# Properties

 Label 4608.2.a.ba.1.3 Level $4608$ Weight $2$ Character 4608.1 Self dual yes Analytic conductor $36.795$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6x^{2} - 4x + 2$$ x^4 - 6*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1536) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.74912$$ of defining polynomial Character $$\chi$$ $$=$$ 4608.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.08402 q^{5} +5.03127 q^{7} +O(q^{10})$$ $$q+2.08402 q^{5} +5.03127 q^{7} -0.828427 q^{11} +2.94725 q^{13} -4.82843 q^{17} -2.82843 q^{19} +4.16804 q^{23} -0.656854 q^{25} +7.97852 q^{29} -5.03127 q^{31} +10.4853 q^{35} +7.11529 q^{37} -8.82843 q^{41} +12.4853 q^{43} +4.16804 q^{47} +18.3137 q^{49} -12.1466 q^{53} -1.72646 q^{55} -1.65685 q^{59} +7.11529 q^{61} +6.14214 q^{65} +2.34315 q^{67} +10.0625 q^{71} +4.00000 q^{73} -4.16804 q^{77} +5.03127 q^{79} +3.17157 q^{83} -10.0625 q^{85} -10.0000 q^{89} +14.8284 q^{91} -5.89450 q^{95} +0.343146 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{11} - 8 q^{17} + 20 q^{25} + 8 q^{35} - 24 q^{41} + 16 q^{43} + 28 q^{49} + 16 q^{59} - 32 q^{65} + 32 q^{67} + 16 q^{73} + 24 q^{83} - 40 q^{89} + 48 q^{91} + 24 q^{97}+O(q^{100})$$ 4 * q + 8 * q^11 - 8 * q^17 + 20 * q^25 + 8 * q^35 - 24 * q^41 + 16 * q^43 + 28 * q^49 + 16 * q^59 - 32 * q^65 + 32 * q^67 + 16 * q^73 + 24 * q^83 - 40 * q^89 + 48 * q^91 + 24 * 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
$$5$$ 2.08402 0.932003 0.466001 0.884784i $$-0.345694\pi$$
0.466001 + 0.884784i $$0.345694\pi$$
$$6$$ 0 0
$$7$$ 5.03127 1.90164 0.950821 0.309740i $$-0.100242\pi$$
0.950821 + 0.309740i $$0.100242\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −0.828427 −0.249780 −0.124890 0.992171i $$-0.539858\pi$$
−0.124890 + 0.992171i $$0.539858\pi$$
$$12$$ 0 0
$$13$$ 2.94725 0.817420 0.408710 0.912664i $$-0.365979\pi$$
0.408710 + 0.912664i $$0.365979\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.82843 −1.17107 −0.585533 0.810649i $$-0.699115\pi$$
−0.585533 + 0.810649i $$0.699115\pi$$
$$18$$ 0 0
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.16804 0.869097 0.434549 0.900648i $$-0.356908\pi$$
0.434549 + 0.900648i $$0.356908\pi$$
$$24$$ 0 0
$$25$$ −0.656854 −0.131371
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.97852 1.48157 0.740787 0.671740i $$-0.234453\pi$$
0.740787 + 0.671740i $$0.234453\pi$$
$$30$$ 0 0
$$31$$ −5.03127 −0.903643 −0.451822 0.892108i $$-0.649226\pi$$
−0.451822 + 0.892108i $$0.649226\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 10.4853 1.77234
$$36$$ 0 0
$$37$$ 7.11529 1.16975 0.584874 0.811124i $$-0.301144\pi$$
0.584874 + 0.811124i $$0.301144\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.82843 −1.37877 −0.689384 0.724396i $$-0.742119\pi$$
−0.689384 + 0.724396i $$0.742119\pi$$
$$42$$ 0 0
$$43$$ 12.4853 1.90399 0.951994 0.306117i $$-0.0990300\pi$$
0.951994 + 0.306117i $$0.0990300\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.16804 0.607972 0.303986 0.952677i $$-0.401682\pi$$
0.303986 + 0.952677i $$0.401682\pi$$
$$48$$ 0 0
$$49$$ 18.3137 2.61624
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −12.1466 −1.66846 −0.834230 0.551417i $$-0.814087\pi$$
−0.834230 + 0.551417i $$0.814087\pi$$
$$54$$ 0 0
$$55$$ −1.72646 −0.232796
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −1.65685 −0.215704 −0.107852 0.994167i $$-0.534397\pi$$
−0.107852 + 0.994167i $$0.534397\pi$$
$$60$$ 0 0
$$61$$ 7.11529 0.911020 0.455510 0.890231i $$-0.349457\pi$$
0.455510 + 0.890231i $$0.349457\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.14214 0.761838
$$66$$ 0 0
$$67$$ 2.34315 0.286261 0.143130 0.989704i $$-0.454283\pi$$
0.143130 + 0.989704i $$0.454283\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.0625 1.19420 0.597102 0.802165i $$-0.296319\pi$$
0.597102 + 0.802165i $$0.296319\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.16804 −0.474993
