# Properties

 Label 2842.2.a.x.1.5 Level $2842$ Weight $2$ Character 2842.1 Self dual yes Analytic conductor $22.693$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2842 = 2 \cdot 7^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2842.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.6934842544$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.1019601.1 Defining polynomial: $$x^{5} - 10x^{3} - x^{2} + 24x + 7$$ x^5 - 10*x^3 - x^2 + 24*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 406) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-0.298978$$ of defining polynomial Character $$\chi$$ $$=$$ 2842.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.44245 q^{3} +1.00000 q^{4} -4.14347 q^{5} +2.44245 q^{6} +1.00000 q^{8} +2.96555 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.44245 q^{3} +1.00000 q^{4} -4.14347 q^{5} +2.44245 q^{6} +1.00000 q^{8} +2.96555 q^{9} -4.14347 q^{10} -4.70697 q^{11} +2.44245 q^{12} +1.91061 q^{13} -10.1202 q^{15} +1.00000 q^{16} -2.76714 q^{17} +2.96555 q^{18} -4.04040 q^{19} -4.14347 q^{20} -4.70697 q^{22} -2.17268 q^{23} +2.44245 q^{24} +12.1683 q^{25} +1.91061 q^{26} -0.0841506 q^{27} -1.00000 q^{29} -10.1202 q^{30} +0.969044 q^{31} +1.00000 q^{32} -11.4965 q^{33} -2.76714 q^{34} +2.96555 q^{36} -1.47340 q^{37} -4.04040 q^{38} +4.66657 q^{39} -4.14347 q^{40} -10.4741 q^{41} +7.84520 q^{43} -4.70697 q^{44} -12.2876 q^{45} -2.17268 q^{46} -11.7294 q^{47} +2.44245 q^{48} +12.1683 q^{50} -6.75860 q^{51} +1.91061 q^{52} -2.78082 q^{53} -0.0841506 q^{54} +19.5032 q^{55} -9.86847 q^{57} -1.00000 q^{58} -3.41673 q^{59} -10.1202 q^{60} -11.4192 q^{61} +0.969044 q^{62} +1.00000 q^{64} -7.91656 q^{65} -11.4965 q^{66} -2.27575 q^{67} -2.76714 q^{68} -5.30667 q^{69} -6.24653 q^{71} +2.96555 q^{72} +13.5230 q^{73} -1.47340 q^{74} +29.7205 q^{75} -4.04040 q^{76} +4.66657 q^{78} -4.46293 q^{79} -4.14347 q^{80} -9.10217 q^{81} -10.4741 q^{82} +6.06003 q^{83} +11.4656 q^{85} +7.84520 q^{86} -2.44245 q^{87} -4.70697 q^{88} -13.2662 q^{89} -12.2876 q^{90} -2.17268 q^{92} +2.36684 q^{93} -11.7294 q^{94} +16.7413 q^{95} +2.44245 q^{96} -16.8452 q^{97} -13.9587 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 - 3 * q^3 + 5 * q^4 - 7 * q^5 - 3 * q^6 + 5 * q^8 + 8 * q^9 $$5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9} - 7 q^{10} - 3 q^{12} - 10 q^{13} - 10 q^{15} + 5 q^{16} - 8 q^{17} + 8 q^{18} - 2 q^{19} - 7 q^{20} + q^{23} - 3 q^{24} + 12 q^{25} - 10 q^{26} - 15 q^{27} - 5 q^{29} - 10 q^{30} - 11 q^{31} + 5 q^{32} - 9 q^{33} - 8 q^{34} + 8 q^{36} - 8 q^{37} - 2 q^{38} + 18 q^{39} - 7 q^{40} - 23 q^{41} - 3 q^{43} - 4 q^{45} + q^{46} - 16 q^{47} - 3 q^{48} + 12 q^{50} + 7 q^{51} - 10 q^{52} + 7 q^{53} - 15 q^{54} - 6 q^{55} - 34 q^{57} - 5 q^{58} + 9 q^{59} - 10 q^{60} - 15 q^{61} - 11 q^{62} + 5 q^{64} + 5 q^{65} - 9 q^{66} - 4 q^{67} - 8 q^{68} - 14 q^{69} - 22 q^{71} + 8 q^{72} - 8 q^{74} + 34 q^{75} - 2 q^{76} + 18 q^{78} - 13 q^{79} - 7 q^{80} + 17 q^{81} - 23 q^{82} - 28 q^{83} - 7 q^{85} - 3 q^{86} + 3 q^{87} - 17 q^{89} - 4 q^{90} + q^{92} + 17 q^{93} - 16 q^{94} - 9 q^{95} - 3 q^{96} - 42 q^{97} - 42 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 - 3 * q^3 + 5 * q^4 - 7 * q^5 - 3 * q^6 + 5 * q^8 + 8 * q^9 - 7 * q^10 - 3 * q^12 - 10 * q^13 - 10 * q^15 + 5 * q^16 - 8 * q^17 + 8 * q^18 - 2 * q^19 - 7 * q^20 + q^23 - 3 * q^24 + 12 * q^25 - 10 * q^26 - 15 * q^27 - 5 * q^29 - 10 * q^30 - 11 * q^31 + 5 * q^32 - 9 * q^33 - 8 * q^34 + 8 * q^36 - 8 * q^37 - 2 * q^38 + 18 * q^39 - 7 * q^40 - 23 * q^41 - 3 * q^43 - 4 * q^45 + q^46 - 16 * q^47 - 3 * q^48 + 12 * q^50 + 7 * q^51 - 10 * q^52 + 7 * q^53 - 15 * q^54 - 6 * q^55 - 34 * q^57 - 5 * q^58 + 9 * q^59 - 10 * q^60 - 15 * q^61 - 11 * q^62 + 5 * q^64 + 5 * q^65 - 9 * q^66 - 4 * q^67 - 8 * q^68 - 14 * q^69 - 22 * q^71 + 8 * q^72 - 8 * q^74 + 34 * q^75 - 2 * q^76 + 18 * q^78 - 13 * q^79 - 7 * q^80 + 17 * q^81 - 23 * q^82 - 28 * q^83 - 7 * q^85 - 3 * q^86 + 3 * q^87 - 17 * q^89 - 4 * q^90 + q^92 + 17 * q^93 - 16 * q^94 - 9 * q^95 - 3 * q^96 - 42 * q^97 - 42 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.44245 1.41015 0.705074 0.709134i $$-0.250914\pi$$
0.705074 + 0.709134i $$0.250914\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −4.14347 −1.85302 −0.926508 0.376276i $$-0.877205\pi$$
−0.926508 + 0.376276i $$0.877205\pi$$
$$6$$ 2.44245 0.997125
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 2.96555 0.988516
$$10$$ −4.14347 −1.31028
$$11$$ −4.70697 −1.41921 −0.709603 0.704602i $$-0.751126\pi$$
−0.709603 + 0.704602i $$0.751126\pi$$
$$12$$ 2.44245 0.705074
$$13$$ 1.91061 0.529908 0.264954 0.964261i $$-0.414643\pi$$
0.264954 + 0.964261i $$0.414643\pi$$
$$14$$ 0 0
$$15$$ −10.1202 −2.61302
$$16$$ 1.00000 0.250000
$$17$$ −2.76714 −0.671131 −0.335565 0.942017i $$-0.608927\pi$$
−0.335565 + 0.942017i $$0.608927\pi$$
$$18$$ 2.96555 0.698986
$$19$$ −4.04040 −0.926932 −0.463466 0.886115i $$-0.653394\pi$$
−0.463466 + 0.886115i $$0.653394\pi$$
$$20$$ −4.14347 −0.926508
$$21$$ 0 0
$$22$$ −4.70697 −1.00353
$$23$$ −2.17268 −0.453036 −0.226518 0.974007i $$-0.572734\pi$$
