# Properties

 Label 1476.2.a.g.1.4 Level $1476$ Weight $2$ Character 1476.1 Self dual yes Analytic conductor $11.786$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1476 = 2^{2} \cdot 3^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1476.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.7859193383$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.25808.1 Defining polynomial: $$x^{4} - 10 x^{2} - 6 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 1.4 Root $$-1.55466$$ of defining polynomial Character $$\chi$$ $$=$$ 1476.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.33307 q^{5} +2.77840 q^{7} +O(q^{10})$$ $$q+3.33307 q^{5} +2.77840 q^{7} -2.02837 q^{11} -3.10932 q^{13} +7.91609 q^{17} +2.17964 q^{19} +4.75004 q^{23} +6.10932 q^{25} -3.85936 q^{29} -10.6661 q^{31} +9.26060 q^{35} +6.58303 q^{37} +1.00000 q^{41} +4.80677 q^{43} -4.03900 q^{47} +0.719526 q^{49} +1.94327 q^{53} -6.76068 q^{55} -4.75004 q^{59} +2.89068 q^{61} -10.3636 q^{65} +5.97163 q^{67} -14.4026 q^{71} -9.24916 q^{73} -5.63562 q^{77} -0.320282 q^{79} +3.69745 q^{83} +26.3849 q^{85} -8.96868 q^{89} -8.63895 q^{91} +7.26489 q^{95} +17.8322 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + O(q^{10})$$ $$4q - 4q^{5} - 4q^{11} + 4q^{17} + 6q^{19} + 12q^{23} + 12q^{25} + 4q^{29} - 8q^{31} + 26q^{35} + 16q^{37} + 4q^{41} + 4q^{43} + 6q^{47} + 16q^{49} + 16q^{53} - 2q^{55} - 12q^{59} + 24q^{61} - 4q^{65} + 28q^{67} + 2q^{71} + 8q^{73} - 8q^{77} - 18q^{79} + 12q^{83} + 32q^{85} - 4q^{89} + 36q^{91} - 14q^{95} + 16q^{97} + O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.33307 1.49059 0.745296 0.666734i $$-0.232308\pi$$
0.745296 + 0.666734i $$0.232308\pi$$
$$6$$ 0 0
$$7$$ 2.77840 1.05014 0.525069 0.851060i $$-0.324040\pi$$
0.525069 + 0.851060i $$0.324040\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.02837 −0.611575 −0.305788 0.952100i $$-0.598920\pi$$
−0.305788 + 0.952100i $$0.598920\pi$$
$$12$$ 0 0
$$13$$ −3.10932 −0.862371 −0.431186 0.902263i $$-0.641905\pi$$
−0.431186 + 0.902263i $$0.641905\pi$$
$$14$$ 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
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.75004 0.990451 0.495226 0.868764i $$-0.335085\pi$$
0.495226 + 0.868764i $$0.335085\pi$$
$$24$$ 0 0
$$25$$ 6.10932 1.22186
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3.85936 −0.716665 −0.358333 0.933594i $$-0.616655\pi$$
−0.358333 + 0.933594i $$0.616655\pi$$
$$30$$ 0 0
$$31$$ −10.6661 −1.91569 −0.957847 0.287280i $$-0.907249\pi$$
−0.957847 + 0.287280i $$0.907249\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 9.26060 1.56533
$$36$$ 0 0
$$37$$ 6.58303 1.08224 0.541122 0.840944i $$-0.318000\pi$$
0.541122 + 0.840944i $$0.318000\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.00000 0.156174
$$42$$ 0 0
$$43$$ 4.80677 0.733025 0.366513 0.930413i $$-0.380552\pi$$
0.366513 + 0.930413i $$0.380552\pi$$
$$44$$ 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
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.94327 0.266928 0.133464 0.991054i $$-0.457390\pi$$
0.133464 + 0.991054i $$0.457390\pi$$
$$54$$ 0 0
$$55$$ −6.76068 −0.911609
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.75004 −0.618402 −0.309201 0.950997i $$-0.600062\pi$$
−0.309201 + 0.950997i $$0.600062\pi$$
$$60$$ 0 0
$$61$$ 2.89068 0.370113 0.185057 0.982728i $$-0.440753\pi$$
0.185057 + 0.982728i $$0.440753\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −10.3636 −1.28544
$$66$$ 0 0
$$67$$ 5.97163 0.729551 0.364776 0.931095i $$-0.381146\pi$$
0.364776 + 0.931095i $$0.381146\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.4026 −1.70927 −0.854636 0.519228i $$-0.826220\pi$$
