# Properties

 Label 4560.2.a.bq.1.2 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1016.1 Defining polynomial: $$x^{3} - x^{2} - 6x + 2$$ x^3 - x^2 - 6*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.85577$$ of defining polynomial Character $$\chi$$ $$=$$ 4560.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -1.29966 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -1.29966 q^{7} +1.00000 q^{9} +5.01121 q^{11} +3.29966 q^{13} +1.00000 q^{15} +5.71155 q^{17} +1.00000 q^{19} +1.29966 q^{21} +3.71155 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.41188 q^{29} -7.71155 q^{31} -5.01121 q^{33} +1.29966 q^{35} +3.29966 q^{37} -3.29966 q^{39} -7.01121 q^{41} -5.29966 q^{43} -1.00000 q^{45} -3.71155 q^{47} -5.31087 q^{49} -5.71155 q^{51} +5.71155 q^{53} -5.01121 q^{55} -1.00000 q^{57} -6.59933 q^{59} +7.11222 q^{61} -1.29966 q^{63} -3.29966 q^{65} -11.4231 q^{67} -3.71155 q^{69} +10.0224 q^{71} +10.0000 q^{73} -1.00000 q^{75} -6.51289 q^{77} +11.4231 q^{79} +1.00000 q^{81} -16.5353 q^{83} -5.71155 q^{85} -4.41188 q^{87} -9.61054 q^{89} -4.28845 q^{91} +7.71155 q^{93} -1.00000 q^{95} +11.2997 q^{97} +5.01121 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 4 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{55} - 3 q^{57} - 12 q^{59} + 14 q^{61} - 6 q^{65} - 4 q^{67} + 4 q^{69} - 8 q^{71} + 30 q^{73} - 3 q^{75} - 20 q^{77} + 4 q^{79} + 3 q^{81} - 12 q^{83} - 2 q^{85} - 2 q^{87} - 2 q^{89} - 28 q^{91} + 8 q^{93} - 3 q^{95} + 30 q^{97} - 4 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9 - 4 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 + 3 * q^19 - 4 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 + 6 * q^37 - 6 * q^39 - 2 * q^41 - 12 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 2 * q^53 + 4 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 - 6 * q^65 - 4 * q^67 + 4 * q^69 - 8 * q^71 + 30 * q^73 - 3 * q^75 - 20 * q^77 + 4 * q^79 + 3 * q^81 - 12 * q^83 - 2 * q^85 - 2 * q^87 - 2 * q^89 - 28 * q^91 + 8 * q^93 - 3 * q^95 + 30 * q^97 - 4 * q^99

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.29966 −0.491227 −0.245613 0.969368i $$-0.578989\pi$$
−0.245613 + 0.969368i $$0.578989\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.01121 1.51094 0.755468 0.655185i $$-0.227409\pi$$
0.755468 + 0.655185i $$0.227409\pi$$
$$12$$ 0 0
$$13$$ 3.29966 0.915162 0.457581 0.889168i $$-0.348716\pi$$
0.457581 + 0.889168i $$0.348716\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 5.71155 1.38525 0.692627 0.721296i $$-0.256453\pi$$
0.692627 + 0.721296i $$0.256453\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.29966 0.283610
$$22$$ 0 0
$$23$$ 3.71155 0.773911 0.386955 0.922098i $$-0.373527\pi$$
0.386955 + 0.922098i $$0.373527\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 4.41188 0.819266 0.409633 0.912250i $$-0.365657\pi$$
0.409633 + 0.912250i $$0.365657\pi$$
$$30$$ 0 0
$$31$$ −7.71155 −1.38503 −0.692517 0.721401i $$-0.743498\pi$$
−0.692517 + 0.721401i $$0.743498\pi$$
$$32$$ 0 0
$$33$$ −5.01121 −0.872340
$$34$$ 0 0
$$35$$ 1.29966 0.219683
$$36$$ 0 0
$$37$$ 3.29966 0.542461 0.271231 0.962514i $$-0.412569\pi$$
0.271231 + 0.962514i $$0.412569\pi$$
$$38$$ 0 0
$$39$$ −3.29966 −0.528369
$$40$$ 0 0
$$41$$ −7.01121 −1.09497 −0.547483 0.836817i $$-0.684414\pi$$
−0.547483 + 0.836817i $$0.684414\pi$$
$$42$$ 0 0
$$43$$ −5.29966 −0.808191 −0.404096 0.914717i $$-0.632414\pi$$
−0.404096 + 0.914717i $$0.632414\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −3.71155 −0.541384 −0.270692 0.962666i $$-0.587253\pi$$
−0.270692 + 0.962666i $$0.587253\pi$$
$$48$$ 0 0
$$49$$ −5.31087 −0.758696
$$50$$ 0 0
$$51$$ −5.71155 −0.799776
$$52$$ 0 0
$$53$$ 5.71155 0.784541 0.392271 0.919850i $$-0.371690\pi$$
0.392271 + 0.919850i $$0.371690\pi$$
$$54$$ 0 0
$$55$$ −5.01121 −0.675711
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −6.59933 −0.859159 −0.429580 0.903029i $$-0.641338\pi$$
−0.429580 + 0.903029i $$0.641338\pi$$
$$60$$ 0 0
$$61$$ 7.11222 0.910626 0.455313 0.890331i $$-0.349527\pi$$
0.455313 + 0.890331i $$0.349527\pi$$
$$62$$ 0 0
$$63$$ −1.29966 −0.163742
$$64$$ 0 0
$$65$$ −3.29966 −0.409273