−0.226518 + 0.974007i $$0.572734\pi$$
$$24$$ 2.44245 0.498562
$$25$$ 12.1683 2.43367
$$26$$ 1.91061 0.374702
$$27$$ −0.0841506 −0.0161948
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ −10.1202 −1.84769
$$31$$ 0.969044 0.174045 0.0870227 0.996206i $$-0.472265\pi$$
0.0870227 + 0.996206i $$0.472265\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −11.4965 −2.00129
$$34$$ −2.76714 −0.474561
$$35$$ 0 0
$$36$$ 2.96555 0.494258
$$37$$ −1.47340 −0.242226 −0.121113 0.992639i $$-0.538646\pi$$
−0.121113 + 0.992639i $$0.538646\pi$$
$$38$$ −4.04040 −0.655440
$$39$$ 4.66657 0.747249
$$40$$ −4.14347 −0.655140
$$41$$ −10.4741 −1.63578 −0.817891 0.575373i $$-0.804857\pi$$
−0.817891 + 0.575373i $$0.804857\pi$$
$$42$$ 0 0
$$43$$ 7.84520 1.19638 0.598191 0.801353i $$-0.295886\pi$$
0.598191 + 0.801353i $$0.295886\pi$$
$$44$$ −4.70697 −0.709603
$$45$$ −12.2876 −1.83173
$$46$$ −2.17268 −0.320345
$$47$$ −11.7294 −1.71091 −0.855453 0.517880i $$-0.826721\pi$$
−0.855453 + 0.517880i $$0.826721\pi$$
$$48$$ 2.44245 0.352537
$$49$$ 0 0
$$50$$ 12.1683 1.72086
$$51$$ −6.75860 −0.946393
$$52$$ 1.91061 0.264954
$$53$$ −2.78082 −0.381975 −0.190988 0.981592i $$-0.561169\pi$$
−0.190988 + 0.981592i $$0.561169\pi$$
$$54$$ −0.0841506 −0.0114514
$$55$$ 19.5032 2.62981
$$56$$ 0 0
$$57$$ −9.86847 −1.30711
$$58$$ −1.00000 −0.131306
$$59$$ −3.41673 −0.444820 −0.222410 0.974953i $$-0.571392\pi$$
−0.222410 + 0.974953i $$0.571392\pi$$
$$60$$ −10.1202 −1.30651
$$61$$ −11.4192 −1.46208 −0.731038 0.682337i $$-0.760964\pi$$
−0.731038 + 0.682337i $$0.760964\pi$$
$$62$$ 0.969044 0.123069
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −7.91656 −0.981929
$$66$$ −11.4965 −1.41512
$$67$$ −2.27575 −0.278027 −0.139014 0.990290i $$-0.544393\pi$$
−0.139014 + 0.990290i $$0.544393\pi$$
$$68$$ −2.76714 −0.335565
$$69$$ −5.30667 −0.638848
$$70$$ 0 0
$$71$$ −6.24653 −0.741327 −0.370664 0.928767i $$-0.620870\pi$$
−0.370664 + 0.928767i $$0.620870\pi$$
$$72$$ 2.96555 0.349493
$$73$$ 13.5230 1.58274 0.791371 0.611336i $$-0.209368\pi$$
0.791371 + 0.611336i $$0.209368\pi$$
$$74$$ −1.47340 −0.171280
$$75$$ 29.7205 3.43183
$$76$$ −4.04040 −0.463466
$$77$$ 0 0
$$78$$ 4.66657 0.528385
$$79$$ −4.46293 −0.502119 −0.251059 0.967972i $$-0.580779\pi$$
−0.251059 + 0.967972i $$0.580779\pi$$
$$80$$ −4.14347 −0.463254
$$81$$ −9.10217 −1.01135
$$82$$ −10.4741 −1.15667
$$83$$ 6.06003 0.665175 0.332587 0.943072i $$-0.392078\pi$$
0.332587 + 0.943072i $$0.392078\pi$$
$$84$$ 0 0
$$85$$ 11.4656 1.24362
$$86$$ 7.84520 0.845970
$$87$$ −2.44245 −0.261858
$$88$$ −4.70697 −0.501765
$$89$$ −13.2662 −1.40621 −0.703105 0.711086i $$-0.748204\pi$$
−0.703105 + 0.711086i $$0.748204\pi$$
$$90$$ −12.2876 −1.29523
$$91$$ 0 0
$$92$$ −2.17268 −0.226518
$$93$$ 2.36684 0.245430
$$94$$ −11.7294 −1.20979
$$95$$ 16.7413 1.71762
$$96$$ 2.44245 0.249281
$$97$$ −16.8452 −1.71037 −0.855186 0.518322i $$-0.826557\pi$$
−0.855186 + 0.518322i $$0.826557\pi$$
$$98$$ 0 0
$$99$$ −13.9587 −1.40291
$$100$$ 12.1683 1.21683
$$101$$ 12.4261 1.23644 0.618219 0.786006i $$-0.287854\pi$$
0.618219 + 0.786006i $$0.287854\pi$$
$$102$$ −6.75860 −0.669201
$$103$$ 16.1718 1.59346 0.796729 0.604337i $$-0.206562\pi$$
0.796729 + 0.604337i $$0.206562\pi$$
$$104$$ 1.91061 0.187351
$$105$$ 0 0
$$106$$ −2.78082 −0.270097
$$107$$ 10.6246 1.02712 0.513558 0.858055i $$-0.328327\pi$$
0.513558 + 0.858055i $$0.328327\pi$$
$$108$$ −0.0841506 −0.00809739
$$109$$ 6.09884 0.584162 0.292081 0.956394i $$-0.405652\pi$$
0.292081 + 0.956394i $$0.405652\pi$$
$$110$$ 19.5032 1.85956
$$111$$ −3.59871 −0.341574
$$112$$ 0 0
$$113$$ 14.5875 1.37228 0.686139 0.727471i $$-0.259304\pi$$
0.686139 + 0.727471i $$0.259304\pi$$
$$114$$ −9.86847 −0.924267
$$115$$ 9.00245 0.839483
$$116$$ −1.00000 −0.0928477
$$117$$ 5.66601 0.523823
$$118$$ −3.41673 −0.314535
$$119$$ 0 0
$$120$$ −10.1202 −0.923844
$$121$$ 11.1556 1.01414
$$122$$ −11.4192 −1.03384
$$123$$ −25.5825 −2.30669
$$124$$ 0.969044 0.0870227
$$125$$ −29.7018 −2.65661
$$126$$ 0 0
$$127$$ 5.46381 0.484835 0.242418 0.970172i $$-0.422060\pi$$
0.242418 + 0.970172i $$0.422060\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 19.1615 1.68708
$$130$$ −7.91656 −0.694328
$$131$$ 13.4182 1.17235 0.586176 0.810184i $$-0.300633\pi$$
0.586176 + 0.810184i $$0.300633\pi$$
$$132$$ −11.4965 −1.00064
$$133$$ 0 0
$$134$$ −2.27575 −0.196595
$$135$$ 0.348675 0.0300092
$$136$$ −2.76714 −0.237281
$$137$$ 17.1032 1.46123 0.730613 0.682791i $$-0.239234\pi$$
0.730613 + 0.682791i $$0.239234\pi$$
$$138$$ −5.30667 −0.451734
$$139$$ 3.90711 0.331397 0.165699 0.986176i $$-0.447012\pi$$
0.165699 + 0.986176i $$0.447012\pi$$
$$140$$ 0 0
$$141$$ −28.6484 −2.41263
$$142$$ −6.24653 −0.524198
$$143$$ −8.99320 −0.752049
$$144$$ 2.96555 0.247129
$$145$$ 4.14347 0.344096
$$146$$ 13.5230 1.11917
$$147$$ 0 0
$$148$$ −1.47340 −0.121113
$$149$$ −7.49827 −0.614282 −0.307141 0.951664i $$-0.599372\pi$$
−0.307141 + 0.951664i $$0.599372\pi$$
$$150$$ 29.7205 2.42667
$$151$$ 9.41817 0.766440 0.383220 0.923657i $$-0.374815\pi$$
0.383220 + 0.923657i $$0.374815\pi$$
$$152$$ −4.04040 −0.327720
$$153$$ −8.20609 −0.663423
