# Properties

 Label 1170.2.a.p.1.1 Level $1170$ Weight $2$ Character 1170.1 Self dual yes Analytic conductor $9.342$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 1170.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.60555 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} +2.60555 q^{14} +1.00000 q^{16} -4.60555 q^{17} +6.60555 q^{19} -1.00000 q^{20} -4.60555 q^{23} +1.00000 q^{25} -1.00000 q^{26} -2.60555 q^{28} +4.60555 q^{29} +2.00000 q^{31} -1.00000 q^{32} +4.60555 q^{34} +2.60555 q^{35} +11.2111 q^{37} -6.60555 q^{38} +1.00000 q^{40} -3.21110 q^{41} +5.21110 q^{43} +4.60555 q^{46} +9.21110 q^{47} -0.211103 q^{49} -1.00000 q^{50} +1.00000 q^{52} +2.60555 q^{56} -4.60555 q^{58} +9.21110 q^{59} -7.21110 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -7.21110 q^{67} -4.60555 q^{68} -2.60555 q^{70} +12.0000 q^{71} +6.60555 q^{73} -11.2111 q^{74} +6.60555 q^{76} -1.21110 q^{79} -1.00000 q^{80} +3.21110 q^{82} +4.60555 q^{85} -5.21110 q^{86} -3.21110 q^{89} -2.60555 q^{91} -4.60555 q^{92} -9.21110 q^{94} -6.60555 q^{95} +6.60555 q^{97} +0.211103 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 2 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} + 8 q^{37} - 6 q^{38} + 2 q^{40} + 8 q^{41} - 4 q^{43} + 2 q^{46} + 4 q^{47} + 14 q^{49} - 2 q^{50} + 2 q^{52} - 2 q^{56} - 2 q^{58} + 4 q^{59} - 4 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{68} + 2 q^{70} + 24 q^{71} + 6 q^{73} - 8 q^{74} + 6 q^{76} + 12 q^{79} - 2 q^{80} - 8 q^{82} + 2 q^{85} + 4 q^{86} + 8 q^{89} + 2 q^{91} - 2 q^{92} - 4 q^{94} - 6 q^{95} + 6 q^{97} - 14 q^{98} + O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.60555 −0.984806 −0.492403 0.870367i $$-0.663881\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 2.60555 0.696363
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.60555 −1.11701 −0.558505 0.829501i $$-0.688625\pi$$
−0.558505 + 0.829501i $$0.688625\pi$$
$$18$$ 0 0
$$19$$ 6.60555 1.51542 0.757709 0.652593i $$-0.226319\pi$$
0.757709 + 0.652593i $$0.226319\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.60555 −0.960324 −0.480162 0.877180i $$-0.659422\pi$$
−0.480162 + 0.877180i $$0.659422\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −2.60555 −0.492403
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 4.60555 0.789846
$$35$$ 2.60555 0.440419
$$36$$ 0 0
$$37$$ 11.2111 1.84309 0.921547 0.388267i $$-0.126926\pi$$
0.921547 + 0.388267i $$0.126926\pi$$
$$38$$ −6.60555 −1.07156
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −3.21110 −0.501490 −0.250745 0.968053i $$-0.580676\pi$$
−0.250745 + 0.968053i $$0.580676\pi$$
$$42$$ 0 0
$$43$$ 5.21110 0.794686 0.397343 0.917670i $$-0.369932\pi$$
0.397343 + 0.917670i $$0.369932\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.60555 0.679051
$$47$$ 9.21110 1.34358 0.671789 0.740743i $$-0.265526\pi$$
0.671789 + 0.740743i $$0.265526\pi$$
$$48$$ 0 0
$$49$$ −0.211103 −0.0301575
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 2.60555 0.348181
$$57$$ 0 0
$$58$$ −4.60555 −0.604739
$$59$$ 9.21110 1.19918 0.599592 0.800306i $$-0.295330\pi$$
0.599592 + 0.800306i $$0.295330\pi$$
$$60$$ 0 0
$$61$$ −7.21110 −0.923287 −0.461644 0.887066i $$-0.652740\pi$$
−0.461644 + 0.887066i $$0.652740\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −7.21110 −0.880976 −0.440488 0.897758i $$-0.645195\pi$$
−0.440488 + 0.897758i $$0.645195\pi$$
$$68$$ −4.60555 −0.558505
$$69$$ 0 0
$$70$$ −2.60555 −0.311423
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 6.60555 0.773121 0.386561 0.922264i $$-0.373663\pi$$
0.386561 + 0.922264i $$0.373663\pi$$
$$74$$ −11.2111 −1.30326
$$75$$ 0 0
$$76$$ 6.60555 0.757709
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.21110 −0.136260 −0.0681298 0.997676i $$-0.521703\pi$$
−0.0681298 + 0.997676i $$0.521703\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 3.21110 0.354607
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 4.60555 0.499542
$$86$$ −5.21110 −0.561928
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.21110 −0.340376 −0.170188 0.985412i $$-0.554438\pi$$
−0.170188 + 0.985412i $$0.554438\pi$$
$$90$$ 0 0
$$91$$ −2.60555 −0.273136
$$92$$ −4.60555 −0.480162