$$66$$ 0 0
$$67$$ −11.4231 −1.39555 −0.697776 0.716316i $$-0.745827\pi$$
−0.697776 + 0.716316i $$0.745827\pi$$
$$68$$ 0 0
$$69$$ −3.71155 −0.446818
$$70$$ 0 0
$$71$$ 10.0224 1.18944 0.594721 0.803932i $$-0.297262\pi$$
0.594721 + 0.803932i $$0.297262\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −6.51289 −0.742213
$$78$$ 0 0
$$79$$ 11.4231 1.28520 0.642599 0.766203i $$-0.277856\pi$$
0.642599 + 0.766203i $$0.277856\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −16.5353 −1.81499 −0.907493 0.420068i $$-0.862006\pi$$
−0.907493 + 0.420068i $$0.862006\pi$$
$$84$$ 0 0
$$85$$ −5.71155 −0.619504
$$86$$ 0 0
$$87$$ −4.41188 −0.473003
$$88$$ 0 0
$$89$$ −9.61054 −1.01871 −0.509357 0.860555i $$-0.670117\pi$$
−0.509357 + 0.860555i $$0.670117\pi$$
$$90$$ 0 0
$$91$$ −4.28845 −0.449552
$$92$$ 0 0
$$93$$ 7.71155 0.799650
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 11.2997 1.14731 0.573654 0.819098i $$-0.305526\pi$$
0.573654 + 0.819098i $$0.305526\pi$$
$$98$$ 0 0
$$99$$ 5.01121 0.503645
$$100$$ 0 0
$$101$$ 4.59933 0.457650 0.228825 0.973468i $$-0.426512\pi$$
0.228825 + 0.973468i $$0.426512\pi$$
$$102$$ 0 0
$$103$$ 11.4231 1.12555 0.562775 0.826610i $$-0.309734\pi$$
0.562775 + 0.826610i $$0.309734\pi$$
$$104$$ 0 0
$$105$$ −1.29966 −0.126834
$$106$$ 0 0
$$107$$ 11.4231 1.10431 0.552156 0.833741i $$-0.313805\pi$$
0.552156 + 0.833741i $$0.313805\pi$$
$$108$$ 0 0
$$109$$ 7.40067 0.708856 0.354428 0.935083i $$-0.384676\pi$$
0.354428 + 0.935083i $$0.384676\pi$$
$$110$$ 0 0
$$111$$ −3.29966 −0.313190
$$112$$ 0 0
$$113$$ 7.11222 0.669061 0.334531 0.942385i $$-0.391422\pi$$
0.334531 + 0.942385i $$0.391422\pi$$
$$114$$ 0 0
$$115$$ −3.71155 −0.346103
$$116$$ 0 0
$$117$$ 3.29966 0.305054
$$118$$ 0 0
$$119$$ −7.42309 −0.680474
$$120$$ 0 0
$$121$$ 14.1122 1.28293
$$122$$ 0 0
$$123$$ 7.01121 0.632179
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 19.4231 1.72352 0.861760 0.507316i $$-0.169362\pi$$
0.861760 + 0.507316i $$0.169362\pi$$
$$128$$ 0 0
$$129$$ 5.29966 0.466609
$$130$$ 0 0
$$131$$ −2.41188 −0.210727 −0.105364 0.994434i $$-0.533601\pi$$
−0.105364 + 0.994434i $$0.533601\pi$$
$$132$$ 0 0
$$133$$ −1.29966 −0.112695
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 7.19866 0.615023 0.307511 0.951544i $$-0.400504\pi$$
0.307511 + 0.951544i $$0.400504\pi$$
$$138$$ 0 0
$$139$$ −22.0224 −1.86792 −0.933959 0.357381i $$-0.883670\pi$$
−0.933959 + 0.357381i $$0.883670\pi$$
$$140$$ 0 0
$$141$$ 3.71155 0.312568
$$142$$ 0 0
$$143$$ 16.5353 1.38275
$$144$$ 0 0
$$145$$ −4.41188 −0.366387
$$146$$ 0 0
$$147$$ 5.31087 0.438033
$$148$$ 0 0
$$149$$ −8.22443 −0.673772 −0.336886 0.941545i $$-0.609374\pi$$
−0.336886 + 0.941545i $$0.609374\pi$$
$$150$$ 0 0
$$151$$ 5.48711 0.446535 0.223267 0.974757i $$-0.428328\pi$$
0.223267 + 0.974757i $$0.428328\pi$$
$$152$$ 0 0
$$153$$ 5.71155 0.461751
$$154$$ 0 0
$$155$$ 7.71155 0.619406
$$156$$ 0 0
$$157$$ −12.0224 −0.959493 −0.479747 0.877407i $$-0.659271\pi$$
−0.479747 + 0.877407i $$0.659271\pi$$
$$158$$ 0 0
$$159$$ −5.71155 −0.452955
$$160$$ 0 0
$$161$$ −4.82376 −0.380166
$$162$$ 0 0
$$163$$ 10.1234 0.792928 0.396464 0.918050i $$-0.370237\pi$$
0.396464 + 0.918050i $$0.370237\pi$$
$$164$$ 0 0
$$165$$ 5.01121 0.390122
$$166$$ 0 0
$$167$$ 6.88778 0.532993 0.266496 0.963836i $$-0.414134\pi$$
0.266496 + 0.963836i $$0.414134\pi$$
$$168$$ 0 0
$$169$$ −2.11222 −0.162478
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 21.5095 1.63534 0.817670 0.575688i $$-0.195266\pi$$
0.817670 + 0.575688i $$0.195266\pi$$
$$174$$ 0 0
$$175$$ −1.29966 −0.0982454
$$176$$ 0 0
$$177$$ 6.59933 0.496036
$$178$$ 0 0
$$179$$ −19.2211 −1.43665 −0.718325 0.695707i $$-0.755091\pi$$
−0.718325 + 0.695707i $$0.755091\pi$$
$$180$$ 0 0
$$181$$ 12.0224 0.893619 0.446810 0.894629i $$-0.352560\pi$$
0.446810 + 0.894629i $$0.352560\pi$$
$$182$$ 0 0
$$183$$ −7.11222 −0.525750
$$184$$ 0 0
$$185$$ −3.29966 −0.242596
$$186$$ 0 0
$$187$$ 28.6217 2.09303
$$188$$ 0 0
$$189$$ 1.29966 0.0945367
$$190$$ 0 0
$$191$$ −4.43430 −0.320855 −0.160427 0.987048i $$-0.551287\pi$$
−0.160427 + 0.987048i $$0.551287\pi$$
$$192$$ 0 0
$$193$$ 24.1234 1.73644 0.868221 0.496178i $$-0.165263\pi$$