$$78$$ 0 0
$$79$$ 5.03127 0.566062 0.283031 0.959111i $$-0.408660\pi$$
0.283031 + 0.959111i $$0.408660\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.17157 0.348125 0.174063 0.984735i $$-0.444310\pi$$
0.174063 + 0.984735i $$0.444310\pi$$
$$84$$ 0 0
$$85$$ −10.0625 −1.09144
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 14.8284 1.55444
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −5.89450 −0.604763
$$96$$ 0 0
$$97$$ 0.343146 0.0348412 0.0174206 0.999848i $$-0.494455\pi$$
0.0174206 + 0.999848i $$0.494455\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.1466 1.20863 0.604314 0.796746i $$-0.293447\pi$$
0.604314 + 0.796746i $$0.293447\pi$$
$$102$$ 0 0
$$103$$ −3.30481 −0.325633 −0.162816 0.986656i $$-0.552058\pi$$
−0.162816 + 0.986656i $$0.552058\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −8.84175 −0.846886 −0.423443 0.905923i $$-0.639179\pi$$
−0.423443 + 0.905923i $$0.639179\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −11.6569 −1.09658 −0.548292 0.836287i $$-0.684722\pi$$
−0.548292 + 0.836287i $$0.684722\pi$$
$$114$$ 0 0
$$115$$ 8.68629 0.810001
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −24.2931 −2.22695
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −11.7890 −1.05444
$$126$$ 0 0
$$127$$ 13.3674 1.18616 0.593081 0.805143i $$-0.297912\pi$$
0.593081 + 0.805143i $$0.297912\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.3137 1.33796 0.668982 0.743278i $$-0.266730\pi$$
0.668982 + 0.743278i $$0.266730\pi$$
$$132$$ 0 0
$$133$$ −14.2306 −1.23395
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −12.1421 −1.03737 −0.518686 0.854965i $$-0.673579\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2.44158 −0.204175
$$144$$ 0 0
$$145$$ 16.6274 1.38083
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.08402 0.170730 0.0853648 0.996350i $$-0.472794\pi$$
0.0853648 + 0.996350i $$0.472794\pi$$
$$150$$ 0 0
$$151$$ −13.3674 −1.08782 −0.543910 0.839143i $$-0.683057\pi$$
−0.543910 + 0.839143i $$0.683057\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −10.4853 −0.842198
$$156$$ 0 0
$$157$$ −15.4514 −1.23315 −0.616577 0.787294i $$-0.711481\pi$$
−0.616577 + 0.787294i $$0.711481\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 20.9706 1.65271
$$162$$ 0 0
$$163$$ 18.1421 1.42100 0.710501 0.703696i $$-0.248468\pi$$
0.710501 + 0.703696i $$0.248468\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −14.2306 −1.10120 −0.550598 0.834771i $$-0.685600\pi$$
−0.550598 + 0.834771i $$0.685600\pi$$
$$168$$ 0 0
$$169$$ −4.31371 −0.331824
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.25206 0.475336 0.237668 0.971346i $$-0.423617\pi$$
0.237668 + 0.971346i $$0.423617\pi$$
$$174$$ 0 0
$$175$$ −3.30481 −0.249820
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −20.9706 −1.56741 −0.783707 0.621131i $$-0.786673\pi$$
−0.783707 + 0.621131i $$0.786673\pi$$
$$180$$ 0 0
$$181$$ 8.84175 0.657202 0.328601 0.944469i $$-0.393423\pi$$
0.328601 + 0.944469i $$0.393423\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 14.8284 1.09021
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 26.0196 1.88271 0.941356 0.337415i $$-0.109553\pi$$
0.941356 + 0.337415i $$0.109553\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −20.4827 −1.45933 −0.729664 0.683806i $$-0.760324\pi$$
−0.729664 + 0.683806i $$0.760324\pi$$
$$198$$ 0 0
$$199$$ 5.03127 0.356657 0.178329 0.983971i $$-0.442931\pi$$
0.178329 + 0.983971i $$0.442931\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 40.1421 2.81743
$$204$$ 0 0
$$205$$ −18.3986 −1.28502
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2.34315 0.162079
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 26.0196 1.77452
$$216$$ 0 0
$$217$$ −25.3137 −1.71841
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −14.2306 −0.957253
$$222$$ 0 0
$$223$$ −13.3674 −0.895145 −0.447572 0.894248i $$-0.647711\pi$$