$$93$$ 0 0
$$94$$ −9.21110 −0.950053
$$95$$ −6.60555 −0.677715
$$96$$ 0 0
$$97$$ 6.60555 0.670692 0.335346 0.942095i $$-0.391147\pi$$
0.335346 + 0.942095i $$0.391147\pi$$
$$98$$ 0.211103 0.0213246
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 13.8167 1.37481 0.687404 0.726275i $$-0.258750\pi$$
0.687404 + 0.726275i $$0.258750\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 18.6056 1.78209 0.891044 0.453916i $$-0.149974\pi$$
0.891044 + 0.453916i $$0.149974\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.60555 −0.246201
$$113$$ −7.39445 −0.695611 −0.347806 0.937567i $$-0.613073\pi$$
−0.347806 + 0.937567i $$0.613073\pi$$
$$114$$ 0 0
$$115$$ 4.60555 0.429470
$$116$$ 4.60555 0.427615
$$117$$ 0 0
$$118$$ −9.21110 −0.847951
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 7.21110 0.652863
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −1.21110 −0.107468 −0.0537340 0.998555i $$-0.517112\pi$$
−0.0537340 + 0.998555i $$0.517112\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 1.00000 0.0877058
$$131$$ 16.6056 1.45083 0.725417 0.688310i $$-0.241647\pi$$
0.725417 + 0.688310i $$0.241647\pi$$
$$132$$ 0 0
$$133$$ −17.2111 −1.49239
$$134$$ 7.21110 0.622944
$$135$$ 0 0
$$136$$ 4.60555 0.394923
$$137$$ −21.6333 −1.84826 −0.924129 0.382080i $$-0.875208\pi$$
−0.924129 + 0.382080i $$0.875208\pi$$
$$138$$ 0 0
$$139$$ 14.4222 1.22328 0.611638 0.791138i $$-0.290511\pi$$
0.611638 + 0.791138i $$0.290511\pi$$
$$140$$ 2.60555 0.220209
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −4.60555 −0.382470
$$146$$ −6.60555 −0.546679
$$147$$ 0 0
$$148$$ 11.2111 0.921547
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.42221 0.685389 0.342695 0.939447i $$-0.388660\pi$$
0.342695 + 0.939447i $$0.388660\pi$$
$$152$$ −6.60555 −0.535781
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2.00000 −0.160644
$$156$$ 0 0
$$157$$ 11.2111 0.894743 0.447372 0.894348i $$-0.352360\pi$$
0.447372 + 0.894348i $$0.352360\pi$$
$$158$$ 1.21110 0.0963501
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −3.21110 −0.250745
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.4222 1.42555 0.712777 0.701391i $$-0.247437\pi$$
0.712777 + 0.701391i $$0.247437\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −4.60555 −0.353230
$$171$$ 0 0
$$172$$ 5.21110 0.397343
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 0 0
$$175$$ −2.60555 −0.196961
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 3.21110 0.240682
$$179$$ −11.0278 −0.824253 −0.412127 0.911127i $$-0.635214\pi$$
−0.412127 + 0.911127i $$0.635214\pi$$
$$180$$ 0 0
$$181$$ −19.2111 −1.42795 −0.713975 0.700171i $$-0.753107\pi$$
−0.713975 + 0.700171i $$0.753107\pi$$
$$182$$ 2.60555 0.193136
$$183$$ 0 0
$$184$$ 4.60555 0.339526
$$185$$ −11.2111 −0.824257
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 9.21110 0.671789
$$189$$ 0 0
$$190$$ 6.60555 0.479217
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −14.6056 −1.05133 −0.525665 0.850691i $$-0.676184\pi$$
−0.525665 + 0.850691i $$0.676184\pi$$
$$194$$ −6.60555 −0.474251
$$195$$ 0 0
$$196$$ −0.211103 −0.0150788
$$197$$ 24.4222 1.74001 0.870005 0.493043i $$-0.164115\pi$$
0.870005 + 0.493043i $$0.164115\pi$$
$$198$$ 0 0
$$199$$ 5.21110 0.369405 0.184703 0.982794i $$-0.440868\pi$$
0.184703 + 0.982794i $$0.440868\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −13.8167 −0.972136
$$203$$ −12.0000 −0.842235
$$204$$ 0 0
$$205$$ 3.21110 0.224273
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −25.2111 −1.73560 −0.867802 0.496910i $$-0.834468\pi$$
−0.867802 + 0.496910i $$0.834468\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −5.21110 −0.355394
$$216$$ 0 0
$$217$$ −5.21110 −0.353753
$$218$$ −18.6056 −1.26013
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.60555 −0.309803
$$222$$ 0 0
$$223$$ −17.3944 −1.16482 −0.582409 0.812896i $$-0.697890\pi$$
−0.582409 + 0.812896i $$0.697890\pi$$
$$224$$ 2.60555 0.174091
$$225$$ 0 0
$$226$$ 7.39445 0.491871
$$227$$ −6.42221 −0.426257 −0.213128 0.977024i $$-0.568365\pi$$
−0.213128 + 0.977024i $$0.568365\pi$$
$$228$$ 0 0
$$229$$ 0.183346 0.0121159 0.00605793 0.999982i $$-0.498072\pi$$
