# Properties

 Label 4100.2.a.c.1.1 Level $4100$ Weight $2$ Character 4100.1 Self dual yes Analytic conductor $32.739$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4100 = 2^{2} \cdot 5^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4100.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.7386648287$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.25808.1 Defining polynomial: $$x^{4} - 10 x^{2} - 6 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 164) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.55466$$ of defining polynomial Character $$\chi$$ $$=$$ 4100.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.92968 q^{3} -2.77840 q^{7} +5.58303 q^{9} +O(q^{10})$$ $$q-2.92968 q^{3} -2.77840 q^{7} +5.58303 q^{9} +2.02837 q^{11} +3.10932 q^{13} +7.91609 q^{17} +2.17964 q^{19} +8.13983 q^{21} +4.75004 q^{23} -7.56744 q^{27} +3.85936 q^{29} -10.6661 q^{31} -5.94246 q^{33} -6.58303 q^{37} -9.10932 q^{39} -1.00000 q^{41} -4.80677 q^{43} -4.03900 q^{47} +0.719526 q^{49} -23.1916 q^{51} +1.94327 q^{53} -6.38566 q^{57} +4.75004 q^{59} +2.89068 q^{61} -15.5119 q^{63} -5.97163 q^{67} -13.9161 q^{69} +14.4026 q^{71} +9.24916 q^{73} -5.63562 q^{77} -0.320282 q^{79} +5.42111 q^{81} +3.69745 q^{83} -11.3067 q^{87} +8.96868 q^{89} -8.63895 q^{91} +31.2484 q^{93} -17.8322 q^{97} +11.3244 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 12q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 12q^{9} + 4q^{11} + 4q^{17} + 6q^{19} + 12q^{23} + 10q^{27} - 4q^{29} - 8q^{31} + 20q^{33} - 16q^{37} - 24q^{39} - 4q^{41} - 4q^{43} + 6q^{47} + 16q^{49} - 4q^{51} + 16q^{53} - 4q^{57} + 12q^{59} + 24q^{61} + 10q^{63} - 28q^{67} - 28q^{69} - 2q^{71} - 8q^{73} - 8q^{77} - 18q^{79} + 28q^{81} + 12q^{83} - 44q^{87} + 4q^{89} + 36q^{91} + 28q^{93} - 16q^{97} + 58q^{99} + O(q^{100})$$

## 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
$$3$$ −2.92968 −1.69145 −0.845726 0.533618i $$-0.820832\pi$$
−0.845726 + 0.533618i $$0.820832\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.77840 −1.05014 −0.525069 0.851060i $$-0.675960\pi$$
−0.525069 + 0.851060i $$0.675960\pi$$
$$8$$ 0 0
$$9$$ 5.58303 1.86101
$$10$$ 0 0
$$11$$ 2.02837 0.611575 0.305788 0.952100i $$-0.401080\pi$$
0.305788 + 0.952100i $$0.401080\pi$$
$$12$$ 0 0
$$13$$ 3.10932 0.862371 0.431186 0.902263i $$-0.358095\pi$$
0.431186 + 0.902263i $$0.358095\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.91609 1.91993 0.959967 0.280112i $$-0.0903717\pi$$
0.959967 + 0.280112i $$0.0903717\pi$$
$$18$$ 0 0
$$19$$ 2.17964 0.500044 0.250022 0.968240i $$-0.419562\pi$$
0.250022 + 0.968240i $$0.419562\pi$$
$$20$$ 0 0
$$21$$ 8.13983 1.77626
$$22$$ 0 0
$$23$$ 4.75004 0.990451 0.495226 0.868764i $$-0.335085\pi$$
0.495226 + 0.868764i $$0.335085\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −7.56744 −1.45636
$$28$$ 0 0
$$29$$ 3.85936 0.716665 0.358333 0.933594i $$-0.383345\pi$$
0.358333 + 0.933594i $$0.383345\pi$$
$$30$$ 0 0
$$31$$ −10.6661 −1.91569 −0.957847 0.287280i $$-0.907249\pi$$
−0.957847 + 0.287280i $$0.907249\pi$$
$$32$$ 0 0
$$33$$ −5.94246 −1.03445
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.58303 −1.08224 −0.541122 0.840944i $$-0.682000\pi$$
−0.541122 + 0.840944i $$0.682000\pi$$
$$38$$ 0 0
$$39$$ −9.10932 −1.45866
$$40$$ 0 0
$$41$$ −1.00000 −0.156174
$$42$$ 0 0
$$43$$ −4.80677 −0.733025 −0.366513 0.930413i $$-0.619448\pi$$
−0.366513 + 0.930413i $$0.619448\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.03900 −0.589149 −0.294575 0.955628i $$-0.595178\pi$$
−0.294575 + 0.955628i $$0.595178\pi$$
$$48$$ 0 0
$$49$$ 0.719526 0.102789
$$50$$ 0 0
$$51$$ −23.1916 −3.24748
$$52$$ 0 0
$$53$$ 1.94327 0.266928 0.133464 0.991054i $$-0.457390\pi$$
0.133464 + 0.991054i $$0.457390\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.38566 −0.845801
$$58$$ 0 0
$$59$$ 4.75004 0.618402 0.309201 0.950997i $$-0.399938\pi$$
0.309201 + 0.950997i $$0.399938\pi$$
$$60$$ 0 0
$$61$$ 2.89068 0.370113 0.185057 0.982728i $$-0.440753\pi$$
0.185057 + 0.982728i $$0.440753\pi$$
$$62$$ 0 0
$$63$$ −15.5119 −1.95432
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −5.97163 −0.729551 −0.364776 0.931095i $$-0.618854\pi$$
