# Properties

 Label 1840.2.a.v.1.4 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.13955077.1 Defining polynomial: $$x^{5} - 14 x^{3} - x^{2} + 32 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.31091$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.31091 q^{3} -1.00000 q^{5} +4.66212 q^{7} -1.28151 q^{9} +O(q^{10})$$ $$q+1.31091 q^{3} -1.00000 q^{5} +4.66212 q^{7} -1.28151 q^{9} -2.23020 q^{11} -2.80072 q^{13} -1.31091 q^{15} +7.63271 q^{17} +1.36222 q^{19} +6.11163 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.61268 q^{27} +8.94362 q^{29} +1.58140 q^{31} -2.92360 q^{33} -4.66212 q^{35} -1.40251 q^{37} -3.67149 q^{39} +10.7134 q^{41} +1.28151 q^{45} +7.26391 q^{47} +14.7353 q^{49} +10.0058 q^{51} -8.38212 q^{53} +2.23020 q^{55} +1.78575 q^{57} +4.88331 q^{59} +4.33282 q^{61} -5.97454 q^{63} +2.80072 q^{65} -8.54355 q^{67} +1.31091 q^{69} -8.81604 q^{71} -5.26391 q^{73} +1.31091 q^{75} -10.3974 q^{77} -7.08222 q^{79} -3.51322 q^{81} +4.59749 q^{83} -7.63271 q^{85} +11.7243 q^{87} -4.70241 q^{89} -13.0573 q^{91} +2.07308 q^{93} -1.36222 q^{95} +16.5160 q^{97} +2.85802 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + O(q^{10})$$ $$5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41} - 13 q^{45} + 11 q^{47} + 25 q^{49} - 19 q^{51} + 3 q^{53} - q^{55} + 48 q^{57} + q^{59} - 5 q^{61} + 41 q^{63} - 4 q^{65} - 9 q^{67} - q^{71} - q^{73} + 18 q^{77} + 2 q^{79} + 57 q^{81} + 45 q^{83} - 4 q^{85} + 9 q^{87} + 6 q^{89} - 11 q^{91} - 39 q^{93} + 7 q^{95} + 25 q^{97} + 65 q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.31091 0.756856 0.378428 0.925631i $$-0.376465\pi$$
0.378428 + 0.925631i $$0.376465\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.66212 1.76211 0.881057 0.473010i $$-0.156832\pi$$
0.881057 + 0.473010i $$0.156832\pi$$
$$8$$ 0 0
$$9$$ −1.28151 −0.427169
$$10$$ 0 0
$$11$$ −2.23020 −0.672430 −0.336215 0.941785i $$-0.609147\pi$$
−0.336215 + 0.941785i $$0.609147\pi$$
$$12$$ 0 0
$$13$$ −2.80072 −0.776779 −0.388389 0.921495i $$-0.626968\pi$$
−0.388389 + 0.921495i $$0.626968\pi$$
$$14$$ 0 0
$$15$$ −1.31091 −0.338476
$$16$$ 0 0
$$17$$ 7.63271 1.85120 0.925602 0.378498i $$-0.123559\pi$$
0.925602 + 0.378498i $$0.123559\pi$$
$$18$$ 0 0
$$19$$ 1.36222 0.312515 0.156258 0.987716i $$-0.450057\pi$$
0.156258 + 0.987716i $$0.450057\pi$$
$$20$$ 0 0
$$21$$ 6.11163 1.33367
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.61268 −1.08016
$$28$$ 0 0
$$29$$ 8.94362 1.66079 0.830395 0.557176i $$-0.188115\pi$$
0.830395 + 0.557176i $$0.188115\pi$$
$$30$$ 0 0
$$31$$ 1.58140 0.284028 0.142014 0.989865i $$-0.454642\pi$$
0.142014 + 0.989865i $$0.454642\pi$$
$$32$$ 0 0
$$33$$ −2.92360 −0.508933
$$34$$ 0 0
$$35$$ −4.66212 −0.788042
$$36$$ 0 0
$$37$$ −1.40251 −0.230572 −0.115286 0.993332i $$-0.536778\pi$$
−0.115286 + 0.993332i $$0.536778\pi$$
$$38$$ 0 0
$$39$$ −3.67149 −0.587909
$$40$$ 0 0
$$41$$ 10.7134 1.67316 0.836578 0.547848i $$-0.184553\pi$$
0.836578 + 0.547848i $$0.184553\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 1.28151 0.191036
$$46$$ 0 0
$$47$$ 7.26391 1.05955 0.529775 0.848138i $$-0.322276\pi$$
0.529775 + 0.848138i $$0.322276\pi$$
$$48$$ 0 0
$$49$$ 14.7353 2.10505
$$50$$ 0 0
$$51$$ 10.0058 1.40109
$$52$$ 0 0
$$53$$ −8.38212 −1.15137 −0.575686 0.817671i $$-0.695265\pi$$
−0.575686 + 0.817671i $$0.695265\pi$$
$$54$$ 0 0
$$55$$ 2.23020 0.300720
$$56$$ 0 0
$$57$$ 1.78575 0.236529
$$58$$ 0 0
$$59$$ 4.88331 0.635752 0.317876 0.948132i $$-0.397030\pi$$
0.317876 + 0.948132i $$0.397030\pi$$
$$60$$ 0 0
$$61$$ 4.33282 0.554760 0.277380 0.960760i $$-0.410534\pi$$
0.277380 + 0.960760i $$0.410534\pi$$
$$62$$ 0 0
$$63$$ −5.97454 −0.752721
$$64$$ 0 0
$$65$$ 2.80072 0.347386
$$66$$ 0 0
$$67$$ −8.54355 −1.04376 −0.521880 0.853019i $$-0.674769\pi$$
−0.521880 + 0.853019i $$0.674769\pi$$
$$68$$ 0 0
$$69$$ 1.31091 0.157815
$$70$$ 0 0
$$71$$ −8.81604 −1.04627 −0.523136 0.852249i $$-0.675238\pi$$
−0.523136 + 0.852249i $$0.675238\pi$$
$$72$$ 0 0
$$73$$ −5.26391 −0.616095 −0.308047 0.951371i $$-0.599675\pi$$
−0.308047 + 0.951371i $$0.599675\pi$$
$$74$$ 0 0
$$75$$ 1.31091 0.151371
$$76$$ 0 0
$$77$$ −10.3974 −1.18490
$$78$$ 0 0