0.868221 + 0.496178i $$0.165263\pi$$
$$194$$ 0 0
$$195$$ 3.29966 0.236294
$$196$$ 0 0
$$197$$ −7.48711 −0.533435 −0.266717 0.963775i $$-0.585939\pi$$
−0.266717 + 0.963775i $$0.585939\pi$$
$$198$$ 0 0
$$199$$ 3.17624 0.225158 0.112579 0.993643i $$-0.464089\pi$$
0.112579 + 0.993643i $$0.464089\pi$$
$$200$$ 0 0
$$201$$ 11.4231 0.805723
$$202$$ 0 0
$$203$$ −5.73396 −0.402445
$$204$$ 0 0
$$205$$ 7.01121 0.489684
$$206$$ 0 0
$$207$$ 3.71155 0.257970
$$208$$ 0 0
$$209$$ 5.01121 0.346633
$$210$$ 0 0
$$211$$ 18.8462 1.29742 0.648712 0.761034i $$-0.275308\pi$$
0.648712 + 0.761034i $$0.275308\pi$$
$$212$$ 0 0
$$213$$ −10.0224 −0.686725
$$214$$ 0 0
$$215$$ 5.29966 0.361434
$$216$$ 0 0
$$217$$ 10.0224 0.680366
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 18.8462 1.26773
$$222$$ 0 0
$$223$$ −3.42309 −0.229227 −0.114614 0.993410i $$-0.536563\pi$$
−0.114614 + 0.993410i $$0.536563\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −2.31087 −0.153378 −0.0766890 0.997055i $$-0.524435\pi$$
−0.0766890 + 0.997055i $$0.524435\pi$$
$$228$$ 0 0
$$229$$ 11.7340 0.775402 0.387701 0.921785i $$-0.373269\pi$$
0.387701 + 0.921785i $$0.373269\pi$$
$$230$$ 0 0
$$231$$ 6.51289 0.428517
$$232$$ 0 0
$$233$$ −18.0448 −1.18216 −0.591078 0.806614i $$-0.701298\pi$$
−0.591078 + 0.806614i $$0.701298\pi$$
$$234$$ 0 0
$$235$$ 3.71155 0.242115
$$236$$ 0 0
$$237$$ −11.4231 −0.742009
$$238$$ 0 0
$$239$$ −5.01121 −0.324148 −0.162074 0.986779i $$-0.551818\pi$$
−0.162074 + 0.986779i $$0.551818\pi$$
$$240$$ 0 0
$$241$$ −18.0448 −1.16237 −0.581185 0.813771i $$-0.697411\pi$$
−0.581185 + 0.813771i $$0.697411\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 5.31087 0.339299
$$246$$ 0 0
$$247$$ 3.29966 0.209953
$$248$$ 0 0
$$249$$ 16.5353 1.04788
$$250$$ 0 0
$$251$$ 7.61054 0.480373 0.240186 0.970727i $$-0.422791\pi$$
0.240186 + 0.970727i $$0.422791\pi$$
$$252$$ 0 0
$$253$$ 18.5993 1.16933
$$254$$ 0 0
$$255$$ 5.71155 0.357671
$$256$$ 0 0
$$257$$ −16.8878 −1.05343 −0.526715 0.850042i $$-0.676577\pi$$
−0.526715 + 0.850042i $$0.676577\pi$$
$$258$$ 0 0
$$259$$ −4.28845 −0.266472
$$260$$ 0 0
$$261$$ 4.41188 0.273089
$$262$$ 0 0
$$263$$ 27.5095 1.69631 0.848155 0.529748i $$-0.177713\pi$$
0.848155 + 0.529748i $$0.177713\pi$$
$$264$$ 0 0
$$265$$ −5.71155 −0.350857
$$266$$ 0 0
$$267$$ 9.61054 0.588155
$$268$$ 0 0
$$269$$ −5.61054 −0.342080 −0.171040 0.985264i $$-0.554713\pi$$
−0.171040 + 0.985264i $$0.554713\pi$$
$$270$$ 0 0
$$271$$ −22.6442 −1.37554 −0.687768 0.725931i $$-0.741409\pi$$
−0.687768 + 0.725931i $$0.741409\pi$$
$$272$$ 0 0
$$273$$ 4.28845 0.259549
$$274$$ 0 0
$$275$$ 5.01121 0.302187
$$276$$ 0 0
$$277$$ −22.0448 −1.32455 −0.662273 0.749263i $$-0.730408\pi$$
−0.662273 + 0.749263i $$0.730408\pi$$
$$278$$ 0 0
$$279$$ −7.71155 −0.461678
$$280$$ 0 0
$$281$$ −17.2356 −1.02819 −0.514096 0.857733i $$-0.671873\pi$$
−0.514096 + 0.857733i $$0.671873\pi$$
$$282$$ 0 0
$$283$$ −2.49832 −0.148510 −0.0742549 0.997239i $$-0.523658\pi$$
−0.0742549 + 0.997239i $$0.523658\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 0 0
$$287$$ 9.11222 0.537877
$$288$$ 0 0
$$289$$ 15.6217 0.918926
$$290$$ 0 0
$$291$$ −11.2997 −0.662398
$$292$$ 0 0
$$293$$ −4.31087 −0.251844 −0.125922 0.992040i $$-0.540189\pi$$
−0.125922 + 0.992040i $$0.540189\pi$$
$$294$$ 0 0
$$295$$ 6.59933 0.384228
$$296$$ 0 0
$$297$$ −5.01121 −0.290780
$$298$$ 0 0
$$299$$ 12.2469 0.708254
$$300$$ 0 0
$$301$$ 6.88778 0.397005
$$302$$ 0 0
$$303$$ −4.59933 −0.264225
$$304$$ 0 0
$$305$$ −7.11222 −0.407244
$$306$$ 0 0
$$307$$ 30.0224 1.71347 0.856735 0.515757i $$-0.172489\pi$$
0.856735 + 0.515757i $$0.172489\pi$$
$$308$$ 0 0
$$309$$ −11.4231 −0.649837
$$310$$ 0 0
$$311$$ 30.6587 1.73850 0.869249 0.494375i $$-0.164603\pi$$
0.869249 + 0.494375i $$0.164603\pi$$
$$312$$ 0 0
$$313$$ 20.0224 1.13173 0.565867 0.824497i $$-0.308542\pi$$
0.565867 + 0.824497i $$0.308542\pi$$
$$314$$ 0 0
$$315$$ 1.29966 0.0732278
$$316$$ 0 0
$$317$$ −20.1089 −1.12943 −0.564713 0.825287i $$-0.691013\pi$$
−0.564713 + 0.825287i $$0.691013\pi$$
$$318$$ 0 0
$$319$$ 22.1089 1.23786
$$320$$ 0 0
$$321$$ −11.4231 −0.637575
$$322$$ 0 0
$$323$$ 5.71155 0.317799
$$324$$ 0 0