−0.364776 + 0.931095i $$0.618854\pi$$
$$68$$ 0 0
$$69$$ −13.9161 −1.67530
$$70$$ 0 0
$$71$$ 14.4026 1.70927 0.854636 0.519228i $$-0.173780\pi$$
0.854636 + 0.519228i $$0.173780\pi$$
$$72$$ 0 0
$$73$$ 9.24916 1.08253 0.541266 0.840851i $$-0.317945\pi$$
0.541266 + 0.840851i $$0.317945\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −5.63562 −0.642238
$$78$$ 0 0
$$79$$ −0.320282 −0.0360345 −0.0180173 0.999838i $$-0.505735\pi$$
−0.0180173 + 0.999838i $$0.505735\pi$$
$$80$$ 0 0
$$81$$ 5.42111 0.602346
$$82$$ 0 0
$$83$$ 3.69745 0.405847 0.202924 0.979195i $$-0.434956\pi$$
0.202924 + 0.979195i $$0.434956\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −11.3067 −1.21220
$$88$$ 0 0
$$89$$ 8.96868 0.950679 0.475339 0.879803i $$-0.342325\pi$$
0.475339 + 0.879803i $$0.342325\pi$$
$$90$$ 0 0
$$91$$ −8.63895 −0.905608
$$92$$ 0 0
$$93$$ 31.2484 3.24030
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.8322 −1.81058 −0.905292 0.424790i $$-0.860348\pi$$
−0.905292 + 0.424790i $$0.860348\pi$$
$$98$$ 0 0
$$99$$ 11.3244 1.13815
$$100$$ 0 0
$$101$$ −13.0254 −1.29608 −0.648039 0.761607i $$-0.724410\pi$$
−0.648039 + 0.761607i $$0.724410\pi$$
$$102$$ 0 0
$$103$$ 8.05673 0.793853 0.396927 0.917850i $$-0.370077\pi$$
0.396927 + 0.917850i $$0.370077\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.41602 0.426912 0.213456 0.976953i $$-0.431528\pi$$
0.213456 + 0.976953i $$0.431528\pi$$
$$108$$ 0 0
$$109$$ 18.5255 1.77442 0.887210 0.461366i $$-0.152640\pi$$
0.887210 + 0.461366i $$0.152640\pi$$
$$110$$ 0 0
$$111$$ 19.2862 1.83056
$$112$$ 0 0
$$113$$ 8.08310 0.760394 0.380197 0.924905i $$-0.375856\pi$$
0.380197 + 0.924905i $$0.375856\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 17.3594 1.60488
$$118$$ 0 0
$$119$$ −21.9941 −2.01620
$$120$$ 0 0
$$121$$ −6.88573 −0.625976
$$122$$ 0 0
$$123$$ 2.92968 0.264160
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.44748 −0.394650 −0.197325 0.980338i $$-0.563225\pi$$
−0.197325 + 0.980338i $$0.563225\pi$$
$$128$$ 0 0
$$129$$ 14.0823 1.23988
$$130$$ 0 0
$$131$$ 9.85936 0.861416 0.430708 0.902491i $$-0.358264\pi$$
0.430708 + 0.902491i $$0.358264\pi$$
$$132$$ 0 0
$$133$$ −6.05593 −0.525115
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.30255 0.196720 0.0983602 0.995151i $$-0.468640\pi$$
0.0983602 + 0.995151i $$0.468640\pi$$
$$138$$ 0 0
$$139$$ 5.16605 0.438179 0.219090 0.975705i $$-0.429691\pi$$
0.219090 + 0.975705i $$0.429691\pi$$
$$140$$ 0 0
$$141$$ 11.8330 0.996517
$$142$$ 0 0
$$143$$ 6.30684 0.527405
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.10798 −0.173863
$$148$$ 0 0
$$149$$ 3.85936 0.316171 0.158086 0.987425i $$-0.449468\pi$$
0.158086 + 0.987425i $$0.449468\pi$$
$$150$$ 0 0
$$151$$ −4.71103 −0.383379 −0.191689 0.981456i $$-0.561397\pi$$
−0.191689 + 0.981456i $$0.561397\pi$$
$$152$$ 0 0
$$153$$ 44.1958 3.57302
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −8.30684 −0.662958 −0.331479 0.943463i $$-0.607548\pi$$
−0.331479 + 0.943463i $$0.607548\pi$$
$$158$$ 0 0
$$159$$ −5.69316 −0.451497
$$160$$ 0 0
$$161$$ −13.1975 −1.04011
$$162$$ 0 0
$$163$$ 13.6348 1.06796 0.533981 0.845497i $$-0.320696\pi$$
0.533981 + 0.845497i $$0.320696\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −9.97163 −0.771628 −0.385814 0.922577i $$-0.626079\pi$$
−0.385814 + 0.922577i $$0.626079\pi$$
$$168$$ 0 0
$$169$$ −3.33211 −0.256316
$$170$$ 0 0
$$171$$ 12.1690 0.930587
$$172$$ 0 0
$$173$$ −9.11361 −0.692895 −0.346448 0.938069i $$-0.612612\pi$$
−0.346448 + 0.938069i $$0.612612\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −13.9161 −1.04600
$$178$$ 0 0
$$179$$ −12.9403 −0.967205 −0.483602 0.875288i $$-0.660672\pi$$
−0.483602 + 0.875288i $$0.660672\pi$$
$$180$$ 0 0
$$181$$ 8.16191 0.606670 0.303335 0.952884i $$-0.401900\pi$$
0.303335 + 0.952884i $$0.401900\pi$$
$$182$$ 0 0
$$183$$ −8.46876 −0.626029
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.0567 1.17418
$$188$$ 0 0
$$189$$ 21.0254 1.52937
$$190$$ 0 0
$$191$$ 4.52829 0.327656 0.163828 0.986489i $$-0.447616\pi$$
0.163828 + 0.986489i $$0.447616\pi$$
$$192$$ 0 0