−0.854636 + 0.519228i $$0.826220\pi$$
$$72$$ 0 0
$$73$$ −9.24916 −1.08253 −0.541266 0.840851i $$-0.682055\pi$$
−0.541266 + 0.840851i $$0.682055\pi$$
$$74$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.69745 0.405847 0.202924 0.979195i $$-0.434956\pi$$
0.202924 + 0.979195i $$0.434956\pi$$
$$84$$ 0 0
$$85$$ 26.3849 2.86184
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.96868 −0.950679 −0.475339 0.879803i $$-0.657675\pi$$
−0.475339 + 0.879803i $$0.657675\pi$$
$$90$$ 0 0
$$91$$ −8.63895 −0.905608
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 7.26489 0.745362
$$96$$ 0 0
$$97$$ 17.8322 1.81058 0.905292 0.424790i $$-0.139652\pi$$
0.905292 + 0.424790i $$0.139652\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.0254 1.29608 0.648039 0.761607i $$-0.275590\pi$$
0.648039 + 0.761607i $$0.275590\pi$$
$$102$$ 0 0
$$103$$ −8.05673 −0.793853 −0.396927 0.917850i $$-0.629923\pi$$
−0.396927 + 0.917850i $$0.629923\pi$$
$$104$$ 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
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 8.08310 0.760394 0.380197 0.924905i $$-0.375856\pi$$
0.380197 + 0.924905i $$0.375856\pi$$
$$114$$ 0 0
$$115$$ 15.8322 1.47636
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 21.9941 2.01620
$$120$$ 0 0
$$121$$ −6.88573 −0.625976
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3.69745 0.330710
$$126$$ 0 0
$$127$$ 4.44748 0.394650 0.197325 0.980338i $$-0.436775\pi$$
0.197325 + 0.980338i $$0.436775\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −9.85936 −0.861416 −0.430708 0.902491i $$-0.641736\pi$$
−0.430708 + 0.902491i $$0.641736\pi$$
$$132$$ 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
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 6.30684 0.527405
$$144$$ 0 0
$$145$$ −12.8635 −1.06826
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −3.85936 −0.316171 −0.158086 0.987425i $$-0.550532\pi$$
−0.158086 + 0.987425i $$0.550532\pi$$
$$150$$ 0 0
$$151$$ −4.71103 −0.383379 −0.191689 0.981456i $$-0.561397\pi$$
−0.191689 + 0.981456i $$0.561397\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −35.5509 −2.85552
$$156$$ 0 0
$$157$$ 8.30684 0.662958 0.331479 0.943463i $$-0.392452\pi$$
0.331479 + 0.943463i $$0.392452\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 13.1975 1.04011
$$162$$ 0 0
$$163$$ −13.6348 −1.06796 −0.533981 0.845497i $$-0.679304\pi$$
−0.533981 + 0.845497i $$0.679304\pi$$
$$164$$ 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
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −9.11361 −0.692895 −0.346448 0.938069i $$-0.612612\pi$$
−0.346448 + 0.938069i $$0.612612\pi$$
$$174$$ 0 0
$$175$$ 16.9742 1.28313
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.9403 0.967205 0.483602 0.875288i $$-0.339328\pi$$
0.483602 + 0.875288i $$0.339328\pi$$
$$180$$ 0 0
$$181$$ 8.16191 0.606670 0.303335 0.952884i $$-0.401900\pi$$
0.303335 + 0.952884i $$0.401900\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 21.9417 1.61318
$$186$$ 0 0
$$187$$ −16.0567 −1.17418
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.52829 −0.327656 −0.163828 0.986489i $$-0.552384\pi$$
−0.163828 + 0.986489i $$0.552384\pi$$
$$192$$ 0 0
$$193$$ −14.1662 −1.01971 −0.509853 0.860262i $$-0.670300\pi$$
−0.509853 + 0.860262i $$0.670300\pi$$
$$194$$ 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
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −10.7229 −0.752597
$$204$$ 0 0
$$205$$ 3.33307 0.232791
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 16.0213 1.09264
$$216$$ 0 0
$$217$$ −29.6348 −2.01174
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −24.6137 −1.65570
$$222$$ 0 0
