# Properties

 Label 1840.2.a.r.1.2 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.43163$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.43163 q^{3} -1.00000 q^{5} -3.08719 q^{7} -0.950444 q^{9} +O(q^{10})$$ $$q-1.43163 q^{3} -1.00000 q^{5} -3.08719 q^{7} -0.950444 q^{9} +6.46926 q^{11} +3.95044 q^{13} +1.43163 q^{15} -3.43163 q^{17} -3.08719 q^{19} +4.41970 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.65556 q^{27} +0.863254 q^{29} +5.95044 q^{31} -9.26157 q^{33} +3.08719 q^{35} -7.03763 q^{37} -5.65556 q^{39} +5.60601 q^{41} -8.00000 q^{43} +0.950444 q^{45} -3.90089 q^{47} +2.53074 q^{49} +4.91281 q^{51} -6.00000 q^{53} -6.46926 q^{55} +4.41970 q^{57} -6.86325 q^{59} -13.5069 q^{61} +2.93420 q^{63} -3.95044 q^{65} +10.0753 q^{67} -1.43163 q^{69} -2.56837 q^{71} +5.90089 q^{73} -1.43163 q^{75} -19.9718 q^{77} -15.8018 q^{79} -5.24533 q^{81} -9.03763 q^{83} +3.43163 q^{85} -1.23586 q^{87} +16.7641 q^{89} -12.1958 q^{91} -8.51882 q^{93} +3.08719 q^{95} -14.2949 q^{97} -6.14867 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{3} - 3q^{5} - 3q^{7} + 10q^{9} + O(q^{10})$$ $$3q - q^{3} - 3q^{5} - 3q^{7} + 10q^{9} - 3q^{11} - q^{13} + q^{15} - 7q^{17} - 3q^{19} - 22q^{21} + 3q^{23} + 3q^{25} + 14q^{27} - 4q^{29} + 5q^{31} - 9q^{33} + 3q^{35} - 2q^{37} - 14q^{39} + q^{41} - 24q^{43} - 10q^{45} + 14q^{47} + 30q^{49} + 21q^{51} - 18q^{53} + 3q^{55} - 22q^{57} - 14q^{59} + q^{61} - 8q^{63} + q^{65} - 8q^{67} - q^{69} - 11q^{71} - 8q^{73} - q^{75} - 24q^{77} + 4q^{79} + 7q^{81} - 8q^{83} + 7q^{85} - 36q^{87} + 18q^{89} - q^{91} - 16q^{93} + 3q^{95} - 33q^{97} - 57q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.43163 −0.826550 −0.413275 0.910606i $$-0.635615\pi$$
−0.413275 + 0.910606i $$0.635615\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.08719 −1.16685 −0.583424 0.812168i $$-0.698287\pi$$
−0.583424 + 0.812168i $$0.698287\pi$$
$$8$$ 0 0
$$9$$ −0.950444 −0.316815
$$10$$ 0 0
$$11$$ 6.46926 1.95056 0.975278 0.220983i $$-0.0709265\pi$$
0.975278 + 0.220983i $$0.0709265\pi$$
$$12$$ 0 0
$$13$$ 3.95044 1.09566 0.547828 0.836591i $$-0.315455\pi$$
0.547828 + 0.836591i $$0.315455\pi$$
$$14$$ 0 0
$$15$$ 1.43163 0.369645
$$16$$ 0 0
$$17$$ −3.43163 −0.832292 −0.416146 0.909298i $$-0.636619\pi$$
−0.416146 + 0.909298i $$0.636619\pi$$
$$18$$ 0 0
$$19$$ −3.08719 −0.708250 −0.354125 0.935198i $$-0.615221\pi$$
−0.354125 + 0.935198i $$0.615221\pi$$
$$20$$ 0 0
$$21$$ 4.41970 0.964459
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.65556 1.08841
$$28$$ 0 0
$$29$$ 0.863254 0.160302 0.0801511 0.996783i $$-0.474460\pi$$
0.0801511 + 0.996783i $$0.474460\pi$$
$$30$$ 0 0
$$31$$ 5.95044 1.06873 0.534366 0.845253i $$-0.320551\pi$$
0.534366 + 0.845253i $$0.320551\pi$$
$$32$$ 0 0
$$33$$ −9.26157 −1.61223
$$34$$ 0 0
$$35$$ 3.08719 0.521830
$$36$$ 0 0
$$37$$ −7.03763 −1.15698 −0.578490 0.815690i $$-0.696358\pi$$
−0.578490 + 0.815690i $$0.696358\pi$$
$$38$$ 0 0
$$39$$ −5.65556 −0.905615
$$40$$ 0 0
$$41$$ 5.60601 0.875511 0.437756 0.899094i $$-0.355774\pi$$
0.437756 + 0.899094i $$0.355774\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0.950444 0.141684
$$46$$ 0 0
$$47$$ −3.90089 −0.569003 −0.284501 0.958676i $$-0.591828\pi$$
−0.284501 + 0.958676i $$0.591828\pi$$
$$48$$ 0 0
$$49$$ 2.53074 0.361534
$$50$$ 0 0
$$51$$ 4.91281 0.687931
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ −6.46926 −0.872315
$$56$$ 0 0
$$57$$ 4.41970 0.585404
$$58$$ 0 0
$$59$$ −6.86325 −0.893520 −0.446760 0.894654i $$-0.647422\pi$$
−0.446760 + 0.894654i $$0.647422\pi$$
$$60$$ 0 0
$$61$$ −13.5069 −1.72938 −0.864690 0.502305i $$-0.832485\pi$$
−0.864690 + 0.502305i $$0.832485\pi$$
$$62$$ 0 0
$$63$$ 2.93420 0.369674
$$64$$ 0 0
$$65$$ −3.95044 −0.489992
$$66$$ 0 0
$$67$$ 10.0753 1.23089 0.615445 0.788180i $$-0.288976\pi$$
0.615445 + 0.788180i $$0.288976\pi$$
$$68$$ 0 0
$$69$$ −1.43163 −0.172348
$$70$$ 0 0
$$71$$ −2.56837 −0.304810 −0.152405 0.988318i $$-0.548702\pi$$
−0.152405 + 0.988318i $$0.548702\pi$$
$$72$$ 0 0
$$73$$ 5.90089 0.690647 0.345323 0.938484i $$-0.387769\pi$$
0.345323 + 0.938484i $$0.387769\pi$$
$$74$$ 0 0
