# Properties

 Label 2070.2.a.w.1.1 Level $2070$ Weight $2$ Character 2070.1 Self dual yes Analytic conductor $16.529$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.5290332184$$ 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: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 2070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.30278 q^{11} -0.302776 q^{13} -0.302776 q^{14} +1.00000 q^{16} +3.90833 q^{17} -4.90833 q^{19} -1.00000 q^{20} +5.30278 q^{22} +1.00000 q^{23} +1.00000 q^{25} -0.302776 q^{26} -0.302776 q^{28} -4.60555 q^{29} +2.90833 q^{31} +1.00000 q^{32} +3.90833 q^{34} +0.302776 q^{35} +8.00000 q^{37} -4.90833 q^{38} -1.00000 q^{40} +9.90833 q^{41} +5.21110 q^{43} +5.30278 q^{44} +1.00000 q^{46} -4.60555 q^{47} -6.90833 q^{49} +1.00000 q^{50} -0.302776 q^{52} -3.21110 q^{53} -5.30278 q^{55} -0.302776 q^{56} -4.60555 q^{58} +10.6056 q^{59} -6.51388 q^{61} +2.90833 q^{62} +1.00000 q^{64} +0.302776 q^{65} -4.00000 q^{67} +3.90833 q^{68} +0.302776 q^{70} +12.6972 q^{71} +15.8167 q^{73} +8.00000 q^{74} -4.90833 q^{76} -1.60555 q^{77} +14.4222 q^{79} -1.00000 q^{80} +9.90833 q^{82} +3.21110 q^{83} -3.90833 q^{85} +5.21110 q^{86} +5.30278 q^{88} +0.0916731 q^{91} +1.00000 q^{92} -4.60555 q^{94} +4.90833 q^{95} +2.69722 q^{97} -6.90833 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98} + O(q^{100})$$

## 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
$$7$$ −0.302776 −0.114438 −0.0572192 0.998362i $$-0.518223\pi$$
−0.0572192 + 0.998362i $$0.518223\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 5.30278 1.59885 0.799424 0.600768i $$-0.205138\pi$$
0.799424 + 0.600768i $$0.205138\pi$$
$$12$$ 0 0
$$13$$ −0.302776 −0.0839749 −0.0419874 0.999118i $$-0.513369\pi$$
−0.0419874 + 0.999118i $$0.513369\pi$$
$$14$$ −0.302776 −0.0809202
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.90833 0.947909 0.473954 0.880549i $$-0.342826\pi$$
0.473954 + 0.880549i $$0.342826\pi$$
$$18$$ 0 0
$$19$$ −4.90833 −1.12605 −0.563024 0.826441i $$-0.690362\pi$$
−0.563024 + 0.826441i $$0.690362\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 5.30278 1.13056
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −0.302776 −0.0593792
$$27$$ 0 0
$$28$$ −0.302776 −0.0572192
$$29$$ −4.60555 −0.855229 −0.427615 0.903961i $$-0.640646\pi$$
−0.427615 + 0.903961i $$0.640646\pi$$
$$30$$ 0 0
$$31$$ 2.90833 0.522351 0.261175 0.965291i $$-0.415890\pi$$
0.261175 + 0.965291i $$0.415890\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 3.90833 0.670273
$$35$$ 0.302776 0.0511784
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ −4.90833 −0.796236
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 9.90833 1.54742 0.773710 0.633540i $$-0.218399\pi$$
0.773710 + 0.633540i $$0.218399\pi$$
$$42$$ 0 0
$$43$$ 5.21110 0.794686 0.397343 0.917670i $$-0.369932\pi$$
0.397343 + 0.917670i $$0.369932\pi$$
$$44$$ 5.30278 0.799424
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −4.60555 −0.671789 −0.335894 0.941900i $$-0.609039\pi$$
−0.335894 + 0.941900i $$0.609039\pi$$
$$48$$ 0 0
$$49$$ −6.90833 −0.986904
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −0.302776 −0.0419874
$$53$$ −3.21110 −0.441079 −0.220539 0.975378i $$-0.570782\pi$$
−0.220539 + 0.975378i $$0.570782\pi$$
$$54$$ 0 0
$$55$$ −5.30278 −0.715026
$$56$$ −0.302776 −0.0404601
$$57$$ 0 0
$$58$$ −4.60555 −0.604739
$$59$$ 10.6056 1.38073 0.690363 0.723464i $$-0.257451\pi$$
0.690363 + 0.723464i $$0.257451\pi$$
$$60$$ 0 0
$$61$$ −6.51388 −0.834017 −0.417008 0.908903i $$-0.636921\pi$$
−0.417008 + 0.908903i $$0.636921\pi$$
$$62$$ 2.90833 0.369358
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0.302776 0.0375547
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 3.90833 0.473954
$$69$$ 0 0
$$70$$ 0.302776 0.0361886
$$71$$ 12.6972 1.50688 0.753442 0.657515i $$-0.228392\pi$$
0.753442 + 0.657515i $$0.228392\pi$$
$$72$$ 0 0
$$73$$ 15.8167 1.85120 0.925600 0.378504i $$-0.123561\pi$$
0.925600 + 0.378504i $$0.123561\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −4.90833 −0.563024
$$77$$ −1.60555 −0.182970
$$78$$ 0 0
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 9.90833 1.09419
$$83$$ 3.21110 0.352464 0.176232 0.984349i $$-0.443609\pi$$
0.176232 + 0.984349i $$0.443609\pi$$
$$84$$ 0 0
$$85$$ −3.90833 −0.423918
$$86$$ 5.21110 0.561928
$$87$$ 0 0
$$88$$ 5.30278 0.565278