$$79$$ −7.08222 −0.796812 −0.398406 0.917209i $$-0.630437\pi$$
−0.398406 + 0.917209i $$0.630437\pi$$
$$80$$ 0 0
$$81$$ −3.51322 −0.390357
$$82$$ 0 0
$$83$$ 4.59749 0.504640 0.252320 0.967644i $$-0.418806\pi$$
0.252320 + 0.967644i $$0.418806\pi$$
$$84$$ 0 0
$$85$$ −7.63271 −0.827884
$$86$$ 0 0
$$87$$ 11.7243 1.25698
$$88$$ 0 0
$$89$$ −4.70241 −0.498454 −0.249227 0.968445i $$-0.580177\pi$$
−0.249227 + 0.968445i $$0.580177\pi$$
$$90$$ 0 0
$$91$$ −13.0573 −1.36877
$$92$$ 0 0
$$93$$ 2.07308 0.214968
$$94$$ 0 0
$$95$$ −1.36222 −0.139761
$$96$$ 0 0
$$97$$ 16.5160 1.67695 0.838474 0.544942i $$-0.183448\pi$$
0.838474 + 0.544942i $$0.183448\pi$$
$$98$$ 0 0
$$99$$ 2.85802 0.287241
$$100$$ 0 0
$$101$$ 10.7267 1.06735 0.533676 0.845689i $$-0.320810\pi$$
0.533676 + 0.845689i $$0.320810\pi$$
$$102$$ 0 0
$$103$$ −1.95300 −0.192435 −0.0962175 0.995360i $$-0.530674\pi$$
−0.0962175 + 0.995360i $$0.530674\pi$$
$$104$$ 0 0
$$105$$ −6.11163 −0.596434
$$106$$ 0 0
$$107$$ 1.97566 0.190994 0.0954972 0.995430i $$-0.469556\pi$$
0.0954972 + 0.995430i $$0.469556\pi$$
$$108$$ 0 0
$$109$$ −8.92360 −0.854725 −0.427363 0.904080i $$-0.640557\pi$$
−0.427363 + 0.904080i $$0.640557\pi$$
$$110$$ 0 0
$$111$$ −1.83857 −0.174510
$$112$$ 0 0
$$113$$ 17.3486 1.63202 0.816008 0.578040i $$-0.196182\pi$$
0.816008 + 0.578040i $$0.196182\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 3.58914 0.331816
$$118$$ 0 0
$$119$$ 35.5846 3.26203
$$120$$ 0 0
$$121$$ −6.02621 −0.547838
$$122$$ 0 0
$$123$$ 14.0444 1.26634
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −19.4872 −1.72921 −0.864603 0.502455i $$-0.832430\pi$$
−0.864603 + 0.502455i $$0.832430\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.68050 −0.496308 −0.248154 0.968721i $$-0.579824\pi$$
−0.248154 + 0.968721i $$0.579824\pi$$
$$132$$ 0 0
$$133$$ 6.35084 0.550687
$$134$$ 0 0
$$135$$ 5.61268 0.483063
$$136$$ 0 0
$$137$$ −8.68645 −0.742134 −0.371067 0.928606i $$-0.621008\pi$$
−0.371067 + 0.928606i $$0.621008\pi$$
$$138$$ 0 0
$$139$$ 9.22569 0.782513 0.391256 0.920282i $$-0.372041\pi$$
0.391256 + 0.920282i $$0.372041\pi$$
$$140$$ 0 0
$$141$$ 9.52236 0.801927
$$142$$ 0 0
$$143$$ 6.24615 0.522329
$$144$$ 0 0
$$145$$ −8.94362 −0.742728
$$146$$ 0 0
$$147$$ 19.3167 1.59322
$$148$$ 0 0
$$149$$ 18.6056 1.52423 0.762115 0.647441i $$-0.224161\pi$$
0.762115 + 0.647441i $$0.224161\pi$$
$$150$$ 0 0
$$151$$ −17.4316 −1.41856 −0.709280 0.704927i $$-0.750980\pi$$
−0.709280 + 0.704927i $$0.750980\pi$$
$$152$$ 0 0
$$153$$ −9.78138 −0.790778
$$154$$ 0 0
$$155$$ −1.58140 −0.127021
$$156$$ 0 0
$$157$$ 0.839497 0.0669992 0.0334996 0.999439i $$-0.489335\pi$$
0.0334996 + 0.999439i $$0.489335\pi$$
$$158$$ 0 0
$$159$$ −10.9882 −0.871423
$$160$$ 0 0
$$161$$ 4.66212 0.367426
$$162$$ 0 0
$$163$$ 14.5673 1.14100 0.570500 0.821297i $$-0.306749\pi$$
0.570500 + 0.821297i $$0.306749\pi$$
$$164$$ 0 0
$$165$$ 2.92360 0.227602
$$166$$ 0 0
$$167$$ 5.03842 0.389884 0.194942 0.980815i $$-0.437548\pi$$
0.194942 + 0.980815i $$0.437548\pi$$
$$168$$ 0 0
$$169$$ −5.15599 −0.396615
$$170$$ 0 0
$$171$$ −1.74570 −0.133497
$$172$$ 0 0
$$173$$ −11.3124 −0.860067 −0.430034 0.902813i $$-0.641498\pi$$
−0.430034 + 0.902813i $$0.641498\pi$$
$$174$$ 0 0
$$175$$ 4.66212 0.352423
$$176$$ 0 0
$$177$$ 6.40159 0.481173
$$178$$ 0 0
$$179$$ 24.3053 1.81667 0.908333 0.418247i $$-0.137355\pi$$
0.908333 + 0.418247i $$0.137355\pi$$
$$180$$ 0 0
$$181$$ −19.4829 −1.44815 −0.724075 0.689721i $$-0.757733\pi$$
−0.724075 + 0.689721i $$0.757733\pi$$
$$182$$ 0 0
$$183$$ 5.67994 0.419874
$$184$$ 0 0
$$185$$ 1.40251 0.103115
$$186$$ 0 0
$$187$$ −17.0225 −1.24481
$$188$$ 0 0
$$189$$ −26.1670 −1.90337
$$190$$ 0 0
$$191$$ −2.62183 −0.189709 −0.0948543 0.995491i $$-0.530239\pi$$
−0.0948543 + 0.995491i $$0.530239\pi$$
$$192$$ 0 0
$$193$$ 17.4332 1.25487 0.627436 0.778668i $$-0.284105\pi$$
0.627436 + 0.778668i $$0.284105\pi$$
$$194$$ 0 0
$$195$$ 3.67149 0.262921
$$196$$ 0 0
$$197$$ −23.4402 −1.67004 −0.835022 0.550217i $$-0.814545\pi$$
−0.835022 + 0.550217i $$0.814545\pi$$
$$198$$ 0 0
$$199$$ 1.29759 0.0919839 0.0459920 0.998942i $$-0.485355\pi$$
0.0459920 + 0.998942i $$0.485355\pi$$
$$200$$ 0 0
$$201$$ −11.1998 −0.789976