−0.447572 + 0.894248i $$0.647711\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 16.1421 1.07139 0.535696 0.844411i $$-0.320049\pi$$
0.535696 + 0.844411i $$0.320049\pi$$
$$228$$ 0 0
$$229$$ 5.38883 0.356104 0.178052 0.984021i $$-0.443020\pi$$
0.178052 + 0.984021i $$0.443020\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.31371 −0.0860639 −0.0430320 0.999074i $$-0.513702\pi$$
−0.0430320 + 0.999074i $$0.513702\pi$$
$$234$$ 0 0
$$235$$ 8.68629 0.566631
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −14.2306 −0.920500 −0.460250 0.887789i $$-0.652240\pi$$
−0.460250 + 0.887789i $$0.652240\pi$$
$$240$$ 0 0
$$241$$ 16.9706 1.09317 0.546585 0.837404i $$-0.315928\pi$$
0.546585 + 0.837404i $$0.315928\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 38.1662 2.43835
$$246$$ 0 0
$$247$$ −8.33609 −0.530412
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 15.1716 0.957621 0.478811 0.877918i $$-0.341068\pi$$
0.478811 + 0.877918i $$0.341068\pi$$
$$252$$ 0 0
$$253$$ −3.45292 −0.217083
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 8.34315 0.520431 0.260216 0.965551i $$-0.416206\pi$$
0.260216 + 0.965551i $$0.416206\pi$$
$$258$$ 0 0
$$259$$ 35.7990 2.22444
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −8.33609 −0.514025 −0.257013 0.966408i $$-0.582738\pi$$
−0.257013 + 0.966408i $$0.582738\pi$$
$$264$$ 0 0
$$265$$ −25.3137 −1.55501
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −7.97852 −0.486459 −0.243230 0.969969i $$-0.578207\pi$$
−0.243230 + 0.969969i $$0.578207\pi$$
$$270$$ 0 0
$$271$$ 3.30481 0.200753 0.100377 0.994950i $$-0.467995\pi$$
0.100377 + 0.994950i $$0.467995\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.544156 0.0328138
$$276$$ 0 0
$$277$$ 25.5139 1.53298 0.766492 0.642254i $$-0.222001\pi$$
0.766492 + 0.642254i $$0.222001\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −11.6569 −0.695390 −0.347695 0.937608i $$-0.613035\pi$$
−0.347695 + 0.937608i $$0.613035\pi$$
$$282$$ 0 0
$$283$$ −4.97056 −0.295469 −0.147735 0.989027i $$-0.547198\pi$$
−0.147735 + 0.989027i $$0.547198\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −44.4182 −2.62193
$$288$$ 0 0
$$289$$ 6.31371 0.371395
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −12.1466 −0.709610 −0.354805 0.934940i $$-0.615453\pi$$
−0.354805 + 0.934940i $$0.615453\pi$$
$$294$$ 0 0
$$295$$ −3.45292 −0.201037
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 12.2843 0.710418
$$300$$ 0 0
$$301$$ 62.8169 3.62070
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 14.8284 0.849073
$$306$$ 0 0
$$307$$ 10.3431 0.590315 0.295157 0.955449i $$-0.404628\pi$$
0.295157 + 0.955449i $$0.404628\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −15.3137 −0.865582 −0.432791 0.901494i $$-0.642471\pi$$
−0.432791 + 0.901494i $$0.642471\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 23.9356 1.34436 0.672178 0.740390i $$-0.265359\pi$$
0.672178 + 0.740390i $$0.265359\pi$$
$$318$$ 0 0
$$319$$ −6.60963 −0.370068
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 13.6569 0.759888
$$324$$ 0 0
$$325$$ −1.93591 −0.107385
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 20.9706 1.15614
$$330$$ 0 0
$$331$$ −21.6569 −1.19037 −0.595184 0.803589i $$-0.702921\pi$$
−0.595184 + 0.803589i $$0.702921\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4.88317 0.266796
$$336$$ 0 0
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.16804 0.225712
$$342$$ 0 0
$$343$$ 56.9224 3.07352
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4.82843 −0.259204 −0.129602 0.991566i $$-0.541370\pi$$
−0.129602 + 0.991566i $$0.541370\pi$$
$$348$$ 0 0
$$349$$ 27.2404 1.45814 0.729072 0.684437i $$-0.239952\pi$$
0.729072 + 0.684437i $$0.239952\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 21.3137 1.13441 0.567207 0.823575i $$-0.308024\pi$$
0.567207 + 0.823575i $$0.308024\pi$$
$$354$$ 0 0
$$355$$ 20.9706 1.11300