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0.0916731 0.00960995
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ −4.60555 −0.475026
$$95$$ 4.90833 0.503584
$$96$$ 0 0
$$97$$ 2.69722 0.273862 0.136931 0.990581i $$-0.456276\pi$$
0.136931 + 0.990581i $$0.456276\pi$$
$$98$$ −6.90833 −0.697846
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 4.60555 0.458269 0.229135 0.973395i $$-0.426410\pi$$
0.229135 + 0.973395i $$0.426410\pi$$
$$102$$ 0 0
$$103$$ −17.1194 −1.68683 −0.843414 0.537265i $$-0.819458\pi$$
−0.843414 + 0.537265i $$0.819458\pi$$
$$104$$ −0.302776 −0.0296896
$$105$$ 0 0
$$106$$ −3.21110 −0.311890
$$107$$ −4.60555 −0.445235 −0.222618 0.974906i $$-0.571460\pi$$
−0.222618 + 0.974906i $$0.571460\pi$$
$$108$$ 0 0
$$109$$ 19.5139 1.86909 0.934545 0.355844i $$-0.115807\pi$$
0.934545 + 0.355844i $$0.115807\pi$$
$$110$$ −5.30278 −0.505600
$$111$$ 0 0
$$112$$ −0.302776 −0.0286096
$$113$$ −12.4222 −1.16858 −0.584291 0.811544i $$-0.698627\pi$$
−0.584291 + 0.811544i $$0.698627\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ −4.60555 −0.427615
$$117$$ 0 0
$$118$$ 10.6056 0.976320
$$119$$ −1.18335 −0.108477
$$120$$ 0 0
$$121$$ 17.1194 1.55631
$$122$$ −6.51388 −0.589739
$$123$$ 0 0
$$124$$ 2.90833 0.261175
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −11.8167 −1.04856 −0.524279 0.851546i $$-0.675665\pi$$
−0.524279 + 0.851546i $$0.675665\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0.302776 0.0265552
$$131$$ −3.21110 −0.280555 −0.140278 0.990112i $$-0.544800\pi$$
−0.140278 + 0.990112i $$0.544800\pi$$
$$132$$ 0 0
$$133$$ 1.48612 0.128863
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 3.90833 0.335136
$$137$$ 6.90833 0.590218 0.295109 0.955464i $$-0.404644\pi$$
0.295109 + 0.955464i $$0.404644\pi$$
$$138$$ 0 0
$$139$$ −5.39445 −0.457551 −0.228776 0.973479i $$-0.573472\pi$$
−0.228776 + 0.973479i $$0.573472\pi$$
$$140$$ 0.302776 0.0255892
$$141$$ 0 0
$$142$$ 12.6972 1.06553
$$143$$ −1.60555 −0.134263
$$144$$ 0 0
$$145$$ 4.60555 0.382470
$$146$$ 15.8167 1.30900
$$147$$ 0 0
$$148$$ 8.00000 0.657596
$$149$$ −9.69722 −0.794428 −0.397214 0.917726i $$-0.630023\pi$$
−0.397214 + 0.917726i $$0.630023\pi$$
$$150$$ 0 0
$$151$$ −1.90833 −0.155297 −0.0776487 0.996981i $$-0.524741\pi$$
−0.0776487 + 0.996981i $$0.524741\pi$$
$$152$$ −4.90833 −0.398118
$$153$$ 0 0
$$154$$ −1.60555 −0.129379
$$155$$ −2.90833 −0.233602
$$156$$ 0 0
$$157$$ −11.3944 −0.909376 −0.454688 0.890651i $$-0.650249\pi$$
−0.454688 + 0.890651i $$0.650249\pi$$
$$158$$ 14.4222 1.14737
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −0.302776 −0.0238621
$$162$$ 0 0
$$163$$ 5.69722 0.446241 0.223121 0.974791i $$-0.428376\pi$$
0.223121 + 0.974791i $$0.428376\pi$$
$$164$$ 9.90833 0.773710
$$165$$ 0 0
$$166$$ 3.21110 0.249230
$$167$$ −21.2111 −1.64136 −0.820682 0.571385i $$-0.806406\pi$$
−0.820682 + 0.571385i $$0.806406\pi$$
$$168$$ 0 0
$$169$$ −12.9083 −0.992948
$$170$$ −3.90833 −0.299755
$$171$$ 0 0
$$172$$ 5.21110 0.397343
$$173$$ −23.3028 −1.77168 −0.885839 0.463993i $$-0.846416\pi$$
−0.885839 + 0.463993i $$0.846416\pi$$
$$174$$ 0 0
$$175$$ −0.302776 −0.0228877
$$176$$ 5.30278 0.399712
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −16.6056 −1.24116 −0.620579 0.784144i $$-0.713102\pi$$
−0.620579 + 0.784144i $$0.713102\pi$$
$$180$$ 0 0
$$181$$ −8.11943 −0.603512 −0.301756 0.953385i $$-0.597573\pi$$
−0.301756 + 0.953385i $$0.597573\pi$$
$$182$$ 0.0916731 0.00679526
$$183$$ 0 0
$$184$$ 1.00000 0.0737210
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 20.7250 1.51556
$$188$$ −4.60555 −0.335894
$$189$$ 0 0
$$190$$ 4.90833 0.356087
$$191$$ 1.39445 0.100899 0.0504494 0.998727i $$-0.483935\pi$$
0.0504494 + 0.998727i $$0.483935\pi$$
$$192$$ 0 0
$$193$$ 3.81665 0.274729 0.137364 0.990521i $$-0.456137\pi$$
0.137364 + 0.990521i $$0.456137\pi$$
$$194$$ 2.69722 0.193649
$$195$$ 0 0
$$196$$ −6.90833 −0.493452
$$197$$ −0.697224 −0.0496752 −0.0248376 0.999691i $$-0.507907\pi$$
−0.0248376 + 0.999691i $$0.507907\pi$$
$$198$$ 0 0
$$199$$ 8.42221 0.597034 0.298517 0.954404i $$-0.403508\pi$$
0.298517 + 0.954404i $$0.403508\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 4.60555 0.324045
$$203$$ 1.39445 0.0978711
$$204$$ 0 0
$$205$$ −9.90833 −0.692028
$$206$$ −17.1194 −1.19277
$$207$$ 0 0
$$208$$ −0.302776 −0.0209937
$$209$$ −26.0278 −1.80038
$$210$$ 0 0
$$211$$ −7.21110 −0.496433 −0.248216 0.968705i $$-0.579844\pi$$