$$75$$ −1.43163 −0.165310
$$76$$ 0 0
$$77$$ −19.9718 −2.27600
$$78$$ 0 0
$$79$$ −15.8018 −1.77784 −0.888919 0.458064i $$-0.848543\pi$$
−0.888919 + 0.458064i $$0.848543\pi$$
$$80$$ 0 0
$$81$$ −5.24533 −0.582814
$$82$$ 0 0
$$83$$ −9.03763 −0.992009 −0.496005 0.868320i $$-0.665200\pi$$
−0.496005 + 0.868320i $$0.665200\pi$$
$$84$$ 0 0
$$85$$ 3.43163 0.372212
$$86$$ 0 0
$$87$$ −1.23586 −0.132498
$$88$$ 0 0
$$89$$ 16.7641 1.77700 0.888498 0.458881i $$-0.151750\pi$$
0.888498 + 0.458881i $$0.151750\pi$$
$$90$$ 0 0
$$91$$ −12.1958 −1.27846
$$92$$ 0 0
$$93$$ −8.51882 −0.883360
$$94$$ 0 0
$$95$$ 3.08719 0.316739
$$96$$ 0 0
$$97$$ −14.2949 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$98$$ 0 0
$$99$$ −6.14867 −0.617964
$$100$$ 0 0
$$101$$ 0.863254 0.0858970 0.0429485 0.999077i $$-0.486325\pi$$
0.0429485 + 0.999077i $$0.486325\pi$$
$$102$$ 0 0
$$103$$ −1.53074 −0.150828 −0.0754141 0.997152i $$-0.524028\pi$$
−0.0754141 + 0.997152i $$0.524028\pi$$
$$104$$ 0 0
$$105$$ −4.41970 −0.431319
$$106$$ 0 0
$$107$$ −17.6274 −1.70410 −0.852052 0.523457i $$-0.824642\pi$$
−0.852052 + 0.523457i $$0.824642\pi$$
$$108$$ 0 0
$$109$$ −2.91281 −0.278997 −0.139498 0.990222i $$-0.544549\pi$$
−0.139498 + 0.990222i $$0.544549\pi$$
$$110$$ 0 0
$$111$$ 10.0753 0.956302
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −3.75467 −0.347120
$$118$$ 0 0
$$119$$ 10.5941 0.971158
$$120$$ 0 0
$$121$$ 30.8513 2.80467
$$122$$ 0 0
$$123$$ −8.02571 −0.723654
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −20.9385 −1.85799 −0.928997 0.370088i $$-0.879327\pi$$
−0.928997 + 0.370088i $$0.879327\pi$$
$$128$$ 0 0
$$129$$ 11.4530 1.00838
$$130$$ 0 0
$$131$$ −10.7641 −0.940467 −0.470234 0.882542i $$-0.655830\pi$$
−0.470234 + 0.882542i $$0.655830\pi$$
$$132$$ 0 0
$$133$$ 9.53074 0.826420
$$134$$ 0 0
$$135$$ −5.65556 −0.486753
$$136$$ 0 0
$$137$$ 3.26157 0.278655 0.139327 0.990246i $$-0.455506\pi$$
0.139327 + 0.990246i $$0.455506\pi$$
$$138$$ 0 0
$$139$$ 16.9385 1.43671 0.718353 0.695678i $$-0.244896\pi$$
0.718353 + 0.695678i $$0.244896\pi$$
$$140$$ 0 0
$$141$$ 5.58462 0.470310
$$142$$ 0 0
$$143$$ 25.5565 2.13714
$$144$$ 0 0
$$145$$ −0.863254 −0.0716894
$$146$$ 0 0
$$147$$ −3.62308 −0.298826
$$148$$ 0 0
$$149$$ 3.26157 0.267198 0.133599 0.991035i $$-0.457347\pi$$
0.133599 + 0.991035i $$0.457347\pi$$
$$150$$ 0 0
$$151$$ −0.294881 −0.0239971 −0.0119986 0.999928i $$-0.503819\pi$$
−0.0119986 + 0.999928i $$0.503819\pi$$
$$152$$ 0 0
$$153$$ 3.26157 0.263682
$$154$$ 0 0
$$155$$ −5.95044 −0.477951
$$156$$ 0 0
$$157$$ 7.13675 0.569574 0.284787 0.958591i $$-0.408077\pi$$
0.284787 + 0.958591i $$0.408077\pi$$
$$158$$ 0 0
$$159$$ 8.58976 0.681212
$$160$$ 0 0
$$161$$ −3.08719 −0.243305
$$162$$ 0 0
$$163$$ 8.12482 0.636385 0.318193 0.948026i $$-0.396924\pi$$
0.318193 + 0.948026i $$0.396924\pi$$
$$164$$ 0 0
$$165$$ 9.26157 0.721012
$$166$$ 0 0
$$167$$ 7.80178 0.603719 0.301860 0.953352i $$-0.402393\pi$$
0.301860 + 0.953352i $$0.402393\pi$$
$$168$$ 0 0
$$169$$ 2.60601 0.200462
$$170$$ 0 0
$$171$$ 2.93420 0.224384
$$172$$ 0 0
$$173$$ −19.3325 −1.46982 −0.734912 0.678163i $$-0.762777\pi$$
−0.734912 + 0.678163i $$0.762777\pi$$
$$174$$ 0 0
$$175$$ −3.08719 −0.233370
$$176$$ 0 0
$$177$$ 9.82562 0.738539
$$178$$ 0 0
$$179$$ −2.17438 −0.162521 −0.0812604 0.996693i $$-0.525895\pi$$
−0.0812604 + 0.996693i $$0.525895\pi$$
$$180$$ 0 0
$$181$$ −7.26157 −0.539748 −0.269874 0.962896i $$-0.586982\pi$$
−0.269874 + 0.962896i $$0.586982\pi$$
$$182$$ 0 0
$$183$$ 19.3368 1.42942
$$184$$ 0 0
$$185$$ 7.03763 0.517417
$$186$$ 0 0
$$187$$ −22.2001 −1.62343
$$188$$ 0 0
$$189$$ −17.4598 −1.27001
$$190$$ 0 0
$$191$$ −8.58976 −0.621533 −0.310767 0.950486i $$-0.600586\pi$$
−0.310767 + 0.950486i $$0.600586\pi$$
$$192$$ 0 0
$$193$$ 2.44787 0.176202 0.0881008 0.996112i $$-0.471920\pi$$
0.0881008 + 0.996112i $$0.471920\pi$$
$$194$$ 0 0
$$195$$ 5.65556 0.405003
$$196$$ 0 0
$$197$$ −10.7428 −0.765389 −0.382695 0.923875i $$-0.625004\pi$$
−0.382695 + 0.923875i $$0.625004\pi$$
$$198$$ 0 0
$$199$$ 11.3111 0.801824 0.400912 0.916116i $$-0.368693\pi$$
0.400912 + 0.916116i $$0.368693\pi$$
$$200$$ 0 0
$$201$$ −14.4240 −1.01739
