# Properties

 Label 4680.2.a.y.1.2 Level $4680$ Weight $2$ Character 4680.1 Self dual yes Analytic conductor $37.370$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 4680.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +1.56155 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +1.56155 q^{7} +5.56155 q^{11} +1.00000 q^{13} +6.68466 q^{17} +3.12311 q^{19} +5.56155 q^{23} +1.00000 q^{25} +2.00000 q^{29} -7.12311 q^{31} -1.56155 q^{35} +9.80776 q^{37} -2.68466 q^{41} -10.2462 q^{43} -7.12311 q^{47} -4.56155 q^{49} -3.56155 q^{53} -5.56155 q^{55} +8.00000 q^{59} -10.6847 q^{61} -1.00000 q^{65} -14.2462 q^{67} -4.68466 q^{71} +16.2462 q^{73} +8.68466 q^{77} +11.8078 q^{79} +8.00000 q^{83} -6.68466 q^{85} +14.6847 q^{89} +1.56155 q^{91} -3.12311 q^{95} -17.8078 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - q^7 $$2 q - 2 q^{5} - q^{7} + 7 q^{11} + 2 q^{13} + q^{17} - 2 q^{19} + 7 q^{23} + 2 q^{25} + 4 q^{29} - 6 q^{31} + q^{35} - q^{37} + 7 q^{41} - 4 q^{43} - 6 q^{47} - 5 q^{49} - 3 q^{53} - 7 q^{55} + 16 q^{59} - 9 q^{61} - 2 q^{65} - 12 q^{67} + 3 q^{71} + 16 q^{73} + 5 q^{77} + 3 q^{79} + 16 q^{83} - q^{85} + 17 q^{89} - q^{91} + 2 q^{95} - 15 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - q^7 + 7 * q^11 + 2 * q^13 + q^17 - 2 * q^19 + 7 * q^23 + 2 * q^25 + 4 * q^29 - 6 * q^31 + q^35 - q^37 + 7 * q^41 - 4 * q^43 - 6 * q^47 - 5 * q^49 - 3 * q^53 - 7 * q^55 + 16 * q^59 - 9 * q^61 - 2 * q^65 - 12 * q^67 + 3 * q^71 + 16 * q^73 + 5 * q^77 + 3 * q^79 + 16 * q^83 - q^85 + 17 * q^89 - q^91 + 2 * q^95 - 15 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.56155 0.590211 0.295106 0.955465i $$-0.404645\pi$$
0.295106 + 0.955465i $$0.404645\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.56155 1.67687 0.838436 0.545001i $$-0.183471\pi$$
0.838436 + 0.545001i $$0.183471\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.68466 1.62127 0.810634 0.585553i $$-0.199123\pi$$
0.810634 + 0.585553i $$0.199123\pi$$
$$18$$ 0 0
$$19$$ 3.12311 0.716490 0.358245 0.933628i $$-0.383375\pi$$
0.358245 + 0.933628i $$0.383375\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.56155 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −7.12311 −1.27935 −0.639674 0.768647i $$-0.720931\pi$$
−0.639674 + 0.768647i $$0.720931\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.56155 −0.263951
$$36$$ 0 0
$$37$$ 9.80776 1.61239 0.806193 0.591652i $$-0.201524\pi$$
0.806193 + 0.591652i $$0.201524\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.68466 −0.419273 −0.209637 0.977779i $$-0.567228\pi$$
−0.209637 + 0.977779i $$0.567228\pi$$
$$42$$ 0 0
$$43$$ −10.2462 −1.56253 −0.781266 0.624198i $$-0.785426\pi$$
−0.781266 + 0.624198i $$0.785426\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.12311 −1.03901 −0.519506 0.854467i $$-0.673884\pi$$
−0.519506 + 0.854467i $$0.673884\pi$$
$$48$$ 0 0
$$49$$ −4.56155 −0.651650
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −3.56155 −0.489217 −0.244608 0.969622i $$-0.578659\pi$$
−0.244608 + 0.969622i $$0.578659\pi$$
$$54$$ 0 0
$$55$$ −5.56155 −0.749920
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −10.6847 −1.36803 −0.684015 0.729468i $$-0.739768\pi$$
−0.684015 + 0.729468i $$0.739768\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −14.2462 −1.74045 −0.870226 0.492653i $$-0.836027\pi$$
−0.870226 + 0.492653i $$0.836027\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.68466 −0.555967 −0.277983 0.960586i $$-0.589666\pi$$
−0.277983 + 0.960586i $$0.589666\pi$$
$$72$$ 0 0
$$73$$ 16.2462 1.90148 0.950738 0.309997i $$-0.100328\pi$$
0.950738 + 0.309997i $$0.100328\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.68466 0.989709
$$78$$ 0 0