$$193$$ 14.1662 1.01971 0.509853 0.860262i $$-0.329700\pi$$
0.509853 + 0.860262i $$0.329700\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.5042 0.890888 0.445444 0.895310i $$-0.353046\pi$$
0.445444 + 0.895310i $$0.353046\pi$$
$$198$$ 0 0
$$199$$ −14.5853 −1.03393 −0.516963 0.856008i $$-0.672938\pi$$
−0.516963 + 0.856008i $$0.672938\pi$$
$$200$$ 0 0
$$201$$ 17.4950 1.23400
$$202$$ 0 0
$$203$$ −10.7229 −0.752597
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 26.5196 1.84324
$$208$$ 0 0
$$209$$ 4.42111 0.305815
$$210$$ 0 0
$$211$$ −26.3860 −1.81649 −0.908245 0.418439i $$-0.862577\pi$$
−0.908245 + 0.418439i $$0.862577\pi$$
$$212$$ 0 0
$$213$$ −42.1950 −2.89115
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 29.6348 2.01174
$$218$$ 0 0
$$219$$ −27.0971 −1.83105
$$220$$ 0 0
$$221$$ 24.6137 1.65570
$$222$$ 0 0
$$223$$ 1.89482 0.126886 0.0634432 0.997985i $$-0.479792\pi$$
0.0634432 + 0.997985i $$0.479792\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 24.9805 1.65801 0.829007 0.559238i $$-0.188906\pi$$
0.829007 + 0.559238i $$0.188906\pi$$
$$228$$ 0 0
$$229$$ −2.52549 −0.166889 −0.0834446 0.996512i $$-0.526592\pi$$
−0.0834446 + 0.996512i $$0.526592\pi$$
$$230$$ 0 0
$$231$$ 16.5106 1.08632
$$232$$ 0 0
$$233$$ −9.24835 −0.605880 −0.302940 0.953010i $$-0.597968\pi$$
−0.302940 + 0.953010i $$0.597968\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0.938323 0.0609507
$$238$$ 0 0
$$239$$ 2.34746 0.151844 0.0759222 0.997114i $$-0.475810\pi$$
0.0759222 + 0.997114i $$0.475810\pi$$
$$240$$ 0 0
$$241$$ 2.89068 0.186205 0.0931024 0.995657i $$-0.470322\pi$$
0.0931024 + 0.995657i $$0.470322\pi$$
$$242$$ 0 0
$$243$$ 6.82021 0.437516
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.77721 0.431224
$$248$$ 0 0
$$249$$ −10.8323 −0.686471
$$250$$ 0 0
$$251$$ 24.9730 1.57628 0.788140 0.615496i $$-0.211044\pi$$
0.788140 + 0.615496i $$0.211044\pi$$
$$252$$ 0 0
$$253$$ 9.63481 0.605736
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −10.7500 −0.670569 −0.335284 0.942117i $$-0.608832\pi$$
−0.335284 + 0.942117i $$0.608832\pi$$
$$258$$ 0 0
$$259$$ 18.2903 1.13650
$$260$$ 0 0
$$261$$ 21.5469 1.33372
$$262$$ 0 0
$$263$$ 4.90250 0.302301 0.151151 0.988511i $$-0.451702\pi$$
0.151151 + 0.988511i $$0.451702\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −26.2754 −1.60803
$$268$$ 0 0
$$269$$ 3.32797 0.202910 0.101455 0.994840i $$-0.467650\pi$$
0.101455 + 0.994840i $$0.467650\pi$$
$$270$$ 0 0
$$271$$ 4.58222 0.278350 0.139175 0.990268i $$-0.455555\pi$$
0.139175 + 0.990268i $$0.455555\pi$$
$$272$$ 0 0
$$273$$ 25.3094 1.53179
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 24.2178 1.45511 0.727555 0.686050i $$-0.240657\pi$$
0.727555 + 0.686050i $$0.240657\pi$$
$$278$$ 0 0
$$279$$ −59.5493 −3.56512
$$280$$ 0 0
$$281$$ 6.61369 0.394540 0.197270 0.980349i $$-0.436792\pi$$
0.197270 + 0.980349i $$0.436792\pi$$
$$282$$ 0 0
$$283$$ −5.16605 −0.307090 −0.153545 0.988142i $$-0.549069\pi$$
−0.153545 + 0.988142i $$0.549069\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.77840 0.164004
$$288$$ 0 0
$$289$$ 45.6645 2.68615
$$290$$ 0 0
$$291$$ 52.2426 3.06252
$$292$$ 0 0
$$293$$ 10.2458 0.598567 0.299284 0.954164i $$-0.403252\pi$$
0.299284 + 0.954164i $$0.403252\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −15.3495 −0.890671
$$298$$ 0 0
$$299$$ 14.7694 0.854137
$$300$$ 0 0
$$301$$ 13.3551 0.769778
$$302$$ 0 0
$$303$$ 38.1603 2.19225
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 18.2754 1.04303 0.521515 0.853242i $$-0.325367\pi$$
0.521515 + 0.853242i $$0.325367\pi$$
$$308$$ 0 0
$$309$$ −23.6036 −1.34276
$$310$$ 0 0
$$311$$ 4.50303 0.255343 0.127672 0.991816i $$-0.459250\pi$$
0.127672 + 0.991816i $$0.459250\pi$$
$$312$$ 0 0
$$313$$ −6.56095 −0.370847 −0.185423 0.982659i $$-0.559366\pi$$
−0.185423 + 0.982659i $$0.559366\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −12.8363 −0.720960 −0.360480 0.932767i $$-0.617387\pi$$
−0.360480 + 0.932767i $$0.617387\pi$$
$$318$$ 0 0
$$319$$ 7.82820 0.438295
$$320$$ 0 0
$$321$$ −12.9375 −0.722102
$$322$$ 0 0
$$323$$ 17.2543 0.960052
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −54.2738 −3.00135
