# Properties

 Label 3800.2.a.p.1.1 Level $3800$ Weight $2$ Character 3800.1 Self dual yes Analytic conductor $30.343$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.414214 q^{3} +0.414214 q^{7} -2.82843 q^{9} +O(q^{10})$$ $$q-0.414214 q^{3} +0.414214 q^{7} -2.82843 q^{9} -1.41421 q^{11} -3.82843 q^{13} +1.00000 q^{17} -1.00000 q^{19} -0.171573 q^{21} -3.24264 q^{23} +2.41421 q^{27} -1.82843 q^{29} +0.585786 q^{31} +0.585786 q^{33} +10.8284 q^{37} +1.58579 q^{39} +7.07107 q^{41} -6.24264 q^{43} +8.00000 q^{47} -6.82843 q^{49} -0.414214 q^{51} +3.82843 q^{53} +0.414214 q^{57} +11.5858 q^{59} +0.585786 q^{61} -1.17157 q^{63} +8.07107 q^{67} +1.34315 q^{69} -11.0711 q^{71} +8.17157 q^{73} -0.585786 q^{77} +4.82843 q^{79} +7.48528 q^{81} +14.4853 q^{83} +0.757359 q^{87} -12.2426 q^{89} -1.58579 q^{91} -0.242641 q^{93} -3.65685 q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{7} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{7} - 2q^{13} + 2q^{17} - 2q^{19} - 6q^{21} + 2q^{23} + 2q^{27} + 2q^{29} + 4q^{31} + 4q^{33} + 16q^{37} + 6q^{39} - 4q^{43} + 16q^{47} - 8q^{49} + 2q^{51} + 2q^{53} - 2q^{57} + 26q^{59} + 4q^{61} - 8q^{63} + 2q^{67} + 14q^{69} - 8q^{71} + 22q^{73} - 4q^{77} + 4q^{79} - 2q^{81} + 12q^{83} + 10q^{87} - 16q^{89} - 6q^{91} + 8q^{93} + 4q^{97} + 8q^{99} + O(q^{100})$$

## 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
$$3$$ −0.414214 −0.239146 −0.119573 0.992825i $$-0.538153\pi$$
−0.119573 + 0.992825i $$0.538153\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.414214 0.156558 0.0782790 0.996931i $$-0.475058\pi$$
0.0782790 + 0.996931i $$0.475058\pi$$
$$8$$ 0 0
$$9$$ −2.82843 −0.942809
$$10$$ 0 0
$$11$$ −1.41421 −0.426401 −0.213201 0.977008i $$-0.568389\pi$$
−0.213201 + 0.977008i $$0.568389\pi$$
$$12$$ 0 0
$$13$$ −3.82843 −1.06181 −0.530907 0.847430i $$-0.678149\pi$$
−0.530907 + 0.847430i $$0.678149\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −0.171573 −0.0374403
$$22$$ 0 0
$$23$$ −3.24264 −0.676137 −0.338069 0.941121i $$-0.609774\pi$$
−0.338069 + 0.941121i $$0.609774\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2.41421 0.464616
$$28$$ 0 0
$$29$$ −1.82843 −0.339530 −0.169765 0.985485i $$-0.554301\pi$$
−0.169765 + 0.985485i $$0.554301\pi$$
$$30$$ 0 0
$$31$$ 0.585786 0.105210 0.0526052 0.998615i $$-0.483248\pi$$
0.0526052 + 0.998615i $$0.483248\pi$$
$$32$$ 0 0
$$33$$ 0.585786 0.101972
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.8284 1.78018 0.890091 0.455782i $$-0.150640\pi$$
0.890091 + 0.455782i $$0.150640\pi$$
$$38$$ 0 0
$$39$$ 1.58579 0.253929
$$40$$ 0 0
$$41$$ 7.07107 1.10432 0.552158 0.833740i $$-0.313805\pi$$
0.552158 + 0.833740i $$0.313805\pi$$
$$42$$ 0 0
$$43$$ −6.24264 −0.951994 −0.475997 0.879447i $$-0.657913\pi$$
−0.475997 + 0.879447i $$0.657913\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −6.82843 −0.975490
$$50$$ 0 0
$$51$$ −0.414214 −0.0580015
$$52$$ 0 0
$$53$$ 3.82843 0.525875 0.262937 0.964813i $$-0.415309\pi$$
0.262937 + 0.964813i $$0.415309\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.414214 0.0548639
$$58$$ 0 0
$$59$$ 11.5858 1.50834 0.754170 0.656679i $$-0.228039\pi$$
0.754170 + 0.656679i $$0.228039\pi$$
$$60$$ 0 0
$$61$$ 0.585786 0.0750023 0.0375011 0.999297i $$-0.488060\pi$$
0.0375011 + 0.999297i $$0.488060\pi$$
$$62$$ 0 0
$$63$$ −1.17157 −0.147604
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.07107 0.986038 0.493019 0.870019i $$-0.335893\pi$$
0.493019 + 0.870019i $$0.335893\pi$$
$$68$$ 0 0
$$69$$ 1.34315 0.161696
$$70$$ 0 0
$$71$$ −11.0711 −1.31389 −0.656947 0.753937i $$-0.728152\pi$$
−0.656947 + 0.753937i $$0.728152\pi$$
$$72$$ 0 0
$$73$$ 8.17157 0.956410 0.478205 0.878248i $$-0.341288\pi$$
0.478205 + 0.878248i $$0.341288\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.585786 −0.0667566
$$78$$ 0 0
$$79$$ 4.82843 0.543240 0.271620 0.962405i $$-0.412441\pi$$
0.271620 + 0.962405i $$0.412441\pi$$
$$80$$ 0 0
$$81$$ 7.48528 0.831698
$$82$$ 0 0