$$202$$ 0 0
$$203$$ −2.66503 −0.187048
$$204$$ 0 0
$$205$$ −5.60601 −0.391540
$$206$$ 0 0
$$207$$ −0.950444 −0.0660604
$$208$$ 0 0
$$209$$ −19.9718 −1.38148
$$210$$ 0 0
$$211$$ −23.1129 −1.59116 −0.795579 0.605850i $$-0.792833\pi$$
−0.795579 + 0.605850i $$0.792833\pi$$
$$212$$ 0 0
$$213$$ 3.67695 0.251941
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −18.3701 −1.24705
$$218$$ 0 0
$$219$$ −8.44787 −0.570854
$$220$$ 0 0
$$221$$ −13.5565 −0.911906
$$222$$ 0 0
$$223$$ 5.72651 0.383475 0.191738 0.981446i $$-0.438588\pi$$
0.191738 + 0.981446i $$0.438588\pi$$
$$224$$ 0 0
$$225$$ −0.950444 −0.0633629
$$226$$ 0 0
$$227$$ 17.6274 1.16997 0.584986 0.811044i $$-0.301100\pi$$
0.584986 + 0.811044i $$0.301100\pi$$
$$228$$ 0 0
$$229$$ −16.0753 −1.06228 −0.531142 0.847283i $$-0.678237\pi$$
−0.531142 + 0.847283i $$0.678237\pi$$
$$230$$ 0 0
$$231$$ 28.5922 1.88123
$$232$$ 0 0
$$233$$ −6.44787 −0.422414 −0.211207 0.977441i $$-0.567739\pi$$
−0.211207 + 0.977441i $$0.567739\pi$$
$$234$$ 0 0
$$235$$ 3.90089 0.254466
$$236$$ 0 0
$$237$$ 22.6222 1.46947
$$238$$ 0 0
$$239$$ −4.34876 −0.281298 −0.140649 0.990060i $$-0.544919\pi$$
−0.140649 + 0.990060i $$0.544919\pi$$
$$240$$ 0 0
$$241$$ 0.764142 0.0492227 0.0246114 0.999697i $$-0.492165\pi$$
0.0246114 + 0.999697i $$0.492165\pi$$
$$242$$ 0 0
$$243$$ −9.45734 −0.606688
$$244$$ 0 0
$$245$$ −2.53074 −0.161683
$$246$$ 0 0
$$247$$ −12.1958 −0.775998
$$248$$ 0 0
$$249$$ 12.9385 0.819945
$$250$$ 0 0
$$251$$ 22.5402 1.42273 0.711363 0.702825i $$-0.248078\pi$$
0.711363 + 0.702825i $$0.248078\pi$$
$$252$$ 0 0
$$253$$ 6.46926 0.406719
$$254$$ 0 0
$$255$$ −4.91281 −0.307652
$$256$$ 0 0
$$257$$ −9.90089 −0.617600 −0.308800 0.951127i $$-0.599927\pi$$
−0.308800 + 0.951127i $$0.599927\pi$$
$$258$$ 0 0
$$259$$ 21.7265 1.35002
$$260$$ 0 0
$$261$$ −0.820475 −0.0507861
$$262$$ 0 0
$$263$$ 7.25725 0.447501 0.223751 0.974646i $$-0.428170\pi$$
0.223751 + 0.974646i $$0.428170\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ −24.0000 −1.46878
$$268$$ 0 0
$$269$$ −6.93852 −0.423049 −0.211525 0.977373i $$-0.567843\pi$$
−0.211525 + 0.977373i $$0.567843\pi$$
$$270$$ 0 0
$$271$$ 10.2992 0.625632 0.312816 0.949814i $$-0.398728\pi$$
0.312816 + 0.949814i $$0.398728\pi$$
$$272$$ 0 0
$$273$$ 17.4598 1.05671
$$274$$ 0 0
$$275$$ 6.46926 0.390111
$$276$$ 0 0
$$277$$ −17.8018 −1.06961 −0.534803 0.844977i $$-0.679614\pi$$
−0.534803 + 0.844977i $$0.679614\pi$$
$$278$$ 0 0
$$279$$ −5.65556 −0.338590
$$280$$ 0 0
$$281$$ −18.9385 −1.12978 −0.564889 0.825167i $$-0.691081\pi$$
−0.564889 + 0.825167i $$0.691081\pi$$
$$282$$ 0 0
$$283$$ −12.6889 −0.754275 −0.377138 0.926157i $$-0.623092\pi$$
−0.377138 + 0.926157i $$0.623092\pi$$
$$284$$ 0 0
$$285$$ −4.41970 −0.261801
$$286$$ 0 0
$$287$$ −17.3068 −1.02159
$$288$$ 0 0
$$289$$ −5.22394 −0.307290
$$290$$ 0 0
$$291$$ 20.4649 1.19968
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 6.86325 0.399594
$$296$$ 0 0
$$297$$ 36.5873 2.12301
$$298$$ 0 0
$$299$$ 3.95044 0.228460
$$300$$ 0 0
$$301$$ 24.6975 1.42354
$$302$$ 0 0
$$303$$ −1.23586 −0.0709982
$$304$$ 0 0
$$305$$ 13.5069 0.773402
$$306$$ 0 0
$$307$$ 22.9599 1.31039 0.655196 0.755459i $$-0.272586\pi$$
0.655196 + 0.755459i $$0.272586\pi$$
$$308$$ 0 0
$$309$$ 2.19145 0.124667
$$310$$ 0 0
$$311$$ −18.0753 −1.02495 −0.512477 0.858701i $$-0.671272\pi$$
−0.512477 + 0.858701i $$0.671272\pi$$
$$312$$ 0 0
$$313$$ 15.1625 0.857033 0.428516 0.903534i $$-0.359036\pi$$
0.428516 + 0.903534i $$0.359036\pi$$
$$314$$ 0 0
$$315$$ −2.93420 −0.165323
$$316$$ 0 0
$$317$$ 14.0257 0.787762 0.393881 0.919161i $$-0.371132\pi$$
0.393881 + 0.919161i $$0.371132\pi$$
$$318$$ 0 0
$$319$$ 5.58462 0.312678
$$320$$ 0 0
$$321$$ 25.2359 1.40853
$$322$$ 0 0
$$323$$ 10.5941 0.589471
$$324$$ 0 0
$$325$$ 3.95044 0.219131
$$326$$ 0 0
$$327$$ 4.17006 0.230605
$$328$$ 0 0
$$329$$ 12.0428 0.663940
$$330$$ 0 0
$$331$$ 24.7403 1.35985 0.679925 0.733282i $$-0.262012\pi$$
0.679925 + 0.733282i $$0.262012\pi$$
$$332$$ 0 0
$$333$$ 6.68888 0.366548
$$334$$ 0 0
$$335$$ −10.0753 −0.550471
$$336$$ 0 0
$$337$$ 14.7146 0.801555 0.400777 0.916176i $$-0.368740\pi$$