$$325$$ 3.29966 0.183032
$$326$$ 0 0
$$327$$ −7.40067 −0.409258
$$328$$ 0 0
$$329$$ 4.82376 0.265943
$$330$$ 0 0
$$331$$ −28.3333 −1.55734 −0.778669 0.627435i $$-0.784105\pi$$
−0.778669 + 0.627435i $$0.784105\pi$$
$$332$$ 0 0
$$333$$ 3.29966 0.180820
$$334$$ 0 0
$$335$$ 11.4231 0.624110
$$336$$ 0 0
$$337$$ 2.92477 0.159322 0.0796612 0.996822i $$-0.474616\pi$$
0.0796612 + 0.996822i $$0.474616\pi$$
$$338$$ 0 0
$$339$$ −7.11222 −0.386283
$$340$$ 0 0
$$341$$ −38.6442 −2.09270
$$342$$ 0 0
$$343$$ 16.0000 0.863919
$$344$$ 0 0
$$345$$ 3.71155 0.199823
$$346$$ 0 0
$$347$$ −1.11222 −0.0597069 −0.0298535 0.999554i $$-0.509504\pi$$
−0.0298535 + 0.999554i $$0.509504\pi$$
$$348$$ 0 0
$$349$$ 24.2244 1.29670 0.648352 0.761341i $$-0.275458\pi$$
0.648352 + 0.761341i $$0.275458\pi$$
$$350$$ 0 0
$$351$$ −3.29966 −0.176123
$$352$$ 0 0
$$353$$ 20.2244 1.07644 0.538219 0.842805i $$-0.319097\pi$$
0.538219 + 0.842805i $$0.319097\pi$$
$$354$$ 0 0
$$355$$ −10.0224 −0.531935
$$356$$ 0 0
$$357$$ 7.42309 0.392872
$$358$$ 0 0
$$359$$ 2.41188 0.127294 0.0636471 0.997972i $$-0.479727\pi$$
0.0636471 + 0.997972i $$0.479727\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −14.1122 −0.740699
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −23.5689 −1.23029 −0.615144 0.788415i $$-0.710902\pi$$
−0.615144 + 0.788415i $$0.710902\pi$$
$$368$$ 0 0
$$369$$ −7.01121 −0.364989
$$370$$ 0 0
$$371$$ −7.42309 −0.385388
$$372$$ 0 0
$$373$$ −4.12343 −0.213503 −0.106751 0.994286i $$-0.534045\pi$$
−0.106751 + 0.994286i $$0.534045\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 14.5577 0.749761
$$378$$ 0 0
$$379$$ 22.5577 1.15871 0.579356 0.815074i $$-0.303304\pi$$
0.579356 + 0.815074i $$0.303304\pi$$
$$380$$ 0 0
$$381$$ −19.4231 −0.995075
$$382$$ 0 0
$$383$$ −19.4679 −0.994765 −0.497382 0.867531i $$-0.665705\pi$$
−0.497382 + 0.867531i $$0.665705\pi$$
$$384$$ 0 0
$$385$$ 6.51289 0.331928
$$386$$ 0 0
$$387$$ −5.29966 −0.269397
$$388$$ 0 0
$$389$$ −23.8204 −1.20774 −0.603871 0.797082i $$-0.706376\pi$$
−0.603871 + 0.797082i $$0.706376\pi$$
$$390$$ 0 0
$$391$$ 21.1987 1.07206
$$392$$ 0 0
$$393$$ 2.41188 0.121663
$$394$$ 0 0
$$395$$ −11.4231 −0.574758
$$396$$ 0 0
$$397$$ −25.4231 −1.27595 −0.637974 0.770058i $$-0.720227\pi$$
−0.637974 + 0.770058i $$0.720227\pi$$
$$398$$ 0 0
$$399$$ 1.29966 0.0650646
$$400$$ 0 0
$$401$$ 28.8316 1.43978 0.719891 0.694087i $$-0.244192\pi$$
0.719891 + 0.694087i $$0.244192\pi$$
$$402$$ 0 0
$$403$$ −25.4455 −1.26753
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 16.5353 0.819625
$$408$$ 0 0
$$409$$ 9.22107 0.455953 0.227976 0.973667i $$-0.426789\pi$$
0.227976 + 0.973667i $$0.426789\pi$$
$$410$$ 0 0
$$411$$ −7.19866 −0.355084
$$412$$ 0 0
$$413$$ 8.57691 0.422042
$$414$$ 0 0
$$415$$ 16.5353 0.811686
$$416$$ 0 0
$$417$$ 22.0224 1.07844
$$418$$ 0 0
$$419$$ −23.0336 −1.12527 −0.562633 0.826707i $$-0.690212\pi$$
−0.562633 + 0.826707i $$0.690212\pi$$
$$420$$ 0 0
$$421$$ −17.4679 −0.851335 −0.425667 0.904880i $$-0.639961\pi$$
−0.425667 + 0.904880i $$0.639961\pi$$
$$422$$ 0 0
$$423$$ −3.71155 −0.180461
$$424$$ 0 0
$$425$$ 5.71155 0.277051
$$426$$ 0 0
$$427$$ −9.24349 −0.447324
$$428$$ 0 0
$$429$$ −16.5353 −0.798332
$$430$$ 0 0
$$431$$ −17.4455 −0.840321 −0.420160 0.907450i $$-0.638026\pi$$
−0.420160 + 0.907450i $$0.638026\pi$$
$$432$$ 0 0
$$433$$ 18.5207 0.890050 0.445025 0.895518i $$-0.353195\pi$$
0.445025 + 0.895518i $$0.353195\pi$$
$$434$$ 0 0
$$435$$ 4.41188 0.211533
$$436$$ 0 0
$$437$$ 3.71155 0.177547
$$438$$ 0 0
$$439$$ 0.621746 0.0296743 0.0148372 0.999890i $$-0.495277\pi$$
0.0148372 + 0.999890i $$0.495277\pi$$
$$440$$ 0 0
$$441$$ −5.31087 −0.252899
$$442$$ 0 0
$$443$$ 8.91020 0.423336 0.211668 0.977342i $$-0.432110\pi$$
0.211668 + 0.977342i $$0.432110\pi$$
$$444$$ 0 0
$$445$$ 9.61054 0.455583
$$446$$ 0 0
$$447$$ 8.22443 0.389002
$$448$$ 0 0
$$449$$ −30.4343 −1.43628 −0.718142 0.695897i $$-0.755007\pi$$
−0.718142 + 0.695897i $$0.755007\pi$$
$$450$$ 0 0
$$451$$ −35.1346 −1.65443
$$452$$ 0 0
$$453$$ −5.48711 −0.257807
$$454$$ 0 0
$$455$$ 4.28845 0.201046
$$456$$ 0 0
$$457$$ 11.4455 0.535398 0.267699 0.963503i $$-0.413737\pi$$
0.267699 + 0.963503i $$0.413737\pi$$