$$83$$ 14.4853 1.58997 0.794983 0.606632i $$-0.207480\pi$$
0.794983 + 0.606632i $$0.207480\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0.757359 0.0811974
$$88$$ 0 0
$$89$$ −12.2426 −1.29772 −0.648859 0.760909i $$-0.724753\pi$$
−0.648859 + 0.760909i $$0.724753\pi$$
$$90$$ 0 0
$$91$$ −1.58579 −0.166236
$$92$$ 0 0
$$93$$ −0.242641 −0.0251607
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.65685 −0.371297 −0.185649 0.982616i $$-0.559439\pi$$
−0.185649 + 0.982616i $$0.559439\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −3.89949 −0.388014 −0.194007 0.981000i $$-0.562148\pi$$
−0.194007 + 0.981000i $$0.562148\pi$$
$$102$$ 0 0
$$103$$ 9.89949 0.975426 0.487713 0.873004i $$-0.337831\pi$$
0.487713 + 0.873004i $$0.337831\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0.414214 0.0400435 0.0200218 0.999800i $$-0.493626\pi$$
0.0200218 + 0.999800i $$0.493626\pi$$
$$108$$ 0 0
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ −4.48528 −0.425724
$$112$$ 0 0
$$113$$ 1.07107 0.100758 0.0503788 0.998730i $$-0.483957\pi$$
0.0503788 + 0.998730i $$0.483957\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 10.8284 1.00109
$$118$$ 0 0
$$119$$ 0.414214 0.0379709
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 0 0
$$123$$ −2.92893 −0.264093
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.8284 1.49328 0.746641 0.665228i $$-0.231665\pi$$
0.746641 + 0.665228i $$0.231665\pi$$
$$128$$ 0 0
$$129$$ 2.58579 0.227666
$$130$$ 0 0
$$131$$ 10.3431 0.903685 0.451842 0.892098i $$-0.350767\pi$$
0.451842 + 0.892098i $$0.350767\pi$$
$$132$$ 0 0
$$133$$ −0.414214 −0.0359169
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.31371 −0.539417 −0.269708 0.962942i $$-0.586927\pi$$
−0.269708 + 0.962942i $$0.586927\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −3.31371 −0.279065
$$142$$ 0 0
$$143$$ 5.41421 0.452759
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.82843 0.233285
$$148$$ 0 0
$$149$$ −6.34315 −0.519651 −0.259825 0.965656i $$-0.583665\pi$$
−0.259825 + 0.965656i $$0.583665\pi$$
$$150$$ 0 0
$$151$$ 16.8284 1.36948 0.684739 0.728788i $$-0.259916\pi$$
0.684739 + 0.728788i $$0.259916\pi$$
$$152$$ 0 0
$$153$$ −2.82843 −0.228665
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.6569 0.930318 0.465159 0.885227i $$-0.345997\pi$$
0.465159 + 0.885227i $$0.345997\pi$$
$$158$$ 0 0
$$159$$ −1.58579 −0.125761
$$160$$ 0 0
$$161$$ −1.34315 −0.105855
$$162$$ 0 0
$$163$$ 17.0711 1.33711 0.668555 0.743663i $$-0.266913\pi$$
0.668555 + 0.743663i $$0.266913\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19.2132 −1.48676 −0.743381 0.668868i $$-0.766779\pi$$
−0.743381 + 0.668868i $$0.766779\pi$$
$$168$$ 0 0
$$169$$ 1.65685 0.127450
$$170$$ 0 0
$$171$$ 2.82843 0.216295
$$172$$ 0 0
$$173$$ 13.1716 1.00142 0.500708 0.865616i $$-0.333073\pi$$
0.500708 + 0.865616i $$0.333073\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.79899 −0.360714
$$178$$ 0 0
$$179$$ 10.9706 0.819978 0.409989 0.912090i $$-0.365532\pi$$
0.409989 + 0.912090i $$0.365532\pi$$
$$180$$ 0 0
$$181$$ −16.4853 −1.22534 −0.612671 0.790338i $$-0.709905\pi$$
−0.612671 + 0.790338i $$0.709905\pi$$
$$182$$ 0 0
$$183$$ −0.242641 −0.0179365
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.41421 −0.103418
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −2.75736 −0.199516 −0.0997578 0.995012i $$-0.531807\pi$$
−0.0997578 + 0.995012i $$0.531807\pi$$
$$192$$ 0 0
$$193$$ −3.65685 −0.263226 −0.131613 0.991301i $$-0.542016\pi$$
−0.131613 + 0.991301i $$0.542016\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.72792 −0.336850 −0.168425 0.985714i $$-0.553868\pi$$
−0.168425 + 0.985714i $$0.553868\pi$$
$$198$$ 0 0
$$199$$ 1.92893 0.136738 0.0683692 0.997660i $$-0.478220\pi$$
0.0683692 + 0.997660i $$0.478220\pi$$
$$200$$ 0 0
$$201$$ −3.34315 −0.235807
$$202$$ 0 0
$$203$$ −0.757359 −0.0531562
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 9.17157 0.637468
$$208$$ 0 0
$$209$$ 1.41421 0.0978232
$$210$$ 0 0
$$211$$ −5.10051 −0.351133 −0.175567 0.984468i $$-0.556176\pi$$