$$223$$ −1.89482 −0.126886 −0.0634432 0.997985i $$-0.520208\pi$$
−0.0634432 + 0.997985i $$0.520208\pi$$
$$224$$ 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
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.24835 −0.605880 −0.302940 0.953010i $$-0.597968\pi$$
−0.302940 + 0.953010i $$0.597968\pi$$
$$234$$ 0 0
$$235$$ −13.4623 −0.878181
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2.34746 −0.151844 −0.0759222 0.997114i $$-0.524190\pi$$
−0.0759222 + 0.997114i $$0.524190\pi$$
$$240$$ 0 0
$$241$$ 2.89068 0.186205 0.0931024 0.995657i $$-0.470322\pi$$
0.0931024 + 0.995657i $$0.470322\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.39823 0.153217
$$246$$ 0 0
$$247$$ −6.77721 −0.431224
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −24.9730 −1.57628 −0.788140 0.615496i $$-0.788956\pi$$
−0.788140 + 0.615496i $$0.788956\pi$$
$$252$$ 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
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4.90250 0.302301 0.151151 0.988511i $$-0.451702\pi$$
0.151151 + 0.988511i $$0.451702\pi$$
$$264$$ 0 0
$$265$$ 6.47704 0.397881
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.32797 −0.202910 −0.101455 0.994840i $$-0.532350\pi$$
−0.101455 + 0.994840i $$0.532350\pi$$
$$270$$ 0 0
$$271$$ 4.58222 0.278350 0.139175 0.990268i $$-0.455555\pi$$
0.139175 + 0.990268i $$0.455555\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.3919 −0.747262
$$276$$ 0 0
$$277$$ −24.2178 −1.45511 −0.727555 0.686050i $$-0.759343\pi$$
−0.727555 + 0.686050i $$0.759343\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.61369 −0.394540 −0.197270 0.980349i $$-0.563208\pi$$
−0.197270 + 0.980349i $$0.563208\pi$$
$$282$$ 0 0
$$283$$ 5.16605 0.307090 0.153545 0.988142i $$-0.450931\pi$$
0.153545 + 0.988142i $$0.450931\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.77840 0.164004
$$288$$ 0 0
$$289$$ 45.6645 2.68615
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 10.2458 0.598567 0.299284 0.954164i $$-0.403252\pi$$
0.299284 + 0.954164i $$0.403252\pi$$
$$294$$ 0 0
$$295$$ −15.8322 −0.921785
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −14.7694 −0.854137
$$300$$ 0 0
$$301$$ 13.3551 0.769778
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 9.63481 0.551688
$$306$$ 0 0
$$307$$ −18.2754 −1.04303 −0.521515 0.853242i $$-0.674633\pi$$
−0.521515 + 0.853242i $$0.674633\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.50303 −0.255343 −0.127672 0.991816i $$-0.540750\pi$$
−0.127672 + 0.991816i $$0.540750\pi$$
$$312$$ 0 0
$$313$$ 6.56095 0.370847 0.185423 0.982659i $$-0.440634\pi$$
0.185423 + 0.982659i $$0.440634\pi$$
$$314$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 17.2543 0.960052
$$324$$ 0 0
$$325$$ −18.9959 −1.05370
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 19.9038 1.08746
$$336$$ 0 0
$$337$$ −35.0502 −1.90930 −0.954652 0.297723i $$-0.903773\pi$$
−0.954652 + 0.297723i $$0.903773\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 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
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.55171 0.0825893 0.0412946 0.999147i $$-0.486852\pi$$
0.0412946 + 0.999147i $$0.486852\pi$$
$$354$$ 0 0
$$355$$ −48.0047 −2.54783
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.28128 0.225957 0.112979 0.993597i $$-0.463961\pi$$
0.112979 + 0.993597i $$0.463961\pi$$
$$360$$ 0 0
$$361$$ −14.2492 −0.749956
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −30.8280 −1.61361
$$366$$ 0 0
$$367$$ −15.3067 −0.799003 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 5.39918 0.280312
$$372$$ 0 0
$$373$$ −5.39489 −0.279337 −0.139668 0.990198i $$-0.544604\pi$$
−0.139668 + 0.990198i $$0.544604\pi$$