−0.248216 + 0.968705i $$0.579844\pi$$
$$212$$ −3.21110 −0.220539
$$213$$ 0 0
$$214$$ −4.60555 −0.314829
$$215$$ −5.21110 −0.355394
$$216$$ 0 0
$$217$$ −0.880571 −0.0597770
$$218$$ 19.5139 1.32165
$$219$$ 0 0
$$220$$ −5.30278 −0.357513
$$221$$ −1.18335 −0.0796005
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −0.302776 −0.0202300
$$225$$ 0 0
$$226$$ −12.4222 −0.826313
$$227$$ 7.39445 0.490787 0.245393 0.969424i $$-0.421083\pi$$
0.245393 + 0.969424i $$0.421083\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ −1.00000 −0.0659380
$$231$$ 0 0
$$232$$ −4.60555 −0.302369
$$233$$ −4.18335 −0.274060 −0.137030 0.990567i $$-0.543756\pi$$
−0.137030 + 0.990567i $$0.543756\pi$$
$$234$$ 0 0
$$235$$ 4.60555 0.300433
$$236$$ 10.6056 0.690363
$$237$$ 0 0
$$238$$ −1.18335 −0.0767049
$$239$$ 9.21110 0.595817 0.297908 0.954594i $$-0.403711\pi$$
0.297908 + 0.954594i $$0.403711\pi$$
$$240$$ 0 0
$$241$$ 14.4222 0.929016 0.464508 0.885569i $$-0.346231\pi$$
0.464508 + 0.885569i $$0.346231\pi$$
$$242$$ 17.1194 1.10048
$$243$$ 0 0
$$244$$ −6.51388 −0.417008
$$245$$ 6.90833 0.441357
$$246$$ 0 0
$$247$$ 1.48612 0.0945597
$$248$$ 2.90833 0.184679
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 5.51388 0.348033 0.174016 0.984743i $$-0.444325\pi$$
0.174016 + 0.984743i $$0.444325\pi$$
$$252$$ 0 0
$$253$$ 5.30278 0.333383
$$254$$ −11.8167 −0.741443
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −19.8167 −1.23613 −0.618064 0.786127i $$-0.712083\pi$$
−0.618064 + 0.786127i $$0.712083\pi$$
$$258$$ 0 0
$$259$$ −2.42221 −0.150509
$$260$$ 0.302776 0.0187773
$$261$$ 0 0
$$262$$ −3.21110 −0.198383
$$263$$ 14.5139 0.894964 0.447482 0.894293i $$-0.352321\pi$$
0.447482 + 0.894293i $$0.352321\pi$$
$$264$$ 0 0
$$265$$ 3.21110 0.197256
$$266$$ 1.48612 0.0911200
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −25.8167 −1.57407 −0.787035 0.616909i $$-0.788385\pi$$
−0.787035 + 0.616909i $$0.788385\pi$$
$$270$$ 0 0
$$271$$ −6.30278 −0.382866 −0.191433 0.981506i $$-0.561314\pi$$
−0.191433 + 0.981506i $$0.561314\pi$$
$$272$$ 3.90833 0.236977
$$273$$ 0 0
$$274$$ 6.90833 0.417347
$$275$$ 5.30278 0.319769
$$276$$ 0 0
$$277$$ −12.7889 −0.768410 −0.384205 0.923248i $$-0.625524\pi$$
−0.384205 + 0.923248i $$0.625524\pi$$
$$278$$ −5.39445 −0.323538
$$279$$ 0 0
$$280$$ 0.302776 0.0180943
$$281$$ 19.3944 1.15698 0.578488 0.815691i $$-0.303643\pi$$
0.578488 + 0.815691i $$0.303643\pi$$
$$282$$ 0 0
$$283$$ 2.00000 0.118888 0.0594438 0.998232i $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ 12.6972 0.753442
$$285$$ 0 0
$$286$$ −1.60555 −0.0949382
$$287$$ −3.00000 −0.177084
$$288$$ 0 0
$$289$$ −1.72498 −0.101469
$$290$$ 4.60555 0.270447
$$291$$ 0 0
$$292$$ 15.8167 0.925600
$$293$$ −8.78890 −0.513453 −0.256726 0.966484i $$-0.582644\pi$$
−0.256726 + 0.966484i $$0.582644\pi$$
$$294$$ 0 0
$$295$$ −10.6056 −0.617479
$$296$$ 8.00000 0.464991
$$297$$ 0 0
$$298$$ −9.69722 −0.561745
$$299$$ −0.302776 −0.0175100
$$300$$ 0 0
$$301$$ −1.57779 −0.0909426
$$302$$ −1.90833 −0.109812
$$303$$ 0 0
$$304$$ −4.90833 −0.281512
$$305$$ 6.51388 0.372984
$$306$$ 0 0
$$307$$ −15.3028 −0.873376 −0.436688 0.899613i $$-0.643849\pi$$
−0.436688 + 0.899613i $$0.643849\pi$$
$$308$$ −1.60555 −0.0914848
$$309$$ 0 0
$$310$$ −2.90833 −0.165182
$$311$$ 6.42221 0.364170 0.182085 0.983283i $$-0.441715\pi$$
0.182085 + 0.983283i $$0.441715\pi$$
$$312$$ 0 0
$$313$$ −12.7250 −0.719258 −0.359629 0.933095i $$-0.617097\pi$$
−0.359629 + 0.933095i $$0.617097\pi$$
$$314$$ −11.3944 −0.643026
$$315$$ 0 0
$$316$$ 14.4222 0.811312
$$317$$ 14.7250 0.827037 0.413519 0.910496i $$-0.364300\pi$$
0.413519 + 0.910496i $$0.364300\pi$$
$$318$$ 0 0
$$319$$ −24.4222 −1.36738
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −0.302776 −0.0168730
$$323$$ −19.1833 −1.06739
$$324$$ 0 0
$$325$$ −0.302776 −0.0167950
$$326$$ 5.69722 0.315540
$$327$$ 0 0
$$328$$ 9.90833 0.547096
$$329$$ 1.39445 0.0768784
$$330$$ 0 0
$$331$$ 9.39445 0.516366 0.258183 0.966096i $$-0.416876\pi$$
0.258183 + 0.966096i $$0.416876\pi$$
$$332$$ 3.21110 0.176232
$$333$$ 0 0
$$334$$ −21.2111 −1.16062
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −4.48612 −0.244375 −0.122187 0.992507i $$-0.538991\pi$$
−0.122187 + 0.992507i $$0.538991\pi$$
$$338$$ −12.9083 −0.702120
$$339$$ 0 0
$$340$$ −3.90833 −0.211959
$$341$$ 15.4222 0.835159
$$342$$ 0 0
$$343$$ 4.21110 0.227378