$$202$$ 0 0
$$203$$ 41.6962 2.92650
$$204$$ 0 0
$$205$$ −10.7134 −0.748258
$$206$$ 0 0
$$207$$ −1.28151 −0.0890709
$$208$$ 0 0
$$209$$ −3.03803 −0.210145
$$210$$ 0 0
$$211$$ −14.7619 −1.01625 −0.508127 0.861282i $$-0.669662\pi$$
−0.508127 + 0.861282i $$0.669662\pi$$
$$212$$ 0 0
$$213$$ −11.5571 −0.791877
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.37268 0.500490
$$218$$ 0 0
$$219$$ −6.90053 −0.466295
$$220$$ 0 0
$$221$$ −21.3771 −1.43798
$$222$$ 0 0
$$223$$ −11.8434 −0.793095 −0.396548 0.918014i $$-0.629792\pi$$
−0.396548 + 0.918014i $$0.629792\pi$$
$$224$$ 0 0
$$225$$ −1.28151 −0.0854338
$$226$$ 0 0
$$227$$ 5.27556 0.350151 0.175075 0.984555i $$-0.443983\pi$$
0.175075 + 0.984555i $$0.443983\pi$$
$$228$$ 0 0
$$229$$ −1.23878 −0.0818611 −0.0409305 0.999162i $$-0.513032\pi$$
−0.0409305 + 0.999162i $$0.513032\pi$$
$$230$$ 0 0
$$231$$ −13.6301 −0.896798
$$232$$ 0 0
$$233$$ 2.66412 0.174533 0.0872663 0.996185i $$-0.472187\pi$$
0.0872663 + 0.996185i $$0.472187\pi$$
$$234$$ 0 0
$$235$$ −7.26391 −0.473845
$$236$$ 0 0
$$237$$ −9.28418 −0.603072
$$238$$ 0 0
$$239$$ −26.2577 −1.69847 −0.849235 0.528014i $$-0.822937\pi$$
−0.849235 + 0.528014i $$0.822937\pi$$
$$240$$ 0 0
$$241$$ −2.22326 −0.143213 −0.0716063 0.997433i $$-0.522813\pi$$
−0.0716063 + 0.997433i $$0.522813\pi$$
$$242$$ 0 0
$$243$$ 12.2325 0.784717
$$244$$ 0 0
$$245$$ −14.7353 −0.941406
$$246$$ 0 0
$$247$$ −3.81520 −0.242755
$$248$$ 0 0
$$249$$ 6.02690 0.381940
$$250$$ 0 0
$$251$$ −13.4829 −0.851031 −0.425515 0.904951i $$-0.639907\pi$$
−0.425515 + 0.904951i $$0.639907\pi$$
$$252$$ 0 0
$$253$$ −2.23020 −0.140211
$$254$$ 0 0
$$255$$ −10.0058 −0.626589
$$256$$ 0 0
$$257$$ 22.0281 1.37408 0.687039 0.726620i $$-0.258910\pi$$
0.687039 + 0.726620i $$0.258910\pi$$
$$258$$ 0 0
$$259$$ −6.53868 −0.406294
$$260$$ 0 0
$$261$$ −11.4613 −0.709438
$$262$$ 0 0
$$263$$ −22.3164 −1.37609 −0.688043 0.725670i $$-0.741530\pi$$
−0.688043 + 0.725670i $$0.741530\pi$$
$$264$$ 0 0
$$265$$ 8.38212 0.514909
$$266$$ 0 0
$$267$$ −6.16445 −0.377258
$$268$$ 0 0
$$269$$ 10.5050 0.640501 0.320251 0.947333i $$-0.396233\pi$$
0.320251 + 0.947333i $$0.396233\pi$$
$$270$$ 0 0
$$271$$ 3.84696 0.233686 0.116843 0.993150i $$-0.462723\pi$$
0.116843 + 0.993150i $$0.462723\pi$$
$$272$$ 0 0
$$273$$ −17.1169 −1.03596
$$274$$ 0 0
$$275$$ −2.23020 −0.134486
$$276$$ 0 0
$$277$$ 14.8653 0.893172 0.446586 0.894741i $$-0.352640\pi$$
0.446586 + 0.894741i $$0.352640\pi$$
$$278$$ 0 0
$$279$$ −2.02658 −0.121328
$$280$$ 0 0
$$281$$ 4.41659 0.263472 0.131736 0.991285i $$-0.457945\pi$$
0.131736 + 0.991285i $$0.457945\pi$$
$$282$$ 0 0
$$283$$ −3.88495 −0.230936 −0.115468 0.993311i $$-0.536837\pi$$
−0.115468 + 0.993311i $$0.536837\pi$$
$$284$$ 0 0
$$285$$ −1.78575 −0.105779
$$286$$ 0 0
$$287$$ 49.9472 2.94829
$$288$$ 0 0
$$289$$ 41.2583 2.42696
$$290$$ 0 0
$$291$$ 21.6511 1.26921
$$292$$ 0 0
$$293$$ −25.0257 −1.46202 −0.731009 0.682368i $$-0.760950\pi$$
−0.731009 + 0.682368i $$0.760950\pi$$
$$294$$ 0 0
$$295$$ −4.88331 −0.284317
$$296$$ 0 0
$$297$$ 12.5174 0.726333
$$298$$ 0 0
$$299$$ −2.80072 −0.161970
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 14.0618 0.807831
$$304$$ 0 0
$$305$$ −4.33282 −0.248096
$$306$$ 0 0
$$307$$ −22.0197 −1.25673 −0.628365 0.777918i $$-0.716276\pi$$
−0.628365 + 0.777918i $$0.716276\pi$$
$$308$$ 0 0
$$309$$ −2.56021 −0.145645
$$310$$ 0 0
$$311$$ −19.6217 −1.11264 −0.556322 0.830967i $$-0.687788\pi$$
−0.556322 + 0.830967i $$0.687788\pi$$
$$312$$ 0 0
$$313$$ −17.9496 −1.01457 −0.507285 0.861778i $$-0.669351\pi$$
−0.507285 + 0.861778i $$0.669351\pi$$
$$314$$ 0 0
$$315$$ 5.97454 0.336627
$$316$$ 0 0
$$317$$ 34.3185 1.92752 0.963761 0.266768i $$-0.0859559\pi$$
0.963761 + 0.266768i $$0.0859559\pi$$
$$318$$ 0 0
$$319$$ −19.9461 −1.11676
$$320$$ 0 0
$$321$$ 2.58992 0.144555
$$322$$ 0 0
$$323$$ 10.3974 0.578529
$$324$$ 0 0
$$325$$ −2.80072 −0.155356
$$326$$ 0 0
$$327$$ −11.6981 −0.646904
$$328$$ 0 0
$$329$$ 33.8652 1.86705
$$330$$ 0 0
$$331$$ 0.299762 0.0164764 0.00823821 0.999966i $$-0.497378\pi$$
0.00823821 + 0.999966i $$0.497378\pi$$
$$332$$ 0 0
$$333$$ 1.79733 0.0984931
$$334$$ 0 0
$$335$$ 8.54355 0.466784
$$336$$ 0 0