$$79$$ 11.8078 1.32848 0.664239 0.747521i $$-0.268756\pi$$
0.664239 + 0.747521i $$0.268756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ −6.68466 −0.725053
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 14.6847 1.55657 0.778285 0.627911i $$-0.216090\pi$$
0.778285 + 0.627911i $$0.216090\pi$$
$$90$$ 0 0
$$91$$ 1.56155 0.163695
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −3.12311 −0.320424
$$96$$ 0 0
$$97$$ −17.8078 −1.80810 −0.904052 0.427422i $$-0.859422\pi$$
−0.904052 + 0.427422i $$0.859422\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1.12311 −0.111753 −0.0558766 0.998438i $$-0.517795\pi$$
−0.0558766 + 0.998438i $$0.517795\pi$$
$$102$$ 0 0
$$103$$ 14.2462 1.40372 0.701860 0.712314i $$-0.252353\pi$$
0.701860 + 0.712314i $$0.252353\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.8078 1.52819 0.764097 0.645101i $$-0.223185\pi$$
0.764097 + 0.645101i $$0.223185\pi$$
$$108$$ 0 0
$$109$$ −13.1231 −1.25697 −0.628483 0.777824i $$-0.716324\pi$$
−0.628483 + 0.777824i $$0.716324\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −5.56155 −0.518617
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 10.4384 0.956891
$$120$$ 0 0
$$121$$ 19.9309 1.81190
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −17.3693 −1.54128 −0.770639 0.637272i $$-0.780063\pi$$
−0.770639 + 0.637272i $$0.780063\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.1231 1.32131 0.660656 0.750689i $$-0.270278\pi$$
0.660656 + 0.750689i $$0.270278\pi$$
$$132$$ 0 0
$$133$$ 4.87689 0.422880
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −16.2462 −1.38801 −0.694004 0.719971i $$-0.744155\pi$$
−0.694004 + 0.719971i $$0.744155\pi$$
$$138$$ 0 0
$$139$$ −9.56155 −0.811000 −0.405500 0.914095i $$-0.632903\pi$$
−0.405500 + 0.914095i $$0.632903\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 5.56155 0.465080
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 12.4384 1.01900 0.509499 0.860471i $$-0.329831\pi$$
0.509499 + 0.860471i $$0.329831\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 7.12311 0.572142
$$156$$ 0 0
$$157$$ 10.4924 0.837386 0.418693 0.908128i $$-0.362488\pi$$
0.418693 + 0.908128i $$0.362488\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.68466 0.684447
$$162$$ 0 0
$$163$$ 13.5616 1.06222 0.531111 0.847302i $$-0.321775\pi$$
0.531111 + 0.847302i $$0.321775\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.75379 0.445242 0.222621 0.974905i $$-0.428539\pi$$
0.222621 + 0.974905i $$0.428539\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 1.56155 0.118042
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 23.1231 1.72830 0.864151 0.503233i $$-0.167856\pi$$
0.864151 + 0.503233i $$0.167856\pi$$
$$180$$ 0 0
$$181$$ −8.93087 −0.663826 −0.331913 0.943310i $$-0.607694\pi$$
−0.331913 + 0.943310i $$0.607694\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −9.80776 −0.721081
$$186$$ 0 0
$$187$$ 37.1771 2.71866
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ 9.31534 0.670533 0.335266 0.942123i $$-0.391174\pi$$
0.335266 + 0.942123i $$0.391174\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.36932 −0.525042 −0.262521 0.964926i $$-0.584554\pi$$
−0.262521 + 0.964926i $$0.584554\pi$$
$$198$$ 0 0
$$199$$ 1.75379 0.124323 0.0621614 0.998066i $$-0.480201\pi$$
0.0621614 + 0.998066i $$0.480201\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 3.12311 0.219199
$$204$$ 0 0
$$205$$ 2.68466 0.187505
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 17.3693 1.20146
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 10.2462 0.698786
$$216$$ 0 0
$$217$$ −11.1231 −0.755086
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.68466 0.449659
$$222$$ 0 0