$$154$$ 0 0
$$155$$ −4.01520 −0.322509
$$156$$ 4.66657 0.373625
$$157$$ −17.8286 −1.42288 −0.711439 0.702747i $$-0.751956\pi$$
−0.711439 + 0.702747i $$0.751956\pi$$
$$158$$ −4.46293 −0.355051
$$159$$ −6.79201 −0.538641
$$160$$ −4.14347 −0.327570
$$161$$ 0 0
$$162$$ −9.10217 −0.715134
$$163$$ −10.8789 −0.852102 −0.426051 0.904699i $$-0.640096\pi$$
−0.426051 + 0.904699i $$0.640096\pi$$
$$164$$ −10.4741 −0.817891
$$165$$ 47.6355 3.70842
$$166$$ 6.06003 0.470350
$$167$$ −16.0326 −1.24064 −0.620318 0.784350i $$-0.712997\pi$$
−0.620318 + 0.784350i $$0.712997\pi$$
$$168$$ 0 0
$$169$$ −9.34956 −0.719197
$$170$$ 11.4656 0.879369
$$171$$ −11.9820 −0.916287
$$172$$ 7.84520 0.598191
$$173$$ 6.18878 0.470524 0.235262 0.971932i $$-0.424405\pi$$
0.235262 + 0.971932i $$0.424405\pi$$
$$174$$ −2.44245 −0.185161
$$175$$ 0 0
$$176$$ −4.70697 −0.354801
$$177$$ −8.34518 −0.627262
$$178$$ −13.2662 −0.994341
$$179$$ 1.20540 0.0900957 0.0450479 0.998985i $$-0.485656\pi$$
0.0450479 + 0.998985i $$0.485656\pi$$
$$180$$ −12.2876 −0.915867
$$181$$ −6.95105 −0.516668 −0.258334 0.966056i $$-0.583173\pi$$
−0.258334 + 0.966056i $$0.583173\pi$$
$$182$$ 0 0
$$183$$ −27.8907 −2.06174
$$184$$ −2.17268 −0.160172
$$185$$ 6.10500 0.448848
$$186$$ 2.36684 0.173545
$$187$$ 13.0249 0.952472
$$188$$ −11.7294 −0.855453
$$189$$ 0 0
$$190$$ 16.7413 1.21454
$$191$$ −6.80847 −0.492644 −0.246322 0.969188i $$-0.579222\pi$$
−0.246322 + 0.969188i $$0.579222\pi$$
$$192$$ 2.44245 0.176268
$$193$$ −3.89664 −0.280486 −0.140243 0.990117i $$-0.544788\pi$$
−0.140243 + 0.990117i $$0.544788\pi$$
$$194$$ −16.8452 −1.20942
$$195$$ −19.3358 −1.38466
$$196$$ 0 0
$$197$$ 15.5624 1.10877 0.554386 0.832260i $$-0.312953\pi$$
0.554386 + 0.832260i $$0.312953\pi$$
$$198$$ −13.9587 −0.992005
$$199$$ −24.5735 −1.74197 −0.870986 0.491308i $$-0.836519\pi$$
−0.870986 + 0.491308i $$0.836519\pi$$
$$200$$ 12.1683 0.860431
$$201$$ −5.55840 −0.392059
$$202$$ 12.4261 0.874294
$$203$$ 0 0
$$204$$ −6.75860 −0.473197
$$205$$ 43.3992 3.03113
$$206$$ 16.1718 1.12674
$$207$$ −6.44320 −0.447833
$$208$$ 1.91061 0.132477
$$209$$ 19.0181 1.31551
$$210$$ 0 0
$$211$$ 8.65646 0.595935 0.297968 0.954576i $$-0.403691\pi$$
0.297968 + 0.954576i $$0.403691\pi$$
$$212$$ −2.78082 −0.190988
$$213$$ −15.2568 −1.04538
$$214$$ 10.6246 0.726281
$$215$$ −32.5064 −2.21691
$$216$$ −0.0841506 −0.00572572
$$217$$ 0 0
$$218$$ 6.09884 0.413065
$$219$$ 33.0291 2.23190
$$220$$ 19.5032 1.31490
$$221$$ −5.28694 −0.355638
$$222$$ −3.59871 −0.241529
$$223$$ 10.9191 0.731194 0.365597 0.930773i $$-0.380865\pi$$
0.365597 + 0.930773i $$0.380865\pi$$
$$224$$ 0 0
$$225$$ 36.0858 2.40572
$$226$$ 14.5875 0.970347
$$227$$ 10.4951 0.696583 0.348292 0.937386i $$-0.386762\pi$$
0.348292 + 0.937386i $$0.386762\pi$$
$$228$$ −9.86847 −0.653555
$$229$$ 7.51525 0.496622 0.248311 0.968680i $$-0.420125\pi$$
0.248311 + 0.968680i $$0.420125\pi$$
$$230$$ 9.00245 0.593604
$$231$$ 0 0
$$232$$ −1.00000 −0.0656532
$$233$$ 5.14262 0.336904 0.168452 0.985710i $$-0.446123\pi$$
0.168452 + 0.985710i $$0.446123\pi$$
$$234$$ 5.66601 0.370399
$$235$$ 48.6003 3.17034
$$236$$ −3.41673 −0.222410
$$237$$ −10.9005 −0.708061
$$238$$ 0 0
$$239$$ 0.749673 0.0484923 0.0242462 0.999706i $$-0.492281\pi$$
0.0242462 + 0.999706i $$0.492281\pi$$
$$240$$ −10.1202 −0.653256
$$241$$ 9.01959 0.581003 0.290501 0.956875i $$-0.406178\pi$$
0.290501 + 0.956875i $$0.406178\pi$$
$$242$$ 11.1556 0.717108
$$243$$ −21.9791 −1.40996
$$244$$ −11.4192 −0.731038
$$245$$ 0 0
$$246$$ −25.5825 −1.63108
$$247$$ −7.71964 −0.491189
$$248$$ 0.969044 0.0615344
$$249$$ 14.8013 0.937994
$$250$$ −29.7018 −1.87850
$$251$$ 29.1169 1.83784 0.918921 0.394441i $$-0.129062\pi$$
0.918921 + 0.394441i $$0.129062\pi$$
$$252$$ 0 0
$$253$$ 10.2268 0.642951
$$254$$ 5.46381 0.342830
$$255$$ 28.0041 1.75368
$$256$$ 1.00000 0.0625000
$$257$$ 23.2275 1.44889 0.724447 0.689330i $$-0.242095\pi$$
0.724447 + 0.689330i $$0.242095\pi$$
$$258$$ 19.1615 1.19294
$$259$$ 0 0
$$260$$ −7.91656 −0.490964
$$261$$ −2.96555 −0.183563
$$262$$ 13.4182 0.828978
$$263$$ −20.5033 −1.26429 −0.632145 0.774850i $$-0.717825\pi$$
−0.632145 + 0.774850i $$0.717825\pi$$
$$264$$ −11.4965 −0.707562
$$265$$ 11.5222 0.707806
$$266$$ 0 0
$$267$$ −32.4019 −1.98296
$$268$$ −2.27575 −0.139014
$$269$$ 11.3952 0.694778 0.347389 0.937721i $$-0.387068\pi$$
0.347389 + 0.937721i $$0.387068\pi$$
$$270$$ 0.348675 0.0212197
$$271$$ −17.9751 −1.09191 −0.545954 0.837815i $$-0.683833\pi$$
−0.545954 + 0.837815i $$0.683833\pi$$
$$272$$ −2.76714 −0.167783
$$273$$ 0 0
$$274$$ 17.1032 1.03324
$$275$$ −57.2760 −3.45387
$$276$$ −5.30667 −0.319424
$$277$$ −2.08032 −0.124994 −0.0624971 0.998045i $$-0.519906\pi$$
−0.0624971 + 0.998045i $$0.519906\pi$$
$$278$$ 3.90711 0.234333
$$279$$ 2.87375 0.172047
$$280$$ 0 0
$$281$$ 7.35741 0.438906 0.219453 0.975623i $$-0.429573\pi$$
0.219453 + 0.975623i $$0.429573\pi$$
$$282$$ −28.6484 −1.70599
$$283$$ 6.08053 0.361450 0.180725 0.983534i $$-0.442156\pi$$
0.180725 + 0.983534i $$0.442156\pi$$
$$284$$ −6.24653 −0.370664
$$285$$ 40.8897 2.42210
$$286$$ −8.99320 −0.531779
$$287$$ 0 0
$$288$$ 2.96555 0.174747