$$458$$ 0 0
$$459$$ −5.71155 −0.266592
$$460$$ 0 0
$$461$$ 22.0448 1.02673 0.513365 0.858170i $$-0.328399\pi$$
0.513365 + 0.858170i $$0.328399\pi$$
$$462$$ 0 0
$$463$$ 16.5207 0.767784 0.383892 0.923378i $$-0.374584\pi$$
0.383892 + 0.923378i $$0.374584\pi$$
$$464$$ 0 0
$$465$$ −7.71155 −0.357614
$$466$$ 0 0
$$467$$ −0.910201 −0.0421191 −0.0210595 0.999778i $$-0.506704\pi$$
−0.0210595 + 0.999778i $$0.506704\pi$$
$$468$$ 0 0
$$469$$ 14.8462 0.685533
$$470$$ 0 0
$$471$$ 12.0224 0.553964
$$472$$ 0 0
$$473$$ −26.5577 −1.22113
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 5.71155 0.261514
$$478$$ 0 0
$$479$$ 9.63296 0.440141 0.220070 0.975484i $$-0.429371\pi$$
0.220070 + 0.975484i $$0.429371\pi$$
$$480$$ 0 0
$$481$$ 10.8878 0.496440
$$482$$ 0 0
$$483$$ 4.82376 0.219489
$$484$$ 0 0
$$485$$ −11.2997 −0.513091
$$486$$ 0 0
$$487$$ 30.5993 1.38659 0.693294 0.720655i $$-0.256159\pi$$
0.693294 + 0.720655i $$0.256159\pi$$
$$488$$ 0 0
$$489$$ −10.1234 −0.457797
$$490$$ 0 0
$$491$$ 26.4119 1.19195 0.595976 0.803002i $$-0.296765\pi$$
0.595976 + 0.803002i $$0.296765\pi$$
$$492$$ 0 0
$$493$$ 25.1987 1.13489
$$494$$ 0 0
$$495$$ −5.01121 −0.225237
$$496$$ 0 0
$$497$$ −13.0258 −0.584286
$$498$$ 0 0
$$499$$ 3.22107 0.144195 0.0720976 0.997398i $$-0.477031\pi$$
0.0720976 + 0.997398i $$0.477031\pi$$
$$500$$ 0 0
$$501$$ −6.88778 −0.307723
$$502$$ 0 0
$$503$$ 31.9584 1.42495 0.712477 0.701695i $$-0.247573\pi$$
0.712477 + 0.701695i $$0.247573\pi$$
$$504$$ 0 0
$$505$$ −4.59933 −0.204667
$$506$$ 0 0
$$507$$ 2.11222 0.0938068
$$508$$ 0 0
$$509$$ −16.4119 −0.727444 −0.363722 0.931508i $$-0.618494\pi$$
−0.363722 + 0.931508i $$0.618494\pi$$
$$510$$ 0 0
$$511$$ −12.9966 −0.574938
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ −11.4231 −0.503361
$$516$$ 0 0
$$517$$ −18.5993 −0.817998
$$518$$ 0 0
$$519$$ −21.5095 −0.944164
$$520$$ 0 0
$$521$$ −7.01121 −0.307167 −0.153583 0.988136i $$-0.549081\pi$$
−0.153583 + 0.988136i $$0.549081\pi$$
$$522$$ 0 0
$$523$$ 17.4007 0.760878 0.380439 0.924806i $$-0.375773\pi$$
0.380439 + 0.924806i $$0.375773\pi$$
$$524$$ 0 0
$$525$$ 1.29966 0.0567220
$$526$$ 0 0
$$527$$ −44.0448 −1.91862
$$528$$ 0 0
$$529$$ −9.22443 −0.401062
$$530$$ 0 0
$$531$$ −6.59933 −0.286386
$$532$$ 0 0
$$533$$ −23.1346 −1.00207
$$534$$ 0 0
$$535$$ −11.4231 −0.493863
$$536$$ 0 0
$$537$$ 19.2211 0.829451
$$538$$ 0 0
$$539$$ −26.6139 −1.14634
$$540$$ 0 0
$$541$$ 20.8462 0.896247 0.448124 0.893972i $$-0.352092\pi$$
0.448124 + 0.893972i $$0.352092\pi$$
$$542$$ 0 0
$$543$$ −12.0224 −0.515931
$$544$$ 0 0
$$545$$ −7.40067 −0.317010
$$546$$ 0 0
$$547$$ 30.0224 1.28367 0.641833 0.766844i $$-0.278174\pi$$
0.641833 + 0.766844i $$0.278174\pi$$
$$548$$ 0 0
$$549$$ 7.11222 0.303542
$$550$$ 0 0
$$551$$ 4.41188 0.187952
$$552$$ 0 0
$$553$$ −14.8462 −0.631324
$$554$$ 0 0
$$555$$ 3.29966 0.140063
$$556$$ 0 0
$$557$$ −34.0448 −1.44253 −0.721263 0.692661i $$-0.756438\pi$$
−0.721263 + 0.692661i $$0.756438\pi$$
$$558$$ 0 0
$$559$$ −17.4871 −0.739626
$$560$$ 0 0
$$561$$ −28.6217 −1.20841
$$562$$ 0 0
$$563$$ 30.9326 1.30365 0.651827 0.758367i $$-0.274003\pi$$
0.651827 + 0.758367i $$0.274003\pi$$
$$564$$ 0 0
$$565$$ −7.11222 −0.299213
$$566$$ 0 0
$$567$$ −1.29966 −0.0545808
$$568$$ 0 0
$$569$$ 39.2581 1.64578 0.822892 0.568198i $$-0.192359\pi$$
0.822892 + 0.568198i $$0.192359\pi$$
$$570$$ 0 0
$$571$$ 41.4903 1.73632 0.868158 0.496287i $$-0.165304\pi$$
0.868158 + 0.496287i $$0.165304\pi$$
$$572$$ 0 0
$$573$$ 4.43430 0.185246
$$574$$ 0 0
$$575$$ 3.71155 0.154782
$$576$$ 0 0
$$577$$ −28.4713 −1.18528 −0.592638 0.805469i $$-0.701913\pi$$
−0.592638 + 0.805469i $$0.701913\pi$$
$$578$$ 0 0
$$579$$ −24.1234 −1.00254
$$580$$ 0 0
$$581$$ 21.4903 0.891570
$$582$$ 0 0
$$583$$ 28.6217 1.18539
$$584$$ 0 0
$$585$$ −3.29966 −0.136424
$$586$$ 0 0
$$587$$ −6.51289 −0.268816 −0.134408 0.990926i $$-0.542913\pi$$
−0.134408 + 0.990926i $$0.542913\pi$$
$$588$$ 0 0
$$589$$ −7.71155 −0.317749
$$590$$ 0 0
$$591$$ 7.48711 0.307979
$$592$$ 0 0
$$593$$ 21.4679 0.881582 0.440791 0.897610i $$-0.354698\pi$$
0.440791 + 0.897610i $$0.354698\pi$$
$$594$$ 0 0
$$595$$ 7.42309 0.304317
$$596$$ 0 0