$$223$$ −2.24621 −0.150417 −0.0752087 0.997168i $$-0.523962\pi$$
−0.0752087 + 0.997168i $$0.523962\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −25.3693 −1.68382 −0.841910 0.539617i $$-0.818569\pi$$
−0.841910 + 0.539617i $$0.818569\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −18.6847 −1.22407 −0.612036 0.790830i $$-0.709649\pi$$
−0.612036 + 0.790830i $$0.709649\pi$$
$$234$$ 0 0
$$235$$ 7.12311 0.464660
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3.31534 0.214452 0.107226 0.994235i $$-0.465803\pi$$
0.107226 + 0.994235i $$0.465803\pi$$
$$240$$ 0 0
$$241$$ 11.7538 0.757128 0.378564 0.925575i $$-0.376418\pi$$
0.378564 + 0.925575i $$0.376418\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 4.56155 0.291427
$$246$$ 0 0
$$247$$ 3.12311 0.198718
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −8.49242 −0.536037 −0.268018 0.963414i $$-0.586369\pi$$
−0.268018 + 0.963414i $$0.586369\pi$$
$$252$$ 0 0
$$253$$ 30.9309 1.94461
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −14.4924 −0.904012 −0.452006 0.892015i $$-0.649292\pi$$
−0.452006 + 0.892015i $$0.649292\pi$$
$$258$$ 0 0
$$259$$ 15.3153 0.951649
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −28.4924 −1.75692 −0.878459 0.477818i $$-0.841428\pi$$
−0.878459 + 0.477818i $$0.841428\pi$$
$$264$$ 0 0
$$265$$ 3.56155 0.218784
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 13.1231 0.800130 0.400065 0.916487i $$-0.368988\pi$$
0.400065 + 0.916487i $$0.368988\pi$$
$$270$$ 0 0
$$271$$ −5.36932 −0.326163 −0.163081 0.986613i $$-0.552143\pi$$
−0.163081 + 0.986613i $$0.552143\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.56155 0.335374
$$276$$ 0 0
$$277$$ −22.4924 −1.35144 −0.675719 0.737159i $$-0.736167\pi$$
−0.675719 + 0.737159i $$0.736167\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.75379 −0.223932 −0.111966 0.993712i $$-0.535715\pi$$
−0.111966 + 0.993712i $$0.535715\pi$$
$$282$$ 0 0
$$283$$ 30.7386 1.82722 0.913611 0.406589i $$-0.133282\pi$$
0.913611 + 0.406589i $$0.133282\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.19224 −0.247460
$$288$$ 0 0
$$289$$ 27.6847 1.62851
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 17.6155 1.02911 0.514555 0.857457i $$-0.327957\pi$$
0.514555 + 0.857457i $$0.327957\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.56155 0.321633
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 10.6847 0.611802
$$306$$ 0 0
$$307$$ 13.5616 0.773999 0.386999 0.922080i $$-0.373512\pi$$
0.386999 + 0.922080i $$0.373512\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 26.7386 1.51621 0.758104 0.652133i $$-0.226126\pi$$
0.758104 + 0.652133i $$0.226126\pi$$
$$312$$ 0 0
$$313$$ −24.7386 −1.39831 −0.699155 0.714970i $$-0.746440\pi$$
−0.699155 + 0.714970i $$0.746440\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.87689 −0.161582 −0.0807912 0.996731i $$-0.525745\pi$$
−0.0807912 + 0.996731i $$0.525745\pi$$
$$318$$ 0 0
$$319$$ 11.1231 0.622774
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 20.8769 1.16162
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −11.1231 −0.613237
$$330$$ 0 0
$$331$$ 9.75379 0.536117 0.268058 0.963403i $$-0.413618\pi$$
0.268058 + 0.963403i $$0.413618\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 14.2462 0.778354
$$336$$ 0 0
$$337$$ −2.87689 −0.156714 −0.0783572 0.996925i $$-0.524967\pi$$
−0.0783572 + 0.996925i $$0.524967\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −39.6155 −2.14530
$$342$$ 0 0
$$343$$ −18.0540 −0.974823
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6.05398 0.324994 0.162497 0.986709i $$-0.448045\pi$$
0.162497 + 0.986709i $$0.448045\pi$$
$$348$$ 0 0
$$349$$ −0.630683 −0.0337597 −0.0168798 0.999858i $$-0.505373\pi$$