$$374$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −21.6675 −1.10716 −0.553578 0.832797i $$-0.686738\pi$$
−0.553578 + 0.832797i $$0.686738\pi$$
$$384$$ 0 0
$$385$$ −18.7839 −0.957315
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −25.7187 −1.30399 −0.651995 0.758223i $$-0.726068\pi$$
−0.651995 + 0.758223i $$0.726068\pi$$
$$390$$ 0 0
$$391$$ 37.6017 1.90160
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −1.06752 −0.0537128
$$396$$ 0 0
$$397$$ 28.1603 1.41333 0.706663 0.707551i $$-0.250200\pi$$
0.706663 + 0.707551i $$0.250200\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.7178 0.535220 0.267610 0.963527i $$-0.413766\pi$$
0.267610 + 0.963527i $$0.413766\pi$$
$$402$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −13.1975 −0.649408
$$414$$ 0 0
$$415$$ 12.3238 0.604953
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7.02542 0.343214 0.171607 0.985165i $$-0.445104\pi$$
0.171607 + 0.985165i $$0.445104\pi$$
$$420$$ 0 0
$$421$$ 16.5992 0.808996 0.404498 0.914539i $$-0.367446\pi$$
0.404498 + 0.914539i $$0.367446\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 48.3620 2.34590
$$426$$ 0 0
$$427$$ 8.03147 0.388670
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 10.7229 0.516502 0.258251 0.966078i $$-0.416854\pi$$
0.258251 + 0.966078i $$0.416854\pi$$
$$432$$ 0 0
$$433$$ 31.1644 1.49767 0.748834 0.662758i $$-0.230614\pi$$
0.748834 + 0.662758i $$0.230614\pi$$
$$434$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 25.8806 1.22963 0.614813 0.788673i $$-0.289231\pi$$
0.614813 + 0.788673i $$0.289231\pi$$
$$444$$ 0 0
$$445$$ −29.8932 −1.41707
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −27.2231 −1.28474 −0.642368 0.766396i $$-0.722048\pi$$
−0.642368 + 0.766396i $$0.722048\pi$$
$$450$$ 0 0
$$451$$ −2.02837 −0.0955120
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −28.7942 −1.34989
$$456$$ 0 0
$$457$$ −6.19767 −0.289915 −0.144957 0.989438i $$-0.546305\pi$$
−0.144957 + 0.989438i $$0.546305\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.9170 0.508458 0.254229 0.967144i $$-0.418178\pi$$
0.254229 + 0.967144i $$0.418178\pi$$
$$462$$ 0 0
$$463$$ 4.07047 0.189171 0.0945854 0.995517i $$-0.469847\pi$$
0.0945854 + 0.995517i $$0.469847\pi$$
$$464$$ 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
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −9.74989 −0.448300
$$474$$ 0 0
$$475$$ 13.3161 0.610986
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 23.5537 1.07620 0.538098 0.842882i $$-0.319143\pi$$
0.538098 + 0.842882i $$0.319143\pi$$
$$480$$ 0 0
$$481$$ −20.4688 −0.933295
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 59.4358 2.69884
$$486$$ 0 0
$$487$$ 19.9102 0.902217 0.451108 0.892469i $$-0.351029\pi$$
0.451108 + 0.892469i $$0.351029\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −29.7585 −1.34298 −0.671490 0.741013i $$-0.734346\pi$$
−0.671490 + 0.741013i $$0.734346\pi$$
$$492$$ 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
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 1.93382 0.0862248 0.0431124 0.999070i $$-0.486273\pi$$
0.0431124 + 0.999070i $$0.486273\pi$$
$$504$$ 0 0
$$505$$ 43.4146 1.93192
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −34.6602 −1.53629 −0.768144 0.640277i $$-0.778819\pi$$
−0.768144 + 0.640277i $$0.778819\pi$$
$$510$$ 0 0
$$511$$ −25.6979 −1.13681
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −26.8536 −1.18331
$$516$$ 0 0
$$517$$ 8.19258 0.360309
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.33402 0.277498 0.138749 0.990328i $$-0.455692\pi$$
0.138749 + 0.990328i $$0.455692\pi$$
$$522$$ 0 0
$$523$$ −29.9460 −1.30944 −0.654722 0.755869i $$-0.727215\pi$$