$$597$$ −3.17624 −0.129995
$$598$$ 0 0
$$599$$ 25.2435 1.03142 0.515711 0.856763i $$-0.327528\pi$$
0.515711 + 0.856763i $$0.327528\pi$$
$$600$$ 0 0
$$601$$ 27.4455 1.11953 0.559763 0.828653i $$-0.310892\pi$$
0.559763 + 0.828653i $$0.310892\pi$$
$$602$$ 0 0
$$603$$ −11.4231 −0.465184
$$604$$ 0 0
$$605$$ −14.1122 −0.573743
$$606$$ 0 0
$$607$$ 35.2211 1.42958 0.714790 0.699340i $$-0.246522\pi$$
0.714790 + 0.699340i $$0.246522\pi$$
$$608$$ 0 0
$$609$$ 5.73396 0.232352
$$610$$ 0 0
$$611$$ −12.2469 −0.495455
$$612$$ 0 0
$$613$$ −11.4455 −0.462280 −0.231140 0.972921i $$-0.574245\pi$$
−0.231140 + 0.972921i $$0.574245\pi$$
$$614$$ 0 0
$$615$$ −7.01121 −0.282719
$$616$$ 0 0
$$617$$ −20.6858 −0.832778 −0.416389 0.909187i $$-0.636704\pi$$
−0.416389 + 0.909187i $$0.636704\pi$$
$$618$$ 0 0
$$619$$ −1.40067 −0.0562978 −0.0281489 0.999604i $$-0.508961\pi$$
−0.0281489 + 0.999604i $$0.508961\pi$$
$$620$$ 0 0
$$621$$ −3.71155 −0.148939
$$622$$ 0 0
$$623$$ 12.4905 0.500420
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −5.01121 −0.200128
$$628$$ 0 0
$$629$$ 18.8462 0.751446
$$630$$ 0 0
$$631$$ 23.5960 0.939341 0.469670 0.882842i $$-0.344373\pi$$
0.469670 + 0.882842i $$0.344373\pi$$
$$632$$ 0 0
$$633$$ −18.8462 −0.749068
$$634$$ 0 0
$$635$$ −19.4231 −0.770782
$$636$$ 0 0
$$637$$ −17.5241 −0.694330
$$638$$ 0 0
$$639$$ 10.0224 0.396481
$$640$$ 0 0
$$641$$ 39.6330 1.56541 0.782704 0.622394i $$-0.213840\pi$$
0.782704 + 0.622394i $$0.213840\pi$$
$$642$$ 0 0
$$643$$ −7.14920 −0.281937 −0.140969 0.990014i $$-0.545022\pi$$
−0.140969 + 0.990014i $$0.545022\pi$$
$$644$$ 0 0
$$645$$ −5.29966 −0.208674
$$646$$ 0 0
$$647$$ 31.9584 1.25641 0.628207 0.778046i $$-0.283789\pi$$
0.628207 + 0.778046i $$0.283789\pi$$
$$648$$ 0 0
$$649$$ −33.0706 −1.29814
$$650$$ 0 0
$$651$$ −10.0224 −0.392810
$$652$$ 0 0
$$653$$ 11.4871 0.449525 0.224763 0.974414i $$-0.427839\pi$$
0.224763 + 0.974414i $$0.427839\pi$$
$$654$$ 0 0
$$655$$ 2.41188 0.0942400
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 24.0448 0.936654 0.468327 0.883555i $$-0.344857\pi$$
0.468327 + 0.883555i $$0.344857\pi$$
$$660$$ 0 0
$$661$$ 6.82376 0.265414 0.132707 0.991155i $$-0.457633\pi$$
0.132707 + 0.991155i $$0.457633\pi$$
$$662$$ 0 0
$$663$$ −18.8462 −0.731925
$$664$$ 0 0
$$665$$ 1.29966 0.0503988
$$666$$ 0 0
$$667$$ 16.3749 0.634038
$$668$$ 0 0
$$669$$ 3.42309 0.132344
$$670$$ 0 0
$$671$$ 35.6408 1.37590
$$672$$ 0 0
$$673$$ −9.69698 −0.373791 −0.186895 0.982380i $$-0.559843\pi$$
−0.186895 + 0.982380i $$0.559843\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −21.0898 −0.810547 −0.405273 0.914196i $$-0.632824\pi$$
−0.405273 + 0.914196i $$0.632824\pi$$
$$678$$ 0 0
$$679$$ −14.6858 −0.563588
$$680$$ 0 0
$$681$$ 2.31087 0.0885529
$$682$$ 0 0
$$683$$ −16.6217 −0.636013 −0.318007 0.948088i $$-0.603013\pi$$
−0.318007 + 0.948088i $$0.603013\pi$$
$$684$$ 0 0
$$685$$ −7.19866 −0.275047
$$686$$ 0 0
$$687$$ −11.7340 −0.447679
$$688$$ 0 0
$$689$$ 18.8462 0.717982
$$690$$ 0 0
$$691$$ −2.55449 −0.0971774 −0.0485887 0.998819i $$-0.515472\pi$$
−0.0485887 + 0.998819i $$0.515472\pi$$
$$692$$ 0 0
$$693$$ −6.51289 −0.247404
$$694$$ 0 0
$$695$$ 22.0224 0.835358
$$696$$ 0 0
$$697$$ −40.0448 −1.51681
$$698$$ 0 0
$$699$$ 18.0448 0.682518
$$700$$ 0 0
$$701$$ −45.0191 −1.70035 −0.850173 0.526503i $$-0.823503\pi$$
−0.850173 + 0.526503i $$0.823503\pi$$
$$702$$ 0 0
$$703$$ 3.29966 0.124449
$$704$$ 0 0
$$705$$ −3.71155 −0.139785
$$706$$ 0 0
$$707$$ −5.97758 −0.224810
$$708$$ 0 0
$$709$$ −25.3815 −0.953222 −0.476611 0.879114i $$-0.658135\pi$$
−0.476611 + 0.879114i $$0.658135\pi$$
$$710$$ 0 0
$$711$$ 11.4231 0.428399
$$712$$ 0 0
$$713$$ −28.6217 −1.07189
$$714$$ 0 0
$$715$$ −16.5353 −0.618385
$$716$$ 0 0
$$717$$ 5.01121 0.187147
$$718$$ 0 0
$$719$$ −9.63296 −0.359249 −0.179624 0.983735i $$-0.557488\pi$$
−0.179624 + 0.983735i $$0.557488\pi$$
$$720$$ 0 0
$$721$$ −14.8462 −0.552901
$$722$$ 0 0
$$723$$ 18.0448 0.671095
$$724$$ 0 0
$$725$$ 4.41188 0.163853
$$726$$ 0 0
$$727$$ −16.5207 −0.612720 −0.306360 0.951916i $$-0.599111\pi$$
−0.306360 + 0.951916i $$0.599111\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −30.2693 −1.11955
$$732$$ 0 0
