$ 0 0
$$546$$ 25.8885 1.10793
$$547$$ 30.8328 1.31832 0.659158 0.752004i $$-0.270913\pi$$
0.659158 + 0.752004i $$0.270913\pi$$
$$548$$ 7.41641 0.316813
$$549$$ 8.47214 0.361582
$$550$$ 0 0
$$551$$ 16.9443 0.721850
$$552$$ −1.00000 −0.0425628
$$553$$ 14.1115 0.600080
$$554$$ −0.583592 −0.0247945
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.88854 0.249506 0.124753 0.992188i $$-0.460186\pi$$
0.124753 + 0.992188i $$0.460186\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 12.4721 0.526105
$$563$$ 25.3050 1.06648 0.533238 0.845965i $$-0.320975\pi$$
0.533238 + 0.845965i $$0.320975\pi$$
$$564$$ −12.9443 −0.545052
$$565$$ 0 0
$$566$$ 24.0000 1.00880
$$567$$ 4.00000 0.167984
$$568$$ −16.4721 −0.691155
$$569$$ −23.3050 −0.976994 −0.488497 0.872565i $$-0.662455\pi$$
−0.488497 + 0.872565i $$0.662455\pi$$
$$570$$ 0 0
$$571$$ −20.8328 −0.871826 −0.435913 0.899989i $$-0.643575\pi$$
−0.435913 + 0.899989i $$0.643575\pi$$
$$572$$ 0 0
$$573$$ 9.88854 0.413100
$$574$$ −27.7771 −1.15939
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −0.944272 −0.0393106 −0.0196553 0.999807i $$-0.506257\pi$$
−0.0196553 + 0.999807i $$0.506257\pi$$
$$578$$ 24.8885 1.03523
$$579$$ 20.0000 0.831172
$$580$$ 0 0
$$581$$ 41.8885 1.73783
$$582$$ 13.4164 0.556128
$$583$$ 0 0
$$584$$ −12.9443 −0.535638
$$585$$ 0 0
$$586$$ 16.9443 0.699961
$$587$$ 5.88854 0.243046 0.121523 0.992589i $$-0.461222\pi$$
0.121523 + 0.992589i $$0.461222\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 1.05573 0.0434269
$$592$$ −8.47214 −0.348203
$$593$$ −11.8885 −0.488204 −0.244102 0.969750i $$-0.578493\pi$$
−0.244102 + 0.969750i $$0.578493\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 13.4164 0.549557
$$597$$ −9.41641 −0.385388
$$598$$ −6.47214 −0.264665
$$599$$ −18.3607 −0.750197 −0.375099 0.926985i $$-0.622391\pi$$
−0.375099 + 0.926985i $$0.622391\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 12.9443 0.527132
$$604$$ −0.944272 −0.0384219
$$605$$ 0 0
$$606$$ −8.47214 −0.344157
$$607$$ −19.4164 −0.788088 −0.394044 0.919092i $$-0.628924\pi$$
−0.394044 + 0.919092i $$0.628924\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 33.8885 1.37323
$$610$$ 0 0
$$611$$ −83.7771 −3.38926
$$612$$ −6.47214 −0.261621
$$613$$ −17.4164 −0.703442 −0.351721 0.936105i $$-0.614403\pi$$
−0.351721 + 0.936105i $$0.614403\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5.52786 0.222543 0.111272 0.993790i $$-0.464508\pi$$
0.111272 + 0.993790i $$0.464508\pi$$
$$618$$ −0.944272 −0.0379842
$$619$$ −12.8328 −0.515794 −0.257897 0.966172i $$-0.583030\pi$$
−0.257897 + 0.966172i $$0.583030\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ −15.5279 −0.622611
$$623$$ −30.1115 −1.20639
$$624$$ 6.47214 0.259093
$$625$$ 0 0
$$626$$ 1.41641 0.0566110
$$627$$ 0 0
$$628$$ −20.4721 −0.816927
$$629$$ 54.8328 2.18633
$$630$$ 0 0
$$631$$ 42.3607 1.68635 0.843176 0.537638i $$-0.180683\pi$$
0.843176 + 0.537638i $$0.180683\pi$$
$$632$$ 3.52786 0.140331
$$633$$ −12.0000 −0.476957
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 8.94427 0.354663
$$637$$ 58.2492 2.30792