−0.654722 + 0.755869i $$0.727215\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −84.4341 −3.67801
$$528$$ 0 0
$$529$$ −0.437142 −0.0190062
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.10932 −0.134680
$$534$$ 0 0
$$535$$ 14.7189 0.636352
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 61.7467 2.64494
$$546$$ 0 0
$$547$$ −20.6527 −0.883045 −0.441523 0.897250i $$-0.645562\pi$$
−0.441523 + 0.897250i $$0.645562\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 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
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −9.80144 −0.413081 −0.206541 0.978438i $$-0.566221\pi$$
−0.206541 + 0.978438i $$0.566221\pi$$
$$564$$ 0 0
$$565$$ 26.9415 1.13344
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −25.1570 −1.05464 −0.527318 0.849668i $$-0.676802\pi$$
−0.527318 + 0.849668i $$0.676802\pi$$
$$570$$ 0 0
$$571$$ 27.1928 1.13798 0.568992 0.822343i $$-0.307334\pi$$
0.568992 + 0.822343i $$0.307334\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 29.0195 1.21020
$$576$$ 0 0
$$577$$ −0.365186 −0.0152029 −0.00760144 0.999971i $$-0.502420\pi$$
−0.00760144 + 0.999971i $$0.502420\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 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
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 45.1333 1.85340 0.926701 0.375800i $$-0.122632\pi$$
0.926701 + 0.375800i $$0.122632\pi$$
$$594$$ 0 0
$$595$$ 73.3078 3.00533
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −15.4982 −0.633238 −0.316619 0.948553i $$-0.602548\pi$$
−0.316619 + 0.948553i $$0.602548\pi$$
$$600$$ 0 0
$$601$$ −5.18924 −0.211674 −0.105837 0.994384i $$-0.533752\pi$$
−0.105837 + 0.994384i $$0.533752\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −22.9506 −0.933074
$$606$$ 0 0
$$607$$ −16.8280 −0.683029 −0.341515 0.939876i $$-0.610940\pi$$
−0.341515 + 0.939876i $$0.610940\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.5586 0.508065
$$612$$ 0 0
$$613$$ 25.1042 1.01395 0.506975 0.861961i $$-0.330764\pi$$
0.506975 + 0.861961i $$0.330764\pi$$
$$614$$ 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
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −24.9186 −0.998344
$$624$$ 0 0
$$625$$ −18.2228 −0.728911
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 14.8238 0.588263
$$636$$ 0 0
$$637$$ −2.23724 −0.0886427
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 25.9984 1.02687 0.513437 0.858127i $$-0.328372\pi$$
0.513437 + 0.858127i $$0.328372\pi$$
$$642$$ 0 0
$$643$$ 11.4402 0.451159 0.225580 0.974225i $$-0.427572\pi$$
0.225580 + 0.974225i $$0.427572\pi$$
$$644$$ 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
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 43.6814 1.70938 0.854692 0.519136i $$-0.173746\pi$$
0.854692 + 0.519136i $$0.173746\pi$$
$$654$$ 0 0
$$655$$ −32.8619 −1.28402
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −25.0900 −0.977367 −0.488684 0.872461i $$-0.662523\pi$$
−0.488684 + 0.872461i $$0.662523\pi$$
$$660$$ 0 0
$$661$$ −37.7392 −1.46788 −0.733942 0.679212i $$-0.762322\pi$$
−0.733942 + 0.679212i $$0.762322\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 20.1848 0.782733
$$666$$ 0 0
$$667$$ −18.3321 −0.709822
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −5.86335 −0.226352
$$672$$ 0 0
$$673$$ 3.63642 0.140174 0.0700869 0.997541i $$-0.477672\pi$$
0.0700869 + 0.997541i $$0.477672\pi$$
$$674$$ 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
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 28.6527 1.09636 0.548182 0.836359i $$-0.315320\pi$$
0.548182 + 0.836359i $$0.315320\pi$$
$$684$$ 0 0
$$685$$ 7.67456 0.293230