−0.0168798 + 0.999858i $$0.505373\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4.24621 0.226003 0.113002 0.993595i $$-0.463954\pi$$
0.113002 + 0.993595i $$0.463954\pi$$
$$354$$ 0 0
$$355$$ 4.68466 0.248636
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 2.24621 0.118550 0.0592752 0.998242i $$-0.481121\pi$$
0.0592752 + 0.998242i $$0.481121\pi$$
$$360$$ 0 0
$$361$$ −9.24621 −0.486643
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −16.2462 −0.850366
$$366$$ 0 0
$$367$$ 14.2462 0.743646 0.371823 0.928304i $$-0.378733\pi$$
0.371823 + 0.928304i $$0.378733\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5.56155 −0.288741
$$372$$ 0 0
$$373$$ −14.8769 −0.770296 −0.385148 0.922855i $$-0.625850\pi$$
−0.385148 + 0.922855i $$0.625850\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 11.1231 0.571356 0.285678 0.958326i $$-0.407781\pi$$
0.285678 + 0.958326i $$0.407781\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.63068 −0.134422 −0.0672108 0.997739i $$-0.521410\pi$$
−0.0672108 + 0.997739i $$0.521410\pi$$
$$384$$ 0 0
$$385$$ −8.68466 −0.442611
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −35.8617 −1.81826 −0.909131 0.416510i $$-0.863253\pi$$
−0.909131 + 0.416510i $$0.863253\pi$$
$$390$$ 0 0
$$391$$ 37.1771 1.88013
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −11.8078 −0.594113
$$396$$ 0 0
$$397$$ 33.8078 1.69676 0.848382 0.529385i $$-0.177577\pi$$
0.848382 + 0.529385i $$0.177577\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.24621 0.212046 0.106023 0.994364i $$-0.466188\pi$$
0.106023 + 0.994364i $$0.466188\pi$$
$$402$$ 0 0
$$403$$ −7.12311 −0.354827
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 54.5464 2.70376
$$408$$ 0 0
$$409$$ −18.4924 −0.914391 −0.457196 0.889366i $$-0.651146\pi$$
−0.457196 + 0.889366i $$0.651146\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 12.4924 0.614712
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −0.876894 −0.0428391 −0.0214195 0.999771i $$-0.506819\pi$$
−0.0214195 + 0.999771i $$0.506819\pi$$
$$420$$ 0 0
$$421$$ 19.8617 0.968002 0.484001 0.875067i $$-0.339183\pi$$
0.484001 + 0.875067i $$0.339183\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6.68466 0.324254
$$426$$ 0 0
$$427$$ −16.6847 −0.807427
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −13.7538 −0.662497 −0.331248 0.943544i $$-0.607470\pi$$
−0.331248 + 0.943544i $$0.607470\pi$$
$$432$$ 0 0
$$433$$ 35.3693 1.69974 0.849870 0.526992i $$-0.176680\pi$$
0.849870 + 0.526992i $$0.176680\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 17.3693 0.830887
$$438$$ 0 0
$$439$$ 6.93087 0.330792 0.165396 0.986227i $$-0.447110\pi$$
0.165396 + 0.986227i $$0.447110\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3.31534 0.157517 0.0787583 0.996894i $$-0.474904\pi$$
0.0787583 + 0.996894i $$0.474904\pi$$
$$444$$ 0 0
$$445$$ −14.6847 −0.696120
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10.6847 −0.504240 −0.252120 0.967696i $$-0.581128\pi$$
−0.252120 + 0.967696i $$0.581128\pi$$
$$450$$ 0 0
$$451$$ −14.9309 −0.703067
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1.56155 −0.0732067
$$456$$ 0 0
$$457$$ −35.5616 −1.66350 −0.831750 0.555151i $$-0.812660\pi$$
−0.831750 + 0.555151i $$0.812660\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 15.5616 0.724774 0.362387 0.932028i $$-0.381962\pi$$
0.362387 + 0.932028i $$0.381962\pi$$
$$462$$ 0 0
$$463$$ −6.43845 −0.299220 −0.149610 0.988745i $$-0.547802\pi$$
−0.149610 + 0.988745i $$0.547802\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −25.1771 −1.16506 −0.582528 0.812811i $$-0.697936\pi$$
−0.582528 + 0.812811i $$0.697936\pi$$
$$468$$ 0 0
$$469$$ −22.2462 −1.02723