0.00605793 + 0.999982i $$0.498072\pi$$
$$230$$ −4.60555 −0.303681
$$231$$ 0 0
$$232$$ −4.60555 −0.302369
$$233$$ 1.81665 0.119013 0.0595065 0.998228i $$-0.481047\pi$$
0.0595065 + 0.998228i $$0.481047\pi$$
$$234$$ 0 0
$$235$$ −9.21110 −0.600866
$$236$$ 9.21110 0.599592
$$237$$ 0 0
$$238$$ −12.0000 −0.777844
$$239$$ −18.4222 −1.19163 −0.595817 0.803120i $$-0.703172\pi$$
−0.595817 + 0.803120i $$0.703172\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 0 0
$$244$$ −7.21110 −0.461644
$$245$$ 0.211103 0.0134868
$$246$$ 0 0
$$247$$ 6.60555 0.420301
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −4.60555 −0.290700 −0.145350 0.989380i $$-0.546431\pi$$
−0.145350 + 0.989380i $$0.546431\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 1.21110 0.0759913
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 11.0278 0.687893 0.343946 0.938989i $$-0.388236\pi$$
0.343946 + 0.938989i $$0.388236\pi$$
$$258$$ 0 0
$$259$$ −29.2111 −1.81509
$$260$$ −1.00000 −0.0620174
$$261$$ 0 0
$$262$$ −16.6056 −1.02589
$$263$$ 11.0278 0.680001 0.340000 0.940425i $$-0.389573\pi$$
0.340000 + 0.940425i $$0.389573\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 17.2111 1.05528
$$267$$ 0 0
$$268$$ −7.21110 −0.440488
$$269$$ −10.1833 −0.620890 −0.310445 0.950591i $$-0.600478\pi$$
−0.310445 + 0.950591i $$0.600478\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ −4.60555 −0.279253
$$273$$ 0 0
$$274$$ 21.6333 1.30692
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −31.2111 −1.87529 −0.937647 0.347590i $$-0.887000\pi$$
−0.937647 + 0.347590i $$0.887000\pi$$
$$278$$ −14.4222 −0.864986
$$279$$ 0 0
$$280$$ −2.60555 −0.155711
$$281$$ 3.21110 0.191558 0.0957792 0.995403i $$-0.469466\pi$$
0.0957792 + 0.995403i $$0.469466\pi$$
$$282$$ 0 0
$$283$$ −1.21110 −0.0719926 −0.0359963 0.999352i $$-0.511460\pi$$
−0.0359963 + 0.999352i $$0.511460\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.36669 0.493870
$$288$$ 0 0
$$289$$ 4.21110 0.247712
$$290$$ 4.60555 0.270447
$$291$$ 0 0
$$292$$ 6.60555 0.386561
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ −9.21110 −0.536291
$$296$$ −11.2111 −0.651632
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ −4.60555 −0.266346
$$300$$ 0 0
$$301$$ −13.5778 −0.782611
$$302$$ −8.42221 −0.484643
$$303$$ 0 0
$$304$$ 6.60555 0.378854
$$305$$ 7.21110 0.412907
$$306$$ 0 0
$$307$$ 4.78890 0.273317 0.136658 0.990618i $$-0.456364\pi$$
0.136658 + 0.990618i $$0.456364\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 2.00000 0.113592
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 11.2111 0.633689 0.316844 0.948478i $$-0.397377\pi$$
0.316844 + 0.948478i $$0.397377\pi$$
$$314$$ −11.2111 −0.632679
$$315$$ 0 0
$$316$$ −1.21110 −0.0681298
$$317$$ 12.4222 0.697701 0.348850 0.937178i $$-0.386572\pi$$
0.348850 + 0.937178i $$0.386572\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −12.0000 −0.668734
$$323$$ −30.4222 −1.69274
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ −2.00000 −0.110770
$$327$$ 0 0
$$328$$ 3.21110 0.177303
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −5.39445 −0.296506 −0.148253 0.988949i $$-0.547365\pi$$
−0.148253 + 0.988949i $$0.547365\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ −18.4222 −1.00802
$$335$$ 7.21110 0.393985
$$336$$ 0 0
$$337$$ 26.8444 1.46231 0.731154 0.682212i $$-0.238982\pi$$
0.731154 + 0.682212i $$0.238982\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 4.60555 0.249771
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 18.7889 1.01451
$$344$$ −5.21110 −0.280964
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ −14.7889 −0.793910 −0.396955 0.917838i $$-0.629933\pi$$
−0.396955 + 0.917838i $$0.629933\pi$$
$$348$$ 0 0
$$349$$ −11.8167 −0.632531 −0.316265 0.948671i $$-0.602429\pi$$
−0.316265 + 0.948671i $$0.602429\pi$$
$$350$$ 2.60555 0.139273
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −21.6333 −1.15142 −0.575712 0.817652i $$-0.695275\pi$$
−0.575712 + 0.817652i $$0.695275\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ −3.21110 −0.170188
$$357$$ 0 0
$$358$$ 11.0278 0.582835
$$359$$ 30.4222 1.60562 0.802811 0.596233i $$-0.203337\pi$$
0.802811 + 0.596233i $$0.203337\pi$$
$$360$$ 0 0