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17.2188 0.653146
$$696$$ 0 0
$$697$$ 7.91609 0.299843
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.79339 −0.105505 −0.0527525 0.998608i $$-0.516799\pi$$
−0.0527525 + 0.998608i $$0.516799\pi$$
$$702$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −50.6645 −1.89740
$$714$$ 0 0
$$715$$ 21.0211 0.786145
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −34.9848 −1.30471 −0.652356 0.757912i $$-0.726219\pi$$
−0.652356 + 0.757912i $$0.726219\pi$$
$$720$$ 0 0
$$721$$ −22.3849 −0.833655
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −23.5781 −0.875668
$$726$$ 0 0
$$727$$ −24.8796 −0.922734 −0.461367 0.887209i $$-0.652641\pi$$
−0.461367 + 0.887209i $$0.652641\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 38.0508 1.40736
$$732$$ 0 0
$$733$$ −28.7737 −1.06278 −0.531390 0.847127i $$-0.678330\pi$$
−0.531390 + 0.847127i $$0.678330\pi$$
$$734$$ 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
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 22.7127 0.833247 0.416624 0.909079i $$-0.363213\pi$$
0.416624 + 0.909079i $$0.363213\pi$$
$$744$$ 0 0
$$745$$ −12.8635 −0.471282
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −15.7022 −0.571461
$$756$$ 0 0
$$757$$ 44.9002 1.63192 0.815962 0.578106i $$-0.196208\pi$$
0.815962 + 0.578106i $$0.196208\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 19.0203 0.689486 0.344743 0.938697i $$-0.387966\pi$$
0.344743 + 0.938697i $$0.387966\pi$$
$$762$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −10.7753 −0.387561 −0.193780 0.981045i $$-0.562075\pi$$
−0.193780 + 0.981045i $$0.562075\pi$$
$$774$$ 0 0
$$775$$ −65.1628 −2.34072
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.17964 0.0780938
$$780$$ 0 0
$$781$$ 29.2137 1.04535
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 27.6873 0.988201
$$786$$ 0 0
$$787$$ 22.8592 0.814843 0.407421 0.913240i $$-0.366428\pi$$
0.407421 + 0.913240i $$0.366428\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 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
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 18.7607 0.662050
$$804$$ 0 0
$$805$$ 43.9882 1.55038
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −7.52135 −0.264437 −0.132218 0.991221i $$-0.542210\pi$$
−0.132218 + 0.991221i $$0.542210\pi$$
$$810$$ 0 0
$$811$$ 29.4103 1.03273 0.516367 0.856367i $$-0.327284\pi$$
0.516367 + 0.856367i $$0.327284\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −45.4457 −1.59189
$$816$$ 0 0
$$817$$ 10.4770 0.366545
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −38.0205 −1.32692 −0.663462 0.748210i $$-0.730913\pi$$
−0.663462 + 0.748210i $$0.730913\pi$$
$$822$$ 0 0
$$823$$ 25.7849 0.898805 0.449403 0.893329i $$-0.351637\pi$$
0.449403 + 0.893329i $$0.351637\pi$$
$$824$$ 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
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 5.69584 0.197349
$$834$$ 0 0
$$835$$ −33.2361 −1.15018
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 13.1105 0.452625 0.226313 0.974055i $$-0.427333\pi$$
0.226313 + 0.974055i $$0.427333\pi$$
$$840$$ 0 0
$$841$$ −14.1053 −0.486391
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −11.1061 −0.382063
$$846$$ 0 0
$$847$$ −19.1313 −0.657361
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 31.2696 1.07191
$$852$$ 0 0
$$853$$ −0.445573 −0.0152561 −0.00762806 0.999971i $$-0.502428\pi$$
−0.00762806 + 0.999971i $$0.502428\pi$$
$$854$$ 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
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 47.9204 1.63123 0.815614 0.578596i $$-0.196399\pi$$
0.815614 + 0.578596i $$0.196399\pi$$