−0.175567 + 0.984468i $$0.556176\pi$$
$$212$$ 0 0
$$213$$ 4.58579 0.314213
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0.242641 0.0164715
$$218$$ 0 0
$$219$$ −3.38478 −0.228722
$$220$$ 0 0
$$221$$ −3.82843 −0.257528
$$222$$ 0 0
$$223$$ 8.82843 0.591195 0.295598 0.955313i $$-0.404481\pi$$
0.295598 + 0.955313i $$0.404481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −8.75736 −0.581246 −0.290623 0.956838i $$-0.593863\pi$$
−0.290623 + 0.956838i $$0.593863\pi$$
$$228$$ 0 0
$$229$$ 6.97056 0.460628 0.230314 0.973116i $$-0.426025\pi$$
0.230314 + 0.973116i $$0.426025\pi$$
$$230$$ 0 0
$$231$$ 0.242641 0.0159646
$$232$$ 0 0
$$233$$ −13.6569 −0.894690 −0.447345 0.894361i $$-0.647630\pi$$
−0.447345 + 0.894361i $$0.647630\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2.00000 −0.129914
$$238$$ 0 0
$$239$$ 27.7279 1.79357 0.896785 0.442466i $$-0.145896\pi$$
0.896785 + 0.442466i $$0.145896\pi$$
$$240$$ 0 0
$$241$$ −5.65685 −0.364390 −0.182195 0.983262i $$-0.558320\pi$$
−0.182195 + 0.983262i $$0.558320\pi$$
$$242$$ 0 0
$$243$$ −10.3431 −0.663513
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.82843 0.243597
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 3.55635 0.224475 0.112237 0.993681i $$-0.464198\pi$$
0.112237 + 0.993681i $$0.464198\pi$$
$$252$$ 0 0
$$253$$ 4.58579 0.288306
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 28.7279 1.79200 0.895999 0.444056i $$-0.146461\pi$$
0.895999 + 0.444056i $$0.146461\pi$$
$$258$$ 0 0
$$259$$ 4.48528 0.278702
$$260$$ 0 0
$$261$$ 5.17157 0.320112
$$262$$ 0 0
$$263$$ −15.6569 −0.965443 −0.482721 0.875774i $$-0.660352\pi$$
−0.482721 + 0.875774i $$0.660352\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 5.07107 0.310344
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 15.3848 0.934559 0.467279 0.884110i $$-0.345234\pi$$
0.467279 + 0.884110i $$0.345234\pi$$
$$272$$ 0 0
$$273$$ 0.656854 0.0397546
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3.31371 −0.199101 −0.0995507 0.995032i $$-0.531741\pi$$
−0.0995507 + 0.995032i $$0.531741\pi$$
$$278$$ 0 0
$$279$$ −1.65685 −0.0991933
$$280$$ 0 0
$$281$$ −3.75736 −0.224145 −0.112073 0.993700i $$-0.535749\pi$$
−0.112073 + 0.993700i $$0.535749\pi$$
$$282$$ 0 0
$$283$$ 9.51472 0.565591 0.282796 0.959180i $$-0.408738\pi$$
0.282796 + 0.959180i $$0.408738\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.92893 0.172889
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 1.51472 0.0887944
$$292$$ 0 0
$$293$$ 10.7990 0.630884 0.315442 0.948945i $$-0.397847\pi$$
0.315442 + 0.948945i $$0.397847\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.41421 −0.198113
$$298$$ 0 0
$$299$$ 12.4142 0.717933
$$300$$ 0 0
$$301$$ −2.58579 −0.149042
$$302$$ 0 0
$$303$$ 1.61522 0.0927922
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.2843 1.84256 0.921280 0.388899i $$-0.127145\pi$$
0.921280 + 0.388899i $$0.127145\pi$$
$$308$$ 0 0
$$309$$ −4.10051 −0.233270
$$310$$ 0 0
$$311$$ 23.2426 1.31797 0.658985 0.752156i $$-0.270986\pi$$
0.658985 + 0.752156i $$0.270986\pi$$
$$312$$ 0 0
$$313$$ 6.65685 0.376268 0.188134 0.982143i $$-0.439756\pi$$
0.188134 + 0.982143i $$0.439756\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.79899 0.381869 0.190935 0.981603i $$-0.438848\pi$$
0.190935 + 0.981603i $$0.438848\pi$$
$$318$$ 0 0
$$319$$ 2.58579 0.144776
$$320$$ 0 0
$$321$$ −0.171573 −0.00957626
$$322$$ 0 0
$$323$$ −1.00000 −0.0556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.07107 0.114530
$$328$$ 0 0
$$329$$ 3.31371 0.182691
$$330$$ 0 0
$$331$$ 19.3848 1.06548 0.532742 0.846278i $$-0.321162\pi$$
0.532742 + 0.846278i $$0.321162\pi$$
$$332$$ 0 0
$$333$$ −30.6274 −1.67837
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 12.7279 0.693334 0.346667 0.937988i $$-0.387313\pi$$
0.346667 + 0.937988i $$0.387313\pi$$
$$338$$ 0 0
$$339$$ −0.443651 −0.0240958
$$340$$ 0 0
$$341$$ −0.828427 −0.0448618
$$342$$ 0 0
$$343$$ −5.72792 −0.309279