$$328$$ 0 0
$$329$$ 11.2220 0.618688
$$330$$ 0 0
$$331$$ −8.99705 −0.494523 −0.247261 0.968949i $$-0.579531\pi$$
−0.247261 + 0.968949i $$0.579531\pi$$
$$332$$ 0 0
$$333$$ −36.7532 −2.01406
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 35.0502 1.90930 0.954652 0.297723i $$-0.0962271\pi$$
0.954652 + 0.297723i $$0.0962271\pi$$
$$338$$ 0 0
$$339$$ −23.6809 −1.28617
$$340$$ 0 0
$$341$$ −21.6348 −1.17159
$$342$$ 0 0
$$343$$ 17.4497 0.942195
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −18.8982 −1.01451 −0.507255 0.861796i $$-0.669340\pi$$
−0.507255 + 0.861796i $$0.669340\pi$$
$$348$$ 0 0
$$349$$ −23.1653 −1.24001 −0.620004 0.784599i $$-0.712869\pi$$
−0.620004 + 0.784599i $$0.712869\pi$$
$$350$$ 0 0
$$351$$ −23.5296 −1.25592
$$352$$ 0 0
$$353$$ 1.55171 0.0825893 0.0412946 0.999147i $$-0.486852\pi$$
0.0412946 + 0.999147i $$0.486852\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 64.4357 3.41030
$$358$$ 0 0
$$359$$ −4.28128 −0.225957 −0.112979 0.993597i $$-0.536039\pi$$
−0.112979 + 0.993597i $$0.536039\pi$$
$$360$$ 0 0
$$361$$ −14.2492 −0.749956
$$362$$ 0 0
$$363$$ 20.1730 1.05881
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 15.3067 0.799003 0.399501 0.916733i $$-0.369183\pi$$
0.399501 + 0.916733i $$0.369183\pi$$
$$368$$ 0 0
$$369$$ −5.58303 −0.290641
$$370$$ 0 0
$$371$$ −5.39918 −0.280312
$$372$$ 0 0
$$373$$ 5.39489 0.279337 0.139668 0.990198i $$-0.455396\pi$$
0.139668 + 0.990198i $$0.455396\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 26.8009 1.37667 0.688334 0.725394i $$-0.258342\pi$$
0.688334 + 0.725394i $$0.258342\pi$$
$$380$$ 0 0
$$381$$ 13.0297 0.667532
$$382$$ 0 0
$$383$$ −21.6675 −1.10716 −0.553578 0.832797i $$-0.686738\pi$$
−0.553578 + 0.832797i $$0.686738\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −26.8363 −1.36417
$$388$$ 0 0
$$389$$ 25.7187 1.30399 0.651995 0.758223i $$-0.273932\pi$$
0.651995 + 0.758223i $$0.273932\pi$$
$$390$$ 0 0
$$391$$ 37.6017 1.90160
$$392$$ 0 0
$$393$$ −28.8848 −1.45704
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −28.1603 −1.41333 −0.706663 0.707551i $$-0.749800\pi$$
−0.706663 + 0.707551i $$0.749800\pi$$
$$398$$ 0 0
$$399$$ 17.7419 0.888208
$$400$$ 0 0
$$401$$ −10.7178 −0.535220 −0.267610 0.963527i $$-0.586234\pi$$
−0.267610 + 0.963527i $$0.586234\pi$$
$$402$$ 0 0
$$403$$ −33.1644 −1.65204
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −13.3528 −0.661873
$$408$$ 0 0
$$409$$ 22.7492 1.12488 0.562439 0.826839i $$-0.309863\pi$$
0.562439 + 0.826839i $$0.309863\pi$$
$$410$$ 0 0
$$411$$ −6.74575 −0.332743
$$412$$ 0 0
$$413$$ −13.1975 −0.649408
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −15.1349 −0.741159
$$418$$ 0 0
$$419$$ −7.02542 −0.343214 −0.171607 0.985165i $$-0.554896\pi$$
−0.171607 + 0.985165i $$0.554896\pi$$
$$420$$ 0 0
$$421$$ 16.5992 0.808996 0.404498 0.914539i $$-0.367446\pi$$
0.404498 + 0.914539i $$0.367446\pi$$
$$422$$ 0 0
$$423$$ −22.5499 −1.09641
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.03147 −0.388670
$$428$$ 0 0
$$429$$ −18.4770 −0.892080
$$430$$ 0 0
$$431$$ −10.7229 −0.516502 −0.258251 0.966078i $$-0.583146\pi$$
−0.258251 + 0.966078i $$0.583146\pi$$
$$432$$ 0 0
$$433$$ −31.1644 −1.49767 −0.748834 0.662758i $$-0.769386\pi$$
−0.748834 + 0.662758i $$0.769386\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 10.3534 0.495270
$$438$$ 0 0
$$439$$ −34.7981 −1.66082 −0.830411 0.557152i $$-0.811894\pi$$
−0.830411 + 0.557152i $$0.811894\pi$$
$$440$$ 0 0
$$441$$ 4.01714 0.191292
$$442$$ 0 0
$$443$$ 25.8806 1.22963 0.614813 0.788673i $$-0.289231\pi$$
0.614813 + 0.788673i $$0.289231\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −11.3067 −0.534788
$$448$$ 0 0
$$449$$ 27.2231 1.28474 0.642368 0.766396i $$-0.277952\pi$$
0.642368 + 0.766396i $$0.277952\pi$$
$$450$$ 0 0
$$451$$ −2.02837 −0.0955120
$$452$$ 0 0
$$453$$ 13.8018 0.648466
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.19767 0.289915 0.144957 0.989438i $$-0.453695\pi$$
0.144957 + 0.989438i $$0.453695\pi$$
$$458$$ 0 0
$$459$$ −59.9046 −2.79611
$$460$$ 0 0
$$461$$ −10.9170 −0.508458 −0.254229 0.967144i $$-0.581822\pi$$
−0.254229 + 0.967144i $$0.581822\pi$$