$$638$$ 0 0
$$639$$ −16.4721 −0.651628
$$640$$ 0 0
$$641$$ −20.4721 −0.808601 −0.404300 0.914626i $$-0.632485\pi$$
−0.404300 + 0.914626i $$0.632485\pi$$
$$642$$ 6.47214 0.255435
$$643$$ −4.94427 −0.194983 −0.0974915 0.995236i $$-0.531082\pi$$
−0.0974915 + 0.995236i $$0.531082\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ −12.9443 −0.509286
$$647$$ −6.11146 −0.240266 −0.120133 0.992758i $$-0.538332\pi$$
−0.120133 + 0.992758i $$0.538332\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 12.9443 0.506937
$$653$$ −32.8328 −1.28485 −0.642424 0.766350i $$-0.722071\pi$$
−0.642424 + 0.766350i $$0.722071\pi$$
$$654$$ −8.47214 −0.331287
$$655$$ 0 0
$$656$$ −6.94427 −0.271128
$$657$$ −12.9443 −0.505004
$$658$$ −51.7771 −2.01848
$$659$$ −17.8885 −0.696839 −0.348419 0.937339i $$-0.613281\pi$$
−0.348419 + 0.937339i $$0.613281\pi$$
$$660$$ 0 0
$$661$$ 44.2492 1.72110 0.860548 0.509370i $$-0.170121\pi$$
0.860548 + 0.509370i $$0.170121\pi$$
$$662$$ 0 0
$$663$$ −41.8885 −1.62682
$$664$$ 10.4721 0.406398
$$665$$ 0 0
$$666$$ −8.47214 −0.328289
$$667$$ −8.47214 −0.328042
$$668$$ 20.9443 0.810358
$$669$$ −15.4164 −0.596033
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 4.00000 0.154303
$$673$$ 14.8328 0.571763 0.285882 0.958265i $$-0.407714\pi$$
0.285882 + 0.958265i $$0.407714\pi$$
$$674$$ −1.41641 −0.0545580
$$675$$ 0 0
$$676$$ 28.8885 1.11110
$$677$$ 9.88854 0.380048 0.190024 0.981779i $$-0.439143\pi$$
0.190024 + 0.981779i $$0.439143\pi$$
$$678$$ −7.41641 −0.284825
$$679$$ 53.6656 2.05950
$$680$$ 0 0
$$681$$ 9.52786 0.365109
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ −1.41641 −0.0540393
$$688$$ 0 0
$$689$$ 57.8885 2.20538
$$690$$ 0 0
$$691$$ 31.7771 1.20886 0.604429 0.796659i $$-0.293401\pi$$
0.604429 + 0.796659i $$0.293401\pi$$
$$692$$ 2.94427 0.111924
$$693$$ 0 0
$$694$$ −8.94427 −0.339520
$$695$$ 0 0
$$696$$ 8.47214 0.321135
$$697$$ 44.9443 1.70239
$$698$$ 7.88854 0.298586
$$699$$ −2.94427 −0.111363
$$700$$ 0 0
$$701$$ 10.5836 0.399737 0.199868 0.979823i $$-0.435949\pi$$
0.199868 + 0.979823i $$0.435949\pi$$
$$702$$ 6.47214 0.244275
$$703$$ −16.9443 −0.639065
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −14.9443 −0.562435
$$707$$ −33.8885 −1.27451
$$708$$ −6.00000 −0.225494
$$709$$ 11.5279 0.432938 0.216469 0.976289i $$-0.430546\pi$$
0.216469 + 0.976289i $$0.430546\pi$$
$$710$$ 0 0
$$711$$ 3.52786 0.132305
$$712$$ −7.52786 −0.282119
$$713$$ 0 0
$$714$$ −25.8885 −0.968854
$$715$$ 0 0
$$716$$ −1.05573 −0.0394544
$$717$$ −3.52786 −0.131750
$$718$$ −15.0557 −0.561875
$$719$$ −44.4721 −1.65853 −0.829265 0.558855i $$-0.811241\pi$$
−0.829265 + 0.558855i $$0.811241\pi$$
$$720$$ 0 0
$$721$$ −3.77709 −0.140666
$$722$$ −15.0000 −0.558242
$$723$$ −2.00000 −0.0743808
$$724$$ −0.472136 −0.0175468
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ −16.9443 −0.628428 −0.314214 0.949352i $$-0.601741\pi$$
−0.314214 + 0.949352i $$0.601741\pi$$
$$728$$ 25.8885 0.959493
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 8.47214 0.313139
$$733$$ 8.47214 0.312925 0.156463 0.987684i $$-0.449991\pi$$