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 6.60963 0.348843 0.174421 0.984671i $$-0.444195\pi$$
0.174421 + 0.984671i $$0.444195\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 8.33609 0.436331
$$366$$ 0 0
$$367$$ −16.8203 −0.878011 −0.439006 0.898484i $$-0.644669\pi$$
−0.439006 + 0.898484i $$0.644669\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −61.1127 −3.17281
$$372$$ 0 0
$$373$$ −18.9043 −0.978828 −0.489414 0.872052i $$-0.662789\pi$$
−0.489414 + 0.872052i $$0.662789\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 23.5147 1.21107
$$378$$ 0 0
$$379$$ −16.4853 −0.846792 −0.423396 0.905945i $$-0.639162\pi$$
−0.423396 + 0.905945i $$0.639162\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5.89450 0.301195 0.150598 0.988595i $$-0.451880\pi$$
0.150598 + 0.988595i $$0.451880\pi$$
$$384$$ 0 0
$$385$$ −8.68629 −0.442694
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −2.08402 −0.105664 −0.0528320 0.998603i $$-0.516825\pi$$
−0.0528320 + 0.998603i $$0.516825\pi$$
$$390$$ 0 0
$$391$$ −20.1251 −1.01777
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 10.4853 0.527572
$$396$$ 0 0
$$397$$ −15.4514 −0.775483 −0.387741 0.921768i $$-0.626745\pi$$
−0.387741 + 0.921768i $$0.626745\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.14214 0.206848 0.103424 0.994637i $$-0.467020\pi$$
0.103424 + 0.994637i $$0.467020\pi$$
$$402$$ 0 0
$$403$$ −14.8284 −0.738657
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −5.89450 −0.292180
$$408$$ 0 0
$$409$$ 7.65685 0.378607 0.189304 0.981919i $$-0.439377\pi$$
0.189304 + 0.981919i $$0.439377\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.33609 −0.410192
$$414$$ 0 0
$$415$$ 6.60963 0.324454
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.8284 1.01754 0.508768 0.860904i $$-0.330101\pi$$
0.508768 + 0.860904i $$0.330101\pi$$
$$420$$ 0 0
$$421$$ 5.38883 0.262636 0.131318 0.991340i $$-0.458079\pi$$
0.131318 + 0.991340i $$0.458079\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3.17157 0.153844
$$426$$ 0 0
$$427$$ 35.7990 1.73243
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.9570 0.768624 0.384312 0.923203i $$-0.374439\pi$$
0.384312 + 0.923203i $$0.374439\pi$$
$$432$$ 0 0
$$433$$ −20.6274 −0.991290 −0.495645 0.868525i $$-0.665068\pi$$
−0.495645 + 0.868525i $$0.665068\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −11.7890 −0.563945
$$438$$ 0 0
$$439$$ 8.48419 0.404928 0.202464 0.979290i $$-0.435105\pi$$
0.202464 + 0.979290i $$0.435105\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1.51472 −0.0719665 −0.0359832 0.999352i $$-0.511456\pi$$
−0.0359832 + 0.999352i $$0.511456\pi$$
$$444$$ 0 0
$$445$$ −20.8402 −0.987921
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −34.4853 −1.62746 −0.813731 0.581242i $$-0.802567\pi$$
−0.813731 + 0.581242i $$0.802567\pi$$
$$450$$ 0 0
$$451$$ 7.31371 0.344389
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 30.9028 1.44874
$$456$$ 0 0
$$457$$ −12.3431 −0.577388 −0.288694 0.957421i $$-0.593221\pi$$
−0.288694 + 0.957421i $$0.593221\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 29.8301 1.38933 0.694663 0.719336i $$-0.255554\pi$$
0.694663 + 0.719336i $$0.255554\pi$$
$$462$$ 0 0
$$463$$ −23.4299 −1.08888 −0.544440 0.838800i $$-0.683258\pi$$
−0.544440 + 0.838800i $$0.683258\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1.79899 −0.0832473 −0.0416237 0.999133i $$-0.513253\pi$$
−0.0416237 + 0.999133i $$0.513253\pi$$
$$468$$ 0 0
$$469$$ 11.7890 0.544366
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −10.3431 −0.475578
$$474$$ 0 0
$$475$$ 1.85786 0.0852447
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 41.9766 1.91796 0.958981 0.283471i $$-0.0914858\pi$$
0.958981 + 0.283471i $$0.0914858\pi$$
$$480$$ 0 0
$$481$$ 20.9706 0.956175
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0.715123 0.0324721
$$486$$ 0 0
$$487$$ −6.75773 −0.306222 −0.153111 0.988209i $$-0.548929\pi$$