0.400777 + 0.916176i $$0.368740\pi$$
$$338$$ 0 0
$$339$$ 8.58976 0.466532
$$340$$ 0 0
$$341$$ 38.4950 2.08462
$$342$$ 0 0
$$343$$ 13.7975 0.744993
$$344$$ 0 0
$$345$$ 1.43163 0.0770762
$$346$$ 0 0
$$347$$ −9.43163 −0.506316 −0.253158 0.967425i $$-0.581469\pi$$
−0.253158 + 0.967425i $$0.581469\pi$$
$$348$$ 0 0
$$349$$ 18.3488 0.982187 0.491093 0.871107i $$-0.336597\pi$$
0.491093 + 0.871107i $$0.336597\pi$$
$$350$$ 0 0
$$351$$ 22.3420 1.19253
$$352$$ 0 0
$$353$$ −33.1129 −1.76242 −0.881211 0.472723i $$-0.843271\pi$$
−0.881211 + 0.472723i $$0.843271\pi$$
$$354$$ 0 0
$$355$$ 2.56837 0.136315
$$356$$ 0 0
$$357$$ −15.1668 −0.802711
$$358$$ 0 0
$$359$$ −33.0376 −1.74366 −0.871830 0.489809i $$-0.837067\pi$$
−0.871830 + 0.489809i $$0.837067\pi$$
$$360$$ 0 0
$$361$$ −9.46926 −0.498382
$$362$$ 0 0
$$363$$ −44.1676 −2.31820
$$364$$ 0 0
$$365$$ −5.90089 −0.308867
$$366$$ 0 0
$$367$$ 2.27349 0.118675 0.0593376 0.998238i $$-0.481101\pi$$
0.0593376 + 0.998238i $$0.481101\pi$$
$$368$$ 0 0
$$369$$ −5.32819 −0.277375
$$370$$ 0 0
$$371$$ 18.5231 0.961673
$$372$$ 0 0
$$373$$ 23.9762 1.24144 0.620719 0.784033i $$-0.286841\pi$$
0.620719 + 0.784033i $$0.286841\pi$$
$$374$$ 0 0
$$375$$ 1.43163 0.0739289
$$376$$ 0 0
$$377$$ 3.41024 0.175636
$$378$$ 0 0
$$379$$ 13.8795 0.712942 0.356471 0.934306i $$-0.383980\pi$$
0.356471 + 0.934306i $$0.383980\pi$$
$$380$$ 0 0
$$381$$ 29.9762 1.53572
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 19.9718 1.01786
$$386$$ 0 0
$$387$$ 7.60355 0.386510
$$388$$ 0 0
$$389$$ −26.6436 −1.35089 −0.675443 0.737412i $$-0.736048\pi$$
−0.675443 + 0.737412i $$0.736048\pi$$
$$390$$ 0 0
$$391$$ −3.43163 −0.173545
$$392$$ 0 0
$$393$$ 15.4102 0.777344
$$394$$ 0 0
$$395$$ 15.8018 0.795074
$$396$$ 0 0
$$397$$ −8.66749 −0.435009 −0.217504 0.976059i $$-0.569792\pi$$
−0.217504 + 0.976059i $$0.569792\pi$$
$$398$$ 0 0
$$399$$ −13.6445 −0.683078
$$400$$ 0 0
$$401$$ 39.9762 1.99631 0.998157 0.0606854i $$-0.0193286\pi$$
0.998157 + 0.0606854i $$0.0193286\pi$$
$$402$$ 0 0
$$403$$ 23.5069 1.17096
$$404$$ 0 0
$$405$$ 5.24533 0.260642
$$406$$ 0 0
$$407$$ −45.5283 −2.25675
$$408$$ 0 0
$$409$$ 30.1248 1.48958 0.744788 0.667301i $$-0.232550\pi$$
0.744788 + 0.667301i $$0.232550\pi$$
$$410$$ 0 0
$$411$$ −4.66935 −0.230322
$$412$$ 0 0
$$413$$ 21.1882 1.04260
$$414$$ 0 0
$$415$$ 9.03763 0.443640
$$416$$ 0 0
$$417$$ −24.2496 −1.18751
$$418$$ 0 0
$$419$$ −25.8770 −1.26418 −0.632088 0.774897i $$-0.717802\pi$$
−0.632088 + 0.774897i $$0.717802\pi$$
$$420$$ 0 0
$$421$$ −10.7146 −0.522197 −0.261098 0.965312i $$-0.584085\pi$$
−0.261098 + 0.965312i $$0.584085\pi$$
$$422$$ 0 0
$$423$$ 3.70757 0.180268
$$424$$ 0 0
$$425$$ −3.43163 −0.166458
$$426$$ 0 0
$$427$$ 41.6983 2.01792
$$428$$ 0 0
$$429$$ −36.5873 −1.76645
$$430$$ 0 0
$$431$$ −32.2496 −1.55341 −0.776705 0.629864i $$-0.783111\pi$$
−0.776705 + 0.629864i $$0.783111\pi$$
$$432$$ 0 0
$$433$$ 20.6393 0.991862 0.495931 0.868362i $$-0.334827\pi$$
0.495931 + 0.868362i $$0.334827\pi$$
$$434$$ 0 0
$$435$$ 1.23586 0.0592549
$$436$$ 0 0
$$437$$ −3.08719 −0.147680
$$438$$ 0 0
$$439$$ 16.2239 0.774326 0.387163 0.922011i $$-0.373455\pi$$
0.387163 + 0.922011i $$0.373455\pi$$
$$440$$ 0 0
$$441$$ −2.40533 −0.114539
$$442$$ 0 0
$$443$$ 15.6770 0.744834 0.372417 0.928065i $$-0.378529\pi$$
0.372417 + 0.928065i $$0.378529\pi$$
$$444$$ 0 0
$$445$$ −16.7641 −0.794697
$$446$$ 0 0
$$447$$ −4.66935 −0.220853
$$448$$ 0 0
$$449$$ 22.5727 1.06527 0.532636 0.846345i $$-0.321202\pi$$
0.532636 + 0.846345i $$0.321202\pi$$
$$450$$ 0 0
$$451$$ 36.2667 1.70773
$$452$$ 0 0
$$453$$ 0.422160 0.0198348
$$454$$ 0 0
$$455$$ 12.1958 0.571746
$$456$$ 0 0
$$457$$ −1.45302 −0.0679693 −0.0339846 0.999422i $$-0.510820\pi$$
−0.0339846 + 0.999422i $$0.510820\pi$$
$$458$$ 0 0
$$459$$ −19.4078 −0.905878
$$460$$ 0 0
$$461$$ −29.7027 −1.38339 −0.691695 0.722189i $$-0.743136\pi$$
−0.691695 + 0.722189i $$0.743136\pi$$
$$462$$ 0 0
$$463$$ −22.9624 −1.06715 −0.533576 0.845752i $$-0.679152\pi$$
−0.533576 + 0.845752i $$0.679152\pi$$
$$464$$ 0 0
$$465$$ 8.51882 0.395051
$$466$$ 0 0
$$467$$ −9.48550 −0.438937 −0.219468 0.975620i $$-0.570432\pi$$
−0.219468 + 0.975620i $$0.570432\pi$$