$$344$$ 5.21110 0.280964
$$345$$ 0 0
$$346$$ −23.3028 −1.25276
$$347$$ 25.5416 1.37115 0.685573 0.728004i $$-0.259552\pi$$
0.685573 + 0.728004i $$0.259552\pi$$
$$348$$ 0 0
$$349$$ −12.7889 −0.684574 −0.342287 0.939595i $$-0.611202\pi$$
−0.342287 + 0.939595i $$0.611202\pi$$
$$350$$ −0.302776 −0.0161840
$$351$$ 0 0
$$352$$ 5.30278 0.282639
$$353$$ −18.4222 −0.980515 −0.490258 0.871578i $$-0.663097\pi$$
−0.490258 + 0.871578i $$0.663097\pi$$
$$354$$ 0 0
$$355$$ −12.6972 −0.673899
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −16.6056 −0.877631
$$359$$ 3.21110 0.169476 0.0847378 0.996403i $$-0.472995\pi$$
0.0847378 + 0.996403i $$0.472995\pi$$
$$360$$ 0 0
$$361$$ 5.09167 0.267983
$$362$$ −8.11943 −0.426748
$$363$$ 0 0
$$364$$ 0.0916731 0.00480498
$$365$$ −15.8167 −0.827881
$$366$$ 0 0
$$367$$ 29.2111 1.52481 0.762404 0.647102i $$-0.224019\pi$$
0.762404 + 0.647102i $$0.224019\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0 0
$$370$$ −8.00000 −0.415900
$$371$$ 0.972244 0.0504764
$$372$$ 0 0
$$373$$ −2.60555 −0.134910 −0.0674552 0.997722i $$-0.521488\pi$$
−0.0674552 + 0.997722i $$0.521488\pi$$
$$374$$ 20.7250 1.07166
$$375$$ 0 0
$$376$$ −4.60555 −0.237513
$$377$$ 1.39445 0.0718178
$$378$$ 0 0
$$379$$ 4.09167 0.210175 0.105088 0.994463i $$-0.466488\pi$$
0.105088 + 0.994463i $$0.466488\pi$$
$$380$$ 4.90833 0.251792
$$381$$ 0 0
$$382$$ 1.39445 0.0713462
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 1.60555 0.0818265
$$386$$ 3.81665 0.194263
$$387$$ 0 0
$$388$$ 2.69722 0.136931
$$389$$ −20.9361 −1.06150 −0.530751 0.847528i $$-0.678090\pi$$
−0.530751 + 0.847528i $$0.678090\pi$$
$$390$$ 0 0
$$391$$ 3.90833 0.197653
$$392$$ −6.90833 −0.348923
$$393$$ 0 0
$$394$$ −0.697224 −0.0351257
$$395$$ −14.4222 −0.725660
$$396$$ 0 0
$$397$$ −21.7250 −1.09035 −0.545173 0.838324i $$-0.683536\pi$$
−0.545173 + 0.838324i $$0.683536\pi$$
$$398$$ 8.42221 0.422167
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 1.39445 0.0696354 0.0348177 0.999394i $$-0.488915\pi$$
0.0348177 + 0.999394i $$0.488915\pi$$
$$402$$ 0 0
$$403$$ −0.880571 −0.0438643
$$404$$ 4.60555 0.229135
$$405$$ 0 0
$$406$$ 1.39445 0.0692053
$$407$$ 42.4222 2.10279
$$408$$ 0 0
$$409$$ −15.0917 −0.746235 −0.373118 0.927784i $$-0.621711\pi$$
−0.373118 + 0.927784i $$0.621711\pi$$
$$410$$ −9.90833 −0.489337
$$411$$ 0 0
$$412$$ −17.1194 −0.843414
$$413$$ −3.21110 −0.158008
$$414$$ 0 0
$$415$$ −3.21110 −0.157627
$$416$$ −0.302776 −0.0148448
$$417$$ 0 0
$$418$$ −26.0278 −1.27306
$$419$$ 39.6333 1.93621 0.968107 0.250538i $$-0.0806073\pi$$
0.968107 + 0.250538i $$0.0806073\pi$$
$$420$$ 0 0
$$421$$ 34.3028 1.67181 0.835907 0.548870i $$-0.184942\pi$$
0.835907 + 0.548870i $$0.184942\pi$$
$$422$$ −7.21110 −0.351031
$$423$$ 0 0
$$424$$ −3.21110 −0.155945
$$425$$ 3.90833 0.189582
$$426$$ 0 0
$$427$$ 1.97224 0.0954436
$$428$$ −4.60555 −0.222618
$$429$$ 0 0
$$430$$ −5.21110 −0.251302
$$431$$ −20.2389 −0.974872 −0.487436 0.873159i $$-0.662068\pi$$
−0.487436 + 0.873159i $$0.662068\pi$$
$$432$$ 0 0
$$433$$ −34.9083 −1.67759 −0.838794 0.544450i $$-0.816739\pi$$
−0.838794 + 0.544450i $$0.816739\pi$$
$$434$$ −0.880571 −0.0422687
$$435$$ 0 0
$$436$$ 19.5139 0.934545
$$437$$ −4.90833 −0.234797
$$438$$ 0 0
$$439$$ −18.3028 −0.873544 −0.436772 0.899572i $$-0.643878\pi$$
−0.436772 + 0.899572i $$0.643878\pi$$
$$440$$ −5.30278 −0.252800
$$441$$ 0 0
$$442$$ −1.18335 −0.0562860
$$443$$ −35.5139 −1.68732 −0.843658 0.536882i $$-0.819602\pi$$
−0.843658 + 0.536882i $$0.819602\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ −0.302776 −0.0143048
$$449$$ 12.9083 0.609182 0.304591 0.952483i $$-0.401480\pi$$
0.304591 + 0.952483i $$0.401480\pi$$
$$450$$ 0 0
$$451$$ 52.5416 2.47409
$$452$$ −12.4222 −0.584291
$$453$$ 0 0
$$454$$ 7.39445 0.347039
$$455$$ −0.0916731 −0.00429770
$$456$$ 0 0
$$457$$ −3.57779 −0.167362 −0.0836811 0.996493i $$-0.526668\pi$$
−0.0836811 + 0.996493i $$0.526668\pi$$
$$458$$ 2.00000 0.0934539
$$459$$ 0 0
$$460$$ −1.00000 −0.0466252
$$461$$ −31.8167 −1.48185 −0.740925 0.671588i $$-0.765612\pi$$
−0.740925 + 0.671588i $$0.765612\pi$$
$$462$$ 0 0
$$463$$ −25.6333 −1.19128 −0.595640 0.803251i $$-0.703102\pi$$
−0.595640 + 0.803251i $$0.703102\pi$$
$$464$$ −4.60555 −0.213807
$$465$$ 0 0
$$466$$ −4.18335 −0.193790
$$467$$ −19.8167 −0.917005 −0.458503 0.888693i $$-0.651614\pi$$
−0.458503 + 0.888693i $$0.651614\pi$$
$$468$$ 0 0