$$462$$ 0 0
$$463$$ −4.07047 −0.189171 −0.0945854 0.995517i $$-0.530153\pi$$
−0.0945854 + 0.995517i $$0.530153\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8.63657 0.399653 0.199826 0.979831i $$-0.435962\pi$$
0.199826 + 0.979831i $$0.435962\pi$$
$$468$$ 0 0
$$469$$ 16.5916 0.766129
$$470$$ 0 0
$$471$$ 24.3364 1.12136
$$472$$ 0 0
$$473$$ −9.74989 −0.448300
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10.8493 0.496756
$$478$$ 0 0
$$479$$ −23.5537 −1.07620 −0.538098 0.842882i $$-0.680857\pi$$
−0.538098 + 0.842882i $$0.680857\pi$$
$$480$$ 0 0
$$481$$ −20.4688 −0.933295
$$482$$ 0 0
$$483$$ 38.6645 1.75930
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −19.9102 −0.902217 −0.451108 0.892469i $$-0.648971\pi$$
−0.451108 + 0.892469i $$0.648971\pi$$
$$488$$ 0 0
$$489$$ −39.9456 −1.80640
$$490$$ 0 0
$$491$$ 29.7585 1.34298 0.671490 0.741013i $$-0.265654\pi$$
0.671490 + 0.741013i $$0.265654\pi$$
$$492$$ 0 0
$$493$$ 30.5511 1.37595
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −40.0162 −1.79497
$$498$$ 0 0
$$499$$ 18.9297 0.847409 0.423704 0.905801i $$-0.360730\pi$$
0.423704 + 0.905801i $$0.360730\pi$$
$$500$$ 0 0
$$501$$ 29.2137 1.30517
$$502$$ 0 0
$$503$$ 1.93382 0.0862248 0.0431124 0.999070i $$-0.486273\pi$$
0.0431124 + 0.999070i $$0.486273\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.76202 0.433546
$$508$$ 0 0
$$509$$ 34.6602 1.53629 0.768144 0.640277i $$-0.221181\pi$$
0.768144 + 0.640277i $$0.221181\pi$$
$$510$$ 0 0
$$511$$ −25.6979 −1.13681
$$512$$ 0 0
$$513$$ −16.4943 −0.728242
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8.19258 −0.360309
$$518$$ 0 0
$$519$$ 26.7000 1.17200
$$520$$ 0 0
$$521$$ −6.33402 −0.277498 −0.138749 0.990328i $$-0.544308\pi$$
−0.138749 + 0.990328i $$0.544308\pi$$
$$522$$ 0 0
$$523$$ 29.9460 1.30944 0.654722 0.755869i $$-0.272785\pi$$
0.654722 + 0.755869i $$0.272785\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −84.4341 −3.67801
$$528$$ 0 0
$$529$$ −0.437142 −0.0190062
$$530$$ 0 0
$$531$$ 26.5196 1.15085
$$532$$ 0 0
$$533$$ −3.10932 −0.134680
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 37.9110 1.63598
$$538$$ 0 0
$$539$$ 1.45946 0.0628635
$$540$$ 0 0
$$541$$ 0.834750 0.0358887 0.0179443 0.999839i $$-0.494288\pi$$
0.0179443 + 0.999839i $$0.494288\pi$$
$$542$$ 0 0
$$543$$ −23.9118 −1.02615
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.6527 0.883045 0.441523 0.897250i $$-0.354438\pi$$
0.441523 + 0.897250i $$0.354438\pi$$
$$548$$ 0 0
$$549$$ 16.1387 0.688784
$$550$$ 0 0
$$551$$ 8.41203 0.358364
$$552$$ 0 0
$$553$$ 0.889872 0.0378412
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 8.72447 0.369668 0.184834 0.982770i $$-0.440825\pi$$
0.184834 + 0.982770i $$0.440825\pi$$
$$558$$ 0 0
$$559$$ −14.9458 −0.632140
$$560$$ 0 0
$$561$$ −47.0411 −1.98608
$$562$$ 0 0
$$563$$ −9.80144 −0.413081 −0.206541 0.978438i $$-0.566221\pi$$
−0.206541 + 0.978438i $$0.566221\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −15.0620 −0.632546
$$568$$ 0 0
$$569$$ 25.1570 1.05464 0.527318 0.849668i $$-0.323198\pi$$
0.527318 + 0.849668i $$0.323198\pi$$
$$570$$ 0 0
$$571$$ 27.1928 1.13798 0.568992 0.822343i $$-0.307334\pi$$
0.568992 + 0.822343i $$0.307334\pi$$
$$572$$ 0 0
$$573$$ −13.2664 −0.554214
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0.365186 0.0152029 0.00760144 0.999971i $$-0.497580\pi$$
0.00760144 + 0.999971i $$0.497580\pi$$
$$578$$ 0 0
$$579$$ −41.5025 −1.72478
$$580$$ 0 0
$$581$$ −10.2730 −0.426196
$$582$$ 0 0
$$583$$ 3.94166 0.163247
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 37.7242 1.55704 0.778522 0.627617i $$-0.215970\pi$$
0.778522 + 0.627617i $$0.215970\pi$$
$$588$$ 0 0
$$589$$ −23.2484 −0.957932
$$590$$ 0 0
$$591$$ −36.6334 −1.50689
$$592$$ 0 0
$$593$$ 45.1333 1.85340 0.926701 0.375800i $$-0.122632\pi$$
0.926701 + 0.375800i $$0.122632\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 42.7303 1.74884
$$598$$ 0 0
$$599$$ 15.4982 0.633238 0.316619 0.948553i $$-0.397452\pi$$
0.316619 + 0.948553i $$0.397452\pi$$
$$600$$ 0 0
$$601$$ −5.18924 −0.211674 −0.105837 0.994384i $$-0.533752\pi$$
−0.105837 + 0.994384i $$0.533752\pi$$