0.156463 + 0.987684i $$0.449991\pi$$
$$734$$ −12.0000 −0.442928
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 0 0
$$738$$ −6.94427 −0.255622
$$739$$ −33.8885 −1.24661 −0.623305 0.781979i $$-0.714211\pi$$
−0.623305 + 0.781979i $$0.714211\pi$$
$$740$$ 0 0
$$741$$ 12.9443 0.475520
$$742$$ 35.7771 1.31342
$$743$$ 37.8885 1.39000 0.694998 0.719012i $$-0.255405\pi$$
0.694998 + 0.719012i $$0.255405\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 5.41641 0.198309
$$747$$ 10.4721 0.383155
$$748$$ 0 0
$$749$$ 25.8885 0.945947
$$750$$ 0 0
$$751$$ 27.5279 1.00451 0.502253 0.864721i $$-0.332505\pi$$
0.502253 + 0.864721i $$0.332505\pi$$
$$752$$ −12.9443 −0.472029
$$753$$ −12.0000 −0.437304
$$754$$ 54.8328 1.99689
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ 18.5836 0.675432 0.337716 0.941248i $$-0.390346\pi$$
0.337716 + 0.941248i $$0.390346\pi$$
$$758$$ −36.8328 −1.33783
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 53.7771 1.94942 0.974709 0.223478i $$-0.0717411\pi$$
0.974709 + 0.223478i $$0.0717411\pi$$
$$762$$ −15.4164 −0.558478
$$763$$ −33.8885 −1.22685
$$764$$ 9.88854 0.357755
$$765$$ 0 0
$$766$$ 21.8885 0.790865
$$767$$ −38.8328 −1.40217
$$768$$ 1.00000 0.0360844
$$769$$ 17.0557 0.615045 0.307523 0.951541i $$-0.400500\pi$$
0.307523 + 0.951541i $$0.400500\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 20.0000 0.719816
$$773$$ −20.9443 −0.753313 −0.376657 0.926353i $$-0.622926\pi$$
−0.376657 + 0.926353i $$0.622926\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 13.4164 0.481621
$$777$$ −33.8885 −1.21574
$$778$$ −4.47214 −0.160334
$$779$$ −13.8885 −0.497609
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 6.47214 0.231443
$$783$$ 8.47214 0.302769
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 10.9443 0.390369
$$787$$ −54.8328 −1.95458 −0.977289 0.211909i $$-0.932032\pi$$
−0.977289 + 0.211909i $$0.932032\pi$$
$$788$$ 1.05573 0.0376088
$$789$$ 3.05573 0.108787
$$790$$ 0 0
$$791$$ −29.6656 −1.05479
$$792$$ 0 0
$$793$$ 54.8328 1.94717
$$794$$ −33.3050 −1.18195
$$795$$ 0 0
$$796$$ −9.41641 −0.333756
$$797$$ −2.11146 −0.0747916 −0.0373958 0.999301i $$-0.511906\pi$$
−0.0373958 + 0.999301i $$0.511906\pi$$
$$798$$ 8.00000 0.283197
$$799$$ 83.7771 2.96382
$$800$$ 0 0
$$801$$ −7.52786 −0.265984
$$802$$ −1.41641 −0.0500151
$$803$$ 0 0
$$804$$ 12.9443 0.456509
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −8.47214 −0.298233
$$808$$ −8.47214 −0.298049
$$809$$ 31.8885 1.12114 0.560571 0.828107i $$-0.310582\pi$$
0.560571 + 0.828107i $$0.310582\pi$$
$$810$$ 0 0
$$811$$ −14.1115 −0.495520 −0.247760 0.968821i $$-0.579694\pi$$
−0.247760 + 0.968821i $$0.579694\pi$$
$$812$$ 33.8885 1.18925
$$813$$ −16.9443 −0.594262
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −6.47214 −0.226570
$$817$$ 0 0
$$818$$ −18.0000 −0.629355
$$819$$ 25.8885 0.904619
$$820$$ 0 0
$$821$$ −7.52786 −0.262724 −0.131362 0.991334i $$-0.541935\pi$$
−0.131362 + 0.991334i $$0.541935\pi$$
$$822$$ 7.41641 0.258677
$$823$$ −24.3607 −0.849160 −0.424580 0.905390i $$-0.639578\pi$$