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 15.1716 0.814453 0.407226 0.913327i $$-0.366496\pi$$
0.407226 + 0.913327i $$0.366496\pi$$
$$348$$ 0 0
$$349$$ 28.6274 1.53239 0.766195 0.642608i $$-0.222148\pi$$
0.766195 + 0.642608i $$0.222148\pi$$
$$350$$ 0 0
$$351$$ −9.24264 −0.493336
$$352$$ 0 0
$$353$$ 36.1127 1.92208 0.961042 0.276401i $$-0.0891417\pi$$
0.961042 + 0.276401i $$0.0891417\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −0.171573 −0.00908060
$$358$$ 0 0
$$359$$ −6.07107 −0.320419 −0.160209 0.987083i $$-0.551217\pi$$
−0.160209 + 0.987083i $$0.551217\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 3.72792 0.195665
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −33.4558 −1.74638 −0.873190 0.487379i $$-0.837953\pi$$
−0.873190 + 0.487379i $$0.837953\pi$$
$$368$$ 0 0
$$369$$ −20.0000 −1.04116
$$370$$ 0 0
$$371$$ 1.58579 0.0823299
$$372$$ 0 0
$$373$$ −27.9706 −1.44826 −0.724130 0.689663i $$-0.757759\pi$$
−0.724130 + 0.689663i $$0.757759\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.00000 0.360518
$$378$$ 0 0
$$379$$ −3.38478 −0.173864 −0.0869321 0.996214i $$-0.527706\pi$$
−0.0869321 + 0.996214i $$0.527706\pi$$
$$380$$ 0 0
$$381$$ −6.97056 −0.357113
$$382$$ 0 0
$$383$$ −20.7279 −1.05915 −0.529574 0.848264i $$-0.677648\pi$$
−0.529574 + 0.848264i $$0.677648\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 17.6569 0.897548
$$388$$ 0 0
$$389$$ 7.89949 0.400520 0.200260 0.979743i $$-0.435821\pi$$
0.200260 + 0.979743i $$0.435821\pi$$
$$390$$ 0 0
$$391$$ −3.24264 −0.163987
$$392$$ 0 0
$$393$$ −4.28427 −0.216113
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.6274 1.13564 0.567819 0.823154i $$-0.307787\pi$$
0.567819 + 0.823154i $$0.307787\pi$$
$$398$$ 0 0
$$399$$ 0.171573 0.00858939
$$400$$ 0 0
$$401$$ −6.10051 −0.304645 −0.152322 0.988331i $$-0.548675\pi$$
−0.152322 + 0.988331i $$0.548675\pi$$
$$402$$ 0 0
$$403$$ −2.24264 −0.111714
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −15.3137 −0.759072
$$408$$ 0 0
$$409$$ 2.58579 0.127859 0.0639295 0.997954i $$-0.479637\pi$$
0.0639295 + 0.997954i $$0.479637\pi$$
$$410$$ 0 0
$$411$$ 2.61522 0.128999
$$412$$ 0 0
$$413$$ 4.79899 0.236143
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 1.65685 0.0811365
$$418$$ 0 0
$$419$$ −27.2132 −1.32945 −0.664726 0.747087i $$-0.731452\pi$$
−0.664726 + 0.747087i $$0.731452\pi$$
$$420$$ 0 0
$$421$$ 13.1421 0.640508 0.320254 0.947332i $$-0.396232\pi$$
0.320254 + 0.947332i $$0.396232\pi$$
$$422$$ 0 0
$$423$$ −22.6274 −1.10018
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.242641 0.0117422
$$428$$ 0 0
$$429$$ −2.24264 −0.108276
$$430$$ 0 0
$$431$$ −5.41421 −0.260793 −0.130397 0.991462i $$-0.541625\pi$$
−0.130397 + 0.991462i $$0.541625\pi$$
$$432$$ 0 0
$$433$$ 5.07107 0.243700 0.121850 0.992549i $$-0.461117\pi$$
0.121850 + 0.992549i $$0.461117\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.24264 0.155117
$$438$$ 0 0
$$439$$ −30.7279 −1.46656 −0.733282 0.679925i $$-0.762012\pi$$
−0.733282 + 0.679925i $$0.762012\pi$$
$$440$$ 0 0
$$441$$ 19.3137 0.919700
$$442$$ 0 0
$$443$$ 15.5563 0.739104 0.369552 0.929210i $$-0.379511\pi$$
0.369552 + 0.929210i $$0.379511\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 2.62742 0.124273
$$448$$ 0 0
$$449$$ −35.9411 −1.69617 −0.848083 0.529863i $$-0.822243\pi$$
−0.848083 + 0.529863i $$0.822243\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ 0 0
$$453$$ −6.97056 −0.327506
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.3137 0.763123 0.381562 0.924343i $$-0.375386\pi$$
0.381562 + 0.924343i $$0.375386\pi$$
$$458$$ 0 0
$$459$$ 2.41421 0.112686
$$460$$ 0 0
$$461$$ −7.27208 −0.338694 −0.169347 0.985556i $$-0.554166\pi$$
−0.169347 + 0.985556i $$0.554166\pi$$
$$462$$ 0 0
$$463$$ 22.1421 1.02903 0.514516 0.857481i $$-0.327972\pi$$
0.514516 + 0.857481i $$0.327972\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −21.8995 −1.01339 −0.506694 0.862126i $$-0.669133\pi$$
−0.506694 + 0.862126i $$0.669133\pi$$
$$468$$ 0 0