$$602$$ 0 0
$$603$$ −33.3398 −1.35770
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.8280 0.683029 0.341515 0.939876i $$-0.389060\pi$$
0.341515 + 0.939876i $$0.389060\pi$$
$$608$$ 0 0
$$609$$ 31.4146 1.27298
$$610$$ 0 0
$$611$$ −12.5586 −0.508065
$$612$$ 0 0
$$613$$ −25.1042 −1.01395 −0.506975 0.861961i $$-0.669236\pi$$
−0.506975 + 0.861961i $$0.669236\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −14.3324 −0.577001 −0.288501 0.957480i $$-0.593157\pi$$
−0.288501 + 0.957480i $$0.593157\pi$$
$$618$$ 0 0
$$619$$ 19.2712 0.774576 0.387288 0.921959i $$-0.373412\pi$$
0.387288 + 0.921959i $$0.373412\pi$$
$$620$$ 0 0
$$621$$ −35.9456 −1.44245
$$622$$ 0 0
$$623$$ −24.9186 −0.998344
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −12.9524 −0.517271
$$628$$ 0 0
$$629$$ −52.1119 −2.07784
$$630$$ 0 0
$$631$$ −30.1856 −1.20167 −0.600834 0.799374i $$-0.705165\pi$$
−0.600834 + 0.799374i $$0.705165\pi$$
$$632$$ 0 0
$$633$$ 77.3027 3.07251
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.23724 0.0886427
$$638$$ 0 0
$$639$$ 80.4100 3.18097
$$640$$ 0 0
$$641$$ −25.9984 −1.02687 −0.513437 0.858127i $$-0.671628\pi$$
−0.513437 + 0.858127i $$0.671628\pi$$
$$642$$ 0 0
$$643$$ −11.4402 −0.451159 −0.225580 0.974225i $$-0.572428\pi$$
−0.225580 + 0.974225i $$0.572428\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −37.6915 −1.48181 −0.740904 0.671611i $$-0.765603\pi$$
−0.740904 + 0.671611i $$0.765603\pi$$
$$648$$ 0 0
$$649$$ 9.63481 0.378200
$$650$$ 0 0
$$651$$ −86.8205 −3.40277
$$652$$ 0 0
$$653$$ 43.6814 1.70938 0.854692 0.519136i $$-0.173746\pi$$
0.854692 + 0.519136i $$0.173746\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 51.6383 2.01460
$$658$$ 0 0
$$659$$ 25.0900 0.977367 0.488684 0.872461i $$-0.337477\pi$$
0.488684 + 0.872461i $$0.337477\pi$$
$$660$$ 0 0
$$661$$ −37.7392 −1.46788 −0.733942 0.679212i $$-0.762322\pi$$
−0.733942 + 0.679212i $$0.762322\pi$$
$$662$$ 0 0
$$663$$ −72.1102 −2.80053
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 18.3321 0.709822
$$668$$ 0 0
$$669$$ −5.55121 −0.214622
$$670$$ 0 0
$$671$$ 5.86335 0.226352
$$672$$ 0 0
$$673$$ −3.63642 −0.140174 −0.0700869 0.997541i $$-0.522328\pi$$
−0.0700869 + 0.997541i $$0.522328\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −29.1948 −1.12205 −0.561024 0.827800i $$-0.689592\pi$$
−0.561024 + 0.827800i $$0.689592\pi$$
$$678$$ 0 0
$$679$$ 49.5450 1.90136
$$680$$ 0 0
$$681$$ −73.1849 −2.80445
$$682$$ 0 0
$$683$$ 28.6527 1.09636 0.548182 0.836359i $$-0.315320\pi$$
0.548182 + 0.836359i $$0.315320\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 7.39888 0.282285
$$688$$ 0 0
$$689$$ 6.04225 0.230191
$$690$$ 0 0
$$691$$ 17.6880 0.672883 0.336442 0.941704i $$-0.390777\pi$$
0.336442 + 0.941704i $$0.390777\pi$$
$$692$$ 0 0
$$693$$ −31.4638 −1.19521
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −7.91609 −0.299843
$$698$$ 0 0
$$699$$ 27.0947 1.02482
$$700$$ 0 0
$$701$$ 2.79339 0.105505 0.0527525 0.998608i $$-0.483201\pi$$
0.0527525 + 0.998608i $$0.483201\pi$$
$$702$$ 0 0
$$703$$ −14.3486 −0.541169
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 36.1899 1.36106
$$708$$ 0 0
$$709$$ 20.4117 0.766578 0.383289 0.923628i $$-0.374791\pi$$
0.383289 + 0.923628i $$0.374791\pi$$
$$710$$ 0 0
$$711$$ −1.78814 −0.0670606
$$712$$ 0 0
$$713$$ −50.6645 −1.89740
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.87730 −0.256838
$$718$$ 0 0
$$719$$ 34.9848 1.30471 0.652356 0.757912i $$-0.273781\pi$$
0.652356 + 0.757912i $$0.273781\pi$$
$$720$$ 0 0
$$721$$ −22.3849 −0.833655
$$722$$ 0 0
$$723$$ −8.46876 −0.314957
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 24.8796 0.922734 0.461367 0.887209i $$-0.347359\pi$$
0.461367 + 0.887209i $$0.347359\pi$$
$$728$$ 0 0
$$729$$ −36.2444 −1.34238
$$730$$ 0 0
$$731$$ −38.0508 −1.40736
$$732$$ 0 0
$$733$$ 28.7737 1.06278 0.531390 0.847127i $$-0.321670\pi$$
0.531390 + 0.847127i $$0.321670\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.1127 −0.446176
$$738$$ 0 0
$$739$$ −30.1856 −1.11039 −0.555197 0.831719i $$-0.687357\pi$$
−0.555197 + 0.831719i $$0.687357\pi$$
$$740$$ 0 0
$$741$$ −19.8551 −0.729394