$$864$$ 0 0
$$865$$ −30.3763 −1.03282
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0.649649 0.0220378
$$870$$ 0 0
$$871$$ −18.5677 −0.629144
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 10.2730 0.347291
$$876$$ 0 0
$$877$$ −50.2738 −1.69762 −0.848812 0.528694i $$-0.822682\pi$$
−0.848812 + 0.528694i $$0.822682\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 38.6019 1.30053 0.650265 0.759707i $$-0.274658\pi$$
0.650265 + 0.759707i $$0.274658\pi$$
$$882$$ 0 0
$$883$$ 3.07860 0.103603 0.0518016 0.998657i $$-0.483504\pi$$
0.0518016 + 0.998657i $$0.483504\pi$$
$$884$$ 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
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −8.80358 −0.294601
$$894$$ 0 0
$$895$$ 43.1309 1.44171
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 41.1644 1.37291
$$900$$ 0 0
$$901$$ 15.3831 0.512485
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 27.2042 0.904298
$$906$$ 0 0
$$907$$ 42.5534 1.41296 0.706482 0.707731i $$-0.250281\pi$$
0.706482 + 0.707731i $$0.250281\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 14.8694 0.492645 0.246323 0.969188i $$-0.420778\pi$$
0.246323 + 0.969188i $$0.420778\pi$$
$$912$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 44.7823 1.47403
$$924$$ 0 0
$$925$$ 40.2178 1.32235
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −2.70749 −0.0888298 −0.0444149 0.999013i $$-0.514142\pi$$
−0.0444149 + 0.999013i $$0.514142\pi$$
$$930$$ 0 0
$$931$$ 1.56831 0.0513993
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −53.5181 −1.75023
$$936$$ 0 0
$$937$$ −36.0067 −1.17629 −0.588143 0.808757i $$-0.700141\pi$$
−0.588143 + 0.808757i $$0.700141\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −11.6135 −0.378591 −0.189295 0.981920i $$-0.560620\pi$$
−0.189295 + 0.981920i $$0.560620\pi$$
$$942$$ 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
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 13.7510 0.445438 0.222719 0.974883i $$-0.428507\pi$$
0.222719 + 0.974883i $$0.428507\pi$$
$$954$$ 0 0
$$955$$ −15.0931 −0.488401
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 6.39742 0.206584
$$960$$ 0 0
$$961$$ 82.7663 2.66988
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −47.2169 −1.51997
$$966$$ 0 0
$$967$$ 7.07935 0.227656 0.113828 0.993500i $$-0.463689\pi$$
0.113828 + 0.993500i $$0.463689\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 18.7382 0.601338 0.300669 0.953729i $$-0.402790\pi$$
0.300669 + 0.953729i $$0.402790\pi$$
$$972$$ 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
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −47.3720 −1.51093 −0.755466 0.655188i $$-0.772590\pi$$
−0.755466 + 0.655188i $$0.772590\pi$$
$$984$$ 0 0
$$985$$ 41.6774 1.32795
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −48.6138 −1.54116
$$996$$ 0 0
$$997$$ −59.1077 −1.87196 −0.935980 0.352053i $$-0.885484\pi$$
−0.935980 + 0.352053i $$0.885484\pi$$
$$998$$ 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 1476.2.a.g.1.4 4
3.2 odd 2 164.2.a.a.1.4 4
4.3 odd 2 5904.2.a.bp.1.4 4
12.11 even 2 656.2.a.i.1.1 4
15.2 even 4 4100.2.d.c.1149.2 8
15.8 even 4 4100.2.d.c.1149.7 8
15.14 odd 2 4100.2.a.c.1.1 4
21.20 even 2 8036.2.a.i.1.1 4
24.5 odd 2 2624.2.a.v.1.1 4
24.11 even 2 2624.2.a.y.1.4 4
123.122 odd 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 3.2 odd 2
656.2.a.i.1.1 4 12.11 even 2
1476.2.a.g.1.4 4 1.1 even 1 trivial
2624.2.a.v.1.1 4 24.5 odd 2
2624.2.a.y.1.4 4 24.11 even 2
4100.2.a.c.1.1 4 15.14 odd 2
4100.2.d.c.1149.2 8 15.2 even 4
4100.2.d.c.1149.7 8 15.8 even 4
5904.2.a.bp.1.4 4 4.3 odd 2
6724.2.a.c.1.1 4 123.122 odd 2
8036.2.a.i.1.1 4 21.20 even 2