$$337$$ −22.2965 −1.21457 −0.607284 0.794485i $$-0.707741\pi$$
−0.607284 + 0.794485i $$0.707741\pi$$
$$338$$ 0 0
$$339$$ 22.7425 1.23520
$$340$$ 0 0
$$341$$ −3.52684 −0.190989
$$342$$ 0 0
$$343$$ 36.0630 1.94722
$$344$$ 0 0
$$345$$ −1.31091 −0.0705772
$$346$$ 0 0
$$347$$ −33.1240 −1.77819 −0.889094 0.457725i $$-0.848664\pi$$
−0.889094 + 0.457725i $$0.848664\pi$$
$$348$$ 0 0
$$349$$ −22.0041 −1.17785 −0.588926 0.808187i $$-0.700449\pi$$
−0.588926 + 0.808187i $$0.700449\pi$$
$$350$$ 0 0
$$351$$ 15.7195 0.839046
$$352$$ 0 0
$$353$$ −15.6085 −0.830759 −0.415379 0.909648i $$-0.636351\pi$$
−0.415379 + 0.909648i $$0.636351\pi$$
$$354$$ 0 0
$$355$$ 8.81604 0.467907
$$356$$ 0 0
$$357$$ 46.6483 2.46889
$$358$$ 0 0
$$359$$ −0.263781 −0.0139218 −0.00696092 0.999976i $$-0.502216\pi$$
−0.00696092 + 0.999976i $$0.502216\pi$$
$$360$$ 0 0
$$361$$ −17.1444 −0.902334
$$362$$ 0 0
$$363$$ −7.89984 −0.414634
$$364$$ 0 0
$$365$$ 5.26391 0.275526
$$366$$ 0 0
$$367$$ −10.8615 −0.566967 −0.283484 0.958977i $$-0.591490\pi$$
−0.283484 + 0.958977i $$0.591490\pi$$
$$368$$ 0 0
$$369$$ −13.7293 −0.714721
$$370$$ 0 0
$$371$$ −39.0784 −2.02885
$$372$$ 0 0
$$373$$ 26.8889 1.39225 0.696127 0.717919i $$-0.254905\pi$$
0.696127 + 0.717919i $$0.254905\pi$$
$$374$$ 0 0
$$375$$ −1.31091 −0.0676952
$$376$$ 0 0
$$377$$ −25.0485 −1.29007
$$378$$ 0 0
$$379$$ 2.15824 0.110861 0.0554306 0.998463i $$-0.482347\pi$$
0.0554306 + 0.998463i $$0.482347\pi$$
$$380$$ 0 0
$$381$$ −25.5460 −1.30876
$$382$$ 0 0
$$383$$ 4.62814 0.236487 0.118244 0.992985i $$-0.462274\pi$$
0.118244 + 0.992985i $$0.462274\pi$$
$$384$$ 0 0
$$385$$ 10.3974 0.529903
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −18.4988 −0.937929 −0.468964 0.883217i $$-0.655373\pi$$
−0.468964 + 0.883217i $$0.655373\pi$$
$$390$$ 0 0
$$391$$ 7.63271 0.386003
$$392$$ 0 0
$$393$$ −7.44664 −0.375634
$$394$$ 0 0
$$395$$ 7.08222 0.356345
$$396$$ 0 0
$$397$$ −10.2685 −0.515360 −0.257680 0.966230i $$-0.582958\pi$$
−0.257680 + 0.966230i $$0.582958\pi$$
$$398$$ 0 0
$$399$$ 8.32539 0.416791
$$400$$ 0 0
$$401$$ −0.885607 −0.0442251 −0.0221125 0.999755i $$-0.507039\pi$$
−0.0221125 + 0.999755i $$0.507039\pi$$
$$402$$ 0 0
$$403$$ −4.42906 −0.220627
$$404$$ 0 0
$$405$$ 3.51322 0.174573
$$406$$ 0 0
$$407$$ 3.12788 0.155043
$$408$$ 0 0
$$409$$ 24.3497 1.20401 0.602006 0.798491i $$-0.294368\pi$$
0.602006 + 0.798491i $$0.294368\pi$$
$$410$$ 0 0
$$411$$ −11.3872 −0.561688
$$412$$ 0 0
$$413$$ 22.7665 1.12027
$$414$$ 0 0
$$415$$ −4.59749 −0.225682
$$416$$ 0 0
$$417$$ 12.0941 0.592249
$$418$$ 0 0
$$419$$ −25.4535 −1.24348 −0.621742 0.783222i $$-0.713575\pi$$
−0.621742 + 0.783222i $$0.713575\pi$$
$$420$$ 0 0
$$421$$ 15.2376 0.742634 0.371317 0.928506i $$-0.378906\pi$$
0.371317 + 0.928506i $$0.378906\pi$$
$$422$$ 0 0
$$423$$ −9.30876 −0.452607
$$424$$ 0 0
$$425$$ 7.63271 0.370241
$$426$$ 0 0
$$427$$ 20.2001 0.977551
$$428$$ 0 0
$$429$$ 8.18816 0.395328
$$430$$ 0 0
$$431$$ 34.6166 1.66742 0.833711 0.552202i $$-0.186212\pi$$
0.833711 + 0.552202i $$0.186212\pi$$
$$432$$ 0 0
$$433$$ 28.4774 1.36854 0.684268 0.729230i $$-0.260122\pi$$
0.684268 + 0.729230i $$0.260122\pi$$
$$434$$ 0 0
$$435$$ −11.7243 −0.562138
$$436$$ 0 0
$$437$$ 1.36222 0.0651639
$$438$$ 0 0
$$439$$ −21.6009 −1.03095 −0.515477 0.856904i $$-0.672385\pi$$
−0.515477 + 0.856904i $$0.672385\pi$$
$$440$$ 0 0
$$441$$ −18.8834 −0.899211
$$442$$ 0 0
$$443$$ −40.3700 −1.91804 −0.959018 0.283346i $$-0.908555\pi$$
−0.959018 + 0.283346i $$0.908555\pi$$
$$444$$ 0 0
$$445$$ 4.70241 0.222915
$$446$$ 0 0
$$447$$ 24.3903 1.15362
$$448$$ 0 0
$$449$$ 5.95484 0.281026 0.140513 0.990079i $$-0.455125\pi$$
0.140513 + 0.990079i $$0.455125\pi$$
$$450$$ 0 0
$$451$$ −23.8931 −1.12508
$$452$$ 0 0
$$453$$ −22.8512 −1.07365
$$454$$ 0 0
$$455$$ 13.0573 0.612134
$$456$$ 0 0
$$457$$ −5.66169 −0.264843 −0.132421 0.991194i $$-0.542275\pi$$
−0.132421 + 0.991194i $$0.542275\pi$$
$$458$$ 0 0
$$459$$ −42.8400 −1.99960
$$460$$ 0 0
$$461$$ −9.26391 −0.431463 −0.215732 0.976453i $$-0.569214\pi$$
−0.215732 + 0.976453i $$0.569214\pi$$
$$462$$ 0 0
$$463$$ −37.9899 −1.76554 −0.882769 0.469807i $$-0.844324\pi$$
−0.882769 + 0.469807i $$0.844324\pi$$