$$469$$ 1.21110 0.0559235
$$470$$ 4.60555 0.212438
$$471$$ 0 0
$$472$$ 10.6056 0.488160
$$473$$ 27.6333 1.27058
$$474$$ 0 0
$$475$$ −4.90833 −0.225209
$$476$$ −1.18335 −0.0542386
$$477$$ 0 0
$$478$$ 9.21110 0.421306
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ −2.42221 −0.110443
$$482$$ 14.4222 0.656913
$$483$$ 0 0
$$484$$ 17.1194 0.778156
$$485$$ −2.69722 −0.122475
$$486$$ 0 0
$$487$$ −11.8167 −0.535464 −0.267732 0.963493i $$-0.586274\pi$$
−0.267732 + 0.963493i $$0.586274\pi$$
$$488$$ −6.51388 −0.294869
$$489$$ 0 0
$$490$$ 6.90833 0.312086
$$491$$ 25.8167 1.16509 0.582545 0.812799i $$-0.302057\pi$$
0.582545 + 0.812799i $$0.302057\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 1.48612 0.0668638
$$495$$ 0 0
$$496$$ 2.90833 0.130588
$$497$$ −3.84441 −0.172445
$$498$$ 0 0
$$499$$ 11.6333 0.520778 0.260389 0.965504i $$-0.416149\pi$$
0.260389 + 0.965504i $$0.416149\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 5.51388 0.246096
$$503$$ 2.72498 0.121501 0.0607504 0.998153i $$-0.480651\pi$$
0.0607504 + 0.998153i $$0.480651\pi$$
$$504$$ 0 0
$$505$$ −4.60555 −0.204944
$$506$$ 5.30278 0.235737
$$507$$ 0 0
$$508$$ −11.8167 −0.524279
$$509$$ −29.4500 −1.30535 −0.652673 0.757639i $$-0.726353\pi$$
−0.652673 + 0.757639i $$0.726353\pi$$
$$510$$ 0 0
$$511$$ −4.78890 −0.211848
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −19.8167 −0.874075
$$515$$ 17.1194 0.754372
$$516$$ 0 0
$$517$$ −24.4222 −1.07409
$$518$$ −2.42221 −0.106426
$$519$$ 0 0
$$520$$ 0.302776 0.0132776
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 8.42221 0.368277 0.184139 0.982900i $$-0.441050\pi$$
0.184139 + 0.982900i $$0.441050\pi$$
$$524$$ −3.21110 −0.140278
$$525$$ 0 0
$$526$$ 14.5139 0.632835
$$527$$ 11.3667 0.495141
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 3.21110 0.139481
$$531$$ 0 0
$$532$$ 1.48612 0.0644316
$$533$$ −3.00000 −0.129944
$$534$$ 0 0
$$535$$ 4.60555 0.199115
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ −25.8167 −1.11303
$$539$$ −36.6333 −1.57791
$$540$$ 0 0
$$541$$ −28.8444 −1.24012 −0.620059 0.784555i $$-0.712891\pi$$
−0.620059 + 0.784555i $$0.712891\pi$$
$$542$$ −6.30278 −0.270727
$$543$$ 0 0
$$544$$ 3.90833 0.167568
$$545$$ −19.5139 −0.835883
$$546$$ 0 0
$$547$$ 7.51388 0.321270 0.160635 0.987014i $$-0.448646\pi$$
0.160635 + 0.987014i $$0.448646\pi$$
$$548$$ 6.90833 0.295109
$$549$$ 0 0
$$550$$ 5.30278 0.226111
$$551$$ 22.6056 0.963029
$$552$$ 0 0
$$553$$ −4.36669 −0.185691
$$554$$ −12.7889 −0.543348
$$555$$ 0 0
$$556$$ −5.39445 −0.228776
$$557$$ 6.42221 0.272118 0.136059 0.990701i $$-0.456556\pi$$
0.136059 + 0.990701i $$0.456556\pi$$
$$558$$ 0 0
$$559$$ −1.57779 −0.0667336
$$560$$ 0.302776 0.0127946
$$561$$ 0 0
$$562$$ 19.3944 0.818105
$$563$$ −39.6333 −1.67034 −0.835172 0.549988i $$-0.814632\pi$$
−0.835172 + 0.549988i $$0.814632\pi$$
$$564$$ 0 0
$$565$$ 12.4222 0.522606
$$566$$ 2.00000 0.0840663
$$567$$ 0 0
$$568$$ 12.6972 0.532764
$$569$$ 0.422205 0.0176998 0.00884988 0.999961i $$-0.497183\pi$$
0.00884988 + 0.999961i $$0.497183\pi$$
$$570$$ 0 0
$$571$$ 9.11943 0.381636 0.190818 0.981625i $$-0.438886\pi$$
0.190818 + 0.981625i $$0.438886\pi$$
$$572$$ −1.60555 −0.0671315
$$573$$ 0 0
$$574$$ −3.00000 −0.125218
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −1.72498 −0.0717497
$$579$$ 0 0
$$580$$ 4.60555 0.191235
$$581$$ −0.972244 −0.0403355
$$582$$ 0 0
$$583$$ −17.0278 −0.705218
$$584$$ 15.8167 0.654498
$$585$$ 0 0
$$586$$ −8.78890 −0.363066
$$587$$ 37.5416 1.54951 0.774755 0.632262i $$-0.217873\pi$$
0.774755 + 0.632262i $$0.217873\pi$$
$$588$$ 0 0
$$589$$ −14.2750 −0.588192
$$590$$ −10.6056 −0.436624
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ −19.8167 −0.813772 −0.406886 0.913479i $$-0.633385\pi$$
−0.406886 + 0.913479i $$0.633385\pi$$
$$594$$ 0 0
$$595$$ 1.18335 0.0485125
$$596$$ −9.69722 −0.397214
$$597$$ 0 0
$$598$$ −0.302776 −0.0123814
$$599$$ −4.33053 −0.176941 −0.0884704 0.996079i $$-0.528198\pi$$
−0.0884704 + 0.996079i $$0.528198\pi$$
$$600$$ 0 0
$$601$$ −3.93608 −0.160556 −0.0802781 0.996773i $$-0.525581\pi$$
−0.0802781 + 0.996773i $$0.525581\pi$$
$$602$$ −1.57779 −0.0643061
$$603$$ 0 0
$$604$$ −1.90833 −0.0776487
$$605$$ −17.1194 −0.696004
$$606$$ 0 0
$$607$$ −26.0555 −1.05756 −0.528780 0.848759i $$-0.677350\pi$$
−0.528780 + 0.848759i $$0.677350\pi$$
$$608$$ −4.90833 −0.199059
$$609$$ 0 0
$$610$$ 6.51388 0.263739