$$733$$ −14.6217 −0.540067 −0.270033 0.962851i $$-0.587035\pi$$
−0.270033 + 0.962851i $$0.587035\pi$$
$$734$$ 0 0
$$735$$ −5.31087 −0.195895
$$736$$ 0 0
$$737$$ −57.2435 −2.10859
$$738$$ 0 0
$$739$$ −34.8462 −1.28184 −0.640919 0.767609i $$-0.721446\pi$$
−0.640919 + 0.767609i $$0.721446\pi$$
$$740$$ 0 0
$$741$$ −3.29966 −0.121216
$$742$$ 0 0
$$743$$ −4.62175 −0.169555 −0.0847777 0.996400i $$-0.527018\pi$$
−0.0847777 + 0.996400i $$0.527018\pi$$
$$744$$ 0 0
$$745$$ 8.22443 0.301320
$$746$$ 0 0
$$747$$ −16.5353 −0.604995
$$748$$ 0 0
$$749$$ −14.8462 −0.542468
$$750$$ 0 0
$$751$$ −5.48711 −0.200228 −0.100114 0.994976i $$-0.531921\pi$$
−0.100114 + 0.994976i $$0.531921\pi$$
$$752$$ 0 0
$$753$$ −7.61054 −0.277343
$$754$$ 0 0
$$755$$ −5.48711 −0.199696
$$756$$ 0 0
$$757$$ 38.2917 1.39174 0.695868 0.718170i $$-0.255020\pi$$
0.695868 + 0.718170i $$0.255020\pi$$
$$758$$ 0 0
$$759$$ −18.5993 −0.675113
$$760$$ 0 0
$$761$$ −19.7756 −0.716864 −0.358432 0.933556i $$-0.616688\pi$$
−0.358432 + 0.933556i $$0.616688\pi$$
$$762$$ 0 0
$$763$$ −9.61839 −0.348209
$$764$$ 0 0
$$765$$ −5.71155 −0.206501
$$766$$ 0 0
$$767$$ −21.7756 −0.786270
$$768$$ 0 0
$$769$$ −9.95516 −0.358992 −0.179496 0.983759i $$-0.557447\pi$$
−0.179496 + 0.983759i $$0.557447\pi$$
$$770$$ 0 0
$$771$$ 16.8878 0.608199
$$772$$ 0 0
$$773$$ 54.3781 1.95585 0.977923 0.208967i $$-0.0670102\pi$$
0.977923 + 0.208967i $$0.0670102\pi$$
$$774$$ 0 0
$$775$$ −7.71155 −0.277007
$$776$$ 0 0
$$777$$ 4.28845 0.153847
$$778$$ 0 0
$$779$$ −7.01121 −0.251203
$$780$$ 0 0
$$781$$ 50.2244 1.79717
$$782$$ 0 0
$$783$$ −4.41188 −0.157668
$$784$$ 0 0
$$785$$ 12.0224 0.429099
$$786$$ 0 0
$$787$$ −13.4455 −0.479281 −0.239640 0.970862i $$-0.577030\pi$$
−0.239640 + 0.970862i $$0.577030\pi$$
$$788$$ 0 0
$$789$$ −27.5095 −0.979365
$$790$$ 0 0
$$791$$ −9.24349 −0.328661
$$792$$ 0 0
$$793$$ 23.4679 0.833371
$$794$$ 0 0
$$795$$ 5.71155 0.202568
$$796$$ 0 0
$$797$$ −19.7340 −0.699013 −0.349506 0.936934i $$-0.613651\pi$$
−0.349506 + 0.936934i $$0.613651\pi$$
$$798$$ 0 0
$$799$$ −21.1987 −0.749955
$$800$$ 0 0
$$801$$ −9.61054 −0.339572
$$802$$ 0 0
$$803$$ 50.1121 1.76842
$$804$$ 0 0
$$805$$ 4.82376 0.170015
$$806$$ 0 0
$$807$$ 5.61054 0.197500
$$808$$ 0 0
$$809$$ 15.7756 0.554639 0.277320 0.960778i $$-0.410554\pi$$
0.277320 + 0.960778i $$0.410554\pi$$
$$810$$ 0 0
$$811$$ 5.64752 0.198311 0.0991557 0.995072i $$-0.468386\pi$$
0.0991557 + 0.995072i $$0.468386\pi$$
$$812$$ 0 0
$$813$$ 22.6442 0.794166
$$814$$ 0 0
$$815$$ −10.1234 −0.354608
$$816$$ 0 0
$$817$$ −5.29966 −0.185412
$$818$$ 0 0
$$819$$ −4.28845 −0.149851
$$820$$ 0 0
$$821$$ −49.6699 −1.73349 −0.866746 0.498749i $$-0.833793\pi$$
−0.866746 + 0.498749i $$0.833793\pi$$
$$822$$ 0 0
$$823$$ 46.7900 1.63100 0.815499 0.578759i $$-0.196463\pi$$
0.815499 + 0.578759i $$0.196463\pi$$
$$824$$ 0 0
$$825$$ −5.01121 −0.174468
$$826$$ 0 0
$$827$$ −16.0864 −0.559380 −0.279690 0.960090i $$-0.590232\pi$$
−0.279690 + 0.960090i $$0.590232\pi$$
$$828$$ 0 0
$$829$$ 18.8686 0.655334 0.327667 0.944793i $$-0.393738\pi$$
0.327667 + 0.944793i $$0.393738\pi$$
$$830$$ 0 0
$$831$$ 22.0448 0.764727
$$832$$ 0 0
$$833$$ −30.3333 −1.05099
$$834$$ 0 0
$$835$$ −6.88778 −0.238362
$$836$$ 0 0
$$837$$ 7.71155 0.266550
$$838$$ 0 0
$$839$$ −18.0224 −0.622203 −0.311101 0.950377i $$-0.600698\pi$$
−0.311101 + 0.950377i $$0.600698\pi$$
$$840$$ 0 0
$$841$$ −9.53531 −0.328804
$$842$$ 0 0
$$843$$ 17.2356 0.593627
$$844$$ 0 0
$$845$$ 2.11222 0.0726625
$$846$$ 0 0
$$847$$ −18.3411 −0.630209
$$848$$ 0 0
$$849$$ 2.49832 0.0857421
$$850$$ 0 0
$$851$$ 12.2469 0.419817
$$852$$ 0 0
$$853$$ 41.0930 1.40700 0.703499 0.710696i $$-0.251620\pi$$
0.703499 + 0.710696i $$0.251620\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ 0.640931 0.0218938 0.0109469 0.999940i $$-0.496515\pi$$
0.0109469 + 0.999940i $$0.496515\pi$$
$$858$$ 0 0
$$859$$ −47.4679 −1.61958 −0.809792 0.586717i $$-0.800420\pi$$
−0.809792 + 0.586717i $$0.800420\pi$$
$$860$$ 0 0
$$861$$ −9.11222 −0.310544
$$862$$ 0 0
$$863$$ −49.0706 −1.67038 −0.835192 0.549959i $$-0.814643\pi$$
−0.835192 + 0.549959i $$0.814643\pi$$
$$864$$ 0 0
$$865$$ −21.5095 −0.731346
$$866$$ 0 0