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −56.9848 −2.62017
$$474$$ 0 0
$$475$$ 3.12311 0.143298
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −34.5464 −1.57847 −0.789233 0.614094i $$-0.789521\pi$$
−0.789233 + 0.614094i $$0.789521\pi$$
$$480$$ 0 0
$$481$$ 9.80776 0.447196
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 17.8078 0.808609
$$486$$ 0 0
$$487$$ 4.68466 0.212282 0.106141 0.994351i $$-0.466150\pi$$
0.106141 + 0.994351i $$0.466150\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.50758 0.338812 0.169406 0.985546i $$-0.445815\pi$$
0.169406 + 0.985546i $$0.445815\pi$$
$$492$$ 0 0
$$493$$ 13.3693 0.602124
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −7.31534 −0.328138
$$498$$ 0 0
$$499$$ −37.8617 −1.69492 −0.847462 0.530856i $$-0.821871\pi$$
−0.847462 + 0.530856i $$0.821871\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −18.7386 −0.835514 −0.417757 0.908559i $$-0.637184\pi$$
−0.417757 + 0.908559i $$0.637184\pi$$
$$504$$ 0 0
$$505$$ 1.12311 0.0499775
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 42.6847 1.89196 0.945982 0.324219i $$-0.105101\pi$$
0.945982 + 0.324219i $$0.105101\pi$$
$$510$$ 0 0
$$511$$ 25.3693 1.12227
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −14.2462 −0.627763
$$516$$ 0 0
$$517$$ −39.6155 −1.74229
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 13.7538 0.601411 0.300706 0.953717i $$-0.402778\pi$$
0.300706 + 0.953717i $$0.402778\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −47.6155 −2.07416
$$528$$ 0 0
$$529$$ 7.93087 0.344820
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.68466 −0.116285
$$534$$ 0 0
$$535$$ −15.8078 −0.683429
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −25.3693 −1.09273
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 13.1231 0.562132
$$546$$ 0 0
$$547$$ −6.73863 −0.288123 −0.144062 0.989569i $$-0.546016\pi$$
−0.144062 + 0.989569i $$0.546016\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.24621 0.266098
$$552$$ 0 0
$$553$$ 18.4384 0.784083
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 34.0000 1.44063 0.720313 0.693649i $$-0.243998\pi$$
0.720313 + 0.693649i $$0.243998\pi$$
$$558$$ 0 0
$$559$$ −10.2462 −0.433369
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 25.5616 1.07729 0.538646 0.842533i $$-0.318936\pi$$
0.538646 + 0.842533i $$0.318936\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.12311 0.0470830 0.0235415 0.999723i $$-0.492506\pi$$
0.0235415 + 0.999723i $$0.492506\pi$$
$$570$$ 0 0
$$571$$ 25.5616 1.06972 0.534859 0.844941i $$-0.320365\pi$$
0.534859 + 0.844941i $$0.320365\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5.56155 0.231933
$$576$$ 0 0
$$577$$ −36.5464 −1.52145 −0.760723 0.649076i $$-0.775156\pi$$
−0.760723 + 0.649076i $$0.775156\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 12.4924 0.518273
$$582$$ 0 0
$$583$$ −19.8078 −0.820354
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 17.3693 0.716908 0.358454 0.933547i $$-0.383304\pi$$
0.358454 + 0.933547i $$0.383304\pi$$
$$588$$ 0 0
$$589$$ −22.2462 −0.916639
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 5.61553 0.230602 0.115301 0.993331i $$-0.463217\pi$$
0.115301 + 0.993331i $$0.463217\pi$$
$$594$$ 0 0
$$595$$ −10.4384 −0.427935
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 18.7386 0.765640 0.382820 0.923823i $$-0.374953\pi$$
0.382820 + 0.923823i $$0.374953\pi$$
$$600$$ 0 0
$$601$$ −3.56155 −0.145279 −0.0726394 0.997358i $$-0.523142\pi$$
−0.0726394 + 0.997358i $$0.523142\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −19.9309 −0.810305
$$606$$ 0 0
$$607$$ 26.7386 1.08529 0.542644 0.839963i $$-0.317423\pi$$