$$464$$ 0 0
$$465$$ −2.07308 −0.0961368
$$466$$ 0 0
$$467$$ 26.2865 1.21640 0.608198 0.793785i $$-0.291893\pi$$
0.608198 + 0.793785i $$0.291893\pi$$
$$468$$ 0 0
$$469$$ −39.8310 −1.83922
$$470$$ 0 0
$$471$$ 1.10051 0.0507087
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.36222 0.0625030
$$476$$ 0 0
$$477$$ 10.7417 0.491831
$$478$$ 0 0
$$479$$ 31.0770 1.41994 0.709971 0.704231i $$-0.248708\pi$$
0.709971 + 0.704231i $$0.248708\pi$$
$$480$$ 0 0
$$481$$ 3.92804 0.179103
$$482$$ 0 0
$$483$$ 6.11163 0.278089
$$484$$ 0 0
$$485$$ −16.5160 −0.749954
$$486$$ 0 0
$$487$$ −14.9591 −0.677860 −0.338930 0.940812i $$-0.610065\pi$$
−0.338930 + 0.940812i $$0.610065\pi$$
$$488$$ 0 0
$$489$$ 19.0965 0.863573
$$490$$ 0 0
$$491$$ −24.1333 −1.08912 −0.544561 0.838721i $$-0.683304\pi$$
−0.544561 + 0.838721i $$0.683304\pi$$
$$492$$ 0 0
$$493$$ 68.2641 3.07446
$$494$$ 0 0
$$495$$ −2.85802 −0.128458
$$496$$ 0 0
$$497$$ −41.1014 −1.84365
$$498$$ 0 0
$$499$$ 17.7266 0.793552 0.396776 0.917915i $$-0.370129\pi$$
0.396776 + 0.917915i $$0.370129\pi$$
$$500$$ 0 0
$$501$$ 6.60492 0.295086
$$502$$ 0 0
$$503$$ 7.69566 0.343133 0.171566 0.985173i $$-0.445117\pi$$
0.171566 + 0.985173i $$0.445117\pi$$
$$504$$ 0 0
$$505$$ −10.7267 −0.477334
$$506$$ 0 0
$$507$$ −6.75906 −0.300180
$$508$$ 0 0
$$509$$ −33.0564 −1.46520 −0.732600 0.680659i $$-0.761693\pi$$
−0.732600 + 0.680659i $$0.761693\pi$$
$$510$$ 0 0
$$511$$ −24.5410 −1.08563
$$512$$ 0 0
$$513$$ −7.64572 −0.337567
$$514$$ 0 0
$$515$$ 1.95300 0.0860595
$$516$$ 0 0
$$517$$ −16.2000 −0.712474
$$518$$ 0 0
$$519$$ −14.8296 −0.650947
$$520$$ 0 0
$$521$$ 10.6047 0.464598 0.232299 0.972644i $$-0.425375\pi$$
0.232299 + 0.972644i $$0.425375\pi$$
$$522$$ 0 0
$$523$$ 34.5128 1.50914 0.754570 0.656219i $$-0.227845\pi$$
0.754570 + 0.656219i $$0.227845\pi$$
$$524$$ 0 0
$$525$$ 6.11163 0.266733
$$526$$ 0 0
$$527$$ 12.0704 0.525794
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −6.25799 −0.271574
$$532$$ 0 0
$$533$$ −30.0053 −1.29967
$$534$$ 0 0
$$535$$ −1.97566 −0.0854153
$$536$$ 0 0
$$537$$ 31.8622 1.37495
$$538$$ 0 0
$$539$$ −32.8627 −1.41550
$$540$$ 0 0
$$541$$ −27.3344 −1.17520 −0.587598 0.809153i $$-0.699926\pi$$
−0.587598 + 0.809153i $$0.699926\pi$$
$$542$$ 0 0
$$543$$ −25.5403 −1.09604
$$544$$ 0 0
$$545$$ 8.92360 0.382245
$$546$$ 0 0
$$547$$ −34.6190 −1.48020 −0.740101 0.672496i $$-0.765222\pi$$
−0.740101 + 0.672496i $$0.765222\pi$$
$$548$$ 0 0
$$549$$ −5.55254 −0.236977
$$550$$ 0 0
$$551$$ 12.1832 0.519022
$$552$$ 0 0
$$553$$ −33.0181 −1.40407
$$554$$ 0 0
$$555$$ 1.83857 0.0780430
$$556$$ 0 0
$$557$$ 25.2490 1.06983 0.534917 0.844905i $$-0.320343\pi$$
0.534917 + 0.844905i $$0.320343\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −22.3150 −0.942139
$$562$$ 0 0
$$563$$ −41.0793 −1.73128 −0.865642 0.500663i $$-0.833090\pi$$
−0.865642 + 0.500663i $$0.833090\pi$$
$$564$$ 0 0
$$565$$ −17.3486 −0.729860
$$566$$ 0 0
$$567$$ −16.3790 −0.687854
$$568$$ 0 0
$$569$$ −28.4230 −1.19155 −0.595776 0.803150i $$-0.703156\pi$$
−0.595776 + 0.803150i $$0.703156\pi$$
$$570$$ 0 0
$$571$$ −13.4690 −0.563659 −0.281830 0.959464i $$-0.590941\pi$$
−0.281830 + 0.959464i $$0.590941\pi$$
$$572$$ 0 0
$$573$$ −3.43698 −0.143582
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 3.69392 0.153780 0.0768899 0.997040i $$-0.475501\pi$$
0.0768899 + 0.997040i $$0.475501\pi$$
$$578$$ 0 0
$$579$$ 22.8534 0.949757
$$580$$ 0 0
$$581$$ 21.4340 0.889233
$$582$$ 0 0
$$583$$ 18.6938 0.774218
$$584$$ 0 0
$$585$$ −3.58914 −0.148393
$$586$$ 0 0
$$587$$ −19.1042 −0.788513 −0.394257 0.919000i $$-0.628998\pi$$
−0.394257 + 0.919000i $$0.628998\pi$$
$$588$$ 0 0
$$589$$ 2.15422 0.0887631
$$590$$ 0 0
$$591$$ −30.7280 −1.26398
$$592$$ 0 0
$$593$$ 13.9194 0.571602 0.285801 0.958289i $$-0.407740\pi$$
0.285801 + 0.958289i $$0.407740\pi$$
$$594$$ 0 0
$$595$$ −35.5846 −1.45883
$$596$$ 0 0
$$597$$ 1.70103 0.0696186
$$598$$ 0 0
$$599$$ −10.9498 −0.447396 −0.223698 0.974659i $$-0.571813\pi$$
−0.223698 + 0.974659i $$0.571813\pi$$
$$600$$ 0 0
$$601$$ 28.7034 1.17084 0.585418 0.810731i $$-0.300930\pi$$
0.585418 + 0.810731i $$0.300930\pi$$
$$602$$ 0 0
$$603$$ 10.9486 0.445862
$$604$$ 0 0