$$611$$ 1.39445 0.0564134
$$612$$ 0 0
$$613$$ 32.4222 1.30952 0.654760 0.755837i $$-0.272770\pi$$
0.654760 + 0.755837i $$0.272770\pi$$
$$614$$ −15.3028 −0.617570
$$615$$ 0 0
$$616$$ −1.60555 −0.0646895
$$617$$ −8.09167 −0.325758 −0.162879 0.986646i $$-0.552078\pi$$
−0.162879 + 0.986646i $$0.552078\pi$$
$$618$$ 0 0
$$619$$ 27.3305 1.09851 0.549253 0.835656i $$-0.314912\pi$$
0.549253 + 0.835656i $$0.314912\pi$$
$$620$$ −2.90833 −0.116801
$$621$$ 0 0
$$622$$ 6.42221 0.257507
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −12.7250 −0.508593
$$627$$ 0 0
$$628$$ −11.3944 −0.454688
$$629$$ 31.2666 1.24668
$$630$$ 0 0
$$631$$ 30.6056 1.21839 0.609194 0.793021i $$-0.291493\pi$$
0.609194 + 0.793021i $$0.291493\pi$$
$$632$$ 14.4222 0.573685
$$633$$ 0 0
$$634$$ 14.7250 0.584804
$$635$$ 11.8167 0.468930
$$636$$ 0 0
$$637$$ 2.09167 0.0828751
$$638$$ −24.4222 −0.966884
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ 0 0
$$643$$ 16.2389 0.640398 0.320199 0.947350i $$-0.396250\pi$$
0.320199 + 0.947350i $$0.396250\pi$$
$$644$$ −0.302776 −0.0119310
$$645$$ 0 0
$$646$$ −19.1833 −0.754759
$$647$$ 30.8444 1.21262 0.606309 0.795229i $$-0.292649\pi$$
0.606309 + 0.795229i $$0.292649\pi$$
$$648$$ 0 0
$$649$$ 56.2389 2.20757
$$650$$ −0.302776 −0.0118758
$$651$$ 0 0
$$652$$ 5.69722 0.223121
$$653$$ −9.27502 −0.362960 −0.181480 0.983395i $$-0.558089\pi$$
−0.181480 + 0.983395i $$0.558089\pi$$
$$654$$ 0 0
$$655$$ 3.21110 0.125468
$$656$$ 9.90833 0.386855
$$657$$ 0 0
$$658$$ 1.39445 0.0543613
$$659$$ 27.6333 1.07644 0.538220 0.842804i $$-0.319097\pi$$
0.538220 + 0.842804i $$0.319097\pi$$
$$660$$ 0 0
$$661$$ −24.0917 −0.937057 −0.468529 0.883448i $$-0.655216\pi$$
−0.468529 + 0.883448i $$0.655216\pi$$
$$662$$ 9.39445 0.365126
$$663$$ 0 0
$$664$$ 3.21110 0.124615
$$665$$ −1.48612 −0.0576293
$$666$$ 0 0
$$667$$ −4.60555 −0.178328
$$668$$ −21.2111 −0.820682
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ −34.5416 −1.33347
$$672$$ 0 0
$$673$$ 5.63331 0.217148 0.108574 0.994088i $$-0.465372\pi$$
0.108574 + 0.994088i $$0.465372\pi$$
$$674$$ −4.48612 −0.172799
$$675$$ 0 0
$$676$$ −12.9083 −0.496474
$$677$$ 12.4222 0.477424 0.238712 0.971090i $$-0.423275\pi$$
0.238712 + 0.971090i $$0.423275\pi$$
$$678$$ 0 0
$$679$$ −0.816654 −0.0313403
$$680$$ −3.90833 −0.149877
$$681$$ 0 0
$$682$$ 15.4222 0.590547
$$683$$ 32.7250 1.25219 0.626093 0.779748i $$-0.284653\pi$$
0.626093 + 0.779748i $$0.284653\pi$$
$$684$$ 0 0
$$685$$ −6.90833 −0.263954
$$686$$ 4.21110 0.160781
$$687$$ 0 0
$$688$$ 5.21110 0.198671
$$689$$ 0.972244 0.0370395
$$690$$ 0 0
$$691$$ 30.1833 1.14823 0.574114 0.818775i $$-0.305347\pi$$
0.574114 + 0.818775i $$0.305347\pi$$
$$692$$ −23.3028 −0.885839
$$693$$ 0 0
$$694$$ 25.5416 0.969547
$$695$$ 5.39445 0.204623
$$696$$ 0 0
$$697$$ 38.7250 1.46681
$$698$$ −12.7889 −0.484067
$$699$$ 0 0
$$700$$ −0.302776 −0.0114438
$$701$$ −42.9083 −1.62063 −0.810313 0.585998i $$-0.800703\pi$$
−0.810313 + 0.585998i $$0.800703\pi$$
$$702$$ 0 0
$$703$$ −39.2666 −1.48097
$$704$$ 5.30278 0.199856
$$705$$ 0 0
$$706$$ −18.4222 −0.693329
$$707$$ −1.39445 −0.0524436
$$708$$ 0 0
$$709$$ −41.1194 −1.54427 −0.772136 0.635457i $$-0.780812\pi$$
−0.772136 + 0.635457i $$0.780812\pi$$
$$710$$ −12.6972 −0.476518
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 2.90833 0.108918
$$714$$ 0 0
$$715$$ 1.60555 0.0600442
$$716$$ −16.6056 −0.620579
$$717$$ 0 0
$$718$$ 3.21110 0.119837
$$719$$ −14.3028 −0.533404 −0.266702 0.963779i $$-0.585934\pi$$
−0.266702 + 0.963779i $$0.585934\pi$$
$$720$$ 0 0
$$721$$ 5.18335 0.193038
$$722$$ 5.09167 0.189492
$$723$$ 0 0
$$724$$ −8.11943 −0.301756
$$725$$ −4.60555 −0.171046
$$726$$ 0 0
$$727$$ −7.90833 −0.293304 −0.146652 0.989188i $$-0.546850\pi$$
−0.146652 + 0.989188i $$0.546850\pi$$
$$728$$ 0.0916731 0.00339763
$$729$$ 0 0
$$730$$ −15.8167 −0.585401
$$731$$ 20.3667 0.753289
$$732$$ 0 0
$$733$$ −13.6333 −0.503558 −0.251779 0.967785i $$-0.581016\pi$$
−0.251779 + 0.967785i $$0.581016\pi$$
$$734$$ 29.2111 1.07820
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ −21.2111 −0.781321
$$738$$ 0 0
$$739$$ −7.63331 −0.280796 −0.140398 0.990095i $$-0.544838\pi$$
−0.140398 + 0.990095i $$0.544838\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 0 0
$$742$$ 0.972244 0.0356922
$$743$$ −7.33053 −0.268931 −0.134466 0.990918i $$-0.542932\pi$$
−0.134466 + 0.990918i $$0.542932\pi$$