0.542644 + 0.839963i $$0.317423\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.12311 −0.288170
$$612$$ 0 0
$$613$$ 4.93087 0.199156 0.0995780 0.995030i $$-0.468251\pi$$
0.0995780 + 0.995030i $$0.468251\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −0.246211 −0.00991209 −0.00495605 0.999988i $$-0.501578\pi$$
−0.00495605 + 0.999988i $$0.501578\pi$$
$$618$$ 0 0
$$619$$ 6.63068 0.266510 0.133255 0.991082i $$-0.457457\pi$$
0.133255 + 0.991082i $$0.457457\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 22.9309 0.918706
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 65.5616 2.61411
$$630$$ 0 0
$$631$$ 26.2462 1.04485 0.522423 0.852687i $$-0.325028\pi$$
0.522423 + 0.852687i $$0.325028\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 17.3693 0.689280
$$636$$ 0 0
$$637$$ −4.56155 −0.180735
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −0.630683 −0.0249105 −0.0124552 0.999922i $$-0.503965\pi$$
−0.0124552 + 0.999922i $$0.503965\pi$$
$$642$$ 0 0
$$643$$ 11.8078 0.465653 0.232826 0.972518i $$-0.425203\pi$$
0.232826 + 0.972518i $$0.425203\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.684658 0.0269167 0.0134584 0.999909i $$-0.495716\pi$$
0.0134584 + 0.999909i $$0.495716\pi$$
$$648$$ 0 0
$$649$$ 44.4924 1.74648
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −23.7538 −0.929558 −0.464779 0.885427i $$-0.653866\pi$$
−0.464779 + 0.885427i $$0.653866\pi$$
$$654$$ 0 0
$$655$$ −15.1231 −0.590909
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −8.49242 −0.330818 −0.165409 0.986225i $$-0.552894\pi$$
−0.165409 + 0.986225i $$0.552894\pi$$
$$660$$ 0 0
$$661$$ −30.8769 −1.20097 −0.600486 0.799635i $$-0.705026\pi$$
−0.600486 + 0.799635i $$0.705026\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.87689 −0.189118
$$666$$ 0 0
$$667$$ 11.1231 0.430688
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −59.4233 −2.29401
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −7.06913 −0.271689 −0.135844 0.990730i $$-0.543375\pi$$
−0.135844 + 0.990730i $$0.543375\pi$$
$$678$$ 0 0
$$679$$ −27.8078 −1.06716
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −4.49242 −0.171898 −0.0859489 0.996300i $$-0.527392\pi$$
−0.0859489 + 0.996300i $$0.527392\pi$$
$$684$$ 0 0
$$685$$ 16.2462 0.620736
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −3.56155 −0.135684
$$690$$ 0 0
$$691$$ 28.8769 1.09853 0.549264 0.835649i $$-0.314908\pi$$
0.549264 + 0.835649i $$0.314908\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9.56155 0.362690
$$696$$ 0 0
$$697$$ −17.9460 −0.679754
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19.3693 0.731569 0.365785 0.930700i $$-0.380801\pi$$
0.365785 + 0.930700i $$0.380801\pi$$
$$702$$ 0 0
$$703$$ 30.6307 1.15526
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.75379 −0.0659580
$$708$$ 0 0
$$709$$ 17.1231 0.643072 0.321536 0.946897i $$-0.395801\pi$$
0.321536 + 0.946897i $$0.395801\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −39.6155 −1.48361
$$714$$ 0 0
$$715$$ −5.56155 −0.207990
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −9.75379 −0.363755 −0.181877 0.983321i $$-0.558217\pi$$
−0.181877 + 0.983321i $$0.558217\pi$$
$$720$$ 0 0
$$721$$ 22.2462 0.828492
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ −22.6307 −0.839326 −0.419663 0.907680i $$-0.637852\pi$$
−0.419663 + 0.907680i $$0.637852\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −68.4924 −2.53328
$$732$$ 0 0
$$733$$ −29.4233 −1.08677 −0.543387 0.839482i $$-0.682858\pi$$
−0.543387 + 0.839482i $$0.682858\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −79.2311 −2.91851
$$738$$ 0 0
$$739$$ 29.8617 1.09848 0.549241 0.835664i $$-0.314917\pi$$