$$361$$ 24.6333 1.29649
$$362$$ 19.2111 1.00971
$$363$$ 0 0
$$364$$ −2.60555 −0.136568
$$365$$ −6.60555 −0.345750
$$366$$ 0 0
$$367$$ −6.78890 −0.354378 −0.177189 0.984177i $$-0.556700\pi$$
−0.177189 + 0.984177i $$0.556700\pi$$
$$368$$ −4.60555 −0.240081
$$369$$ 0 0
$$370$$ 11.2111 0.582837
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −0.788897 −0.0408476 −0.0204238 0.999791i $$-0.506502\pi$$
−0.0204238 + 0.999791i $$0.506502\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.21110 −0.475026
$$377$$ 4.60555 0.237198
$$378$$ 0 0
$$379$$ −26.6056 −1.36664 −0.683318 0.730121i $$-0.739464\pi$$
−0.683318 + 0.730121i $$0.739464\pi$$
$$380$$ −6.60555 −0.338858
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ −18.4222 −0.941331 −0.470665 0.882312i $$-0.655986\pi$$
−0.470665 + 0.882312i $$0.655986\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.6056 0.743403
$$387$$ 0 0
$$388$$ 6.60555 0.335346
$$389$$ 1.81665 0.0921080 0.0460540 0.998939i $$-0.485335\pi$$
0.0460540 + 0.998939i $$0.485335\pi$$
$$390$$ 0 0
$$391$$ 21.2111 1.07269
$$392$$ 0.211103 0.0106623
$$393$$ 0 0
$$394$$ −24.4222 −1.23037
$$395$$ 1.21110 0.0609372
$$396$$ 0 0
$$397$$ 35.2111 1.76719 0.883597 0.468248i $$-0.155114\pi$$
0.883597 + 0.468248i $$0.155114\pi$$
$$398$$ −5.21110 −0.261209
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −33.6333 −1.67957 −0.839784 0.542921i $$-0.817318\pi$$
−0.839784 + 0.542921i $$0.817318\pi$$
$$402$$ 0 0
$$403$$ 2.00000 0.0996271
$$404$$ 13.8167 0.687404
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −13.6333 −0.674124 −0.337062 0.941483i $$-0.609433\pi$$
−0.337062 + 0.941483i $$0.609433\pi$$
$$410$$ −3.21110 −0.158585
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 29.4500 1.43872 0.719362 0.694635i $$-0.244434\pi$$
0.719362 + 0.694635i $$0.244434\pi$$
$$420$$ 0 0
$$421$$ −9.02776 −0.439986 −0.219993 0.975501i $$-0.570603\pi$$
−0.219993 + 0.975501i $$0.570603\pi$$
$$422$$ 25.2111 1.22726
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −4.60555 −0.223402
$$426$$ 0 0
$$427$$ 18.7889 0.909258
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 5.21110 0.251302
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ −22.8444 −1.09783 −0.548916 0.835877i $$-0.684959\pi$$
−0.548916 + 0.835877i $$0.684959\pi$$
$$434$$ 5.21110 0.250141
$$435$$ 0 0
$$436$$ 18.6056 0.891044
$$437$$ −30.4222 −1.45529
$$438$$ 0 0
$$439$$ 32.8444 1.56758 0.783789 0.621027i $$-0.213284\pi$$
0.783789 + 0.621027i $$0.213284\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4.60555 0.219064
$$443$$ 39.6333 1.88304 0.941518 0.336964i $$-0.109400\pi$$
0.941518 + 0.336964i $$0.109400\pi$$
$$444$$ 0 0
$$445$$ 3.21110 0.152221
$$446$$ 17.3944 0.823651
$$447$$ 0 0
$$448$$ −2.60555 −0.123101
$$449$$ 9.63331 0.454624 0.227312 0.973822i $$-0.427006\pi$$
0.227312 + 0.973822i $$0.427006\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −7.39445 −0.347806
$$453$$ 0 0
$$454$$ 6.42221 0.301409
$$455$$ 2.60555 0.122150
$$456$$ 0 0
$$457$$ −17.3944 −0.813678 −0.406839 0.913500i $$-0.633369\pi$$
−0.406839 + 0.913500i $$0.633369\pi$$
$$458$$ −0.183346 −0.00856720
$$459$$ 0 0
$$460$$ 4.60555 0.214735
$$461$$ 3.21110 0.149556 0.0747780 0.997200i $$-0.476175\pi$$
0.0747780 + 0.997200i $$0.476175\pi$$
$$462$$ 0 0
$$463$$ 12.1833 0.566208 0.283104 0.959089i $$-0.408636\pi$$
0.283104 + 0.959089i $$0.408636\pi$$
$$464$$ 4.60555 0.213807
$$465$$ 0 0
$$466$$ −1.81665 −0.0841549
$$467$$ 5.57779 0.258110 0.129055 0.991637i $$-0.458806\pi$$
0.129055 + 0.991637i $$0.458806\pi$$
$$468$$ 0 0
$$469$$ 18.7889 0.867591
$$470$$ 9.21110 0.424876
$$471$$ 0 0
$$472$$ −9.21110 −0.423975
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 6.60555 0.303083
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ 18.4222 0.842612
$$479$$ −18.4222 −0.841732 −0.420866 0.907123i $$-0.638274\pi$$
−0.420866 + 0.907123i $$0.638274\pi$$
$$480$$ 0 0
$$481$$ 11.2111 0.511182
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ −6.60555 −0.299943
$$486$$ 0 0
$$487$$ 2.97224 0.134685 0.0673426 0.997730i $$-0.478548\pi$$
0.0673426 + 0.997730i $$0.478548\pi$$
$$488$$ 7.21110 0.326431
$$489$$ 0 0
$$490$$ −0.211103 −0.00953664