$$289$$ −9.34292 −0.549583
$$290$$ 4.14347 0.243313
$$291$$ −41.1435 −2.41188
$$292$$ 13.5230 0.791371
$$293$$ −13.7159 −0.801291 −0.400646 0.916233i $$-0.631214\pi$$
−0.400646 + 0.916233i $$0.631214\pi$$
$$294$$ 0 0
$$295$$ 14.1571 0.824259
$$296$$ −1.47340 −0.0856398
$$297$$ 0.396094 0.0229837
$$298$$ −7.49827 −0.434363
$$299$$ −4.15116 −0.240068
$$300$$ 29.7205 1.71591
$$301$$ 0 0
$$302$$ 9.41817 0.541955
$$303$$ 30.3500 1.74356
$$304$$ −4.04040 −0.231733
$$305$$ 47.3150 2.70925
$$306$$ −8.20609 −0.469111
$$307$$ −7.64494 −0.436320 −0.218160 0.975913i $$-0.570005\pi$$
−0.218160 + 0.975913i $$0.570005\pi$$
$$308$$ 0 0
$$309$$ 39.4988 2.24701
$$310$$ −4.01520 −0.228048
$$311$$ −24.4252 −1.38502 −0.692512 0.721406i $$-0.743496\pi$$
−0.692512 + 0.721406i $$0.743496\pi$$
$$312$$ 4.66657 0.264192
$$313$$ 18.2645 1.03237 0.516185 0.856477i $$-0.327352\pi$$
0.516185 + 0.856477i $$0.327352\pi$$
$$314$$ −17.8286 −1.00613
$$315$$ 0 0
$$316$$ −4.46293 −0.251059
$$317$$ −10.4217 −0.585342 −0.292671 0.956213i $$-0.594544\pi$$
−0.292671 + 0.956213i $$0.594544\pi$$
$$318$$ −6.79201 −0.380877
$$319$$ 4.70697 0.263540
$$320$$ −4.14347 −0.231627
$$321$$ 25.9500 1.44839
$$322$$ 0 0
$$323$$ 11.1804 0.622093
$$324$$ −9.10217 −0.505676
$$325$$ 23.2490 1.28962
$$326$$ −10.8789 −0.602527
$$327$$ 14.8961 0.823755
$$328$$ −10.4741 −0.578336
$$329$$ 0 0
$$330$$ 47.6355 2.62225
$$331$$ 4.68351 0.257429 0.128715 0.991682i $$-0.458915\pi$$
0.128715 + 0.991682i $$0.458915\pi$$
$$332$$ 6.06003 0.332587
$$333$$ −4.36944 −0.239444
$$334$$ −16.0326 −0.877262
$$335$$ 9.42950 0.515189
$$336$$ 0 0
$$337$$ −25.3719 −1.38210 −0.691048 0.722809i $$-0.742851\pi$$
−0.691048 + 0.722809i $$0.742851\pi$$
$$338$$ −9.34956 −0.508549
$$339$$ 35.6292 1.93511
$$340$$ 11.4656 0.621808
$$341$$ −4.56126 −0.247006
$$342$$ −11.9820 −0.647913
$$343$$ 0 0
$$344$$ 7.84520 0.422985
$$345$$ 21.9880 1.18379
$$346$$ 6.18878 0.332711
$$347$$ −1.93057 −0.103639 −0.0518193 0.998656i $$-0.516502\pi$$
−0.0518193 + 0.998656i $$0.516502\pi$$
$$348$$ −2.44245 −0.130929
$$349$$ −1.95782 −0.104800 −0.0523998 0.998626i $$-0.516687\pi$$
−0.0523998 + 0.998626i $$0.516687\pi$$
$$350$$ 0 0
$$351$$ −0.160779 −0.00858175
$$352$$ −4.70697 −0.250882
$$353$$ 16.3330 0.869317 0.434658 0.900595i $$-0.356869\pi$$
0.434658 + 0.900595i $$0.356869\pi$$
$$354$$ −8.34518 −0.443541
$$355$$ 25.8823 1.37369
$$356$$ −13.2662 −0.703105
$$357$$ 0 0
$$358$$ 1.20540 0.0637073
$$359$$ −11.7801 −0.621729 −0.310865 0.950454i $$-0.600619\pi$$
−0.310865 + 0.950454i $$0.600619\pi$$
$$360$$ −12.2876 −0.647616
$$361$$ −2.67514 −0.140797
$$362$$ −6.95105 −0.365339
$$363$$ 27.2469 1.43009
$$364$$ 0 0
$$365$$ −56.0320 −2.93285
$$366$$ −27.8907 −1.45787
$$367$$ −35.6438 −1.86059 −0.930295 0.366813i $$-0.880449\pi$$
−0.930295 + 0.366813i $$0.880449\pi$$
$$368$$ −2.17268 −0.113259
$$369$$ −31.0615 −1.61700
$$370$$ 6.10500 0.317384
$$371$$ 0 0
$$372$$ 2.36684 0.122715
$$373$$ 17.7483 0.918971 0.459486 0.888185i $$-0.348034\pi$$
0.459486 + 0.888185i $$0.348034\pi$$
$$374$$ 13.0249 0.673500
$$375$$ −72.5450 −3.74621
$$376$$ −11.7294 −0.604897
$$377$$ −1.91061 −0.0984015
$$378$$ 0 0
$$379$$ −15.2263 −0.782124 −0.391062 0.920364i $$-0.627892\pi$$
−0.391062 + 0.920364i $$0.627892\pi$$
$$380$$ 16.7413 0.858810
$$381$$ 13.3451 0.683689
$$382$$ −6.80847 −0.348352
$$383$$ 15.8118 0.807943 0.403971 0.914772i $$-0.367629\pi$$
0.403971 + 0.914772i $$0.367629\pi$$
$$384$$ 2.44245 0.124641
$$385$$ 0 0
$$386$$ −3.89664 −0.198334
$$387$$ 23.2653 1.18264
$$388$$ −16.8452 −0.855186
$$389$$ −21.7172 −1.10110 −0.550552 0.834801i $$-0.685583\pi$$
−0.550552 + 0.834801i $$0.685583\pi$$
$$390$$ −19.3358 −0.979105
$$391$$ 6.01213 0.304047
$$392$$ 0 0
$$393$$ 32.7732 1.65319
$$394$$ 15.5624 0.784020
$$395$$ 18.4920 0.930434
$$396$$ −13.9587 −0.701453
$$397$$ −1.26033 −0.0632543 −0.0316272 0.999500i $$-0.510069\pi$$
−0.0316272 + 0.999500i $$0.510069\pi$$
$$398$$ −24.5735 −1.23176
$$399$$ 0 0
$$400$$ 12.1683 0.608417
$$401$$ −1.91181 −0.0954713 −0.0477356 0.998860i $$-0.515201\pi$$
−0.0477356 + 0.998860i $$0.515201\pi$$
$$402$$ −5.55840 −0.277228
$$403$$ 1.85147 0.0922282
$$404$$ 12.4261 0.618219
$$405$$ 37.7146 1.87405
$$406$$ 0 0
$$407$$ 6.93526 0.343768
$$408$$ −6.75860 −0.334601
$$409$$ 3.01815 0.149238 0.0746189 0.997212i $$-0.476226\pi$$
0.0746189 + 0.997212i $$0.476226\pi$$
$$410$$ 43.3992 2.14333
$$411$$ 41.7737 2.06054
$$412$$ 16.1718 0.796729
$$413$$ 0 0
$$414$$ −6.44320 −0.316666
$$415$$ −25.1095 −1.23258
$$416$$ 1.91061 0.0936755
$$417$$ 9.54292 0.467319
$$418$$ 19.0181 0.930204
$$419$$ −4.64504 −0.226925 −0.113463 0.993542i $$-0.536194\pi$$
−0.113463 + 0.993542i $$0.536194\pi$$
$$420$$ 0 0
$$421$$ −26.6202 −1.29739 −0.648695 0.761048i $$-0.724685\pi$$
−0.648695 + 0.761048i $$0.724685\pi$$
$$422$$ 8.65646 0.421390
$$423$$ −34.7840 −1.69126
$$424$$ −2.78082 −0.135049
$$425$$ −33.6715 −1.63331
$$426$$ −15.2568 −0.739196
$$427$$ 0 0
$$428$$ 10.6246 0.513558
$$429$$ −21.9654 −1.06050
$$430$$ −32.5064 −1.56760
$$431$$ −9.63869 −0.464279 −0.232140 0.972682i $$-0.574573\pi$$