$$605$$ 6.02621 0.245000
$$606$$ 0 0
$$607$$ 43.7745 1.77675 0.888376 0.459117i $$-0.151834\pi$$
0.888376 + 0.459117i $$0.151834\pi$$
$$608$$ 0 0
$$609$$ 54.6601 2.21494
$$610$$ 0 0
$$611$$ −20.3442 −0.823036
$$612$$ 0 0
$$613$$ 6.59492 0.266366 0.133183 0.991091i $$-0.457480\pi$$
0.133183 + 0.991091i $$0.457480\pi$$
$$614$$ 0 0
$$615$$ −14.0444 −0.566324
$$616$$ 0 0
$$617$$ 20.3939 0.821029 0.410514 0.911854i $$-0.365349\pi$$
0.410514 + 0.911854i $$0.365349\pi$$
$$618$$ 0 0
$$619$$ −0.513389 −0.0206348 −0.0103174 0.999947i $$-0.503284\pi$$
−0.0103174 + 0.999947i $$0.503284\pi$$
$$620$$ 0 0
$$621$$ −5.61268 −0.225229
$$622$$ 0 0
$$623$$ −21.9232 −0.878333
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −3.98259 −0.159049
$$628$$ 0 0
$$629$$ −10.7050 −0.426835
$$630$$ 0 0
$$631$$ −16.3428 −0.650596 −0.325298 0.945612i $$-0.605465\pi$$
−0.325298 + 0.945612i $$0.605465\pi$$
$$632$$ 0 0
$$633$$ −19.3516 −0.769157
$$634$$ 0 0
$$635$$ 19.4872 0.773325
$$636$$ 0 0
$$637$$ −41.2695 −1.63516
$$638$$ 0 0
$$639$$ 11.2978 0.446935
$$640$$ 0 0
$$641$$ −33.7798 −1.33422 −0.667110 0.744959i $$-0.732469\pi$$
−0.667110 + 0.744959i $$0.732469\pi$$
$$642$$ 0 0
$$643$$ 2.18090 0.0860062 0.0430031 0.999075i $$-0.486307\pi$$
0.0430031 + 0.999075i $$0.486307\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 34.7346 1.36556 0.682778 0.730625i $$-0.260771\pi$$
0.682778 + 0.730625i $$0.260771\pi$$
$$648$$ 0 0
$$649$$ −10.8907 −0.427499
$$650$$ 0 0
$$651$$ 9.66494 0.378799
$$652$$ 0 0
$$653$$ −6.68309 −0.261530 −0.130765 0.991413i $$-0.541743\pi$$
−0.130765 + 0.991413i $$0.541743\pi$$
$$654$$ 0 0
$$655$$ 5.68050 0.221956
$$656$$ 0 0
$$657$$ 6.74575 0.263177
$$658$$ 0 0
$$659$$ 45.8903 1.78763 0.893815 0.448435i $$-0.148018\pi$$
0.893815 + 0.448435i $$0.148018\pi$$
$$660$$ 0 0
$$661$$ 13.6251 0.529957 0.264978 0.964254i $$-0.414635\pi$$
0.264978 + 0.964254i $$0.414635\pi$$
$$662$$ 0 0
$$663$$ −28.0234 −1.08834
$$664$$ 0 0
$$665$$ −6.35084 −0.246275
$$666$$ 0 0
$$667$$ 8.94362 0.346299
$$668$$ 0 0
$$669$$ −15.5257 −0.600259
$$670$$ 0 0
$$671$$ −9.66304 −0.373038
$$672$$ 0 0
$$673$$ −25.3475 −0.977075 −0.488537 0.872543i $$-0.662469\pi$$
−0.488537 + 0.872543i $$0.662469\pi$$
$$674$$ 0 0
$$675$$ −5.61268 −0.216032
$$676$$ 0 0
$$677$$ −17.2279 −0.662123 −0.331062 0.943609i $$-0.607407\pi$$
−0.331062 + 0.943609i $$0.607407\pi$$
$$678$$ 0 0
$$679$$ 76.9996 2.95497
$$680$$ 0 0
$$681$$ 6.91579 0.265014
$$682$$ 0 0
$$683$$ −39.5454 −1.51316 −0.756582 0.653899i $$-0.773132\pi$$
−0.756582 + 0.653899i $$0.773132\pi$$
$$684$$ 0 0
$$685$$ 8.68645 0.331892
$$686$$ 0 0
$$687$$ −1.62394 −0.0619570
$$688$$ 0 0
$$689$$ 23.4759 0.894361
$$690$$ 0 0
$$691$$ 29.0501 1.10512 0.552558 0.833474i $$-0.313652\pi$$
0.552558 + 0.833474i $$0.313652\pi$$
$$692$$ 0 0
$$693$$ 13.3244 0.506152
$$694$$ 0 0
$$695$$ −9.22569 −0.349950
$$696$$ 0 0
$$697$$ 81.7725 3.09735
$$698$$ 0 0
$$699$$ 3.49244 0.132096
$$700$$ 0 0
$$701$$ −17.7735 −0.671298 −0.335649 0.941987i $$-0.608956\pi$$
−0.335649 + 0.941987i $$0.608956\pi$$
$$702$$ 0 0
$$703$$ −1.91053 −0.0720571
$$704$$ 0 0
$$705$$ −9.52236 −0.358633
$$706$$ 0 0
$$707$$ 50.0093 1.88079
$$708$$ 0 0
$$709$$ −14.8606 −0.558103 −0.279052 0.960276i $$-0.590020\pi$$
−0.279052 + 0.960276i $$0.590020\pi$$
$$710$$ 0 0
$$711$$ 9.07592 0.340374
$$712$$ 0 0
$$713$$ 1.58140 0.0592240
$$714$$ 0 0
$$715$$ −6.24615 −0.233593
$$716$$ 0 0
$$717$$ −34.4216 −1.28550
$$718$$ 0 0
$$719$$ −27.4365 −1.02321 −0.511604 0.859221i $$-0.670949\pi$$
−0.511604 + 0.859221i $$0.670949\pi$$
$$720$$ 0 0
$$721$$ −9.10512 −0.339092
$$722$$ 0 0
$$723$$ −2.91449 −0.108391
$$724$$ 0 0
$$725$$ 8.94362 0.332158
$$726$$ 0 0
$$727$$ 23.3674 0.866649 0.433325 0.901238i $$-0.357340\pi$$
0.433325 + 0.901238i $$0.357340\pi$$
$$728$$ 0 0
$$729$$ 26.5754 0.984275
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −2.53381 −0.0935884 −0.0467942 0.998905i $$-0.514900\pi$$
−0.0467942 + 0.998905i $$0.514900\pi$$
$$734$$ 0 0
$$735$$ −19.3167 −0.712508
$$736$$ 0 0
$$737$$ 19.0538 0.701856
$$738$$ 0 0
$$739$$ −25.9318 −0.953918 −0.476959 0.878925i $$-0.658261\pi$$
−0.476959 + 0.878925i $$0.658261\pi$$
$$740$$ 0 0