$$491$$ −32.2389 −1.45492 −0.727460 0.686150i $$-0.759299\pi$$
−0.727460 + 0.686150i $$0.759299\pi$$
$$492$$ 0 0
$$493$$ −21.2111 −0.955300
$$494$$ −6.60555 −0.297198
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ −31.2666 −1.40250
$$498$$ 0 0
$$499$$ 27.8167 1.24524 0.622622 0.782523i $$-0.286067\pi$$
0.622622 + 0.782523i $$0.286067\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 4.60555 0.205556
$$503$$ −20.2389 −0.902406 −0.451203 0.892421i $$-0.649005\pi$$
−0.451203 + 0.892421i $$0.649005\pi$$
$$504$$ 0 0
$$505$$ −13.8167 −0.614833
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −1.21110 −0.0537340
$$509$$ −9.63331 −0.426989 −0.213494 0.976944i $$-0.568485\pi$$
−0.213494 + 0.976944i $$0.568485\pi$$
$$510$$ 0 0
$$511$$ −17.2111 −0.761374
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −11.0278 −0.486413
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 29.2111 1.28346
$$519$$ 0 0
$$520$$ 1.00000 0.0438529
$$521$$ 21.2111 0.929275 0.464638 0.885501i $$-0.346185\pi$$
0.464638 + 0.885501i $$0.346185\pi$$
$$522$$ 0 0
$$523$$ 38.4222 1.68009 0.840043 0.542520i $$-0.182530\pi$$
0.840043 + 0.542520i $$0.182530\pi$$
$$524$$ 16.6056 0.725417
$$525$$ 0 0
$$526$$ −11.0278 −0.480833
$$527$$ −9.21110 −0.401242
$$528$$ 0 0
$$529$$ −1.78890 −0.0777781
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −17.2111 −0.746196
$$533$$ −3.21110 −0.139088
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 7.21110 0.311472
$$537$$ 0 0
$$538$$ 10.1833 0.439035
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9.02776 −0.388134 −0.194067 0.980988i $$-0.562168\pi$$
−0.194067 + 0.980988i $$0.562168\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ 0 0
$$544$$ 4.60555 0.197461
$$545$$ −18.6056 −0.796974
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −21.6333 −0.924129
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 30.4222 1.29603
$$552$$ 0 0
$$553$$ 3.15559 0.134189
$$554$$ 31.2111 1.32603
$$555$$ 0 0
$$556$$ 14.4222 0.611638
$$557$$ 18.8444 0.798463 0.399232 0.916850i $$-0.369277\pi$$
0.399232 + 0.916850i $$0.369277\pi$$
$$558$$ 0 0
$$559$$ 5.21110 0.220406
$$560$$ 2.60555 0.110105
$$561$$ 0 0
$$562$$ −3.21110 −0.135452
$$563$$ 15.6333 0.658865 0.329433 0.944179i $$-0.393143\pi$$
0.329433 + 0.944179i $$0.393143\pi$$
$$564$$ 0 0
$$565$$ 7.39445 0.311087
$$566$$ 1.21110 0.0509064
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ 26.7889 1.12305 0.561525 0.827460i $$-0.310215\pi$$
0.561525 + 0.827460i $$0.310215\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −8.36669 −0.349219
$$575$$ −4.60555 −0.192065
$$576$$ 0 0
$$577$$ 27.8167 1.15802 0.579011 0.815320i $$-0.303439\pi$$
0.579011 + 0.815320i $$0.303439\pi$$
$$578$$ −4.21110 −0.175159
$$579$$ 0 0
$$580$$ −4.60555 −0.191235
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −6.60555 −0.273340
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 6.42221 0.265073 0.132536 0.991178i $$-0.457688\pi$$
0.132536 + 0.991178i $$0.457688\pi$$
$$588$$ 0 0
$$589$$ 13.2111 0.544354
$$590$$ 9.21110 0.379215
$$591$$ 0 0
$$592$$ 11.2111 0.460773
$$593$$ −20.7889 −0.853698 −0.426849 0.904323i $$-0.640376\pi$$
−0.426849 + 0.904323i $$0.640376\pi$$
$$594$$ 0 0
$$595$$ −12.0000 −0.491952
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 4.60555 0.188335
$$599$$ 36.8444 1.50542 0.752711 0.658351i $$-0.228746\pi$$
0.752711 + 0.658351i $$0.228746\pi$$
$$600$$ 0 0
$$601$$ 17.6333 0.719278 0.359639 0.933092i $$-0.382900\pi$$
0.359639 + 0.933092i $$0.382900\pi$$
$$602$$ 13.5778 0.553390
$$603$$ 0 0
$$604$$ 8.42221 0.342695
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ 41.2111 1.67271 0.836354 0.548190i $$-0.184683\pi$$
0.836354 + 0.548190i $$0.184683\pi$$
$$608$$ −6.60555 −0.267890
$$609$$ 0 0
$$610$$ −7.21110 −0.291969
$$611$$ 9.21110 0.372641
$$612$$ 0 0
$$613$$ 5.63331 0.227527 0.113764 0.993508i $$-0.463709\pi$$
0.113764 + 0.993508i $$0.463709\pi$$
$$614$$ −4.78890 −0.193264
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.8444 0.758647 0.379324 0.925264i $$-0.376157\pi$$
0.379324 + 0.925264i $$0.376157\pi$$
$$618$$ 0 0
$$619$$ −2.60555 −0.104726 −0.0523630 0.998628i $$-0.516675\pi$$