−0.232140 + 0.972682i $$0.574573\pi$$
$$432$$ −0.0841506 −0.00404870
$$433$$ 3.16466 0.152084 0.0760420 0.997105i $$-0.475772\pi$$
0.0760420 + 0.997105i $$0.475772\pi$$
$$434$$ 0 0
$$435$$ 10.1202 0.485227
$$436$$ 6.09884 0.292081
$$437$$ 8.77852 0.419934
$$438$$ 33.0291 1.57819
$$439$$ −13.8900 −0.662933 −0.331467 0.943467i $$-0.607543\pi$$
−0.331467 + 0.943467i $$0.607543\pi$$
$$440$$ 19.5032 0.929778
$$441$$ 0 0
$$442$$ −5.28694 −0.251474
$$443$$ 20.3708 0.967845 0.483923 0.875111i $$-0.339212\pi$$
0.483923 + 0.875111i $$0.339212\pi$$
$$444$$ −3.59871 −0.170787
$$445$$ 54.9679 2.60573
$$446$$ 10.9191 0.517032
$$447$$ −18.3141 −0.866228
$$448$$ 0 0
$$449$$ 33.1442 1.56417 0.782086 0.623170i $$-0.214156\pi$$
0.782086 + 0.623170i $$0.214156\pi$$
$$450$$ 36.0858 1.70110
$$451$$ 49.3014 2.32151
$$452$$ 14.5875 0.686139
$$453$$ 23.0034 1.08079
$$454$$ 10.4951 0.492559
$$455$$ 0 0
$$456$$ −9.86847 −0.462133
$$457$$ −18.2087 −0.851767 −0.425883 0.904778i $$-0.640037\pi$$
−0.425883 + 0.904778i $$0.640037\pi$$
$$458$$ 7.51525 0.351165
$$459$$ 0.232857 0.0108688
$$460$$ 9.00245 0.419741
$$461$$ 3.52484 0.164168 0.0820841 0.996625i $$-0.473842\pi$$
0.0820841 + 0.996625i $$0.473842\pi$$
$$462$$ 0 0
$$463$$ 10.7653 0.500304 0.250152 0.968207i $$-0.419519\pi$$
0.250152 + 0.968207i $$0.419519\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ −9.80692 −0.454785
$$466$$ 5.14262 0.238227
$$467$$ −12.0199 −0.556214 −0.278107 0.960550i $$-0.589707\pi$$
−0.278107 + 0.960550i $$0.589707\pi$$
$$468$$ 5.66601 0.261911
$$469$$ 0 0
$$470$$ 48.6003 2.24177
$$471$$ −43.5455 −2.00647
$$472$$ −3.41673 −0.157268
$$473$$ −36.9271 −1.69791
$$474$$ −10.9005 −0.500675
$$475$$ −49.1650 −2.25584
$$476$$ 0 0
$$477$$ −8.24666 −0.377588
$$478$$ 0.749673 0.0342893
$$479$$ −8.29202 −0.378872 −0.189436 0.981893i $$-0.560666\pi$$
−0.189436 + 0.981893i $$0.560666\pi$$
$$480$$ −10.1202 −0.461922
$$481$$ −2.81510 −0.128358
$$482$$ 9.01959 0.410831
$$483$$ 0 0
$$484$$ 11.1556 0.507072
$$485$$ 69.7976 3.16934
$$486$$ −21.9791 −0.996993
$$487$$ 7.89345 0.357687 0.178843 0.983878i $$-0.442764\pi$$
0.178843 + 0.983878i $$0.442764\pi$$
$$488$$ −11.4192 −0.516922
$$489$$ −26.5711 −1.20159
$$490$$ 0 0
$$491$$ −37.7250 −1.70250 −0.851252 0.524757i $$-0.824156\pi$$
−0.851252 + 0.524757i $$0.824156\pi$$
$$492$$ −25.5825 −1.15335
$$493$$ 2.76714 0.124626
$$494$$ −7.71964 −0.347323
$$495$$ 57.8376 2.59961
$$496$$ 0.969044 0.0435114
$$497$$ 0 0
$$498$$ 14.8013 0.663262
$$499$$ −8.47942 −0.379591 −0.189796 0.981824i $$-0.560783\pi$$
−0.189796 + 0.981824i $$0.560783\pi$$
$$500$$ −29.7018 −1.32830
$$501$$ −39.1587 −1.74948
$$502$$ 29.1169 1.29955
$$503$$ 25.2796 1.12716 0.563581 0.826061i $$-0.309423\pi$$
0.563581 + 0.826061i $$0.309423\pi$$
$$504$$ 0 0
$$505$$ −51.4870 −2.29114
$$506$$ 10.2268 0.454635
$$507$$ −22.8358 −1.01417
$$508$$ 5.46381 0.242418
$$509$$ −4.96964 −0.220275 −0.110138 0.993916i $$-0.535129\pi$$
−0.110138 + 0.993916i $$0.535129\pi$$
$$510$$ 28.0041 1.24004
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0.340002 0.0150115
$$514$$ 23.2275 1.02452
$$515$$ −67.0075 −2.95270
$$516$$ 19.1615 0.843538
$$517$$ 55.2099 2.42813
$$518$$ 0 0
$$519$$ 15.1158 0.663508
$$520$$ −7.91656 −0.347164
$$521$$ −26.9628 −1.18126 −0.590631 0.806942i $$-0.701121\pi$$
−0.590631 + 0.806942i $$0.701121\pi$$
$$522$$ −2.96555 −0.129798
$$523$$ 10.4899 0.458691 0.229345 0.973345i $$-0.426341\pi$$
0.229345 + 0.973345i $$0.426341\pi$$
$$524$$ 13.4182 0.586176
$$525$$ 0 0
$$526$$ −20.5033 −0.893988
$$527$$ −2.68148 −0.116807
$$528$$ −11.4965 −0.500322
$$529$$ −18.2794 −0.794758
$$530$$ 11.5222 0.500494
$$531$$ −10.1325 −0.439712
$$532$$ 0 0
$$533$$ −20.0120 −0.866815
$$534$$ −32.4019 −1.40217
$$535$$ −44.0226 −1.90326
$$536$$ −2.27575 −0.0982975
$$537$$ 2.94412 0.127048
$$538$$ 11.3952 0.491282
$$539$$ 0 0
$$540$$ 0.348675 0.0150046
$$541$$ −44.5694 −1.91619 −0.958093 0.286456i $$-0.907523\pi$$
−0.958093 + 0.286456i $$0.907523\pi$$
$$542$$ −17.9751 −0.772095
$$543$$ −16.9776 −0.728578
$$544$$ −2.76714 −0.118640
$$545$$ −25.2703 −1.08246
$$546$$ 0 0
$$547$$ 34.8005 1.48796 0.743980 0.668202i $$-0.232936\pi$$
0.743980 + 0.668202i $$0.232936\pi$$
$$548$$ 17.1032 0.730613
$$549$$ −33.8641 −1.44529
$$550$$ −57.2760 −2.44226
$$551$$ 4.04040 0.172127
$$552$$ −5.30667 −0.225867
$$553$$ 0 0
$$554$$ −2.08032 −0.0883843
$$555$$ 14.9111 0.632942
$$556$$ 3.90711 0.165699
$$557$$ −33.7665 −1.43073 −0.715365 0.698751i $$-0.753740\pi$$
−0.715365 + 0.698751i $$0.753740\pi$$
$$558$$ 2.87375 0.121655
$$559$$ 14.9891 0.633973
$$560$$ 0 0
$$561$$ 31.8125 1.34313
$$562$$ 7.35741 0.310354
$$563$$ −41.2408 −1.73809 −0.869047 0.494730i $$-0.835267\pi$$
−0.869047 + 0.494730i $$0.835267\pi$$
$$564$$ −28.6484 −1.20632
$$565$$ −60.4429 −2.54285
$$566$$ 6.08053 0.255584
$$567$$ 0 0
$$568$$ −6.24653 −0.262099
$$569$$ −25.9185 −1.08656 −0.543281 0.839551i $$-0.682818\pi$$
−0.543281 + 0.839551i $$0.682818\pi$$
$$570$$ 40.8897 1.71268
$$571$$ 15.8953 0.665196 0.332598 0.943069i $$-0.392075\pi$$
0.332598 + 0.943069i $$0.392075\pi$$
$$572$$ −8.99320 −0.376024
$$573$$ −16.6293 −0.694700