−0.424580 + 0.905390i $$0.639578\pi$$
$$824$$ −0.944272 −0.0328953
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ −33.5279 −1.16588 −0.582939 0.812516i $$-0.698097\pi$$
−0.582939 + 0.812516i $$0.698097\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ 13.0557 0.453444 0.226722 0.973959i $$-0.427199\pi$$
0.226722 + 0.973959i $$0.427199\pi$$
$$830$$ 0 0
$$831$$ −0.583592 −0.0202446
$$832$$ 6.47214 0.224381
$$833$$ −58.2492 −2.01822
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −2.11146 −0.0729390
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 42.7771 1.47507
$$842$$ −10.5836 −0.364735
$$843$$ 12.4721 0.429563
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −12.9443 −0.445033
$$847$$ −44.0000 −1.51186
$$848$$ 8.94427 0.307148
$$849$$ 24.0000 0.823678
$$850$$ 0 0
$$851$$ 8.47214 0.290421
$$852$$ −16.4721 −0.564326
$$853$$ 18.4721 0.632474 0.316237 0.948680i $$-0.397581\pi$$
0.316237 + 0.948680i $$0.397581\pi$$
$$854$$ 33.8885 1.15964
$$855$$ 0 0
$$856$$ 6.47214 0.221213
$$857$$ 11.8885 0.406105 0.203052 0.979168i $$-0.434914\pi$$
0.203052 + 0.979168i $$0.434914\pi$$
$$858$$ 0 0
$$859$$ 31.7771 1.08422 0.542110 0.840307i $$-0.317626\pi$$
0.542110 + 0.840307i $$0.317626\pi$$
$$860$$ 0 0
$$861$$ −27.7771 −0.946641
$$862$$ 0 0
$$863$$ 20.9443 0.712951 0.356476 0.934305i $$-0.383978\pi$$
0.356476 + 0.934305i $$0.383978\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 5.41641 0.184057
$$867$$ 24.8885 0.845259
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 83.7771 2.83868
$$872$$ −8.47214 −0.286903
$$873$$ 13.4164 0.454077
$$874$$ −2.00000 −0.0676510
$$875$$ 0 0
$$876$$ −12.9443 −0.437346
$$877$$ −24.3607 −0.822602 −0.411301 0.911500i $$-0.634925\pi$$
−0.411301 + 0.911500i $$0.634925\pi$$
$$878$$ 26.8328 0.905564
$$879$$ 16.9443 0.571516
$$880$$ 0 0
$$881$$ 4.47214 0.150670 0.0753350 0.997158i $$-0.475997\pi$$
0.0753350 + 0.997158i $$0.475997\pi$$
$$882$$ 9.00000 0.303046
$$883$$ 51.7771 1.74244 0.871219 0.490895i $$-0.163330\pi$$
0.871219 + 0.490895i $$0.163330\pi$$
$$884$$ −41.8885 −1.40886
$$885$$ 0 0
$$886$$ 8.94427 0.300489
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −8.47214 −0.284306
$$889$$ −61.6656 −2.06820
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −15.4164 −0.516180
$$893$$ −25.8885 −0.866327
$$894$$ 13.4164 0.448712
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ −6.47214 −0.216098
$$898$$ −17.0557 −0.569157
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −57.8885 −1.92855
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −7.41641 −0.246666
$$905$$ 0 0
$$906$$ −0.944272 −0.0313713
$$907$$ −38.8328 −1.28942 −0.644711 0.764426i $$-0.723022\pi$$
−0.644711 + 0.764426i $$0.723022\pi$$
$$908$$ 9.52786 0.316193
$$909$$ −8.47214 −0.281003
$$910$$ 0 0
$$911$$ 7.05573 0.233767 0.116883 0.993146i $$-0.462710\pi$$
0.116883 + 0.993146i $$0.462710\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 0 0
$$914$$ −1.41641 −0.0468506
$$915$$ 0 0
$$916$$ −1.41641 −0.0467994
$$917$$ 43.7771 1.44565
$$918$$ −6.47214 −0.213612