$$469$$ 3.34315 0.154372
$$470$$ 0 0
$$471$$ −4.82843 −0.222482
$$472$$ 0 0
$$473$$ 8.82843 0.405932
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.8284 −0.495800
$$478$$ 0 0
$$479$$ 33.4558 1.52864 0.764318 0.644839i $$-0.223076\pi$$
0.764318 + 0.644839i $$0.223076\pi$$
$$480$$ 0 0
$$481$$ −41.4558 −1.89022
$$482$$ 0 0
$$483$$ 0.556349 0.0253148
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.17157 0.324975 0.162487 0.986711i $$-0.448048\pi$$
0.162487 + 0.986711i $$0.448048\pi$$
$$488$$ 0 0
$$489$$ −7.07107 −0.319765
$$490$$ 0 0
$$491$$ 17.7574 0.801378 0.400689 0.916214i $$-0.368771\pi$$
0.400689 + 0.916214i $$0.368771\pi$$
$$492$$ 0 0
$$493$$ −1.82843 −0.0823482
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −4.58579 −0.205701
$$498$$ 0 0
$$499$$ 18.9289 0.847375 0.423688 0.905808i $$-0.360735\pi$$
0.423688 + 0.905808i $$0.360735\pi$$
$$500$$ 0 0
$$501$$ 7.95837 0.355554
$$502$$ 0 0
$$503$$ 32.8995 1.46692 0.733458 0.679735i $$-0.237905\pi$$
0.733458 + 0.679735i $$0.237905\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −0.686292 −0.0304793
$$508$$ 0 0
$$509$$ 9.65685 0.428033 0.214016 0.976830i $$-0.431345\pi$$
0.214016 + 0.976830i $$0.431345\pi$$
$$510$$ 0 0
$$511$$ 3.38478 0.149734
$$512$$ 0 0
$$513$$ −2.41421 −0.106590
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −11.3137 −0.497576
$$518$$ 0 0
$$519$$ −5.45584 −0.239485
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ −20.2132 −0.883862 −0.441931 0.897049i $$-0.645706\pi$$
−0.441931 + 0.897049i $$0.645706\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0.585786 0.0255173
$$528$$ 0 0
$$529$$ −12.4853 −0.542838
$$530$$ 0 0
$$531$$ −32.7696 −1.42208
$$532$$ 0 0
$$533$$ −27.0711 −1.17258
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4.54416 −0.196095
$$538$$ 0 0
$$539$$ 9.65685 0.415950
$$540$$ 0 0
$$541$$ 30.0416 1.29159 0.645795 0.763511i $$-0.276526\pi$$
0.645795 + 0.763511i $$0.276526\pi$$
$$542$$ 0 0
$$543$$ 6.82843 0.293036
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −23.9411 −1.02365 −0.511824 0.859090i $$-0.671030\pi$$
−0.511824 + 0.859090i $$0.671030\pi$$
$$548$$ 0 0
$$549$$ −1.65685 −0.0707128
$$550$$ 0 0
$$551$$ 1.82843 0.0778936
$$552$$ 0 0
$$553$$ 2.00000 0.0850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −43.3137 −1.83526 −0.917630 0.397435i $$-0.869900\pi$$
−0.917630 + 0.397435i $$0.869900\pi$$
$$558$$ 0 0
$$559$$ 23.8995 1.01084
$$560$$ 0 0
$$561$$ 0.585786 0.0247319
$$562$$ 0 0
$$563$$ −44.9706 −1.89528 −0.947642 0.319336i $$-0.896540\pi$$
−0.947642 + 0.319336i $$0.896540\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3.10051 0.130209
$$568$$ 0 0
$$569$$ −8.97056 −0.376066 −0.188033 0.982163i $$-0.560211\pi$$
−0.188033 + 0.982163i $$0.560211\pi$$
$$570$$ 0 0
$$571$$ 47.6985 1.99612 0.998060 0.0622637i $$-0.0198320\pi$$
0.998060 + 0.0622637i $$0.0198320\pi$$
$$572$$ 0 0
$$573$$ 1.14214 0.0477134
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −25.0000 −1.04076 −0.520382 0.853934i $$-0.674210\pi$$
−0.520382 + 0.853934i $$0.674210\pi$$
$$578$$ 0 0
$$579$$ 1.51472 0.0629496
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ −5.41421 −0.224234
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −32.3848 −1.33666 −0.668331 0.743864i $$-0.732991\pi$$
−0.668331 + 0.743864i $$0.732991\pi$$
$$588$$ 0 0
$$589$$ −0.585786 −0.0241369
$$590$$ 0 0
$$591$$ 1.95837 0.0805566
$$592$$ 0 0
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −0.798990 −0.0327005
$$598$$ 0 0
$$599$$ −33.5563 −1.37108 −0.685538 0.728037i $$-0.740433\pi$$
−0.685538 + 0.728037i $$0.740433\pi$$
$$600$$ 0 0
$$601$$ 0.443651 0.0180969 0.00904845 0.999959i $$-0.497120\pi$$
0.00904845 + 0.999959i $$0.497120\pi$$
$$602$$ 0 0
$$603$$ −22.8284 −0.929645
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 30.4853 1.23736 0.618680 0.785643i $$-0.287668\pi$$
0.618680 + 0.785643i $$0.287668\pi$$
$$608$$ 0 0
$$609$$ 0.313708 0.0127121
$$610$$ 0 0
$$611$$ −30.6274 −1.23905
$$612$$ 0 0