$$867$$ −15.6217 −0.530542
$$868$$ 0 0
$$869$$ 57.2435 1.94185
$$870$$ 0 0
$$871$$ −37.6924 −1.27716
$$872$$ 0 0
$$873$$ 11.2997 0.382436
$$874$$ 0 0
$$875$$ 1.29966 0.0439367
$$876$$ 0 0
$$877$$ 49.1940 1.66116 0.830582 0.556896i $$-0.188008\pi$$
0.830582 + 0.556896i $$0.188008\pi$$
$$878$$ 0 0
$$879$$ 4.31087 0.145402
$$880$$ 0 0
$$881$$ 25.2211 0.849720 0.424860 0.905259i $$-0.360323\pi$$
0.424860 + 0.905259i $$0.360323\pi$$
$$882$$ 0 0
$$883$$ −7.12007 −0.239609 −0.119805 0.992797i $$-0.538227\pi$$
−0.119805 + 0.992797i $$0.538227\pi$$
$$884$$ 0 0
$$885$$ −6.59933 −0.221834
$$886$$ 0 0
$$887$$ −10.8013 −0.362674 −0.181337 0.983421i $$-0.558042\pi$$
−0.181337 + 0.983421i $$0.558042\pi$$
$$888$$ 0 0
$$889$$ −25.2435 −0.846640
$$890$$ 0 0
$$891$$ 5.01121 0.167882
$$892$$ 0 0
$$893$$ −3.71155 −0.124202
$$894$$ 0 0
$$895$$ 19.2211 0.642490
$$896$$ 0 0
$$897$$ −12.2469 −0.408910
$$898$$ 0 0
$$899$$ −34.0224 −1.13471
$$900$$ 0 0
$$901$$ 32.6217 1.08679
$$902$$ 0 0
$$903$$ −6.88778 −0.229211
$$904$$ 0 0
$$905$$ −12.0224 −0.399639
$$906$$ 0 0
$$907$$ 3.62511 0.120370 0.0601848 0.998187i $$-0.480831\pi$$
0.0601848 + 0.998187i $$0.480831\pi$$
$$908$$ 0 0
$$909$$ 4.59933 0.152550
$$910$$ 0 0
$$911$$ 5.77557 0.191353 0.0956765 0.995412i $$-0.469499\pi$$
0.0956765 + 0.995412i $$0.469499\pi$$
$$912$$ 0 0
$$913$$ −82.8619 −2.74233
$$914$$ 0 0
$$915$$ 7.11222 0.235123
$$916$$ 0 0
$$917$$ 3.13464 0.103515
$$918$$ 0 0
$$919$$ −40.6666 −1.34147 −0.670733 0.741699i $$-0.734021\pi$$
−0.670733 + 0.741699i $$0.734021\pi$$
$$920$$ 0 0
$$921$$ −30.0224 −0.989272
$$922$$ 0 0
$$923$$ 33.0706 1.08853
$$924$$ 0 0
$$925$$ 3.29966 0.108492
$$926$$ 0 0
$$927$$ 11.4231 0.375184
$$928$$ 0 0
$$929$$ 30.4197 0.998039 0.499020 0.866591i $$-0.333694\pi$$
0.499020 + 0.866591i $$0.333694\pi$$
$$930$$ 0 0
$$931$$ −5.31087 −0.174057
$$932$$ 0 0
$$933$$ −30.6587 −1.00372
$$934$$ 0 0
$$935$$ −28.6217 −0.936031
$$936$$ 0 0
$$937$$ −25.6699 −0.838600 −0.419300 0.907848i $$-0.637725\pi$$
−0.419300 + 0.907848i $$0.637725\pi$$
$$938$$ 0 0
$$939$$ −20.0224 −0.653407
$$940$$ 0 0
$$941$$ −51.3029 −1.67243 −0.836213 0.548404i $$-0.815236\pi$$
−0.836213 + 0.548404i $$0.815236\pi$$
$$942$$ 0 0
$$943$$ −26.0224 −0.847407
$$944$$ 0 0
$$945$$ −1.29966 −0.0422781
$$946$$ 0 0
$$947$$ −6.31087 −0.205076 −0.102538 0.994729i $$-0.532696\pi$$
−0.102538 + 0.994729i $$0.532696\pi$$
$$948$$ 0 0
$$949$$ 32.9966 1.07112
$$950$$ 0 0
$$951$$ 20.1089 0.652074
$$952$$ 0 0
$$953$$ 58.0032 1.87891 0.939455 0.342674i $$-0.111332\pi$$
0.939455 + 0.342674i $$0.111332\pi$$
$$954$$ 0 0
$$955$$ 4.43430 0.143491
$$956$$ 0 0
$$957$$ −22.1089 −0.714678
$$958$$ 0 0
$$959$$ −9.35584 −0.302116
$$960$$ 0 0
$$961$$ 28.4679 0.918320
$$962$$ 0 0
$$963$$ 11.4231 0.368104
$$964$$ 0 0
$$965$$ −24.1234 −0.776561
$$966$$ 0 0
$$967$$ 27.6970 0.890675 0.445337 0.895363i $$-0.353084\pi$$
0.445337 + 0.895363i $$0.353084\pi$$
$$968$$ 0 0
$$969$$ −5.71155 −0.183481
$$970$$ 0 0
$$971$$ −30.3973 −0.975496 −0.487748 0.872984i $$-0.662182\pi$$
−0.487748 + 0.872984i $$0.662182\pi$$
$$972$$ 0 0
$$973$$ 28.6217 0.917571
$$974$$ 0 0
$$975$$ −3.29966 −0.105674
$$976$$ 0 0
$$977$$ −9.17947 −0.293677 −0.146839 0.989160i $$-0.546910\pi$$
−0.146839 + 0.989160i $$0.546910\pi$$
$$978$$ 0 0
$$979$$ −48.1604 −1.53921
$$980$$ 0 0
$$981$$ 7.40067 0.236285
$$982$$ 0 0
$$983$$ −60.5801 −1.93221 −0.966103 0.258156i $$-0.916885\pi$$
−0.966103 + 0.258156i $$0.916885\pi$$
$$984$$ 0 0
$$985$$ 7.48711 0.238559
$$986$$ 0 0
$$987$$ −4.82376 −0.153542
$$988$$ 0 0
$$989$$ −19.6699 −0.625468
$$990$$ 0 0
$$991$$ 43.4231 1.37938 0.689690 0.724105i $$-0.257747\pi$$
0.689690 + 0.724105i $$0.257747\pi$$
$$992$$ 0 0
$$993$$ 28.3333 0.899130
$$994$$ 0 0
$$995$$ −3.17624 −0.100694
$$996$$ 0 0
$$997$$ −25.6251 −0.811555 −0.405778 0.913972i $$-0.632999\pi$$
−0.405778 + 0.913972i $$0.632999\pi$$
$$998$$ 0 0
$$999$$ −3.29966 −0.104397
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 4560.2.a.bq.1.2 3
4.3 odd 2 2280.2.a.t.1.2 3
12.11 even 2 6840.2.a.bn.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.t.1.2 3 4.3 odd 2
4560.2.a.bq.1.2 3 1.1 even 1 trivial
6840.2.a.bn.1.2 3 12.11 even 2