$$468$$ 0 0
$$469$$ −31.1043 −1.43626
$$470$$ 0 0
$$471$$ −10.2172 −0.470782
$$472$$ 0 0
$$473$$ −51.7541 −2.37966
$$474$$ 0 0
$$475$$ −3.08719 −0.141650
$$476$$ 0 0
$$477$$ 5.70266 0.261107
$$478$$ 0 0
$$479$$ 30.5659 1.39659 0.698296 0.715809i $$-0.253942\pi$$
0.698296 + 0.715809i $$0.253942\pi$$
$$480$$ 0 0
$$481$$ −27.8018 −1.26765
$$482$$ 0 0
$$483$$ 4.41970 0.201104
$$484$$ 0 0
$$485$$ 14.2949 0.649097
$$486$$ 0 0
$$487$$ 21.1367 0.957797 0.478899 0.877870i $$-0.341036\pi$$
0.478899 + 0.877870i $$0.341036\pi$$
$$488$$ 0 0
$$489$$ −11.6317 −0.526004
$$490$$ 0 0
$$491$$ 28.0514 1.26594 0.632971 0.774175i $$-0.281835\pi$$
0.632971 + 0.774175i $$0.281835\pi$$
$$492$$ 0 0
$$493$$ −2.96237 −0.133418
$$494$$ 0 0
$$495$$ 6.14867 0.276362
$$496$$ 0 0
$$497$$ 7.92905 0.355667
$$498$$ 0 0
$$499$$ −3.80178 −0.170191 −0.0850954 0.996373i $$-0.527120\pi$$
−0.0850954 + 0.996373i $$0.527120\pi$$
$$500$$ 0 0
$$501$$ −11.1692 −0.499005
$$502$$ 0 0
$$503$$ −19.0872 −0.851056 −0.425528 0.904945i $$-0.639912\pi$$
−0.425528 + 0.904945i $$0.639912\pi$$
$$504$$ 0 0
$$505$$ −0.863254 −0.0384143
$$506$$ 0 0
$$507$$ −3.73083 −0.165692
$$508$$ 0 0
$$509$$ −9.41024 −0.417101 −0.208551 0.978012i $$-0.566875\pi$$
−0.208551 + 0.978012i $$0.566875\pi$$
$$510$$ 0 0
$$511$$ −18.2172 −0.805880
$$512$$ 0 0
$$513$$ −17.4598 −0.770869
$$514$$ 0 0
$$515$$ 1.53074 0.0674524
$$516$$ 0 0
$$517$$ −25.2359 −1.10987
$$518$$ 0 0
$$519$$ 27.6770 1.21488
$$520$$ 0 0
$$521$$ 25.8018 1.13040 0.565198 0.824955i $$-0.308800\pi$$
0.565198 + 0.824955i $$0.308800\pi$$
$$522$$ 0 0
$$523$$ 19.1129 0.835749 0.417874 0.908505i $$-0.362775\pi$$
0.417874 + 0.908505i $$0.362775\pi$$
$$524$$ 0 0
$$525$$ 4.41970 0.192892
$$526$$ 0 0
$$527$$ −20.4197 −0.889496
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 6.52314 0.283080
$$532$$ 0 0
$$533$$ 22.1462 0.959259
$$534$$ 0 0
$$535$$ 17.6274 0.762099
$$536$$ 0 0
$$537$$ 3.11290 0.134332
$$538$$ 0 0
$$539$$ 16.3720 0.705193
$$540$$ 0 0
$$541$$ 30.8394 1.32589 0.662945 0.748668i $$-0.269306\pi$$
0.662945 + 0.748668i $$0.269306\pi$$
$$542$$ 0 0
$$543$$ 10.3959 0.446129
$$544$$ 0 0
$$545$$ 2.91281 0.124771
$$546$$ 0 0
$$547$$ −13.8513 −0.592240 −0.296120 0.955151i $$-0.595693\pi$$
−0.296120 + 0.955151i $$0.595693\pi$$
$$548$$ 0 0
$$549$$ 12.8375 0.547893
$$550$$ 0 0
$$551$$ −2.66503 −0.113534
$$552$$ 0 0
$$553$$ 48.7831 2.07447
$$554$$ 0 0
$$555$$ −10.0753 −0.427671
$$556$$ 0 0
$$557$$ −28.7641 −1.21878 −0.609388 0.792872i $$-0.708585\pi$$
−0.609388 + 0.792872i $$0.708585\pi$$
$$558$$ 0 0
$$559$$ −31.6036 −1.33669
$$560$$ 0 0
$$561$$ 31.7823 1.34185
$$562$$ 0 0
$$563$$ −36.1505 −1.52356 −0.761782 0.647834i $$-0.775675\pi$$
−0.761782 + 0.647834i $$0.775675\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ 0 0
$$567$$ 16.1933 0.680055
$$568$$ 0 0
$$569$$ −10.3488 −0.433843 −0.216921 0.976189i $$-0.569601\pi$$
−0.216921 + 0.976189i $$0.569601\pi$$
$$570$$ 0 0
$$571$$ −19.6060 −0.820486 −0.410243 0.911976i $$-0.634556\pi$$
−0.410243 + 0.911976i $$0.634556\pi$$
$$572$$ 0 0
$$573$$ 12.2973 0.513729
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −34.1505 −1.42171 −0.710853 0.703341i $$-0.751691\pi$$
−0.710853 + 0.703341i $$0.751691\pi$$
$$578$$ 0 0
$$579$$ −3.50444 −0.145639
$$580$$ 0 0
$$581$$ 27.9009 1.15752
$$582$$ 0 0
$$583$$ −38.8156 −1.60758
$$584$$ 0 0
$$585$$ 3.75467 0.155237
$$586$$ 0 0
$$587$$ 5.19062 0.214240 0.107120 0.994246i $$-0.465837\pi$$
0.107120 + 0.994246i $$0.465837\pi$$
$$588$$ 0 0
$$589$$ −18.3701 −0.756929
$$590$$ 0 0
$$591$$ 15.3796 0.632633
$$592$$ 0 0
$$593$$ −2.09911 −0.0862002 −0.0431001 0.999071i $$-0.513723\pi$$
−0.0431001 + 0.999071i $$0.513723\pi$$
$$594$$ 0 0
$$595$$ −10.5941 −0.434315
$$596$$ 0 0
$$597$$ −16.1933 −0.662748
$$598$$ 0 0
$$599$$ −11.1581 −0.455909 −0.227955 0.973672i $$-0.573204\pi$$
−0.227955 + 0.973672i $$0.573204\pi$$
$$600$$ 0 0
$$601$$ 9.57784 0.390688 0.195344 0.980735i $$-0.437418\pi$$
0.195344 + 0.980735i $$0.437418\pi$$
$$602$$ 0 0
$$603$$ −9.57597 −0.389964
$$604$$ 0 0
$$605$$ −30.8513 −1.25428
$$606$$ 0 0
$$607$$ 24.6975 1.00244 0.501221 0.865320i $$-0.332885\pi$$