$$613$$ −2.72792 −0.110180 −0.0550899 0.998481i $$-0.517545\pi$$
−0.0550899 + 0.998481i $$0.517545\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −31.7990 −1.28018 −0.640090 0.768300i $$-0.721103\pi$$
−0.640090 + 0.768300i $$0.721103\pi$$
$$618$$ 0 0
$$619$$ 6.38478 0.256626 0.128313 0.991734i $$-0.459044\pi$$
0.128313 + 0.991734i $$0.459044\pi$$
$$620$$ 0 0
$$621$$ −7.82843 −0.314144
$$622$$ 0 0
$$623$$ −5.07107 −0.203168
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −0.585786 −0.0233941
$$628$$ 0 0
$$629$$ 10.8284 0.431758
$$630$$ 0 0
$$631$$ −5.02944 −0.200219 −0.100109 0.994976i $$-0.531919\pi$$
−0.100109 + 0.994976i $$0.531919\pi$$
$$632$$ 0 0
$$633$$ 2.11270 0.0839722
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 26.1421 1.03579
$$638$$ 0 0
$$639$$ 31.3137 1.23875
$$640$$ 0 0
$$641$$ 3.21320 0.126914 0.0634570 0.997985i $$-0.479787\pi$$
0.0634570 + 0.997985i $$0.479787\pi$$
$$642$$ 0 0
$$643$$ 36.8284 1.45237 0.726186 0.687499i $$-0.241291\pi$$
0.726186 + 0.687499i $$0.241291\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −2.07107 −0.0814221 −0.0407110 0.999171i $$-0.512962\pi$$
−0.0407110 + 0.999171i $$0.512962\pi$$
$$648$$ 0 0
$$649$$ −16.3848 −0.643159
$$650$$ 0 0
$$651$$ −0.100505 −0.00393910
$$652$$ 0 0
$$653$$ 24.2843 0.950317 0.475158 0.879900i $$-0.342391\pi$$
0.475158 + 0.879900i $$0.342391\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −23.1127 −0.901712
$$658$$ 0 0
$$659$$ 10.8995 0.424584 0.212292 0.977206i $$-0.431907\pi$$
0.212292 + 0.977206i $$0.431907\pi$$
$$660$$ 0 0
$$661$$ 26.5147 1.03130 0.515652 0.856798i $$-0.327550\pi$$
0.515652 + 0.856798i $$0.327550\pi$$
$$662$$ 0 0
$$663$$ 1.58579 0.0615868
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.92893 0.229569
$$668$$ 0 0
$$669$$ −3.65685 −0.141382
$$670$$ 0 0
$$671$$ −0.828427 −0.0319811
$$672$$ 0 0
$$673$$ 28.0000 1.07932 0.539660 0.841883i $$-0.318553\pi$$
0.539660 + 0.841883i $$0.318553\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −32.6569 −1.25510 −0.627552 0.778574i $$-0.715943\pi$$
−0.627552 + 0.778574i $$0.715943\pi$$
$$678$$ 0 0
$$679$$ −1.51472 −0.0581296
$$680$$ 0 0
$$681$$ 3.62742 0.139003
$$682$$ 0 0
$$683$$ 0.686292 0.0262602 0.0131301 0.999914i $$-0.495820\pi$$
0.0131301 + 0.999914i $$0.495820\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −2.88730 −0.110157
$$688$$ 0 0
$$689$$ −14.6569 −0.558382
$$690$$ 0 0
$$691$$ −43.4558 −1.65314 −0.826569 0.562835i $$-0.809711\pi$$
−0.826569 + 0.562835i $$0.809711\pi$$
$$692$$ 0 0
$$693$$ 1.65685 0.0629387
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 7.07107 0.267836
$$698$$ 0 0
$$699$$ 5.65685 0.213962
$$700$$ 0 0
$$701$$ −0.970563 −0.0366576 −0.0183288 0.999832i $$-0.505835\pi$$
−0.0183288 + 0.999832i $$0.505835\pi$$
$$702$$ 0 0
$$703$$ −10.8284 −0.408402
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.61522 −0.0607467
$$708$$ 0 0
$$709$$ 33.3137 1.25112 0.625561 0.780175i $$-0.284870\pi$$
0.625561 + 0.780175i $$0.284870\pi$$
$$710$$ 0 0
$$711$$ −13.6569 −0.512172
$$712$$ 0 0
$$713$$ −1.89949 −0.0711366
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −11.4853 −0.428926
$$718$$ 0 0
$$719$$ 27.5858 1.02878 0.514388 0.857557i $$-0.328019\pi$$
0.514388 + 0.857557i $$0.328019\pi$$
$$720$$ 0 0
$$721$$ 4.10051 0.152711
$$722$$ 0 0
$$723$$ 2.34315 0.0871425
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −44.2132 −1.63978 −0.819888 0.572523i $$-0.805965\pi$$
−0.819888 + 0.572523i $$0.805965\pi$$
$$728$$ 0 0
$$729$$ −18.1716 −0.673021
$$730$$ 0 0
$$731$$ −6.24264 −0.230892
$$732$$ 0 0
$$733$$ 34.6274 1.27899 0.639496 0.768794i $$-0.279143\pi$$
0.639496 + 0.768794i $$0.279143\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −11.4142 −0.420448
$$738$$ 0 0
$$739$$ 4.58579 0.168691 0.0843454 0.996437i $$-0.473120\pi$$
0.0843454 + 0.996437i $$0.473120\pi$$
$$740$$ 0 0
$$741$$ −1.58579 −0.0582553
$$742$$ 0 0
$$743$$ 5.75736 0.211217 0.105609 0.994408i $$-0.466321\pi$$