−0.0523630 + 0.998628i $$0.516675\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 8.36669 0.335204
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −11.2111 −0.448086
$$627$$ 0 0
$$628$$ 11.2111 0.447372
$$629$$ −51.6333 −2.05875
$$630$$ 0 0
$$631$$ 23.2111 0.924019 0.462010 0.886875i $$-0.347129\pi$$
0.462010 + 0.886875i $$0.347129\pi$$
$$632$$ 1.21110 0.0481751
$$633$$ 0 0
$$634$$ −12.4222 −0.493349
$$635$$ 1.21110 0.0480611
$$636$$ 0 0
$$637$$ −0.211103 −0.00836419
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ 48.0555 1.89512 0.947562 0.319571i $$-0.103539\pi$$
0.947562 + 0.319571i $$0.103539\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 30.4222 1.19695
$$647$$ 35.0278 1.37708 0.688542 0.725197i $$-0.258251\pi$$
0.688542 + 0.725197i $$0.258251\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −1.00000 −0.0392232
$$651$$ 0 0
$$652$$ 2.00000 0.0783260
$$653$$ 2.78890 0.109138 0.0545690 0.998510i $$-0.482622\pi$$
0.0545690 + 0.998510i $$0.482622\pi$$
$$654$$ 0 0
$$655$$ −16.6056 −0.648833
$$656$$ −3.21110 −0.125372
$$657$$ 0 0
$$658$$ 24.0000 0.935617
$$659$$ −13.8167 −0.538220 −0.269110 0.963109i $$-0.586730\pi$$
−0.269110 + 0.963109i $$0.586730\pi$$
$$660$$ 0 0
$$661$$ −21.0278 −0.817885 −0.408942 0.912560i $$-0.634102\pi$$
−0.408942 + 0.912560i $$0.634102\pi$$
$$662$$ 5.39445 0.209661
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 17.2111 0.667418
$$666$$ 0 0
$$667$$ −21.2111 −0.821297
$$668$$ 18.4222 0.712777
$$669$$ 0 0
$$670$$ −7.21110 −0.278589
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −19.2111 −0.740534 −0.370267 0.928925i $$-0.620734\pi$$
−0.370267 + 0.928925i $$0.620734\pi$$
$$674$$ −26.8444 −1.03401
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −9.21110 −0.354011 −0.177006 0.984210i $$-0.556641\pi$$
−0.177006 + 0.984210i $$0.556641\pi$$
$$678$$ 0 0
$$679$$ −17.2111 −0.660501
$$680$$ −4.60555 −0.176615
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −5.57779 −0.213428 −0.106714 0.994290i $$-0.534033\pi$$
−0.106714 + 0.994290i $$0.534033\pi$$
$$684$$ 0 0
$$685$$ 21.6333 0.826566
$$686$$ −18.7889 −0.717363
$$687$$ 0 0
$$688$$ 5.21110 0.198671
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −23.8167 −0.906028 −0.453014 0.891503i $$-0.649651\pi$$
−0.453014 + 0.891503i $$0.649651\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 14.7889 0.561379
$$695$$ −14.4222 −0.547065
$$696$$ 0 0
$$697$$ 14.7889 0.560169
$$698$$ 11.8167 0.447267
$$699$$ 0 0
$$700$$ −2.60555 −0.0984806
$$701$$ −16.6056 −0.627183 −0.313592 0.949558i $$-0.601532\pi$$
−0.313592 + 0.949558i $$0.601532\pi$$
$$702$$ 0 0
$$703$$ 74.0555 2.79306
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 21.6333 0.814180
$$707$$ −36.0000 −1.35392
$$708$$ 0 0
$$709$$ 31.4500 1.18113 0.590564 0.806991i $$-0.298905\pi$$
0.590564 + 0.806991i $$0.298905\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 0 0
$$712$$ 3.21110 0.120341
$$713$$ −9.21110 −0.344959
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −11.0278 −0.412127
$$717$$ 0 0
$$718$$ −30.4222 −1.13535
$$719$$ −21.2111 −0.791041 −0.395520 0.918457i $$-0.629436\pi$$
−0.395520 + 0.918457i $$0.629436\pi$$
$$720$$ 0 0
$$721$$ 10.4222 0.388143
$$722$$ −24.6333 −0.916757
$$723$$ 0 0
$$724$$ −19.2111 −0.713975
$$725$$ 4.60555 0.171046
$$726$$ 0 0
$$727$$ −43.6333 −1.61827 −0.809135 0.587623i $$-0.800064\pi$$
−0.809135 + 0.587623i $$0.800064\pi$$
$$728$$ 2.60555 0.0965682
$$729$$ 0 0
$$730$$ 6.60555 0.244482
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 41.6333 1.53776 0.768881 0.639392i $$-0.220814\pi$$
0.768881 + 0.639392i $$0.220814\pi$$
$$734$$ 6.78890 0.250583
$$735$$ 0 0
$$736$$ 4.60555 0.169763
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −26.6056 −0.978701 −0.489351 0.872087i $$-0.662766\pi$$
−0.489351 + 0.872087i $$0.662766\pi$$
$$740$$ −11.2111 −0.412128
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 46.0555 1.68961 0.844806 0.535072i $$-0.179716\pi$$
0.844806 + 0.535072i $$0.179716\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0.788897 0.0288836
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −31.2666 −1.14246
$$750$$ 0 0