0.549241 + 0.835664i $$0.314917\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8.87689 0.325662 0.162831 0.986654i $$-0.447938\pi$$
0.162831 + 0.986654i $$0.447938\pi$$
$$744$$ 0 0
$$745$$ −12.4384 −0.455709
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 24.6847 0.901958
$$750$$ 0 0
$$751$$ −7.31534 −0.266941 −0.133470 0.991053i $$-0.542612\pi$$
−0.133470 + 0.991053i $$0.542612\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ −14.8769 −0.540710 −0.270355 0.962761i $$-0.587141\pi$$
−0.270355 + 0.962761i $$0.587141\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.4924 −0.815350 −0.407675 0.913127i $$-0.633660\pi$$
−0.407675 + 0.913127i $$0.633660\pi$$
$$762$$ 0 0
$$763$$ −20.4924 −0.741876
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ 25.6155 0.923720 0.461860 0.886953i $$-0.347182\pi$$
0.461860 + 0.886953i $$0.347182\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 21.1231 0.759745 0.379873 0.925039i $$-0.375968\pi$$
0.379873 + 0.925039i $$0.375968\pi$$
$$774$$ 0 0
$$775$$ −7.12311 −0.255870
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8.38447 −0.300405
$$780$$ 0 0
$$781$$ −26.0540 −0.932285
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −10.4924 −0.374491
$$786$$ 0 0
$$787$$ −18.7386 −0.667960 −0.333980 0.942580i $$-0.608392\pi$$
−0.333980 + 0.942580i $$0.608392\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −3.12311 −0.111045
$$792$$ 0 0
$$793$$ −10.6847 −0.379423
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −41.4233 −1.46729 −0.733644 0.679534i $$-0.762182\pi$$
−0.733644 + 0.679534i $$0.762182\pi$$
$$798$$ 0 0
$$799$$ −47.6155 −1.68452
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 90.3542 3.18853
$$804$$ 0 0
$$805$$ −8.68466 −0.306094
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 18.4924 0.650159 0.325079 0.945687i $$-0.394609\pi$$
0.325079 + 0.945687i $$0.394609\pi$$
$$810$$ 0 0
$$811$$ 40.9848 1.43917 0.719586 0.694403i $$-0.244331\pi$$
0.719586 + 0.694403i $$0.244331\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −13.5616 −0.475040
$$816$$ 0 0
$$817$$ −32.0000 −1.11954
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 33.3153 1.16271 0.581357 0.813649i $$-0.302522\pi$$
0.581357 + 0.813649i $$0.302522\pi$$
$$822$$ 0 0
$$823$$ −11.5076 −0.401129 −0.200564 0.979681i $$-0.564278\pi$$
−0.200564 + 0.979681i $$0.564278\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −22.6307 −0.786946 −0.393473 0.919336i $$-0.628727\pi$$
−0.393473 + 0.919336i $$0.628727\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −30.4924 −1.05650
$$834$$ 0 0
$$835$$ −5.75379 −0.199118
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 31.8078 1.09813 0.549063 0.835781i $$-0.314985\pi$$
0.549063 + 0.835781i $$0.314985\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 31.1231 1.06940
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 54.5464 1.86983
$$852$$ 0 0
$$853$$ −39.5616 −1.35456 −0.677281 0.735725i $$-0.736842\pi$$
−0.677281 + 0.735725i $$0.736842\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 34.7926 1.18849 0.594246 0.804283i $$-0.297450\pi$$
0.594246 + 0.804283i $$0.297450\pi$$
$$858$$ 0 0
$$859$$ 32.1922 1.09838 0.549192 0.835696i $$-0.314935\pi$$
0.549192 + 0.835696i $$0.314935\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −1.86174 −0.0633743 −0.0316872 0.999498i $$-0.510088\pi$$
−0.0316872 + 0.999498i $$0.510088\pi$$
$$864$$ 0 0
$$865$$ 14.0000 0.476014
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 65.6695 2.22769
$$870$$ 0 0
$$871$$ −14.2462 −0.482714
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1.56155 −0.0527901
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −8.24621 −0.277822 −0.138911 0.990305i $$-0.544360\pi$$