0.105609 + 0.994408i $$0.466321\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −40.9706 −1.49903
$$748$$ 0 0
$$749$$ 0.171573 0.00626914
$$750$$ 0 0
$$751$$ 38.2426 1.39549 0.697747 0.716344i $$-0.254186\pi$$
0.697747 + 0.716344i $$0.254186\pi$$
$$752$$ 0 0
$$753$$ −1.47309 −0.0536823
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −27.6569 −1.00521 −0.502603 0.864517i $$-0.667624\pi$$
−0.502603 + 0.864517i $$0.667624\pi$$
$$758$$ 0 0
$$759$$ −1.89949 −0.0689473
$$760$$ 0 0
$$761$$ −37.9706 −1.37643 −0.688216 0.725506i $$-0.741606\pi$$
−0.688216 + 0.725506i $$0.741606\pi$$
$$762$$ 0 0
$$763$$ −2.07107 −0.0749777
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −44.3553 −1.60158
$$768$$ 0 0
$$769$$ 8.79899 0.317300 0.158650 0.987335i $$-0.449286\pi$$
0.158650 + 0.987335i $$0.449286\pi$$
$$770$$ 0 0
$$771$$ −11.8995 −0.428550
$$772$$ 0 0
$$773$$ 9.00000 0.323708 0.161854 0.986815i $$-0.448253\pi$$
0.161854 + 0.986815i $$0.448253\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1.85786 −0.0666505
$$778$$ 0 0
$$779$$ −7.07107 −0.253347
$$780$$ 0 0
$$781$$ 15.6569 0.560246
$$782$$ 0 0
$$783$$ −4.41421 −0.157751
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −31.2426 −1.11368 −0.556840 0.830620i $$-0.687986\pi$$
−0.556840 + 0.830620i $$0.687986\pi$$
$$788$$ 0 0
$$789$$ 6.48528 0.230882
$$790$$ 0 0
$$791$$ 0.443651 0.0157744
$$792$$ 0 0
$$793$$ −2.24264 −0.0796385
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.85786 0.242918 0.121459 0.992596i $$-0.461243\pi$$
0.121459 + 0.992596i $$0.461243\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 34.6274 1.22350
$$802$$ 0 0
$$803$$ −11.5563 −0.407815
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 4.14214 0.145810
$$808$$ 0 0
$$809$$ −25.4853 −0.896015 −0.448007 0.894030i $$-0.647866\pi$$
−0.448007 + 0.894030i $$0.647866\pi$$
$$810$$ 0 0
$$811$$ 40.0122 1.40502 0.702509 0.711675i $$-0.252063\pi$$
0.702509 + 0.711675i $$0.252063\pi$$
$$812$$ 0 0
$$813$$ −6.37258 −0.223496
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 6.24264 0.218402
$$818$$ 0 0
$$819$$ 4.48528 0.156728
$$820$$ 0 0
$$821$$ 27.5980 0.963176 0.481588 0.876398i $$-0.340060\pi$$
0.481588 + 0.876398i $$0.340060\pi$$
$$822$$ 0 0
$$823$$ 17.5269 0.610950 0.305475 0.952200i $$-0.401185\pi$$
0.305475 + 0.952200i $$0.401185\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 19.8701 0.690950 0.345475 0.938428i $$-0.387718\pi$$
0.345475 + 0.938428i $$0.387718\pi$$
$$828$$ 0 0
$$829$$ −47.7696 −1.65911 −0.829553 0.558429i $$-0.811404\pi$$
−0.829553 + 0.558429i $$0.811404\pi$$
$$830$$ 0 0
$$831$$ 1.37258 0.0476144
$$832$$ 0 0
$$833$$ −6.82843 −0.236591
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.41421 0.0488824
$$838$$ 0 0
$$839$$ 17.9411 0.619396 0.309698 0.950835i $$-0.399772\pi$$
0.309698 + 0.950835i $$0.399772\pi$$
$$840$$ 0 0
$$841$$ −25.6569 −0.884719
$$842$$ 0 0
$$843$$ 1.55635 0.0536035
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.72792 −0.128093
$$848$$ 0 0
$$849$$ −3.94113 −0.135259
$$850$$ 0 0
$$851$$ −35.1127 −1.20365
$$852$$ 0 0
$$853$$ 47.9411 1.64147 0.820736 0.571307i $$-0.193563\pi$$
0.820736 + 0.571307i $$0.193563\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.0000 0.546550 0.273275 0.961936i $$-0.411893\pi$$
0.273275 + 0.961936i $$0.411893\pi$$
$$858$$ 0 0
$$859$$ −37.2132 −1.26970 −0.634849 0.772636i $$-0.718938\pi$$
−0.634849 + 0.772636i $$0.718938\pi$$
$$860$$ 0 0
$$861$$ −1.21320 −0.0413459
$$862$$ 0 0
$$863$$ −20.9289 −0.712429 −0.356215 0.934404i $$-0.615933\pi$$
−0.356215 + 0.934404i $$0.615933\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 6.62742 0.225079
$$868$$ 0 0
$$869$$ −6.82843 −0.231639
$$870$$ 0 0
$$871$$ −30.8995 −1.04699
$$872$$ 0 0
$$873$$ 10.3431 0.350062
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12.5147 −0.422592 −0.211296 0.977422i $$-0.567768\pi$$
−0.211296 + 0.977422i $$0.567768\pi$$
$$878$$ 0 0
$$879$$ −4.47309 −0.150874
$$880$$ 0 0
$$881$$ −44.2843 −1.49198 −0.745988 0.665960i $$-0.768022\pi$$