$$742$$ 0 0
$$743$$ 22.7127 0.833247 0.416624 0.909079i $$-0.363213\pi$$
0.416624 + 0.909079i $$0.363213\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 20.6429 0.755286
$$748$$ 0 0
$$749$$ −12.2695 −0.448317
$$750$$ 0 0
$$751$$ −0.0200854 −0.000732927 0 −0.000366463 1.00000i $$-0.500117\pi$$
−0.000366463 1.00000i $$0.500117\pi$$
$$752$$ 0 0
$$753$$ −73.1628 −2.66620
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −44.9002 −1.63192 −0.815962 0.578106i $$-0.803792\pi$$
−0.815962 + 0.578106i $$0.803792\pi$$
$$758$$ 0 0
$$759$$ −28.2269 −1.02457
$$760$$ 0 0
$$761$$ −19.0203 −0.689486 −0.344743 0.938697i $$-0.612034\pi$$
−0.344743 + 0.938697i $$0.612034\pi$$
$$762$$ 0 0
$$763$$ −51.4713 −1.86339
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 14.7694 0.533292
$$768$$ 0 0
$$769$$ 42.1220 1.51896 0.759480 0.650531i $$-0.225454\pi$$
0.759480 + 0.650531i $$0.225454\pi$$
$$770$$ 0 0
$$771$$ 31.4942 1.13424
$$772$$ 0 0
$$773$$ −10.7753 −0.387561 −0.193780 0.981045i $$-0.562075\pi$$
−0.193780 + 0.981045i $$0.562075\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −53.5848 −1.92234
$$778$$ 0 0
$$779$$ −2.17964 −0.0780938
$$780$$ 0 0
$$781$$ 29.2137 1.04535
$$782$$ 0 0
$$783$$ −29.2055 −1.04372
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −22.8592 −0.814843 −0.407421 0.913240i $$-0.633572\pi$$
−0.407421 + 0.913240i $$0.633572\pi$$
$$788$$ 0 0
$$789$$ −14.3628 −0.511328
$$790$$ 0 0
$$791$$ −22.4581 −0.798519
$$792$$ 0 0
$$793$$ 8.98805 0.319175
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 5.00444 0.177266 0.0886332 0.996064i $$-0.471750\pi$$
0.0886332 + 0.996064i $$0.471750\pi$$
$$798$$ 0 0
$$799$$ −31.9731 −1.13113
$$800$$ 0 0
$$801$$ 50.0724 1.76922
$$802$$ 0 0
$$803$$ 18.7607 0.662050
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −9.74989 −0.343212
$$808$$ 0 0
$$809$$ 7.52135 0.264437 0.132218 0.991221i $$-0.457790\pi$$
0.132218 + 0.991221i $$0.457790\pi$$
$$810$$ 0 0
$$811$$ 29.4103 1.03273 0.516367 0.856367i $$-0.327284\pi$$
0.516367 + 0.856367i $$0.327284\pi$$
$$812$$ 0 0
$$813$$ −13.4244 −0.470816
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −10.4770 −0.366545
$$818$$ 0 0
$$819$$ −48.2315 −1.68535
$$820$$ 0 0
$$821$$ 38.0205 1.32692 0.663462 0.748210i $$-0.269087\pi$$
0.663462 + 0.748210i $$0.269087\pi$$
$$822$$ 0 0
$$823$$ −25.7849 −0.898805 −0.449403 0.893329i $$-0.648363\pi$$
−0.449403 + 0.893329i $$0.648363\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.00295 −0.243516 −0.121758 0.992560i $$-0.538853\pi$$
−0.121758 + 0.992560i $$0.538853\pi$$
$$828$$ 0 0
$$829$$ 0.166709 0.00579003 0.00289501 0.999996i $$-0.499078\pi$$
0.00289501 + 0.999996i $$0.499078\pi$$
$$830$$ 0 0
$$831$$ −70.9505 −2.46125
$$832$$ 0 0
$$833$$ 5.69584 0.197349
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 80.7154 2.78993
$$838$$ 0 0
$$839$$ −13.1105 −0.452625 −0.226313 0.974055i $$-0.572667\pi$$
−0.226313 + 0.974055i $$0.572667\pi$$
$$840$$ 0 0
$$841$$ −14.1053 −0.486391
$$842$$ 0 0
$$843$$ −19.3760 −0.667345
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 19.1313 0.657361
$$848$$ 0 0
$$849$$ 15.1349 0.519428
$$850$$ 0 0
$$851$$ −31.2696 −1.07191
$$852$$ 0 0
$$853$$ 0.445573 0.0152561 0.00762806 0.999971i $$-0.497572\pi$$
0.00762806 + 0.999971i $$0.497572\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 20.5519 0.702038 0.351019 0.936368i $$-0.385835\pi$$
0.351019 + 0.936368i $$0.385835\pi$$
$$858$$ 0 0
$$859$$ −18.8891 −0.644487 −0.322243 0.946657i $$-0.604437\pi$$
−0.322243 + 0.946657i $$0.604437\pi$$
$$860$$ 0 0
$$861$$ −8.13983 −0.277405
$$862$$ 0 0
$$863$$ 47.9204 1.63123 0.815614 0.578596i $$-0.196399\pi$$
0.815614 + 0.578596i $$0.196399\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −133.782 −4.54349
$$868$$ 0 0
$$869$$ −0.649649 −0.0220378
$$870$$ 0 0
$$871$$ −18.5677 −0.629144
$$872$$ 0 0
$$873$$ −99.5576 −3.36951
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 50.2738 1.69762 0.848812 0.528694i $$-0.177318\pi$$
0.848812 + 0.528694i $$0.177318\pi$$
$$878$$ 0 0
$$879$$ −30.0170 −1.01245
$$880$$ 0 0
$$881$$ −38.6019 −1.30053 −0.650265 0.759707i $$-0.725342\pi$$