$$574$$ 0 0
$$575$$ −26.4380 −1.10254
$$576$$ 2.96555 0.123564
$$577$$ 10.2723 0.427639 0.213820 0.976873i $$-0.431409\pi$$
0.213820 + 0.976873i $$0.431409\pi$$
$$578$$ −9.34292 −0.388614
$$579$$ −9.51734 −0.395527
$$580$$ 4.14347 0.172048
$$581$$ 0 0
$$582$$ −41.1435 −1.70545
$$583$$ 13.0892 0.542101
$$584$$ 13.5230 0.559584
$$585$$ −23.4769 −0.970652
$$586$$ −13.7159 −0.566599
$$587$$ 23.9902 0.990183 0.495091 0.868841i $$-0.335135\pi$$
0.495091 + 0.868841i $$0.335135\pi$$
$$588$$ 0 0
$$589$$ −3.91533 −0.161328
$$590$$ 14.1571 0.582839
$$591$$ 38.0102 1.56353
$$592$$ −1.47340 −0.0605565
$$593$$ 15.5367 0.638014 0.319007 0.947752i $$-0.396651\pi$$
0.319007 + 0.947752i $$0.396651\pi$$
$$594$$ 0.396094 0.0162519
$$595$$ 0 0
$$596$$ −7.49827 −0.307141
$$597$$ −60.0196 −2.45644
$$598$$ −4.15116 −0.169753
$$599$$ −20.8308 −0.851122 −0.425561 0.904930i $$-0.639923\pi$$
−0.425561 + 0.904930i $$0.639923\pi$$
$$600$$ 29.7205 1.21333
$$601$$ −17.7149 −0.722606 −0.361303 0.932448i $$-0.617668\pi$$
−0.361303 + 0.932448i $$0.617668\pi$$
$$602$$ 0 0
$$603$$ −6.74884 −0.274834
$$604$$ 9.41817 0.383220
$$605$$ −46.2228 −1.87922
$$606$$ 30.3500 1.23288
$$607$$ 2.44517 0.0992466 0.0496233 0.998768i $$-0.484198\pi$$
0.0496233 + 0.998768i $$0.484198\pi$$
$$608$$ −4.04040 −0.163860
$$609$$ 0 0
$$610$$ 47.3150 1.91573
$$611$$ −22.4103 −0.906624
$$612$$ −8.20609 −0.331712
$$613$$ −1.80610 −0.0729477 −0.0364739 0.999335i $$-0.511613\pi$$
−0.0364739 + 0.999335i $$0.511613\pi$$
$$614$$ −7.64494 −0.308525
$$615$$ 106.000 4.27434
$$616$$ 0 0
$$617$$ −15.0917 −0.607567 −0.303784 0.952741i $$-0.598250\pi$$
−0.303784 + 0.952741i $$0.598250\pi$$
$$618$$ 39.4988 1.58888
$$619$$ 44.5910 1.79227 0.896133 0.443786i $$-0.146365\pi$$
0.896133 + 0.443786i $$0.146365\pi$$
$$620$$ −4.01520 −0.161254
$$621$$ 0.182833 0.00733682
$$622$$ −24.4252 −0.979360
$$623$$ 0 0
$$624$$ 4.66657 0.186812
$$625$$ 62.2267 2.48907
$$626$$ 18.2645 0.729996
$$627$$ 46.4506 1.85506
$$628$$ −17.8286 −0.711439
$$629$$ 4.07712 0.162565
$$630$$ 0 0
$$631$$ −14.3265 −0.570328 −0.285164 0.958479i $$-0.592048\pi$$
−0.285164 + 0.958479i $$0.592048\pi$$
$$632$$ −4.46293 −0.177526
$$633$$ 21.1429 0.840357
$$634$$ −10.4217 −0.413899
$$635$$ −22.6391 −0.898407
$$636$$ −6.79201 −0.269321
$$637$$ 0 0
$$638$$ 4.70697 0.186351
$$639$$ −18.5244 −0.732814
$$640$$ −4.14347 −0.163785
$$641$$ −17.4354 −0.688658 −0.344329 0.938849i $$-0.611894\pi$$
−0.344329 + 0.938849i $$0.611894\pi$$
$$642$$ 25.9500 1.02416
$$643$$ 6.99067 0.275685 0.137843 0.990454i $$-0.455983\pi$$
0.137843 + 0.990454i $$0.455983\pi$$
$$644$$ 0 0
$$645$$ −79.3950 −3.12618
$$646$$ 11.1804 0.439886
$$647$$ 16.2341 0.638227 0.319113 0.947717i $$-0.396615\pi$$
0.319113 + 0.947717i $$0.396615\pi$$
$$648$$ −9.10217 −0.357567
$$649$$ 16.0824 0.631291
$$650$$ 23.2490 0.911899
$$651$$ 0 0
$$652$$ −10.8789 −0.426051
$$653$$ 22.5201 0.881281 0.440641 0.897684i $$-0.354751\pi$$
0.440641 + 0.897684i $$0.354751\pi$$
$$654$$ 14.8961 0.582483
$$655$$ −55.5978 −2.17239
$$656$$ −10.4741 −0.408945
$$657$$ 40.1030 1.56457
$$658$$ 0 0
$$659$$ −6.71682 −0.261650 −0.130825 0.991405i $$-0.541763\pi$$
−0.130825 + 0.991405i $$0.541763\pi$$
$$660$$ 47.6355 1.85421
$$661$$ 35.3608 1.37538 0.687689 0.726006i $$-0.258625\pi$$
0.687689 + 0.726006i $$0.258625\pi$$
$$662$$ 4.68351 0.182030
$$663$$ −12.9131 −0.501502
$$664$$ 6.06003 0.235175
$$665$$ 0 0
$$666$$ −4.36944 −0.169313
$$667$$ 2.17268 0.0841267
$$668$$ −16.0326 −0.620318
$$669$$ 26.6692 1.03109
$$670$$ 9.42950 0.364293
$$671$$ 53.7498 2.07499
$$672$$ 0 0
$$673$$ −23.1388 −0.891935 −0.445967 0.895049i $$-0.647140\pi$$
−0.445967 + 0.895049i $$0.647140\pi$$
$$674$$ −25.3719 −0.977290
$$675$$ −1.02397 −0.0394127
$$676$$ −9.34956 −0.359599
$$677$$ −22.7344 −0.873755 −0.436878 0.899521i $$-0.643916\pi$$
−0.436878 + 0.899521i $$0.643916\pi$$
$$678$$ 35.6292 1.36833
$$679$$ 0 0
$$680$$ 11.4656 0.439685
$$681$$ 25.6337 0.982285
$$682$$ −4.56126 −0.174660
$$683$$ 20.2928 0.776483 0.388242 0.921558i $$-0.373083\pi$$
0.388242 + 0.921558i $$0.373083\pi$$
$$684$$ −11.9820 −0.458143
$$685$$ −70.8667 −2.70768
$$686$$ 0 0
$$687$$ 18.3556 0.700310
$$688$$ 7.84520 0.299096
$$689$$ −5.31307 −0.202412
$$690$$ 21.9880 0.837069
$$691$$ −2.01180 −0.0765324 −0.0382662 0.999268i $$-0.512183\pi$$
−0.0382662 + 0.999268i $$0.512183\pi$$
$$692$$ 6.18878 0.235262
$$693$$ 0 0
$$694$$ −1.93057 −0.0732836
$$695$$ −16.1890 −0.614084
$$696$$ −2.44245 −0.0925807
$$697$$ 28.9834 1.09782
$$698$$ −1.95782 −0.0741045
$$699$$ 12.5606 0.475084
$$700$$ 0 0
$$701$$ −3.02432 −0.114227 −0.0571136 0.998368i $$-0.518190\pi$$
−0.0571136 + 0.998368i $$0.518190\pi$$
$$702$$ −0.160779 −0.00606822
$$703$$ 5.95314 0.224527
$$704$$ −4.70697 −0.177401
$$705$$ 118.704 4.47064
$$706$$ 16.3330 0.614700
$$707$$ 0 0
$$708$$ −8.34518 −0.313631
$$709$$ −45.4669 −1.70754 −0.853772 0.520647i $$-0.825691\pi$$
−0.853772 + 0.520647i $$0.825691\pi$$
$$710$$ 25.8823 0.971346
$$711$$ −13.2350 −0.496352
$$712$$ −13.2662 −0.497170
$$713$$ −2.10543 −0.0788489
$$714$$ 0 0
$$715$$ 37.2630 1.39356
$$716$$ 1.20540 0.0450479