0.501221 + 0.865320i $$0.332885\pi$$
$$608$$ 0 0
$$609$$ 3.81533 0.154605
$$610$$ 0 0
$$611$$ −15.4102 −0.623431
$$612$$ 0 0
$$613$$ −26.3488 −1.06422 −0.532108 0.846676i $$-0.678600\pi$$
−0.532108 + 0.846676i $$0.678600\pi$$
$$614$$ 0 0
$$615$$ 8.02571 0.323628
$$616$$ 0 0
$$617$$ 5.94612 0.239382 0.119691 0.992811i $$-0.461810\pi$$
0.119691 + 0.992811i $$0.461810\pi$$
$$618$$ 0 0
$$619$$ −39.6856 −1.59510 −0.797549 0.603254i $$-0.793871\pi$$
−0.797549 + 0.603254i $$0.793871\pi$$
$$620$$ 0 0
$$621$$ 5.65556 0.226950
$$622$$ 0 0
$$623$$ −51.7541 −2.07348
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 28.5922 1.14186
$$628$$ 0 0
$$629$$ 24.1505 0.962945
$$630$$ 0 0
$$631$$ 23.0138 0.916164 0.458082 0.888910i $$-0.348537\pi$$
0.458082 + 0.888910i $$0.348537\pi$$
$$632$$ 0 0
$$633$$ 33.0891 1.31517
$$634$$ 0 0
$$635$$ 20.9385 0.830920
$$636$$ 0 0
$$637$$ 9.99754 0.396117
$$638$$ 0 0
$$639$$ 2.44109 0.0965682
$$640$$ 0 0
$$641$$ 25.4617 1.00568 0.502838 0.864381i $$-0.332289\pi$$
0.502838 + 0.864381i $$0.332289\pi$$
$$642$$ 0 0
$$643$$ 16.4479 0.648641 0.324320 0.945947i $$-0.394864\pi$$
0.324320 + 0.945947i $$0.394864\pi$$
$$644$$ 0 0
$$645$$ −11.4530 −0.450962
$$646$$ 0 0
$$647$$ 34.9147 1.37264 0.686319 0.727301i $$-0.259226\pi$$
0.686319 + 0.727301i $$0.259226\pi$$
$$648$$ 0 0
$$649$$ −44.4002 −1.74286
$$650$$ 0 0
$$651$$ 26.2992 1.03075
$$652$$ 0 0
$$653$$ −34.2754 −1.34130 −0.670649 0.741775i $$-0.733984\pi$$
−0.670649 + 0.741775i $$0.733984\pi$$
$$654$$ 0 0
$$655$$ 10.7641 0.420590
$$656$$ 0 0
$$657$$ −5.60846 −0.218807
$$658$$ 0 0
$$659$$ −35.2548 −1.37333 −0.686666 0.726973i $$-0.740926\pi$$
−0.686666 + 0.726973i $$0.740926\pi$$
$$660$$ 0 0
$$661$$ 28.5684 1.11118 0.555590 0.831456i $$-0.312492\pi$$
0.555590 + 0.831456i $$0.312492\pi$$
$$662$$ 0 0
$$663$$ 19.4078 0.753736
$$664$$ 0 0
$$665$$ −9.53074 −0.369586
$$666$$ 0 0
$$667$$ 0.863254 0.0334253
$$668$$ 0 0
$$669$$ −8.19822 −0.316962
$$670$$ 0 0
$$671$$ −87.3796 −3.37325
$$672$$ 0 0
$$673$$ −23.8770 −0.920392 −0.460196 0.887817i $$-0.652221\pi$$
−0.460196 + 0.887817i $$0.652221\pi$$
$$674$$ 0 0
$$675$$ 5.65556 0.217683
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 44.1310 1.69359
$$680$$ 0 0
$$681$$ −25.2359 −0.967040
$$682$$ 0 0
$$683$$ −34.6480 −1.32577 −0.662884 0.748722i $$-0.730668\pi$$
−0.662884 + 0.748722i $$0.730668\pi$$
$$684$$ 0 0
$$685$$ −3.26157 −0.124618
$$686$$ 0 0
$$687$$ 23.0138 0.878031
$$688$$ 0 0
$$689$$ −23.7027 −0.903000
$$690$$ 0 0
$$691$$ 18.1744 0.691386 0.345693 0.938348i $$-0.387644\pi$$
0.345693 + 0.938348i $$0.387644\pi$$
$$692$$ 0 0
$$693$$ 18.9821 0.721071
$$694$$ 0 0
$$695$$ −16.9385 −0.642515
$$696$$ 0 0
$$697$$ −19.2377 −0.728681
$$698$$ 0 0
$$699$$ 9.23095 0.349146
$$700$$ 0 0
$$701$$ −40.6907 −1.53687 −0.768434 0.639929i $$-0.778964\pi$$
−0.768434 + 0.639929i $$0.778964\pi$$
$$702$$ 0 0
$$703$$ 21.7265 0.819431
$$704$$ 0 0
$$705$$ −5.58462 −0.210329
$$706$$ 0 0
$$707$$ −2.66503 −0.100229
$$708$$ 0 0
$$709$$ −26.4454 −0.993178 −0.496589 0.867986i $$-0.665414\pi$$
−0.496589 + 0.867986i $$0.665414\pi$$
$$710$$ 0 0
$$711$$ 15.0187 0.563245
$$712$$ 0 0
$$713$$ 5.95044 0.222846
$$714$$ 0 0
$$715$$ −25.5565 −0.955757
$$716$$ 0 0
$$717$$ 6.22580 0.232507
$$718$$ 0 0
$$719$$ 2.24778 0.0838281 0.0419140 0.999121i $$-0.486654\pi$$
0.0419140 + 0.999121i $$0.486654\pi$$
$$720$$ 0 0
$$721$$ 4.72568 0.175994
$$722$$ 0 0
$$723$$ −1.09397 −0.0406850
$$724$$ 0 0
$$725$$ 0.863254 0.0320605
$$726$$ 0 0
$$727$$ −21.4830 −0.796762 −0.398381 0.917220i $$-0.630428\pi$$
−0.398381 + 0.917220i $$0.630428\pi$$
$$728$$ 0 0
$$729$$ 29.2754 1.08427
$$730$$ 0 0
$$731$$ 27.4530 1.01539
$$732$$ 0 0
$$733$$ 11.3778 0.420247 0.210123 0.977675i $$-0.432613\pi$$
0.210123 + 0.977675i $$0.432613\pi$$
$$734$$ 0 0
$$735$$ 3.62308 0.133639
$$736$$ 0 0
$$737$$ 65.1795 2.40092
$$738$$ 0 0
$$739$$ −21.8770 −0.804760 −0.402380 0.915473i $$-0.631817\pi$$
−0.402380 + 0.915473i $$0.631817\pi$$
$$740$$ 0 0
$$741$$ 17.4598 0.641402
$$742$$ 0 0
$$743$$ −5.36068 −0.196664 −0.0983322 0.995154i $$-0.531351\pi$$
−0.0983322 + 0.995154i $$0.531351\pi$$