$$741$$ −5.00139 −0.183731
$$742$$ 0 0
$$743$$ 2.11370 0.0775442 0.0387721 0.999248i $$-0.487655\pi$$
0.0387721 + 0.999248i $$0.487655\pi$$
$$744$$ 0 0
$$745$$ −18.6056 −0.681657
$$746$$ 0 0
$$747$$ −5.89172 −0.215567
$$748$$ 0 0
$$749$$ 9.21077 0.336554
$$750$$ 0 0
$$751$$ −2.54986 −0.0930459 −0.0465229 0.998917i $$-0.514814\pi$$
−0.0465229 + 0.998917i $$0.514814\pi$$
$$752$$ 0 0
$$753$$ −17.6749 −0.644107
$$754$$ 0 0
$$755$$ 17.4316 0.634399
$$756$$ 0 0
$$757$$ 30.6897 1.11544 0.557718 0.830030i $$-0.311677\pi$$
0.557718 + 0.830030i $$0.311677\pi$$
$$758$$ 0 0
$$759$$ −2.92360 −0.106120
$$760$$ 0 0
$$761$$ 19.3112 0.700032 0.350016 0.936744i $$-0.386176\pi$$
0.350016 + 0.936744i $$0.386176\pi$$
$$762$$ 0 0
$$763$$ −41.6028 −1.50612
$$764$$ 0 0
$$765$$ 9.78138 0.353646
$$766$$ 0 0
$$767$$ −13.6767 −0.493839
$$768$$ 0 0
$$769$$ −8.52624 −0.307464 −0.153732 0.988113i $$-0.549129\pi$$
−0.153732 + 0.988113i $$0.549129\pi$$
$$770$$ 0 0
$$771$$ 28.8770 1.03998
$$772$$ 0 0
$$773$$ 29.5107 1.06143 0.530713 0.847551i $$-0.321924\pi$$
0.530713 + 0.847551i $$0.321924\pi$$
$$774$$ 0 0
$$775$$ 1.58140 0.0568056
$$776$$ 0 0
$$777$$ −8.57164 −0.307506
$$778$$ 0 0
$$779$$ 14.5941 0.522887
$$780$$ 0 0
$$781$$ 19.6615 0.703545
$$782$$ 0 0
$$783$$ −50.1977 −1.79392
$$784$$ 0 0
$$785$$ −0.839497 −0.0299629
$$786$$ 0 0
$$787$$ 37.1733 1.32508 0.662542 0.749025i $$-0.269478\pi$$
0.662542 + 0.749025i $$0.269478\pi$$
$$788$$ 0 0
$$789$$ −29.2548 −1.04150
$$790$$ 0 0
$$791$$ 80.8811 2.87580
$$792$$ 0 0
$$793$$ −12.1350 −0.430926
$$794$$ 0 0
$$795$$ 10.9882 0.389712
$$796$$ 0 0
$$797$$ 16.9954 0.602008 0.301004 0.953623i $$-0.402678\pi$$
0.301004 + 0.953623i $$0.402678\pi$$
$$798$$ 0 0
$$799$$ 55.4434 1.96144
$$800$$ 0 0
$$801$$ 6.02617 0.212924
$$802$$ 0 0
$$803$$ 11.7396 0.414281
$$804$$ 0 0
$$805$$ −4.66212 −0.164318
$$806$$ 0 0
$$807$$ 13.7711 0.484767
$$808$$ 0 0
$$809$$ 16.8409 0.592095 0.296047 0.955173i $$-0.404331\pi$$
0.296047 + 0.955173i $$0.404331\pi$$
$$810$$ 0 0
$$811$$ −4.44804 −0.156192 −0.0780959 0.996946i $$-0.524884\pi$$
−0.0780959 + 0.996946i $$0.524884\pi$$
$$812$$ 0 0
$$813$$ 5.04303 0.176867
$$814$$ 0 0
$$815$$ −14.5673 −0.510271
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 16.7330 0.584698
$$820$$ 0 0
$$821$$ −16.7361 −0.584093 −0.292047 0.956404i $$-0.594336\pi$$
−0.292047 + 0.956404i $$0.594336\pi$$
$$822$$ 0 0
$$823$$ 43.6605 1.52191 0.760955 0.648805i $$-0.224731\pi$$
0.760955 + 0.648805i $$0.224731\pi$$
$$824$$ 0 0
$$825$$ −2.92360 −0.101787
$$826$$ 0 0
$$827$$ 37.9750 1.32052 0.660260 0.751037i $$-0.270446\pi$$
0.660260 + 0.751037i $$0.270446\pi$$
$$828$$ 0 0
$$829$$ 17.9046 0.621851 0.310925 0.950434i $$-0.399361\pi$$
0.310925 + 0.950434i $$0.399361\pi$$
$$830$$ 0 0
$$831$$ 19.4872 0.676002
$$832$$ 0 0
$$833$$ 112.471 3.89687
$$834$$ 0 0
$$835$$ −5.03842 −0.174362
$$836$$ 0 0
$$837$$ −8.87591 −0.306796
$$838$$ 0 0
$$839$$ 42.4503 1.46555 0.732773 0.680473i $$-0.238226\pi$$
0.732773 + 0.680473i $$0.238226\pi$$
$$840$$ 0 0
$$841$$ 50.9884 1.75822
$$842$$ 0 0
$$843$$ 5.78976 0.199410
$$844$$ 0 0
$$845$$ 5.15599 0.177372
$$846$$ 0 0
$$847$$ −28.0949 −0.965353
$$848$$ 0 0
$$849$$ −5.09283 −0.174785
$$850$$ 0 0
$$851$$ −1.40251 −0.0480775
$$852$$ 0 0
$$853$$ −22.9952 −0.787339 −0.393670 0.919252i $$-0.628795\pi$$
−0.393670 + 0.919252i $$0.628795\pi$$
$$854$$ 0 0
$$855$$ 1.74570 0.0597016
$$856$$ 0 0
$$857$$ −9.46053 −0.323166 −0.161583 0.986859i $$-0.551660\pi$$
−0.161583 + 0.986859i $$0.551660\pi$$
$$858$$ 0 0
$$859$$ 6.20043 0.211556 0.105778 0.994390i $$-0.466267\pi$$
0.105778 + 0.994390i $$0.466267\pi$$
$$860$$ 0 0
$$861$$ 65.4765 2.23143
$$862$$ 0 0
$$863$$ 9.68754 0.329768 0.164884 0.986313i $$-0.447275\pi$$
0.164884 + 0.986313i $$0.447275\pi$$
$$864$$ 0 0
$$865$$ 11.3124 0.384634
$$866$$ 0 0
$$867$$ 54.0860 1.83686
$$868$$ 0 0
$$869$$ 15.7948 0.535801
$$870$$ 0 0
$$871$$ 23.9280 0.810771
$$872$$ 0 0
$$873$$ −21.1654 −0.716340
$$874$$ 0 0
$$875$$ −4.66212 −0.157608
$$876$$ 0 0
$$877$$ 51.5663 1.74127 0.870636 0.491928i $$-0.163708\pi$$
0.870636 + 0.491928i $$0.163708\pi$$
$$878$$ 0 0
$$879$$ −32.8065 −1.10654
$$880$$ 0 0