−0.153111 + 0.988209i $$0.548929\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9.65685 −0.435808 −0.217904 0.975970i $$-0.569922\pi$$
−0.217904 + 0.975970i $$0.569922\pi$$
$$492$$ 0 0
$$493$$ −38.5237 −1.73502
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 50.6274 2.27095
$$498$$ 0 0
$$499$$ 41.6569 1.86482 0.932408 0.361406i $$-0.117703\pi$$
0.932408 + 0.361406i $$0.117703\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 18.3986 0.820354 0.410177 0.912006i $$-0.365467\pi$$
0.410177 + 0.912006i $$0.365467\pi$$
$$504$$ 0 0
$$505$$ 25.3137 1.12645
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 16.3146 0.723132 0.361566 0.932346i $$-0.382242\pi$$
0.361566 + 0.932346i $$0.382242\pi$$
$$510$$ 0 0
$$511$$ 20.1251 0.890282
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −6.88730 −0.303491
$$516$$ 0 0
$$517$$ −3.45292 −0.151859
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −20.8284 −0.912510 −0.456255 0.889849i $$-0.650810\pi$$
−0.456255 + 0.889849i $$0.650810\pi$$
$$522$$ 0 0
$$523$$ −18.8284 −0.823310 −0.411655 0.911340i $$-0.635049\pi$$
−0.411655 + 0.911340i $$0.635049\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.2931 1.05823
$$528$$ 0 0
$$529$$ −5.62742 −0.244670
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −26.0196 −1.12703
$$534$$ 0 0
$$535$$ −8.33609 −0.360400
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −15.1716 −0.653486
$$540$$ 0 0
$$541$$ −11.2833 −0.485109 −0.242554 0.970138i $$-0.577985\pi$$
−0.242554 + 0.970138i $$0.577985\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −18.4264 −0.789301
$$546$$ 0 0
$$547$$ −31.7990 −1.35963 −0.679813 0.733385i $$-0.737939\pi$$
−0.679813 + 0.733385i $$0.737939\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −22.5667 −0.961373
$$552$$ 0 0
$$553$$ 25.3137 1.07645
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14.5882 −0.618120 −0.309060 0.951043i $$-0.600014\pi$$
−0.309060 + 0.951043i $$0.600014\pi$$
$$558$$ 0 0
$$559$$ 36.7973 1.55636
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −0.828427 −0.0349140 −0.0174570 0.999848i $$-0.505557\pi$$
−0.0174570 + 0.999848i $$0.505557\pi$$
$$564$$ 0 0
$$565$$ −24.2931 −1.02202
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −23.4558 −0.983320 −0.491660 0.870787i $$-0.663610\pi$$
−0.491660 + 0.870787i $$0.663610\pi$$
$$570$$ 0 0
$$571$$ −23.3137 −0.975648 −0.487824 0.872942i $$-0.662209\pi$$
−0.487824 + 0.872942i $$0.662209\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.73780 −0.114174
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 15.9570 0.662010
$$582$$ 0 0
$$583$$ 10.0625 0.416748
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −4.97056 −0.205157 −0.102579 0.994725i $$-0.532709\pi$$
−0.102579 + 0.994725i $$0.532709\pi$$
$$588$$ 0 0
$$589$$ 14.2306 0.586361
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 2.97056 0.121986 0.0609932 0.998138i $$-0.480573\pi$$
0.0609932 + 0.998138i $$0.480573\pi$$
$$594$$ 0 0
$$595$$ −50.6274 −2.07552
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −0.715123 −0.0292191 −0.0146096 0.999893i $$-0.504651\pi$$
−0.0146096 + 0.999893i $$0.504651\pi$$
$$600$$ 0 0
$$601$$ −4.68629 −0.191158 −0.0955789 0.995422i $$-0.530470\pi$$
−0.0955789 + 0.995422i $$0.530470\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −21.4940 −0.873855
$$606$$ 0 0
$$607$$ 25.1564 1.02107 0.510533 0.859858i $$-0.329448\pi$$
0.510533 + 0.859858i $$0.329448\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.2843 0.496968
$$612$$ 0 0
$$613$$ −10.5682 −0.426846 −0.213423 0.976960i $$-0.568461\pi$$
−0.213423 + 0.976960i $$0.568461\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 43.9411 1.76900 0.884502 0.466537i $$-0.154499\pi$$
0.884502 + 0.466537i $$0.154499\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −50.3127 −2.01574
$$624$$ 0 0
$$625$$ −21.2843 −0.851371