−0.745988 + 0.665960i $$0.768022\pi$$
$$882$$ 0 0
$$883$$ −35.4558 −1.19318 −0.596592 0.802545i $$-0.703479\pi$$
−0.596592 + 0.802545i $$0.703479\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 15.0711 0.506037 0.253018 0.967461i $$-0.418577\pi$$
0.253018 + 0.967461i $$0.418577\pi$$
$$888$$ 0 0
$$889$$ 6.97056 0.233785
$$890$$ 0 0
$$891$$ −10.5858 −0.354637
$$892$$ 0 0
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.14214 −0.171691
$$898$$ 0 0
$$899$$ −1.07107 −0.0357221
$$900$$ 0 0
$$901$$ 3.82843 0.127543
$$902$$ 0 0
$$903$$ 1.07107 0.0356429
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −22.5563 −0.748971 −0.374486 0.927233i $$-0.622181\pi$$
−0.374486 + 0.927233i $$0.622181\pi$$
$$908$$ 0 0
$$909$$ 11.0294 0.365823
$$910$$ 0 0
$$911$$ 1.65685 0.0548940 0.0274470 0.999623i $$-0.491262\pi$$
0.0274470 + 0.999623i $$0.491262\pi$$
$$912$$ 0 0
$$913$$ −20.4853 −0.677964
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4.28427 0.141479
$$918$$ 0 0
$$919$$ −40.8406 −1.34721 −0.673604 0.739093i $$-0.735255\pi$$
−0.673604 + 0.739093i $$0.735255\pi$$
$$920$$ 0 0
$$921$$ −13.3726 −0.440642
$$922$$ 0 0
$$923$$ 42.3848 1.39511
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −28.0000 −0.919641
$$928$$ 0 0
$$929$$ 8.51472 0.279359 0.139679 0.990197i $$-0.455393\pi$$
0.139679 + 0.990197i $$0.455393\pi$$
$$930$$ 0 0
$$931$$ 6.82843 0.223793
$$932$$ 0 0
$$933$$ −9.62742 −0.315187
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −0.313708 −0.0102484 −0.00512420 0.999987i $$-0.501631\pi$$
−0.00512420 + 0.999987i $$0.501631\pi$$
$$938$$ 0 0
$$939$$ −2.75736 −0.0899830
$$940$$ 0 0
$$941$$ 28.8579 0.940739 0.470370 0.882469i $$-0.344121\pi$$
0.470370 + 0.882469i $$0.344121\pi$$
$$942$$ 0 0
$$943$$ −22.9289 −0.746669
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 49.2548 1.60057 0.800284 0.599622i $$-0.204682\pi$$
0.800284 + 0.599622i $$0.204682\pi$$
$$948$$ 0 0
$$949$$ −31.2843 −1.01553
$$950$$ 0 0
$$951$$ −2.81623 −0.0913226
$$952$$ 0 0
$$953$$ −23.2132 −0.751949 −0.375975 0.926630i $$-0.622692\pi$$
−0.375975 + 0.926630i $$0.622692\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −1.07107 −0.0346227
$$958$$ 0 0
$$959$$ −2.61522 −0.0844500
$$960$$ 0 0
$$961$$ −30.6569 −0.988931
$$962$$ 0 0
$$963$$ −1.17157 −0.0377534
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 21.8579 0.702902 0.351451 0.936206i $$-0.385688\pi$$
0.351451 + 0.936206i $$0.385688\pi$$
$$968$$ 0 0
$$969$$ 0.414214 0.0133065
$$970$$ 0 0
$$971$$ 43.5980 1.39913 0.699563 0.714571i $$-0.253378\pi$$
0.699563 + 0.714571i $$0.253378\pi$$
$$972$$ 0 0
$$973$$ −1.65685 −0.0531163
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −38.3848 −1.22804 −0.614019 0.789291i $$-0.710448\pi$$
−0.614019 + 0.789291i $$0.710448\pi$$
$$978$$ 0 0
$$979$$ 17.3137 0.553349
$$980$$ 0 0
$$981$$ 14.1421 0.451524
$$982$$ 0 0
$$983$$ −14.2010 −0.452942 −0.226471 0.974018i $$-0.572719\pi$$
−0.226471 + 0.974018i $$0.572719\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −1.37258 −0.0436898
$$988$$ 0 0
$$989$$ 20.2426 0.643679
$$990$$ 0 0
$$991$$ −54.5269 −1.73210 −0.866052 0.499954i $$-0.833350\pi$$
−0.866052 + 0.499954i $$0.833350\pi$$
$$992$$ 0 0
$$993$$ −8.02944 −0.254806
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −3.55635 −0.112631 −0.0563154 0.998413i $$-0.517935\pi$$
−0.0563154 + 0.998413i $$0.517935\pi$$
$$998$$ 0 0
$$999$$ 26.1421 0.827101
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 3800.2.a.p.1.1 2
4.3 odd 2 7600.2.a.x.1.2 2
5.2 odd 4 760.2.d.c.609.3 yes 4
5.3 odd 4 760.2.d.c.609.2 4
5.4 even 2 3800.2.a.l.1.2 2
20.3 even 4 1520.2.d.d.609.3 4
20.7 even 4 1520.2.d.d.609.2 4
20.19 odd 2 7600.2.a.bc.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.c.609.2 4 5.3 odd 4
760.2.d.c.609.3 yes 4 5.2 odd 4
1520.2.d.d.609.2 4 20.7 even 4
1520.2.d.d.609.3 4 20.3 even 4
3800.2.a.l.1.2 2 5.4 even 2
3800.2.a.p.1.1 2 1.1 even 1 trivial
7600.2.a.x.1.2 2 4.3 odd 2
7600.2.a.bc.1.1 2 20.19 odd 2