$$744$$ 0 0
$$745$$ −3.26157 −0.119495
$$746$$ 0 0
$$747$$ 8.58976 0.314283
$$748$$ 0 0
$$749$$ 54.4191 1.98843
$$750$$ 0 0
$$751$$ −24.3915 −0.890060 −0.445030 0.895516i $$-0.646807\pi$$
−0.445030 + 0.895516i $$0.646807\pi$$
$$752$$ 0 0
$$753$$ −32.2692 −1.17595
$$754$$ 0 0
$$755$$ 0.294881 0.0107318
$$756$$ 0 0
$$757$$ 41.9437 1.52447 0.762234 0.647301i $$-0.224102\pi$$
0.762234 + 0.647301i $$0.224102\pi$$
$$758$$ 0 0
$$759$$ −9.26157 −0.336174
$$760$$ 0 0
$$761$$ −18.3745 −0.666074 −0.333037 0.942914i $$-0.608073\pi$$
−0.333037 + 0.942914i $$0.608073\pi$$
$$762$$ 0 0
$$763$$ 8.99240 0.325547
$$764$$ 0 0
$$765$$ −3.26157 −0.117922
$$766$$ 0 0
$$767$$ −27.1129 −0.978990
$$768$$ 0 0
$$769$$ 42.4993 1.53256 0.766282 0.642505i $$-0.222105\pi$$
0.766282 + 0.642505i $$0.222105\pi$$
$$770$$ 0 0
$$771$$ 14.1744 0.510478
$$772$$ 0 0
$$773$$ −15.0376 −0.540866 −0.270433 0.962739i $$-0.587167\pi$$
−0.270433 + 0.962739i $$0.587167\pi$$
$$774$$ 0 0
$$775$$ 5.95044 0.213746
$$776$$ 0 0
$$777$$ −31.1043 −1.11586
$$778$$ 0 0
$$779$$ −17.3068 −0.620081
$$780$$ 0 0
$$781$$ −16.6155 −0.594548
$$782$$ 0 0
$$783$$ 4.88219 0.174475
$$784$$ 0 0
$$785$$ −7.13675 −0.254721
$$786$$ 0 0
$$787$$ 8.39154 0.299126 0.149563 0.988752i $$-0.452213\pi$$
0.149563 + 0.988752i $$0.452213\pi$$
$$788$$ 0 0
$$789$$ −10.3897 −0.369882
$$790$$ 0 0
$$791$$ 18.5231 0.658607
$$792$$ 0 0
$$793$$ −53.3582 −1.89481
$$794$$ 0 0
$$795$$ −8.58976 −0.304647
$$796$$ 0 0
$$797$$ −22.0514 −0.781101 −0.390551 0.920581i $$-0.627715\pi$$
−0.390551 + 0.920581i $$0.627715\pi$$
$$798$$ 0 0
$$799$$ 13.3864 0.473576
$$800$$ 0 0
$$801$$ −15.9334 −0.562978
$$802$$ 0 0
$$803$$ 38.1744 1.34714
$$804$$ 0 0
$$805$$ 3.08719 0.108809
$$806$$ 0 0
$$807$$ 9.93337 0.349671
$$808$$ 0 0
$$809$$ 2.04524 0.0719066 0.0359533 0.999353i $$-0.488553\pi$$
0.0359533 + 0.999353i $$0.488553\pi$$
$$810$$ 0 0
$$811$$ −4.24965 −0.149225 −0.0746126 0.997213i $$-0.523772\pi$$
−0.0746126 + 0.997213i $$0.523772\pi$$
$$812$$ 0 0
$$813$$ −14.7446 −0.517116
$$814$$ 0 0
$$815$$ −8.12482 −0.284600
$$816$$ 0 0
$$817$$ 24.6975 0.864057
$$818$$ 0 0
$$819$$ 11.5914 0.405036
$$820$$ 0 0
$$821$$ −29.2548 −1.02100 −0.510500 0.859878i $$-0.670540\pi$$
−0.510500 + 0.859878i $$0.670540\pi$$
$$822$$ 0 0
$$823$$ −13.5846 −0.473530 −0.236765 0.971567i $$-0.576087\pi$$
−0.236765 + 0.971567i $$0.576087\pi$$
$$824$$ 0 0
$$825$$ −9.26157 −0.322446
$$826$$ 0 0
$$827$$ −24.7880 −0.861963 −0.430981 0.902361i $$-0.641833\pi$$
−0.430981 + 0.902361i $$0.641833\pi$$
$$828$$ 0 0
$$829$$ 26.1505 0.908246 0.454123 0.890939i $$-0.349953\pi$$
0.454123 + 0.890939i $$0.349953\pi$$
$$830$$ 0 0
$$831$$ 25.4855 0.884082
$$832$$ 0 0
$$833$$ −8.68455 −0.300902
$$834$$ 0 0
$$835$$ −7.80178 −0.269992
$$836$$ 0 0
$$837$$ 33.6531 1.16322
$$838$$ 0 0
$$839$$ −6.37260 −0.220007 −0.110003 0.993931i $$-0.535086\pi$$
−0.110003 + 0.993931i $$0.535086\pi$$
$$840$$ 0 0
$$841$$ −28.2548 −0.974303
$$842$$ 0 0
$$843$$ 27.1129 0.933818
$$844$$ 0 0
$$845$$ −2.60601 −0.0896493
$$846$$ 0 0
$$847$$ −95.2439 −3.27262
$$848$$ 0 0
$$849$$ 18.1657 0.623447
$$850$$ 0 0
$$851$$ −7.03763 −0.241247
$$852$$ 0 0
$$853$$ −33.6293 −1.15144 −0.575722 0.817646i $$-0.695279\pi$$
−0.575722 + 0.817646i $$0.695279\pi$$
$$854$$ 0 0
$$855$$ −2.93420 −0.100348
$$856$$ 0 0
$$857$$ 11.1795 0.381885 0.190943 0.981601i $$-0.438846\pi$$
0.190943 + 0.981601i $$0.438846\pi$$
$$858$$ 0 0
$$859$$ −47.9009 −1.63436 −0.817179 0.576385i $$-0.804463\pi$$
−0.817179 + 0.576385i $$0.804463\pi$$
$$860$$ 0 0
$$861$$ 24.7769 0.844394
$$862$$ 0 0
$$863$$ 30.1180 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$864$$ 0 0
$$865$$ 19.3325 0.657325
$$866$$ 0 0
$$867$$ 7.47873 0.253991
$$868$$ 0 0
$$869$$ −102.226 −3.46777
$$870$$ 0 0
$$871$$ 39.8018 1.34863
$$872$$ 0 0
$$873$$ 13.5865 0.459833
$$874$$ 0 0
$$875$$ 3.08719 0.104366
$$876$$ 0 0
$$877$$ −22.3940 −0.756191 −0.378096 0.925767i $$-0.623421\pi$$
−0.378096 + 0.925767i $$0.623421\pi$$
$$878$$ 0 0
$$879$$ 8.58976 0.289726
$$880$$ 0 0
$$881$$ 21.1129 0.711312 0.355656 0.934617i $$-0.384258\pi$$
0.355656 + 0.934617i $$0.384258\pi$$