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −34.3557 −1.36985
$$630$$ 0 0
$$631$$ −35.2189 −1.40204 −0.701021 0.713140i $$-0.747272\pi$$
−0.701021 + 0.713140i $$0.747272\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 27.8579 1.10551
$$636$$ 0 0
$$637$$ 53.9751 2.13857
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −32.8284 −1.29664 −0.648322 0.761366i $$-0.724529\pi$$
−0.648322 + 0.761366i $$0.724529\pi$$
$$642$$ 0 0
$$643$$ −3.51472 −0.138607 −0.0693035 0.997596i $$-0.522078\pi$$
−0.0693035 + 0.997596i $$0.522078\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −44.4182 −1.74626 −0.873130 0.487487i $$-0.837914\pi$$
−0.873130 + 0.487487i $$0.837914\pi$$
$$648$$ 0 0
$$649$$ 1.37258 0.0538786
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.5452 −1.19533 −0.597663 0.801747i $$-0.703904\pi$$
−0.597663 + 0.801747i $$0.703904\pi$$
$$654$$ 0 0
$$655$$ 31.9141 1.24699
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9.65685 −0.376178 −0.188089 0.982152i $$-0.560229\pi$$
−0.188089 + 0.982152i $$0.560229\pi$$
$$660$$ 0 0
$$661$$ −27.2404 −1.05953 −0.529764 0.848145i $$-0.677720\pi$$
−0.529764 + 0.848145i $$0.677720\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −29.6569 −1.15004
$$666$$ 0 0
$$667$$ 33.2548 1.28763
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −5.89450 −0.227555
$$672$$ 0 0
$$673$$ 37.3137 1.43834 0.719169 0.694835i $$-0.244523\pi$$
0.719169 + 0.694835i $$0.244523\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 22.2091 0.853566 0.426783 0.904354i $$-0.359647\pi$$
0.426783 + 0.904354i $$0.359647\pi$$
$$678$$ 0 0
$$679$$ 1.72646 0.0662555
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −22.4853 −0.860375 −0.430188 0.902739i $$-0.641553\pi$$
−0.430188 + 0.902739i $$0.641553\pi$$
$$684$$ 0 0
$$685$$ −25.3045 −0.966834
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −35.7990 −1.36383
$$690$$ 0 0
$$691$$ −33.1716 −1.26191 −0.630953 0.775821i $$-0.717336\pi$$
−0.630953 + 0.775821i $$0.717336\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 16.6722 0.632412
$$696$$ 0 0
$$697$$ 42.6274 1.61463
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −4.52560 −0.170930 −0.0854649 0.996341i $$-0.527238\pi$$
−0.0854649 + 0.996341i $$0.527238\pi$$
$$702$$ 0 0
$$703$$ −20.1251 −0.759032
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 61.1127 2.29838
$$708$$ 0 0
$$709$$ −19.6194 −0.736823 −0.368411 0.929663i $$-0.620098\pi$$
−0.368411 + 0.929663i $$0.620098\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −20.9706 −0.785354
$$714$$ 0 0
$$715$$ −5.08831 −0.190292
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −6.60963 −0.246497 −0.123249 0.992376i $$-0.539331\pi$$
−0.123249 + 0.992376i $$0.539331\pi$$
$$720$$ 0 0
$$721$$ −16.6274 −0.619237
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −5.24073 −0.194636
$$726$$ 0 0
$$727$$ −36.9454 −1.37023 −0.685114 0.728436i $$-0.740248\pi$$
−0.685114 + 0.728436i $$0.740248\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −60.2843 −2.22969
$$732$$ 0 0
$$733$$ −43.1974 −1.59553 −0.797767 0.602966i $$-0.793985\pi$$
−0.797767 + 0.602966i $$0.793985\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.94113 −0.0715023
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 17.6835 0.648745 0.324373 0.945929i $$-0.394847\pi$$
0.324373 + 0.945929i $$0.394847\pi$$
$$744$$ 0 0
$$745$$ 4.34315 0.159121
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −20.1251 −0.735355
$$750$$ 0 0
$$751$$ 5.03127 0.183594 0.0917969 0.995778i $$-0.470739\pi$$
0.0917969 + 0.995778i $$0.470739\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −27.8579 −1.01385
$$756$$ 0 0
$$757$$ 17.1778 0.624339 0.312170 0.950026i $$-0.398944\pi$$
0.312170 + 0.950026i $$0.398944\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.7696 1.55040 0.775198 0.631719i $$-0.217650\pi$$
0.775198 + 0.631719i $$0.217650\pi$$