$$717$$ 1.83104 0.0683813
$$718$$ −11.7801 −0.439629
$$719$$ −30.1441 −1.12418 −0.562092 0.827075i $$-0.690003\pi$$
−0.562092 + 0.827075i $$0.690003\pi$$
$$720$$ −12.2876 −0.457934
$$721$$ 0 0
$$722$$ −2.67514 −0.0995585
$$723$$ 22.0299 0.819300
$$724$$ −6.95105 −0.258334
$$725$$ −12.1683 −0.451921
$$726$$ 27.2469 1.01123
$$727$$ −45.6599 −1.69343 −0.846717 0.532044i $$-0.821424\pi$$
−0.846717 + 0.532044i $$0.821424\pi$$
$$728$$ 0 0
$$729$$ −26.3763 −0.976901
$$730$$ −56.0320 −2.07384
$$731$$ −21.7088 −0.802929
$$732$$ −27.8907 −1.03087
$$733$$ 27.3458 1.01004 0.505020 0.863107i $$-0.331485\pi$$
0.505020 + 0.863107i $$0.331485\pi$$
$$734$$ −35.6438 −1.31564
$$735$$ 0 0
$$736$$ −2.17268 −0.0800862
$$737$$ 10.7119 0.394578
$$738$$ −31.0615 −1.14339
$$739$$ 49.9166 1.83621 0.918106 0.396335i $$-0.129718\pi$$
0.918106 + 0.396335i $$0.129718\pi$$
$$740$$ 6.10500 0.224424
$$741$$ −18.8548 −0.692649
$$742$$ 0 0
$$743$$ 28.6655 1.05164 0.525818 0.850597i $$-0.323759\pi$$
0.525818 + 0.850597i $$0.323759\pi$$
$$744$$ 2.36684 0.0867725
$$745$$ 31.0688 1.13827
$$746$$ 17.7483 0.649811
$$747$$ 17.9713 0.657536
$$748$$ 13.0249 0.476236
$$749$$ 0 0
$$750$$ −72.5450 −2.64897
$$751$$ −19.0316 −0.694473 −0.347236 0.937778i $$-0.612880\pi$$
−0.347236 + 0.937778i $$0.612880\pi$$
$$752$$ −11.7294 −0.427727
$$753$$ 71.1165 2.59163
$$754$$ −1.91061 −0.0695804
$$755$$ −39.0239 −1.42023
$$756$$ 0 0
$$757$$ −9.39694 −0.341538 −0.170769 0.985311i $$-0.554625\pi$$
−0.170769 + 0.985311i $$0.554625\pi$$
$$758$$ −15.2263 −0.553045
$$759$$ 24.9783 0.906656
$$760$$ 16.7413 0.607270
$$761$$ 10.7493 0.389661 0.194831 0.980837i $$-0.437584\pi$$
0.194831 + 0.980837i $$0.437584\pi$$
$$762$$ 13.3451 0.483441
$$763$$ 0 0
$$764$$ −6.80847 −0.246322
$$765$$ 34.0017 1.22933
$$766$$ 15.8118 0.571302
$$767$$ −6.52804 −0.235714
$$768$$ 2.44245 0.0881342
$$769$$ −39.3774 −1.41998 −0.709992 0.704209i $$-0.751302\pi$$
−0.709992 + 0.704209i $$0.751302\pi$$
$$770$$ 0 0
$$771$$ 56.7321 2.04316
$$772$$ −3.89664 −0.140243
$$773$$ 19.0471 0.685078 0.342539 0.939504i $$-0.388713\pi$$
0.342539 + 0.939504i $$0.388713\pi$$
$$774$$ 23.2653 0.836254
$$775$$ 11.7917 0.423569
$$776$$ −16.8452 −0.604708
$$777$$ 0 0
$$778$$ −21.7172 −0.778598
$$779$$ 42.3196 1.51626
$$780$$ −19.3358 −0.692332
$$781$$ 29.4023 1.05210
$$782$$ 6.01213 0.214993
$$783$$ 0.0841506 0.00300730
$$784$$ 0 0
$$785$$ 73.8723 2.63662
$$786$$ 32.7732 1.16898
$$787$$ −26.4446 −0.942650 −0.471325 0.881960i $$-0.656224\pi$$
−0.471325 + 0.881960i $$0.656224\pi$$
$$788$$ 15.5624 0.554386
$$789$$ −50.0783 −1.78283
$$790$$ 18.4920 0.657916
$$791$$ 0 0
$$792$$ −13.9587 −0.496002
$$793$$ −21.8176 −0.774767
$$794$$ −1.26033 −0.0447275
$$795$$ 28.1425 0.998111
$$796$$ −24.5735 −0.870986
$$797$$ 42.8772 1.51879 0.759395 0.650630i $$-0.225495\pi$$
0.759395 + 0.650630i $$0.225495\pi$$
$$798$$ 0 0
$$799$$ 32.4569 1.14824
$$800$$ 12.1683 0.430216
$$801$$ −39.3414 −1.39006
$$802$$ −1.91181 −0.0675084
$$803$$ −63.6522 −2.24624
$$804$$ −5.55840 −0.196030
$$805$$ 0 0
$$806$$ 1.85147 0.0652152
$$807$$ 27.8322 0.979739
$$808$$ 12.4261 0.437147
$$809$$ −1.72392 −0.0606097 −0.0303048 0.999541i $$-0.509648\pi$$
−0.0303048 + 0.999541i $$0.509648\pi$$
$$810$$ 37.7146 1.32515
$$811$$ −7.23311 −0.253989 −0.126995 0.991903i $$-0.540533\pi$$
−0.126995 + 0.991903i $$0.540533\pi$$
$$812$$ 0 0
$$813$$ −43.9031 −1.53975
$$814$$ 6.93526 0.243081
$$815$$ 45.0764 1.57896
$$816$$ −6.75860 −0.236598
$$817$$ −31.6978 −1.10896
$$818$$ 3.01815 0.105527
$$819$$ 0 0
$$820$$ 43.3992 1.51556
$$821$$ 30.7040 1.07158 0.535789 0.844352i $$-0.320014\pi$$
0.535789 + 0.844352i $$0.320014\pi$$
$$822$$ 41.7737 1.45703
$$823$$ −9.58578 −0.334139 −0.167070 0.985945i $$-0.553430\pi$$
−0.167070 + 0.985945i $$0.553430\pi$$
$$824$$ 16.1718 0.563372
$$825$$ −139.894 −4.87047
$$826$$ 0 0
$$827$$ −6.26986 −0.218025 −0.109012 0.994040i $$-0.534769\pi$$
−0.109012 + 0.994040i $$0.534769\pi$$
$$828$$ −6.44320 −0.223917
$$829$$ −11.2898 −0.392110 −0.196055 0.980593i $$-0.562813\pi$$
−0.196055 + 0.980593i $$0.562813\pi$$
$$830$$ −25.1095 −0.871565
$$831$$ −5.08107 −0.176260
$$832$$ 1.91061 0.0662386
$$833$$ 0 0
$$834$$ 9.54292 0.330444
$$835$$ 66.4304 2.29892
$$836$$ 19.0181 0.657753
$$837$$ −0.0815456 −0.00281863
$$838$$ −4.64504 −0.160460
$$839$$ 44.0101 1.51940 0.759698 0.650276i $$-0.225347\pi$$
0.759698 + 0.650276i $$0.225347\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −26.6202 −0.917394
$$843$$ 17.9701 0.618923
$$844$$ 8.65646 0.297968
$$845$$ 38.7396 1.33268
$$846$$ −34.7840 −1.19590
$$847$$ 0 0
$$848$$ −2.78082 −0.0954938
$$849$$ 14.8514 0.509698
$$850$$ −33.6715 −1.15492
$$851$$ 3.20124 0.109737
$$852$$ −15.2568 −0.522690
$$853$$ −10.9272 −0.374141 −0.187070 0.982347i $$-0.559899\pi$$
−0.187070 + 0.982347i $$0.559899\pi$$
$$854$$ 0 0
$$855$$ 49.6471 1.69789
$$856$$ 10.6246 0.363141
$$857$$ −29.2754 −1.00003 −0.500015 0.866017i $$-0.666672\pi$$
−0.500015 + 0.866017i $$0.666672\pi$$
$$858$$ −21.9654 −0.749887
$$859$$ 43.3222 1.47813 0.739067 0.673632i $$-0.235267\pi$$
0.739067 + 0.673632i $$0.235267\pi$$
$$860$$ −32.5064 −1.10846