$$751$$ −35.2666 −1.28690 −0.643449 0.765489i $$-0.722497\pi$$
−0.643449 + 0.765489i $$0.722497\pi$$
$$752$$ 9.21110 0.335894
$$753$$ 0 0
$$754$$ −4.60555 −0.167724
$$755$$ −8.42221 −0.306515
$$756$$ 0 0
$$757$$ −34.8444 −1.26644 −0.633221 0.773971i $$-0.718268\pi$$
−0.633221 + 0.773971i $$0.718268\pi$$
$$758$$ 26.6056 0.966357
$$759$$ 0 0
$$760$$ 6.60555 0.239609
$$761$$ 9.63331 0.349207 0.174604 0.984639i $$-0.444136\pi$$
0.174604 + 0.984639i $$0.444136\pi$$
$$762$$ 0 0
$$763$$ −48.4777 −1.75501
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 18.4222 0.665621
$$767$$ 9.21110 0.332594
$$768$$ 0 0
$$769$$ −34.8444 −1.25652 −0.628261 0.778003i $$-0.716233\pi$$
−0.628261 + 0.778003i $$0.716233\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.6056 −0.525665
$$773$$ 12.4222 0.446796 0.223398 0.974727i $$-0.428285\pi$$
0.223398 + 0.974727i $$0.428285\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ −6.60555 −0.237125
$$777$$ 0 0
$$778$$ −1.81665 −0.0651302
$$779$$ −21.2111 −0.759967
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −21.2111 −0.758507
$$783$$ 0 0
$$784$$ −0.211103 −0.00753938
$$785$$ −11.2111 −0.400141
$$786$$ 0 0
$$787$$ 38.0000 1.35455 0.677277 0.735728i $$-0.263160\pi$$
0.677277 + 0.735728i $$0.263160\pi$$
$$788$$ 24.4222 0.870005
$$789$$ 0 0
$$790$$ −1.21110 −0.0430891
$$791$$ 19.2666 0.685042
$$792$$ 0 0
$$793$$ −7.21110 −0.256074
$$794$$ −35.2111 −1.24960
$$795$$ 0 0
$$796$$ 5.21110 0.184703
$$797$$ −42.4222 −1.50267 −0.751336 0.659920i $$-0.770590\pi$$
−0.751336 + 0.659920i $$0.770590\pi$$
$$798$$ 0 0
$$799$$ −42.4222 −1.50079
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 33.6333 1.18763
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −12.0000 −0.422944
$$806$$ −2.00000 −0.0704470
$$807$$ 0 0
$$808$$ −13.8167 −0.486068
$$809$$ −6.42221 −0.225793 −0.112896 0.993607i $$-0.536013\pi$$
−0.112896 + 0.993607i $$0.536013\pi$$
$$810$$ 0 0
$$811$$ 49.0278 1.72160 0.860799 0.508946i $$-0.169965\pi$$
0.860799 + 0.508946i $$0.169965\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −2.00000 −0.0700569
$$816$$ 0 0
$$817$$ 34.4222 1.20428
$$818$$ 13.6333 0.476677
$$819$$ 0 0
$$820$$ 3.21110 0.112137
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 0.972244 0.0336862
$$834$$ 0 0
$$835$$ −18.4222 −0.637527
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −29.4500 −1.01733
$$839$$ 42.4222 1.46458 0.732289 0.680994i $$-0.238452\pi$$
0.732289 + 0.680994i $$0.238452\pi$$
$$840$$ 0 0
$$841$$ −7.78890 −0.268583
$$842$$ 9.02776 0.311117
$$843$$ 0 0
$$844$$ −25.2111 −0.867802
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 28.6611 0.984806
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 4.60555 0.157969
$$851$$ −51.6333 −1.76997
$$852$$ 0 0
$$853$$ 24.0555 0.823645 0.411823 0.911264i $$-0.364892\pi$$
0.411823 + 0.911264i $$0.364892\pi$$
$$854$$ −18.7889 −0.642943
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 29.4500 1.00599 0.502996 0.864289i $$-0.332231\pi$$
0.502996 + 0.864289i $$0.332231\pi$$
$$858$$ 0 0
$$859$$ 4.36669 0.148990 0.0744948 0.997221i $$-0.476266\pi$$
0.0744948 + 0.997221i $$0.476266\pi$$
$$860$$ −5.21110 −0.177697
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 42.4222 1.44407 0.722034 0.691857i $$-0.243207\pi$$
0.722034 + 0.691857i $$0.243207\pi$$
$$864$$ 0 0
$$865$$ 12.0000 0.408012
$$866$$ 22.8444 0.776285
$$867$$ 0 0
$$868$$ −5.21110 −0.176876
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −7.21110 −0.244339
$$872$$ −18.6056 −0.630063
$$873$$ 0 0
$$874$$ 30.4222 1.02905
$$875$$ 2.60555 0.0880837
$$876$$ 0 0
$$877$$ 26.0000 0.877958 0.438979 0.898497i $$-0.355340\pi$$
0.438979 + 0.898497i $$0.355340\pi$$
$$878$$ −32.8444 −1.10845
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −26.7889 −0.902541 −0.451270 0.892387i $$-0.649029\pi$$
−0.451270 + 0.892387i $$0.649029\pi$$
$$882$$ 0 0
$$883$$ 14.4222 0.485346 0.242673 0.970108i $$-0.421976\pi$$
0.242673 + 0.970108i $$0.421976\pi$$
$$884$$ −4.60555 −0.154901
$$885$$ 0 0
$$886$$ −39.6333 −1.33151
$$887$$ −4.60555 −0.154639 −0.0773196 0.997006i $$-0.524636\pi$$
−0.0773196 + 0.997006i $$0.524636\pi$$
$$888$$ 0 0
$$889$$ 3.15559 0.105835