$$762$$ 0 0
$$763$$ −44.4853 −1.61048
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −4.88317 −0.176321
$$768$$ 0 0
$$769$$ 12.0000 0.432731 0.216366 0.976312i $$-0.430580\pi$$
0.216366 + 0.976312i $$0.430580\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 32.2717 1.16073 0.580365 0.814356i $$-0.302910\pi$$
0.580365 + 0.814356i $$0.302910\pi$$
$$774$$ 0 0
$$775$$ 3.30481 0.118712
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.9706 0.894663
$$780$$ 0 0
$$781$$ −8.33609 −0.298289
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −32.2010 −1.14930
$$786$$ 0 0
$$787$$ −52.7696 −1.88103 −0.940516 0.339750i $$-0.889657\pi$$
−0.940516 + 0.339750i $$0.889657\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −58.6488 −2.08531
$$792$$ 0 0
$$793$$ 20.9706 0.744687
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 30.5452 1.08197 0.540983 0.841033i $$-0.318052\pi$$
0.540983 + 0.841033i $$0.318052\pi$$
$$798$$ 0 0
$$799$$ −20.1251 −0.711975
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3.31371 −0.116938
$$804$$ 0 0
$$805$$ 43.7031 1.54033
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 3.45584 0.121501 0.0607505 0.998153i $$-0.480651\pi$$
0.0607505 + 0.998153i $$0.480651\pi$$
$$810$$ 0 0
$$811$$ −49.4558 −1.73663 −0.868315 0.496014i $$-0.834797\pi$$
−0.868315 + 0.496014i $$0.834797\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 37.8086 1.32438
$$816$$ 0 0
$$817$$ −35.3137 −1.23547
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −36.4397 −1.27175 −0.635877 0.771790i $$-0.719362\pi$$
−0.635877 + 0.771790i $$0.719362\pi$$
$$822$$ 0 0
$$823$$ 13.3674 0.465957 0.232978 0.972482i $$-0.425153\pi$$
0.232978 + 0.972482i $$0.425153\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −53.9411 −1.87572 −0.937858 0.347018i $$-0.887194\pi$$
−0.937858 + 0.347018i $$0.887194\pi$$
$$828$$ 0 0
$$829$$ 2.94725 0.102362 0.0511811 0.998689i $$-0.483701\pi$$
0.0511811 + 0.998689i $$0.483701\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −88.4264 −3.06379
$$834$$ 0 0
$$835$$ −29.6569 −1.02632
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 41.9766 1.44919 0.724597 0.689172i $$-0.242026\pi$$
0.724597 + 0.689172i $$0.242026\pi$$
$$840$$ 0 0
$$841$$ 34.6569 1.19506
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −8.98986 −0.309261
$$846$$ 0 0
$$847$$ −51.8911 −1.78300
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 29.6569 1.01662
$$852$$ 0 0
$$853$$ −15.4514 −0.529045 −0.264523 0.964379i $$-0.585214\pi$$
−0.264523 + 0.964379i $$0.585214\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −30.4853 −1.04136 −0.520679 0.853753i $$-0.674321\pi$$
−0.520679 + 0.853753i $$0.674321\pi$$
$$858$$ 0 0
$$859$$ −21.4558 −0.732064 −0.366032 0.930602i $$-0.619284\pi$$
−0.366032 + 0.930602i $$0.619284\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 14.2306 0.484415 0.242207 0.970224i $$-0.422129\pi$$
0.242207 + 0.970224i $$0.422129\pi$$
$$864$$ 0 0
$$865$$ 13.0294 0.443014
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.16804 −0.141391
$$870$$ 0 0
$$871$$ 6.90584 0.233995
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −59.3137 −2.00517
$$876$$ 0 0
$$877$$ 22.3572 0.754950 0.377475 0.926020i $$-0.376792\pi$$
0.377475 + 0.926020i $$0.376792\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 38.9706 1.31295 0.656476 0.754347i $$-0.272046\pi$$
0.656476 + 0.754347i $$0.272046\pi$$
$$882$$ 0 0
$$883$$ −24.7696 −0.833562 −0.416781 0.909007i $$-0.636842\pi$$
−0.416781 + 0.909007i $$0.636842\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −8.33609 −0.279898 −0.139949 0.990159i $$-0.544694\pi$$
−0.139949 + 0.990159i $$0.544694\pi$$
$$888$$ 0 0
$$889$$ 67.2548 2.25565
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −11.7890 −0.394504
$$894$$ 0 0
$$895$$ −43.7031 −1.46083
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −40.1421 −1.33882
$$900$$ 0 0