−0.138911 + 0.990305i $$0.544360\pi$$
$$882$$ 0 0
$$883$$ 32.4924 1.09346 0.546729 0.837310i $$-0.315873\pi$$
0.546729 + 0.837310i $$0.315873\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −26.0540 −0.874807 −0.437403 0.899265i $$-0.644102\pi$$
−0.437403 + 0.899265i $$0.644102\pi$$
$$888$$ 0 0
$$889$$ −27.1231 −0.909680
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −22.2462 −0.744441
$$894$$ 0 0
$$895$$ −23.1231 −0.772920
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −14.2462 −0.475138
$$900$$ 0 0
$$901$$ −23.8078 −0.793152
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 8.93087 0.296872
$$906$$ 0 0
$$907$$ −47.1231 −1.56470 −0.782349 0.622841i $$-0.785978\pi$$
−0.782349 + 0.622841i $$0.785978\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −17.7538 −0.588209 −0.294105 0.955773i $$-0.595021\pi$$
−0.294105 + 0.955773i $$0.595021\pi$$
$$912$$ 0 0
$$913$$ 44.4924 1.47248
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 23.6155 0.779853
$$918$$ 0 0
$$919$$ −12.1922 −0.402185 −0.201092 0.979572i $$-0.564449\pi$$
−0.201092 + 0.979572i $$0.564449\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −4.68466 −0.154197
$$924$$ 0 0
$$925$$ 9.80776 0.322477
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −18.3002 −0.600410 −0.300205 0.953875i $$-0.597055\pi$$
−0.300205 + 0.953875i $$0.597055\pi$$
$$930$$ 0 0
$$931$$ −14.2462 −0.466901
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −37.1771 −1.21582
$$936$$ 0 0
$$937$$ −31.7538 −1.03735 −0.518676 0.854971i $$-0.673575\pi$$
−0.518676 + 0.854971i $$0.673575\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −20.5464 −0.669793 −0.334897 0.942255i $$-0.608701\pi$$
−0.334897 + 0.942255i $$0.608701\pi$$
$$942$$ 0 0
$$943$$ −14.9309 −0.486216
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 50.7386 1.64878 0.824392 0.566019i $$-0.191517\pi$$
0.824392 + 0.566019i $$0.191517\pi$$
$$948$$ 0 0
$$949$$ 16.2462 0.527374
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −29.8078 −0.965568 −0.482784 0.875739i $$-0.660374\pi$$
−0.482784 + 0.875739i $$0.660374\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −25.3693 −0.819218
$$960$$ 0 0
$$961$$ 19.7386 0.636730
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −9.31534 −0.299871
$$966$$ 0 0
$$967$$ −42.2462 −1.35855 −0.679273 0.733885i $$-0.737705\pi$$
−0.679273 + 0.733885i $$0.737705\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 8.49242 0.272535 0.136267 0.990672i $$-0.456489\pi$$
0.136267 + 0.990672i $$0.456489\pi$$
$$972$$ 0 0
$$973$$ −14.9309 −0.478662
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 43.8617 1.40326 0.701631 0.712541i $$-0.252456\pi$$
0.701631 + 0.712541i $$0.252456\pi$$
$$978$$ 0 0
$$979$$ 81.6695 2.61017
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 52.9848 1.68995 0.844977 0.534803i $$-0.179614\pi$$
0.844977 + 0.534803i $$0.179614\pi$$
$$984$$ 0 0
$$985$$ 7.36932 0.234806
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −56.9848 −1.81201
$$990$$ 0 0
$$991$$ −34.0540 −1.08176 −0.540880 0.841100i $$-0.681909\pi$$
−0.540880 + 0.841100i $$0.681909\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.75379 −0.0555988
$$996$$ 0 0
$$997$$ 10.8769 0.344475 0.172237 0.985055i $$-0.444900\pi$$
0.172237 + 0.985055i $$0.444900\pi$$
$$998$$ 0 0
$$999$$ 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 4680.2.a.y.1.2 2
3.2 odd 2 1560.2.a.o.1.2 2
4.3 odd 2 9360.2.a.ci.1.1 2
12.11 even 2 3120.2.a.bg.1.1 2
15.14 odd 2 7800.2.a.bd.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.o.1.2 2 3.2 odd 2
3120.2.a.bg.1.1 2 12.11 even 2
4680.2.a.y.1.2 2 1.1 even 1 trivial
7800.2.a.bd.1.1 2 15.14 odd 2
9360.2.a.ci.1.1 2 4.3 odd 2