$$881$$ −33.5969 −1.13191 −0.565954 0.824437i $$-0.691492\pi$$
−0.565954 + 0.824437i $$0.691492\pi$$
$$882$$ 0 0
$$883$$ −11.7914 −0.396814 −0.198407 0.980120i $$-0.563577\pi$$
−0.198407 + 0.980120i $$0.563577\pi$$
$$884$$ 0 0
$$885$$ −6.40159 −0.215187
$$886$$ 0 0
$$887$$ 54.9578 1.84530 0.922652 0.385634i $$-0.126017\pi$$
0.922652 + 0.385634i $$0.126017\pi$$
$$888$$ 0 0
$$889$$ −90.8515 −3.04706
$$890$$ 0 0
$$891$$ 7.83517 0.262488
$$892$$ 0 0
$$893$$ 9.89506 0.331126
$$894$$ 0 0
$$895$$ −24.3053 −0.812438
$$896$$ 0 0
$$897$$ −3.67149 −0.122588
$$898$$ 0 0
$$899$$ 14.1435 0.471711
$$900$$ 0 0
$$901$$ −63.9783 −2.13143
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.4829 0.647632
$$906$$ 0 0
$$907$$ −8.41429 −0.279392 −0.139696 0.990194i $$-0.544613\pi$$
−0.139696 + 0.990194i $$0.544613\pi$$
$$908$$ 0 0
$$909$$ −13.7464 −0.455940
$$910$$ 0 0
$$911$$ 26.4143 0.875146 0.437573 0.899183i $$-0.355838\pi$$
0.437573 + 0.899183i $$0.355838\pi$$
$$912$$ 0 0
$$913$$ −10.2533 −0.339335
$$914$$ 0 0
$$915$$ −5.67994 −0.187773
$$916$$ 0 0
$$917$$ −26.4832 −0.874551
$$918$$ 0 0
$$919$$ −59.2734 −1.95525 −0.977624 0.210359i $$-0.932537\pi$$
−0.977624 + 0.210359i $$0.932537\pi$$
$$920$$ 0 0
$$921$$ −28.8659 −0.951164
$$922$$ 0 0
$$923$$ 24.6912 0.812722
$$924$$ 0 0
$$925$$ −1.40251 −0.0461143
$$926$$ 0 0
$$927$$ 2.50279 0.0822023
$$928$$ 0 0
$$929$$ −17.2475 −0.565871 −0.282935 0.959139i $$-0.591308\pi$$
−0.282935 + 0.959139i $$0.591308\pi$$
$$930$$ 0 0
$$931$$ 20.0728 0.657859
$$932$$ 0 0
$$933$$ −25.7223 −0.842111
$$934$$ 0 0
$$935$$ 17.0225 0.556694
$$936$$ 0 0
$$937$$ −21.6918 −0.708642 −0.354321 0.935124i $$-0.615288\pi$$
−0.354321 + 0.935124i $$0.615288\pi$$
$$938$$ 0 0
$$939$$ −23.5303 −0.767883
$$940$$ 0 0
$$941$$ −11.1158 −0.362365 −0.181182 0.983449i $$-0.557992\pi$$
−0.181182 + 0.983449i $$0.557992\pi$$
$$942$$ 0 0
$$943$$ 10.7134 0.348877
$$944$$ 0 0
$$945$$ 26.1670 0.851212
$$946$$ 0 0
$$947$$ 40.8093 1.32613 0.663063 0.748564i $$-0.269256\pi$$
0.663063 + 0.748564i $$0.269256\pi$$
$$948$$ 0 0
$$949$$ 14.7427 0.478569
$$950$$ 0 0
$$951$$ 44.9886 1.45886
$$952$$ 0 0
$$953$$ 11.8237 0.383009 0.191504 0.981492i $$-0.438663\pi$$
0.191504 + 0.981492i $$0.438663\pi$$
$$954$$ 0 0
$$955$$ 2.62183 0.0848403
$$956$$ 0 0
$$957$$ −26.1475 −0.845230
$$958$$ 0 0
$$959$$ −40.4973 −1.30772
$$960$$ 0 0
$$961$$ −28.4992 −0.919328
$$962$$ 0 0
$$963$$ −2.53183 −0.0815869
$$964$$ 0 0
$$965$$ −17.4332 −0.561195
$$966$$ 0 0
$$967$$ −27.8536 −0.895710 −0.447855 0.894106i $$-0.647812\pi$$
−0.447855 + 0.894106i $$0.647812\pi$$
$$968$$ 0 0
$$969$$ 13.6301 0.437863
$$970$$ 0 0
$$971$$ 16.5249 0.530309 0.265155 0.964206i $$-0.414577\pi$$
0.265155 + 0.964206i $$0.414577\pi$$
$$972$$ 0 0
$$973$$ 43.0112 1.37888
$$974$$ 0 0
$$975$$ −3.67149 −0.117582
$$976$$ 0 0
$$977$$ −50.0059 −1.59983 −0.799915 0.600114i $$-0.795122\pi$$
−0.799915 + 0.600114i $$0.795122\pi$$
$$978$$ 0 0
$$979$$ 10.4873 0.335176
$$980$$ 0 0
$$981$$ 11.4357 0.365112
$$982$$ 0 0
$$983$$ −22.7894 −0.726869 −0.363434 0.931620i $$-0.618396\pi$$
−0.363434 + 0.931620i $$0.618396\pi$$
$$984$$ 0 0
$$985$$ 23.4402 0.746866
$$986$$ 0 0
$$987$$ 44.3943 1.41309
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −33.9822 −1.07948 −0.539740 0.841832i $$-0.681478\pi$$
−0.539740 + 0.841832i $$0.681478\pi$$
$$992$$ 0 0
$$993$$ 0.392962 0.0124703
$$994$$ 0 0
$$995$$ −1.29759 −0.0411365
$$996$$ 0 0
$$997$$ −17.7258 −0.561382 −0.280691 0.959798i $$-0.590564\pi$$
−0.280691 + 0.959798i $$0.590564\pi$$
$$998$$ 0 0
$$999$$ 7.87186 0.249055
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1840.2.a.v.1.4 5
4.3 odd 2 920.2.a.j.1.2 5
5.4 even 2 9200.2.a.cu.1.2 5
8.3 odd 2 7360.2.a.co.1.4 5
8.5 even 2 7360.2.a.cp.1.2 5
12.11 even 2 8280.2.a.bs.1.1 5
20.3 even 4 4600.2.e.u.4049.4 10
20.7 even 4 4600.2.e.u.4049.7 10
20.19 odd 2 4600.2.a.be.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.2 5 4.3 odd 2
1840.2.a.v.1.4 5 1.1 even 1 trivial
4600.2.a.be.1.4 5 20.19 odd 2
4600.2.e.u.4049.4 10 20.3 even 4
4600.2.e.u.4049.7 10 20.7 even 4
7360.2.a.co.1.4 5 8.3 odd 2
7360.2.a.cp.1.2 5 8.5 even 2
8280.2.a.bs.1.1 5 12.11 even 2
9200.2.a.cu.1.2 5 5.4 even 2