$$861$$ 0 0
$$862$$ −9.63869 −0.328295
$$863$$ −43.8381 −1.49227 −0.746133 0.665797i $$-0.768092\pi$$
−0.746133 + 0.665797i $$0.768092\pi$$
$$864$$ −0.0841506 −0.00286286
$$865$$ −25.6430 −0.871888
$$866$$ 3.16466 0.107540
$$867$$ −22.8196 −0.774994
$$868$$ 0 0
$$869$$ 21.0069 0.712609
$$870$$ 10.1202 0.343107
$$871$$ −4.34808 −0.147329
$$872$$ 6.09884 0.206533
$$873$$ −49.9552 −1.69073
$$874$$ 8.77852 0.296938
$$875$$ 0 0
$$876$$ 33.0291 1.11595
$$877$$ 29.5072 0.996388 0.498194 0.867066i $$-0.333997\pi$$
0.498194 + 0.867066i $$0.333997\pi$$
$$878$$ −13.8900 −0.468764
$$879$$ −33.5004 −1.12994
$$880$$ 19.5032 0.657452
$$881$$ 13.1060 0.441554 0.220777 0.975324i $$-0.429141\pi$$
0.220777 + 0.975324i $$0.429141\pi$$
$$882$$ 0 0
$$883$$ −7.44435 −0.250522 −0.125261 0.992124i $$-0.539977\pi$$
−0.125261 + 0.992124i $$0.539977\pi$$
$$884$$ −5.28694 −0.177819
$$885$$ 34.5780 1.16233
$$886$$ 20.3708 0.684370
$$887$$ −38.6833 −1.29886 −0.649430 0.760422i $$-0.724992\pi$$
−0.649430 + 0.760422i $$0.724992\pi$$
$$888$$ −3.59871 −0.120765
$$889$$ 0 0
$$890$$ 54.9679 1.84253
$$891$$ 42.8437 1.43532
$$892$$ 10.9191 0.365597
$$893$$ 47.3914 1.58589
$$894$$ −18.3141 −0.612516
$$895$$ −4.99453 −0.166949
$$896$$ 0 0
$$897$$ −10.1390 −0.338531
$$898$$ 33.1442 1.10604
$$899$$ −0.969044 −0.0323194
$$900$$ 36.0858 1.20286
$$901$$ 7.69493 0.256355
$$902$$ 49.3014 1.64156
$$903$$ 0 0
$$904$$ 14.5875 0.485173
$$905$$ 28.8015 0.957393
$$906$$ 23.0034 0.764236
$$907$$ 2.42625 0.0805624 0.0402812 0.999188i $$-0.487175\pi$$
0.0402812 + 0.999188i $$0.487175\pi$$
$$908$$ 10.4951 0.348292
$$909$$ 36.8500 1.22224
$$910$$ 0 0
$$911$$ −1.86661 −0.0618435 −0.0309218 0.999522i $$-0.509844\pi$$
−0.0309218 + 0.999522i $$0.509844\pi$$
$$912$$ −9.86847 −0.326778
$$913$$ −28.5244 −0.944020
$$914$$ −18.2087 −0.602290
$$915$$ 115.564 3.82044
$$916$$ 7.51525 0.248311
$$917$$ 0 0
$$918$$ 0.232857 0.00768542
$$919$$ 27.4057 0.904031 0.452015 0.892010i $$-0.350705\pi$$
0.452015 + 0.892010i $$0.350705\pi$$
$$920$$ 9.00245 0.296802
$$921$$ −18.6724 −0.615275
$$922$$ 3.52484 0.116084
$$923$$ −11.9347 −0.392836
$$924$$ 0 0
$$925$$ −17.9289 −0.589497
$$926$$ 10.7653 0.353768
$$927$$ 47.9583 1.57516
$$928$$ −1.00000 −0.0328266
$$929$$ −43.9595 −1.44227 −0.721133 0.692797i $$-0.756378\pi$$
−0.721133 + 0.692797i $$0.756378\pi$$
$$930$$ −9.80692 −0.321582
$$931$$ 0 0
$$932$$ 5.14262 0.168452
$$933$$ −59.6572 −1.95309
$$934$$ −12.0199 −0.393303
$$935$$ −53.9681 −1.76495
$$936$$ 5.66601 0.185199
$$937$$ 29.9446 0.978246 0.489123 0.872215i $$-0.337317\pi$$
0.489123 + 0.872215i $$0.337317\pi$$
$$938$$ 0 0
$$939$$ 44.6101 1.45579
$$940$$ 48.6003 1.58517
$$941$$ −21.5723 −0.703236 −0.351618 0.936144i $$-0.614368\pi$$
−0.351618 + 0.936144i $$0.614368\pi$$
$$942$$ −43.5455 −1.41879
$$943$$ 22.7570 0.741068
$$944$$ −3.41673 −0.111205
$$945$$ 0 0
$$946$$ −36.9271 −1.20061
$$947$$ 4.39212 0.142725 0.0713624 0.997450i $$-0.477265\pi$$
0.0713624 + 0.997450i $$0.477265\pi$$
$$948$$ −10.9005 −0.354031
$$949$$ 25.8371 0.838709
$$950$$ −49.1650 −1.59512
$$951$$ −25.4545 −0.825418
$$952$$ 0 0
$$953$$ 32.9052 1.06591 0.532953 0.846145i $$-0.321082\pi$$
0.532953 + 0.846145i $$0.321082\pi$$
$$954$$ −8.24666 −0.266995
$$955$$ 28.2107 0.912876
$$956$$ 0.749673 0.0242462
$$957$$ 11.4965 0.371630
$$958$$ −8.29202 −0.267903
$$959$$ 0 0
$$960$$ −10.1202 −0.326628
$$961$$ −30.0610 −0.969708
$$962$$ −2.81510 −0.0907625
$$963$$ 31.5077 1.01532
$$964$$ 9.01959 0.290501
$$965$$ 16.1456 0.519745
$$966$$ 0 0
$$967$$ 4.50365 0.144827 0.0724137 0.997375i $$-0.476930\pi$$
0.0724137 + 0.997375i $$0.476930\pi$$
$$968$$ 11.1556 0.358554
$$969$$ 27.3075 0.877242
$$970$$ 69.7976 2.24107
$$971$$ −30.4189 −0.976187 −0.488094 0.872791i $$-0.662308\pi$$
−0.488094 + 0.872791i $$0.662308\pi$$
$$972$$ −21.9791 −0.704981
$$973$$ 0 0
$$974$$ 7.89345 0.252923
$$975$$ 56.7844 1.81856
$$976$$ −11.4192 −0.365519
$$977$$ 37.5202 1.20038 0.600189 0.799858i $$-0.295092\pi$$
0.600189 + 0.799858i $$0.295092\pi$$
$$978$$ −26.5711 −0.849652
$$979$$ 62.4434 1.99570
$$980$$ 0 0
$$981$$ 18.0864 0.577454
$$982$$ −37.7250 −1.20385
$$983$$ −43.4878 −1.38705 −0.693523 0.720435i $$-0.743942\pi$$
−0.693523 + 0.720435i $$0.743942\pi$$
$$984$$ −25.5825 −0.815539
$$985$$ −64.4821 −2.05457
$$986$$ 2.76714 0.0881238
$$987$$ 0 0
$$988$$ −7.71964 −0.245595
$$989$$ −17.0452 −0.542004
$$990$$ 57.8376 1.83820
$$991$$ −55.4198 −1.76047 −0.880233 0.474541i $$-0.842614\pi$$
−0.880233 + 0.474541i $$0.842614\pi$$
$$992$$ 0.969044 0.0307672
$$993$$ 11.4392 0.363013
$$994$$ 0 0
$$995$$ 101.820 3.22790
$$996$$ 14.8013 0.468997
$$997$$ 23.8889 0.756570 0.378285 0.925689i $$-0.376514\pi$$
0.378285 + 0.925689i $$0.376514\pi$$
$$998$$ −8.47942 −0.268411
$$999$$ 0.123988 0.00392280
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 2842.2.a.x.1.5 5
7.3 odd 6 406.2.e.a.233.5 10
7.5 odd 6 406.2.e.a.291.5 yes 10
7.6 odd 2 2842.2.a.z.1.1 5

By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.a.233.5 10 7.3 odd 6
406.2.e.a.291.5 yes 10 7.5 odd 6
2842.2.a.x.1.5 5 1.1 even 1 trivial
2842.2.a.z.1.1 5 7.6 odd 2