$$882$$ 0 0
$$883$$ −49.1061 −1.65255 −0.826276 0.563265i $$-0.809545\pi$$
−0.826276 + 0.563265i $$0.809545\pi$$
$$884$$ 0 0
$$885$$ −9.82562 −0.330285
$$886$$ 0 0
$$887$$ 43.0566 1.44570 0.722849 0.691006i $$-0.242832\pi$$
0.722849 + 0.691006i $$0.242832\pi$$
$$888$$ 0 0
$$889$$ 64.6412 2.16800
$$890$$ 0 0
$$891$$ −33.9334 −1.13681
$$892$$ 0 0
$$893$$ 12.0428 0.402996
$$894$$ 0 0
$$895$$ 2.17438 0.0726815
$$896$$ 0 0
$$897$$ −5.65556 −0.188834
$$898$$ 0 0
$$899$$ 5.13675 0.171320
$$900$$ 0 0
$$901$$ 20.5898 0.685944
$$902$$ 0 0
$$903$$ −35.3576 −1.17663
$$904$$ 0 0
$$905$$ 7.26157 0.241383
$$906$$ 0 0
$$907$$ 37.9762 1.26098 0.630489 0.776198i $$-0.282854\pi$$
0.630489 + 0.776198i $$0.282854\pi$$
$$908$$ 0 0
$$909$$ −0.820475 −0.0272134
$$910$$ 0 0
$$911$$ 15.7504 0.521833 0.260916 0.965361i $$-0.415975\pi$$
0.260916 + 0.965361i $$0.415975\pi$$
$$912$$ 0 0
$$913$$ −58.4668 −1.93497
$$914$$ 0 0
$$915$$ −19.3368 −0.639256
$$916$$ 0 0
$$917$$ 33.2309 1.09738
$$918$$ 0 0
$$919$$ −30.4240 −1.00360 −0.501798 0.864985i $$-0.667328\pi$$
−0.501798 + 0.864985i $$0.667328\pi$$
$$920$$ 0 0
$$921$$ −32.8700 −1.08310
$$922$$ 0 0
$$923$$ −10.1462 −0.333967
$$924$$ 0 0
$$925$$ −7.03763 −0.231396
$$926$$ 0 0
$$927$$ 1.45488 0.0477846
$$928$$ 0 0
$$929$$ −37.8018 −1.24024 −0.620118 0.784509i $$-0.712915\pi$$
−0.620118 + 0.784509i $$0.712915\pi$$
$$930$$ 0 0
$$931$$ −7.81287 −0.256057
$$932$$ 0 0
$$933$$ 25.8770 0.847176
$$934$$ 0 0
$$935$$ 22.2001 0.726021
$$936$$ 0 0
$$937$$ −32.6907 −1.06796 −0.533980 0.845497i $$-0.679304\pi$$
−0.533980 + 0.845497i $$0.679304\pi$$
$$938$$ 0 0
$$939$$ −21.7070 −0.708381
$$940$$ 0 0
$$941$$ −46.4882 −1.51547 −0.757736 0.652561i $$-0.773694\pi$$
−0.757736 + 0.652561i $$0.773694\pi$$
$$942$$ 0 0
$$943$$ 5.60601 0.182557
$$944$$ 0 0
$$945$$ 17.4598 0.567967
$$946$$ 0 0
$$947$$ 37.2052 1.20901 0.604504 0.796602i $$-0.293371\pi$$
0.604504 + 0.796602i $$0.293371\pi$$
$$948$$ 0 0
$$949$$ 23.3111 0.756711
$$950$$ 0 0
$$951$$ −20.0796 −0.651125
$$952$$ 0 0
$$953$$ 49.1104 1.59084 0.795422 0.606056i $$-0.207249\pi$$
0.795422 + 0.606056i $$0.207249\pi$$
$$954$$ 0 0
$$955$$ 8.58976 0.277958
$$956$$ 0 0
$$957$$ −7.99509 −0.258445
$$958$$ 0 0
$$959$$ −10.0691 −0.325148
$$960$$ 0 0
$$961$$ 4.40778 0.142187
$$962$$ 0 0
$$963$$ 16.7538 0.539885
$$964$$ 0 0
$$965$$ −2.44787 −0.0787997
$$966$$ 0 0
$$967$$ −1.96751 −0.0632709 −0.0316355 0.999499i $$-0.510072\pi$$
−0.0316355 + 0.999499i $$0.510072\pi$$
$$968$$ 0 0
$$969$$ −15.1668 −0.487227
$$970$$ 0 0
$$971$$ −26.1205 −0.838247 −0.419123 0.907929i $$-0.637663\pi$$
−0.419123 + 0.907929i $$0.637663\pi$$
$$972$$ 0 0
$$973$$ −52.2924 −1.67642
$$974$$ 0 0
$$975$$ −5.65556 −0.181123
$$976$$ 0 0
$$977$$ −0.639319 −0.0204536 −0.0102268 0.999948i $$-0.503255\pi$$
−0.0102268 + 0.999948i $$0.503255\pi$$
$$978$$ 0 0
$$979$$ 108.452 3.46613
$$980$$ 0 0
$$981$$ 2.76846 0.0883902
$$982$$ 0 0
$$983$$ 6.46926 0.206337 0.103169 0.994664i $$-0.467102\pi$$
0.103169 + 0.994664i $$0.467102\pi$$
$$984$$ 0 0
$$985$$ 10.7428 0.342293
$$986$$ 0 0
$$987$$ −17.2408 −0.548780
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −37.2334 −1.18276 −0.591379 0.806394i $$-0.701416\pi$$
−0.591379 + 0.806394i $$0.701416\pi$$
$$992$$ 0 0
$$993$$ −35.4189 −1.12398
$$994$$ 0 0
$$995$$ −11.3111 −0.358587
$$996$$ 0 0
$$997$$ 9.65124 0.305658 0.152829 0.988253i $$-0.451162\pi$$
0.152829 + 0.988253i $$0.451162\pi$$
$$998$$ 0 0
$$999$$ −39.8018 −1.25927
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 1840.2.a.r.1.2 3
4.3 odd 2 230.2.a.d.1.2 3
5.4 even 2 9200.2.a.cf.1.2 3
8.3 odd 2 7360.2.a.bz.1.2 3
8.5 even 2 7360.2.a.ce.1.2 3
12.11 even 2 2070.2.a.z.1.2 3
20.3 even 4 1150.2.b.j.599.2 6
20.7 even 4 1150.2.b.j.599.5 6
20.19 odd 2 1150.2.a.q.1.2 3
92.91 even 2 5290.2.a.r.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.2 3 4.3 odd 2
1150.2.a.q.1.2 3 20.19 odd 2
1150.2.b.j.599.2 6 20.3 even 4
1150.2.b.j.599.5 6 20.7 even 4
1840.2.a.r.1.2 3 1.1 even 1 trivial
2070.2.a.z.1.2 3 12.11 even 2
5290.2.a.r.1.2 3 92.91 even 2
7360.2.a.bz.1.2 3 8.3 odd 2
7360.2.a.ce.1.2 3 8.5 even 2
9200.2.a.cf.1.2 3 5.4 even 2