$$919$$ 17.4164 0.574514 0.287257 0.957854i $$-0.407257\pi$$
0.287257 + 0.957854i $$0.407257\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 30.3607 0.999876
$$923$$ −106.610 −3.50911
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 11.4164 0.375166
$$927$$ −0.944272 −0.0310140
$$928$$ 8.47214 0.278111
$$929$$ 6.94427 0.227834 0.113917 0.993490i $$-0.463660\pi$$
0.113917 + 0.993490i $$0.463660\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ −2.94427 −0.0964428
$$933$$ −15.5279 −0.508359
$$934$$ 17.5279 0.573529
$$935$$ 0 0
$$936$$ 6.47214 0.211548
$$937$$ 33.4164 1.09167 0.545833 0.837894i $$-0.316213\pi$$
0.545833 + 0.837894i $$0.316213\pi$$
$$938$$ 51.7771 1.69058
$$939$$ 1.41641 0.0462227
$$940$$ 0 0
$$941$$ 27.5279 0.897383 0.448691 0.893687i $$-0.351890\pi$$
0.448691 + 0.893687i $$0.351890\pi$$
$$942$$ −20.4721 −0.667018
$$943$$ 6.94427 0.226137
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 3.52786 0.114580
$$949$$ −83.7771 −2.71952
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ −25.8885 −0.839053
$$953$$ 32.3607 1.04827 0.524133 0.851637i $$-0.324390\pi$$
0.524133 + 0.851637i $$0.324390\pi$$
$$954$$ 8.94427 0.289581
$$955$$ 0 0
$$956$$ −3.52786 −0.114099
$$957$$ 0 0
$$958$$ 0.944272 0.0305080
$$959$$ 29.6656 0.957953
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −54.8328 −1.76788
$$963$$ 6.47214 0.208562
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ −4.00000 −0.128698
$$967$$ 28.5836 0.919186 0.459593 0.888130i $$-0.347995\pi$$
0.459593 + 0.888130i $$0.347995\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ −12.9443 −0.415830
$$970$$ 0 0
$$971$$ −53.8885 −1.72937 −0.864683 0.502318i $$-0.832481\pi$$
−0.864683 + 0.502318i $$0.832481\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −4.58359 −0.146868
$$975$$ 0 0
$$976$$ 8.47214 0.271186
$$977$$ 10.4721 0.335033 0.167517 0.985869i $$-0.446425\pi$$
0.167517 + 0.985869i $$0.446425\pi$$
$$978$$ 12.9443 0.413912
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −8.47214 −0.270494
$$982$$ −7.88854 −0.251734
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ −6.94427 −0.221375
$$985$$ 0 0
$$986$$ −54.8328 −1.74623
$$987$$ −51.7771 −1.64808
$$988$$ 12.9443 0.411812
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −53.6656 −1.70474 −0.852372 0.522935i $$-0.824837\pi$$
−0.852372 + 0.522935i $$0.824837\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −65.8885 −2.08986
$$995$$ 0 0
$$996$$ 10.4721 0.331822
$$997$$ 19.4164 0.614924 0.307462 0.951560i $$-0.400520\pi$$
0.307462 + 0.951560i $$0.400520\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ −8.47214 −0.268047
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.bn.1.2 2
5.2 odd 4 690.2.d.a.139.3 yes 4
5.3 odd 4 690.2.d.a.139.1 4
5.4 even 2 3450.2.a.bc.1.1 2
15.2 even 4 2070.2.d.b.829.2 4
15.8 even 4 2070.2.d.b.829.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.a.139.1 4 5.3 odd 4
690.2.d.a.139.3 yes 4 5.2 odd 4
2070.2.d.b.829.2 4 15.2 even 4
2070.2.d.b.829.4 4 15.8 even 4
3450.2.a.bc.1.1 2 5.4 even 2
3450.2.a.bn.1.2 2 1.1 even 1 trivial