$$744$$ 0 0
$$745$$ 9.69722 0.355279
$$746$$ −2.60555 −0.0953960
$$747$$ 0 0
$$748$$ 20.7250 0.757780
$$749$$ 1.39445 0.0509520
$$750$$ 0 0
$$751$$ 0.183346 0.00669040 0.00334520 0.999994i $$-0.498935\pi$$
0.00334520 + 0.999994i $$0.498935\pi$$
$$752$$ −4.60555 −0.167947
$$753$$ 0 0
$$754$$ 1.39445 0.0507828
$$755$$ 1.90833 0.0694511
$$756$$ 0 0
$$757$$ −1.21110 −0.0440183 −0.0220091 0.999758i $$-0.507006\pi$$
−0.0220091 + 0.999758i $$0.507006\pi$$
$$758$$ 4.09167 0.148616
$$759$$ 0 0
$$760$$ 4.90833 0.178044
$$761$$ −4.54163 −0.164634 −0.0823171 0.996606i $$-0.526232\pi$$
−0.0823171 + 0.996606i $$0.526232\pi$$
$$762$$ 0 0
$$763$$ −5.90833 −0.213896
$$764$$ 1.39445 0.0504494
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −3.21110 −0.115946
$$768$$ 0 0
$$769$$ −41.2666 −1.48811 −0.744056 0.668117i $$-0.767100\pi$$
−0.744056 + 0.668117i $$0.767100\pi$$
$$770$$ 1.60555 0.0578601
$$771$$ 0 0
$$772$$ 3.81665 0.137364
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 0 0
$$775$$ 2.90833 0.104470
$$776$$ 2.69722 0.0968247
$$777$$ 0 0
$$778$$ −20.9361 −0.750595
$$779$$ −48.6333 −1.74247
$$780$$ 0 0
$$781$$ 67.3305 2.40928
$$782$$ 3.90833 0.139761
$$783$$ 0 0
$$784$$ −6.90833 −0.246726
$$785$$ 11.3944 0.406685
$$786$$ 0 0
$$787$$ −27.4500 −0.978485 −0.489243 0.872148i $$-0.662727\pi$$
−0.489243 + 0.872148i $$0.662727\pi$$
$$788$$ −0.697224 −0.0248376
$$789$$ 0 0
$$790$$ −14.4222 −0.513119
$$791$$ 3.76114 0.133731
$$792$$ 0 0
$$793$$ 1.97224 0.0700364
$$794$$ −21.7250 −0.770991
$$795$$ 0 0
$$796$$ 8.42221 0.298517
$$797$$ 19.8167 0.701942 0.350971 0.936386i $$-0.385852\pi$$
0.350971 + 0.936386i $$0.385852\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 1.39445 0.0492397
$$803$$ 83.8722 2.95978
$$804$$ 0 0
$$805$$ 0.302776 0.0106714
$$806$$ −0.880571 −0.0310168
$$807$$ 0 0
$$808$$ 4.60555 0.162023
$$809$$ −18.2750 −0.642515 −0.321258 0.946992i $$-0.604106\pi$$
−0.321258 + 0.946992i $$0.604106\pi$$
$$810$$ 0 0
$$811$$ −4.97224 −0.174599 −0.0872995 0.996182i $$-0.527824\pi$$
−0.0872995 + 0.996182i $$0.527824\pi$$
$$812$$ 1.39445 0.0489356
$$813$$ 0 0
$$814$$ 42.4222 1.48690
$$815$$ −5.69722 −0.199565
$$816$$ 0 0
$$817$$ −25.5778 −0.894854
$$818$$ −15.0917 −0.527668
$$819$$ 0 0
$$820$$ −9.90833 −0.346014
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ −0.788897 −0.0274992 −0.0137496 0.999905i $$-0.504377\pi$$
−0.0137496 + 0.999905i $$0.504377\pi$$
$$824$$ −17.1194 −0.596384
$$825$$ 0 0
$$826$$ −3.21110 −0.111729
$$827$$ −35.4500 −1.23272 −0.616358 0.787466i $$-0.711393\pi$$
−0.616358 + 0.787466i $$0.711393\pi$$
$$828$$ 0 0
$$829$$ 16.7889 0.583103 0.291551 0.956555i $$-0.405829\pi$$
0.291551 + 0.956555i $$0.405829\pi$$
$$830$$ −3.21110 −0.111459
$$831$$ 0 0
$$832$$ −0.302776 −0.0104969
$$833$$ −27.0000 −0.935495
$$834$$ 0 0
$$835$$ 21.2111 0.734040
$$836$$ −26.0278 −0.900189
$$837$$ 0 0
$$838$$ 39.6333 1.36911
$$839$$ 22.1833 0.765854 0.382927 0.923779i $$-0.374916\pi$$
0.382927 + 0.923779i $$0.374916\pi$$
$$840$$ 0 0
$$841$$ −7.78890 −0.268583
$$842$$ 34.3028 1.18215
$$843$$ 0 0
$$844$$ −7.21110 −0.248216
$$845$$ 12.9083 0.444060
$$846$$ 0 0
$$847$$ −5.18335 −0.178102
$$848$$ −3.21110 −0.110270
$$849$$ 0 0
$$850$$ 3.90833 0.134055
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ 10.7250 0.367216 0.183608 0.983000i $$-0.441222\pi$$
0.183608 + 0.983000i $$0.441222\pi$$
$$854$$ 1.97224 0.0674888
$$855$$ 0 0
$$856$$ −4.60555 −0.157415
$$857$$ 33.6333 1.14889 0.574446 0.818543i $$-0.305218\pi$$
0.574446 + 0.818543i $$0.305218\pi$$
$$858$$ 0 0
$$859$$ −14.1833 −0.483930 −0.241965 0.970285i $$-0.577792\pi$$
−0.241965 + 0.970285i $$0.577792\pi$$
$$860$$ −5.21110 −0.177697
$$861$$ 0 0
$$862$$ −20.2389 −0.689338
$$863$$ −23.4500 −0.798246 −0.399123 0.916897i $$-0.630685\pi$$
−0.399123 + 0.916897i $$0.630685\pi$$
$$864$$ 0 0
$$865$$ 23.3028 0.792318
$$866$$ −34.9083 −1.18623
$$867$$ 0 0
$$868$$ −0.880571 −0.0298885
$$869$$ 76.4777 2.59433
$$870$$ 0 0
$$871$$ 1.21110 0.0410366
$$872$$ 19.5139 0.660823
$$873$$ 0 0
$$874$$ −4.90833 −0.166027
$$875$$ 0.302776 0.0102357
$$876$$ 0 0
$$877$$ 49.1749 1.66052 0.830260 0.557376i $$-0.188192\pi$$
0.830260 + 0.557376i $$0.188192\pi$$
$$878$$ −18.3028 −0.617689
$$879$$ 0 0
$$880$$ −5.30278 −0.178757
$$881$$ 31.2666 1.05340 0.526700 0.850052i $$-0.323429\pi$$
0.526700 + 0.850052i $$0.323429\pi$$
$$882$$ 0 0