$$901$$ 58.6488 1.95388
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.4264 0.612514
$$906$$ 0 0
$$907$$ 23.7990 0.790232 0.395116 0.918631i $$-0.370704\pi$$
0.395116 + 0.918631i $$0.370704\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −34.3557 −1.13825 −0.569127 0.822249i $$-0.692719\pi$$
−0.569127 + 0.822249i $$0.692719\pi$$
$$912$$ 0 0
$$913$$ −2.62742 −0.0869548
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 77.0474 2.54433
$$918$$ 0 0
$$919$$ −15.0938 −0.497899 −0.248950 0.968516i $$-0.580085\pi$$
−0.248950 + 0.968516i $$0.580085\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 29.6569 0.976167
$$924$$ 0 0
$$925$$ −4.67371 −0.153671
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 17.7990 0.583966 0.291983 0.956424i $$-0.405685\pi$$
0.291983 + 0.956424i $$0.405685\pi$$
$$930$$ 0 0
$$931$$ −51.7990 −1.69764
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 8.33609 0.272619
$$936$$ 0 0
$$937$$ 4.62742 0.151171 0.0755856 0.997139i $$-0.475917\pi$$
0.0755856 + 0.997139i $$0.475917\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 51.3854 1.67512 0.837558 0.546348i $$-0.183982\pi$$
0.837558 + 0.546348i $$0.183982\pi$$
$$942$$ 0 0
$$943$$ −36.7973 −1.19828
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.97056 0.161522 0.0807608 0.996734i $$-0.474265\pi$$
0.0807608 + 0.996734i $$0.474265\pi$$
$$948$$ 0 0
$$949$$ 11.7890 0.382687
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.79899 0.317420 0.158710 0.987325i $$-0.449266\pi$$
0.158710 + 0.987325i $$0.449266\pi$$
$$954$$ 0 0
$$955$$ 54.2254 1.75469
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −61.0904 −1.97271
$$960$$ 0 0
$$961$$ −5.68629 −0.183429
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −8.33609 −0.268348
$$966$$ 0 0
$$967$$ 18.5467 0.596423 0.298211 0.954500i $$-0.403610\pi$$
0.298211 + 0.954500i $$0.403610\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 35.4558 1.13783 0.568916 0.822396i $$-0.307363\pi$$
0.568916 + 0.822396i $$0.307363\pi$$
$$972$$ 0 0
$$973$$ 40.2502 1.29036
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −30.7696 −0.984405 −0.492203 0.870481i $$-0.663808\pi$$
−0.492203 + 0.870481i $$0.663808\pi$$
$$978$$ 0 0
$$979$$ 8.28427 0.264766
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −3.45292 −0.110131 −0.0550655 0.998483i $$-0.517537\pi$$
−0.0550655 + 0.998483i $$0.517537\pi$$
$$984$$ 0 0
$$985$$ −42.6863 −1.36010
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 52.0392 1.65475
$$990$$ 0 0
$$991$$ 28.6093 0.908804 0.454402 0.890797i $$-0.349853\pi$$
0.454402 + 0.890797i $$0.349853\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 10.4853 0.332406
$$996$$ 0 0
$$997$$ −9.55688 −0.302669 −0.151335 0.988483i $$-0.548357\pi$$
−0.151335 + 0.988483i $$0.548357\pi$$
$$998$$ 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 4608.2.a.ba.1.3 4
3.2 odd 2 1536.2.a.m.1.2 4
4.3 odd 2 4608.2.a.t.1.3 4
8.3 odd 2 inner 4608.2.a.ba.1.2 4
8.5 even 2 4608.2.a.t.1.2 4
12.11 even 2 1536.2.a.n.1.2 yes 4
16.3 odd 4 4608.2.d.p.2305.4 8
16.5 even 4 4608.2.d.p.2305.5 8
16.11 odd 4 4608.2.d.p.2305.6 8
16.13 even 4 4608.2.d.p.2305.3 8
24.5 odd 2 1536.2.a.n.1.3 yes 4
24.11 even 2 1536.2.a.m.1.3 yes 4
48.5 odd 4 1536.2.d.g.769.6 8
48.11 even 4 1536.2.d.g.769.2 8
48.29 odd 4 1536.2.d.g.769.3 8
48.35 even 4 1536.2.d.g.769.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.m.1.2 4 3.2 odd 2
1536.2.a.m.1.3 yes 4 24.11 even 2
1536.2.a.n.1.2 yes 4 12.11 even 2
1536.2.a.n.1.3 yes 4 24.5 odd 2
1536.2.d.g.769.2 8 48.11 even 4
1536.2.d.g.769.3 8 48.29 odd 4
1536.2.d.g.769.6 8 48.5 odd 4
1536.2.d.g.769.7 8 48.35 even 4
4608.2.a.t.1.2 4 8.5 even 2
4608.2.a.t.1.3 4 4.3 odd 2
4608.2.a.ba.1.2 4 8.3 odd 2 inner
4608.2.a.ba.1.3 4 1.1 even 1 trivial
4608.2.d.p.2305.3 8 16.13 even 4
4608.2.d.p.2305.4 8 16.3 odd 4
4608.2.d.p.2305.5 8 16.5 even 4
4608.2.d.p.2305.6 8 16.11 odd 4