−0.650265 + 0.759707i $$0.725342\pi$$
$$882$$ 0 0
$$883$$ −3.07860 −0.103603 −0.0518016 0.998657i $$-0.516496\pi$$
−0.0518016 + 0.998657i $$0.516496\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 31.0770 1.04346 0.521731 0.853110i $$-0.325286\pi$$
0.521731 + 0.853110i $$0.325286\pi$$
$$888$$ 0 0
$$889$$ 12.3569 0.414437
$$890$$ 0 0
$$891$$ 10.9960 0.368380
$$892$$ 0 0
$$893$$ −8.80358 −0.294601
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −43.2696 −1.44473
$$898$$ 0 0
$$899$$ −41.1644 −1.37291
$$900$$ 0 0
$$901$$ 15.3831 0.512485
$$902$$ 0 0
$$903$$ −39.1263 −1.30204
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −42.5534 −1.41296 −0.706482 0.707731i $$-0.749719\pi$$
−0.706482 + 0.707731i $$0.749719\pi$$
$$908$$ 0 0
$$909$$ −72.7213 −2.41201
$$910$$ 0 0
$$911$$ −14.8694 −0.492645 −0.246323 0.969188i $$-0.579222\pi$$
−0.246323 + 0.969188i $$0.579222\pi$$
$$912$$ 0 0
$$913$$ 7.49977 0.248206
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −27.3933 −0.904606
$$918$$ 0 0
$$919$$ 26.5539 0.875931 0.437965 0.898992i $$-0.355699\pi$$
0.437965 + 0.898992i $$0.355699\pi$$
$$920$$ 0 0
$$921$$ −53.5410 −1.76424
$$922$$ 0 0
$$923$$ 44.7823 1.47403
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 44.9810 1.47737
$$928$$ 0 0
$$929$$ 2.70749 0.0888298 0.0444149 0.999013i $$-0.485858\pi$$
0.0444149 + 0.999013i $$0.485858\pi$$
$$930$$ 0 0
$$931$$ 1.56831 0.0513993
$$932$$ 0 0
$$933$$ −13.1924 −0.431901
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 36.0067 1.17629 0.588143 0.808757i $$-0.299859\pi$$
0.588143 + 0.808757i $$0.299859\pi$$
$$938$$ 0 0
$$939$$ 19.2215 0.627269
$$940$$ 0 0
$$941$$ 11.6135 0.378591 0.189295 0.981920i $$-0.439380\pi$$
0.189295 + 0.981920i $$0.439380\pi$$
$$942$$ 0 0
$$943$$ −4.75004 −0.154683
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 20.9061 0.679355 0.339678 0.940542i $$-0.389682\pi$$
0.339678 + 0.940542i $$0.389682\pi$$
$$948$$ 0 0
$$949$$ 28.7586 0.933544
$$950$$ 0 0
$$951$$ 37.6063 1.21947
$$952$$ 0 0
$$953$$ 13.7510 0.445438 0.222719 0.974883i $$-0.428507\pi$$
0.222719 + 0.974883i $$0.428507\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −22.9341 −0.741355
$$958$$ 0 0
$$959$$ −6.39742 −0.206584
$$960$$ 0 0
$$961$$ 82.7663 2.66988
$$962$$ 0 0
$$963$$ 24.6547 0.794488
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −7.07935 −0.227656 −0.113828 0.993500i $$-0.536311\pi$$
−0.113828 + 0.993500i $$0.536311\pi$$
$$968$$ 0 0
$$969$$ −50.5494 −1.62388
$$970$$ 0 0
$$971$$ −18.7382 −0.601338 −0.300669 0.953729i $$-0.597210\pi$$
−0.300669 + 0.953729i $$0.597210\pi$$
$$972$$ 0 0
$$973$$ −14.3534 −0.460148
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −60.6756 −1.94118 −0.970592 0.240729i $$-0.922613\pi$$
−0.970592 + 0.240729i $$0.922613\pi$$
$$978$$ 0 0
$$979$$ 18.1918 0.581412
$$980$$ 0 0
$$981$$ 103.428 3.30221
$$982$$ 0 0
$$983$$ −47.3720 −1.51093 −0.755466 0.655188i $$-0.772590\pi$$
−0.755466 + 0.655188i $$0.772590\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −32.8768 −1.04648
$$988$$ 0 0
$$989$$ −22.8323 −0.726026
$$990$$ 0 0
$$991$$ −19.2110 −0.610256 −0.305128 0.952311i $$-0.598699\pi$$
−0.305128 + 0.952311i $$0.598699\pi$$
$$992$$ 0 0
$$993$$ 26.3585 0.836461
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 59.1077 1.87196 0.935980 0.352053i $$-0.114516\pi$$
0.935980 + 0.352053i $$0.114516\pi$$
$$998$$ 0 0
$$999$$ 49.8167 1.57613
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 4100.2.a.c.1.1 4
5.2 odd 4 4100.2.d.c.1149.7 8
5.3 odd 4 4100.2.d.c.1149.2 8
5.4 even 2 164.2.a.a.1.4 4
15.14 odd 2 1476.2.a.g.1.4 4
20.19 odd 2 656.2.a.i.1.1 4
35.34 odd 2 8036.2.a.i.1.1 4
40.19 odd 2 2624.2.a.y.1.4 4
40.29 even 2 2624.2.a.v.1.1 4
60.59 even 2 5904.2.a.bp.1.4 4
205.204 even 2 6724.2.a.c.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.4 4 5.4 even 2
656.2.a.i.1.1 4 20.19 odd 2
1476.2.a.g.1.4 4 15.14 odd 2
2624.2.a.v.1.1 4 40.29 even 2
2624.2.a.y.1.4 4 40.19 odd 2
4100.2.a.c.1.1 4 1.1 even 1 trivial
4100.2.d.c.1149.2 8 5.3 odd 4
4100.2.d.c.1149.7 8 5.2 odd 4
5904.2.a.bp.1.4 4 60.59 even 2
6724.2.a.c.1.1 4 205.204 even 2
8036.2.a.i.1.1 4 35.34 odd 2