$$883$$ 40.7250 1.37050 0.685252 0.728306i $$-0.259692\pi$$
0.685252 + 0.728306i $$0.259692\pi$$
$$884$$ −1.18335 −0.0398002
$$885$$ 0 0
$$886$$ −35.5139 −1.19311
$$887$$ 15.6333 0.524915 0.262458 0.964944i $$-0.415467\pi$$
0.262458 + 0.964944i $$0.415467\pi$$
$$888$$ 0 0
$$889$$ 3.57779 0.119995
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ 22.6056 0.756466
$$894$$ 0 0
$$895$$ 16.6056 0.555062
$$896$$ −0.302776 −0.0101150
$$897$$ 0 0
$$898$$ 12.9083 0.430756
$$899$$ −13.3944 −0.446730
$$900$$ 0 0
$$901$$ −12.5500 −0.418102
$$902$$ 52.5416 1.74945
$$903$$ 0 0
$$904$$ −12.4222 −0.413156
$$905$$ 8.11943 0.269899
$$906$$ 0 0
$$907$$ −30.6611 −1.01808 −0.509042 0.860742i $$-0.670000\pi$$
−0.509042 + 0.860742i $$0.670000\pi$$
$$908$$ 7.39445 0.245393
$$909$$ 0 0
$$910$$ −0.0916731 −0.00303893
$$911$$ 25.8167 0.855344 0.427672 0.903934i $$-0.359334\pi$$
0.427672 + 0.903934i $$0.359334\pi$$
$$912$$ 0 0
$$913$$ 17.0278 0.563536
$$914$$ −3.57779 −0.118343
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ 0.972244 0.0321063
$$918$$ 0 0
$$919$$ 44.0000 1.45143 0.725713 0.687998i $$-0.241510\pi$$
0.725713 + 0.687998i $$0.241510\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ −31.8167 −1.04783
$$923$$ −3.84441 −0.126540
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ −25.6333 −0.842363
$$927$$ 0 0
$$928$$ −4.60555 −0.151185
$$929$$ 57.6333 1.89089 0.945444 0.325785i $$-0.105629\pi$$
0.945444 + 0.325785i $$0.105629\pi$$
$$930$$ 0 0
$$931$$ 33.9083 1.11130
$$932$$ −4.18335 −0.137030
$$933$$ 0 0
$$934$$ −19.8167 −0.648421
$$935$$ −20.7250 −0.677779
$$936$$ 0 0
$$937$$ −44.9638 −1.46890 −0.734452 0.678660i $$-0.762561\pi$$
−0.734452 + 0.678660i $$0.762561\pi$$
$$938$$ 1.21110 0.0395439
$$939$$ 0 0
$$940$$ 4.60555 0.150217
$$941$$ 20.9361 0.682497 0.341248 0.939973i $$-0.389150\pi$$
0.341248 + 0.939973i $$0.389150\pi$$
$$942$$ 0 0
$$943$$ 9.90833 0.322660
$$944$$ 10.6056 0.345181
$$945$$ 0 0
$$946$$ 27.6333 0.898436
$$947$$ −41.9361 −1.36274 −0.681370 0.731939i $$-0.738615\pi$$
−0.681370 + 0.731939i $$0.738615\pi$$
$$948$$ 0 0
$$949$$ −4.78890 −0.155454
$$950$$ −4.90833 −0.159247
$$951$$ 0 0
$$952$$ −1.18335 −0.0383525
$$953$$ 1.66947 0.0540794 0.0270397 0.999634i $$-0.491392\pi$$
0.0270397 + 0.999634i $$0.491392\pi$$
$$954$$ 0 0
$$955$$ −1.39445 −0.0451233
$$956$$ 9.21110 0.297908
$$957$$ 0 0
$$958$$ 30.0000 0.969256
$$959$$ −2.09167 −0.0675436
$$960$$ 0 0
$$961$$ −22.5416 −0.727150
$$962$$ −2.42221 −0.0780950
$$963$$ 0 0
$$964$$ 14.4222 0.464508
$$965$$ −3.81665 −0.122862
$$966$$ 0 0
$$967$$ −5.39445 −0.173474 −0.0867369 0.996231i $$-0.527644\pi$$
−0.0867369 + 0.996231i $$0.527644\pi$$
$$968$$ 17.1194 0.550239
$$969$$ 0 0
$$970$$ −2.69722 −0.0866027
$$971$$ 27.9083 0.895621 0.447810 0.894129i $$-0.352204\pi$$
0.447810 + 0.894129i $$0.352204\pi$$
$$972$$ 0 0
$$973$$ 1.63331 0.0523614
$$974$$ −11.8167 −0.378630
$$975$$ 0 0
$$976$$ −6.51388 −0.208504
$$977$$ 11.5139 0.368362 0.184181 0.982892i $$-0.441037\pi$$
0.184181 + 0.982892i $$0.441037\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 6.90833 0.220678
$$981$$ 0 0
$$982$$ 25.8167 0.823843
$$983$$ 19.5416 0.623281 0.311641 0.950200i $$-0.399121\pi$$
0.311641 + 0.950200i $$0.399121\pi$$
$$984$$ 0 0
$$985$$ 0.697224 0.0222154
$$986$$ −18.0000 −0.573237
$$987$$ 0 0
$$988$$ 1.48612 0.0472798
$$989$$ 5.21110 0.165703
$$990$$ 0 0
$$991$$ 24.3305 0.772885 0.386442 0.922314i $$-0.373704\pi$$
0.386442 + 0.922314i $$0.373704\pi$$
$$992$$ 2.90833 0.0923395
$$993$$ 0 0
$$994$$ −3.84441 −0.121937
$$995$$ −8.42221 −0.267002
$$996$$ 0 0
$$997$$ −31.2111 −0.988466 −0.494233 0.869330i $$-0.664551\pi$$
−0.494233 + 0.869330i $$0.664551\pi$$
$$998$$ 11.6333 0.368246
$$999$$ 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 2070.2.a.w.1.1 2
3.2 odd 2 230.2.a.b.1.2 2
12.11 even 2 1840.2.a.j.1.1 2
15.2 even 4 1150.2.b.f.599.1 4
15.8 even 4 1150.2.b.f.599.4 4
15.14 odd 2 1150.2.a.m.1.1 2
24.5 odd 2 7360.2.a.bc.1.1 2
24.11 even 2 7360.2.a.bu.1.2 2
60.59 even 2 9200.2.a.ca.1.2 2
69.68 even 2 5290.2.a.j.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 2 3.2 odd 2
1150.2.a.m.1.1 2 15.14 odd 2
1150.2.b.f.599.1 4 15.2 even 4
1150.2.b.f.599.4 4 15.8 even 4
1840.2.a.j.1.1 2 12.11 even 2
2070.2.a.w.1.1 2 1.1 even 1 trivial
5290.2.a.j.1.2 2 69.68 even 2
7360.2.a.bc.1.1 2 24.5 odd 2
7360.2.a.bu.1.2 2 24.11 even 2
9200.2.a.ca.1.2 2 60.59 even 2