$$890$$ −3.21110 −0.107636
$$891$$ 0 0
$$892$$ −17.3944 −0.582409
$$893$$ 60.8444 2.03608
$$894$$ 0 0
$$895$$ 11.0278 0.368617
$$896$$ 2.60555 0.0870454
$$897$$ 0 0
$$898$$ −9.63331 −0.321468
$$899$$ 9.21110 0.307207
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 7.39445 0.245936
$$905$$ 19.2111 0.638599
$$906$$ 0 0
$$907$$ −12.3667 −0.410629 −0.205315 0.978696i $$-0.565822\pi$$
−0.205315 + 0.978696i $$0.565822\pi$$
$$908$$ −6.42221 −0.213128
$$909$$ 0 0
$$910$$ −2.60555 −0.0863732
$$911$$ −15.6333 −0.517955 −0.258977 0.965883i $$-0.583385\pi$$
−0.258977 + 0.965883i $$0.583385\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 17.3944 0.575357
$$915$$ 0 0
$$916$$ 0.183346 0.00605793
$$917$$ −43.2666 −1.42879
$$918$$ 0 0
$$919$$ −52.0000 −1.71532 −0.857661 0.514216i $$-0.828083\pi$$
−0.857661 + 0.514216i $$0.828083\pi$$
$$920$$ −4.60555 −0.151841
$$921$$ 0 0
$$922$$ −3.21110 −0.105752
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ 11.2111 0.368619
$$926$$ −12.1833 −0.400370
$$927$$ 0 0
$$928$$ −4.60555 −0.151185
$$929$$ 40.0555 1.31418 0.657089 0.753813i $$-0.271787\pi$$
0.657089 + 0.753813i $$0.271787\pi$$
$$930$$ 0 0
$$931$$ −1.39445 −0.0457012
$$932$$ 1.81665 0.0595065
$$933$$ 0 0
$$934$$ −5.57779 −0.182511
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −13.6333 −0.445381 −0.222690 0.974889i $$-0.571484\pi$$
−0.222690 + 0.974889i $$0.571484\pi$$
$$938$$ −18.7889 −0.613479
$$939$$ 0 0
$$940$$ −9.21110 −0.300433
$$941$$ −21.6333 −0.705226 −0.352613 0.935769i $$-0.614707\pi$$
−0.352613 + 0.935769i $$0.614707\pi$$
$$942$$ 0 0
$$943$$ 14.7889 0.481593
$$944$$ 9.21110 0.299796
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −42.4222 −1.37854 −0.689268 0.724506i $$-0.742068\pi$$
−0.689268 + 0.724506i $$0.742068\pi$$
$$948$$ 0 0
$$949$$ 6.60555 0.214425
$$950$$ −6.60555 −0.214312
$$951$$ 0 0
$$952$$ −12.0000 −0.388922
$$953$$ −16.6056 −0.537907 −0.268953 0.963153i $$-0.586678\pi$$
−0.268953 + 0.963153i $$0.586678\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ −18.4222 −0.595817
$$957$$ 0 0
$$958$$ 18.4222 0.595194
$$959$$ 56.3667 1.82018
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −11.2111 −0.361460
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 14.6056 0.470169
$$966$$ 0 0
$$967$$ −26.6056 −0.855577 −0.427788 0.903879i $$-0.640707\pi$$
−0.427788 + 0.903879i $$0.640707\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 6.60555 0.212091
$$971$$ 19.3944 0.622397 0.311199 0.950345i $$-0.399270\pi$$
0.311199 + 0.950345i $$0.399270\pi$$
$$972$$ 0 0
$$973$$ −37.5778 −1.20469
$$974$$ −2.97224 −0.0952368
$$975$$ 0 0
$$976$$ −7.21110 −0.230822
$$977$$ −57.6333 −1.84385 −0.921926 0.387365i $$-0.873385\pi$$
−0.921926 + 0.387365i $$0.873385\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0.211103 0.00674342
$$981$$ 0 0
$$982$$ 32.2389 1.02878
$$983$$ −27.6333 −0.881366 −0.440683 0.897663i $$-0.645264\pi$$
−0.440683 + 0.897663i $$0.645264\pi$$
$$984$$ 0 0
$$985$$ −24.4222 −0.778156
$$986$$ 21.2111 0.675499
$$987$$ 0 0
$$988$$ 6.60555 0.210151
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −31.6333 −1.00487 −0.502433 0.864616i $$-0.667561\pi$$
−0.502433 + 0.864616i $$0.667561\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 31.2666 0.991717
$$995$$ −5.21110 −0.165203
$$996$$ 0 0
$$997$$ 4.78890 0.151666 0.0758330 0.997121i $$-0.475838\pi$$
0.0758330 + 0.997121i $$0.475838\pi$$
$$998$$ −27.8167 −0.880521
$$999$$ 0 0
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 1170.2.a.p.1.1 2
3.2 odd 2 1170.2.a.q.1.1 yes 2
4.3 odd 2 9360.2.a.cf.1.2 2
5.2 odd 4 5850.2.e.bl.5149.1 4
5.3 odd 4 5850.2.e.bl.5149.4 4
5.4 even 2 5850.2.a.ck.1.2 2
12.11 even 2 9360.2.a.cn.1.2 2
15.2 even 4 5850.2.e.bj.5149.3 4
15.8 even 4 5850.2.e.bj.5149.2 4
15.14 odd 2 5850.2.a.ce.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1170.2.a.p.1.1 2 1.1 even 1 trivial
1170.2.a.q.1.1 yes 2 3.2 odd 2
5850.2.a.ce.1.2 2 15.14 odd 2
5850.2.a.ck.1.2 2 5.4 even 2
5850.2.e.bj.5149.2 4 15.8 even 4
5850.2.e.bj.5149.3 4 15.2 even 4
5850.2.e.bl.5149.1 4 5.2 odd 4
5850.2.e.bl.5149.4 4 5.3 odd 4
9360.2.a.cf.1.2 2 4.3 odd 2
9360.2.a.cn.1.2 2 12.11 even 2