# Properties

 Label 1600.2.f.c.1249.4 Level $1600$ Weight $2$ Character 1600.1249 Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1600.1249 Dual form 1600.2.f.c.1249.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +3.46410i q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +3.46410i q^{7} -2.00000 q^{9} +3.00000i q^{11} +3.46410 q^{13} +3.00000i q^{17} -1.00000i q^{19} -3.46410i q^{21} +5.00000 q^{27} -10.3923i q^{29} -6.92820 q^{31} -3.00000i q^{33} -10.3923 q^{37} -3.46410 q^{39} -9.00000 q^{41} +4.00000 q^{43} +10.3923i q^{47} -5.00000 q^{49} -3.00000i q^{51} +1.00000i q^{57} +12.0000i q^{59} -3.46410i q^{61} -6.92820i q^{63} +11.0000 q^{67} -10.3923 q^{71} -7.00000i q^{73} -10.3923 q^{77} -10.3923 q^{79} +1.00000 q^{81} -15.0000 q^{83} +10.3923i q^{87} +3.00000 q^{89} +12.0000i q^{91} +6.92820 q^{93} +14.0000i q^{97} -6.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 8 q^{9} + 20 q^{27} - 36 q^{41} + 16 q^{43} - 20 q^{49} + 44 q^{67} + 4 q^{81} - 60 q^{83} + 12 q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.46410i 1.30931i 0.755929 + 0.654654i $$0.227186\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000i 0.904534i 0.891883 + 0.452267i $$0.149385\pi$$
−0.891883 + 0.452267i $$0.850615\pi$$
$$12$$ 0 0
$$13$$ 3.46410 0.960769 0.480384 0.877058i $$-0.340497\pi$$
0.480384 + 0.877058i $$0.340497\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 0 0
$$19$$ − 1.00000i − 0.229416i −0.993399 0.114708i $$-0.963407\pi$$
0.993399 0.114708i $$-0.0365932\pi$$
$$20$$ 0 0
$$21$$ − 3.46410i − 0.755929i
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ − 10.3923i − 1.92980i −0.262613 0.964901i $$-0.584584\pi$$
0.262613 0.964901i $$-0.415416\pi$$
$$30$$ 0 0
$$31$$ −6.92820 −1.24434 −0.622171 0.782881i $$-0.713749\pi$$
−0.622171 + 0.782881i $$0.713749\pi$$
$$32$$ 0 0
$$33$$ − 3.00000i − 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.3923 −1.70848 −0.854242 0.519875i $$-0.825978\pi$$
−0.854242 + 0.519875i $$0.825978\pi$$
$$38$$ 0 0
$$39$$ −3.46410 −0.554700
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.3923i 1.51587i 0.652328 + 0.757937i $$0.273792\pi$$
−0.652328 + 0.757937i $$0.726208\pi$$
$$48$$ 0 0
$$49$$ −5.00000 −0.714286
$$50$$ 0 0
$$51$$ − 3.00000i − 0.420084i
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000i 0.132453i
$$58$$ 0 0
$$59$$ 12.0000i 1.56227i 0.624364 + 0.781133i $$0.285358\pi$$
−0.624364 + 0.781133i $$0.714642\pi$$
$$60$$ 0 0
$$61$$ − 3.46410i − 0.443533i −0.975100 0.221766i $$-0.928818\pi$$
0.975100 0.221766i $$-0.0711822\pi$$
$$62$$ 0 0
$$63$$ − 6.92820i − 0.872872i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.3923 −1.23334 −0.616670 0.787222i $$-0.711519\pi$$
−0.616670 + 0.787222i $$0.711519\pi$$
$$72$$ 0 0
$$73$$ − 7.00000i − 0.819288i −0.912245 0.409644i $$-0.865653\pi$$
0.912245 0.409644i $$-0.134347\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −10.3923 −1.18431
$$78$$ 0 0
$$79$$ −10.3923 −1.16923 −0.584613 0.811312i $$-0.698754\pi$$
−0.584613 + 0.811312i $$0.698754\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −15.0000 −1.64646 −0.823232 0.567705i $$-0.807831\pi$$
−0.823232 + 0.567705i $$0.807831\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.3923i 1.11417i
$$88$$ 0 0
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 12.0000i 1.25794i
$$92$$ 0 0
$$93$$ 6.92820 0.718421
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 0 0
$$99$$ − 6.00000i − 0.603023i
$$100$$ 0 0
$$101$$ − 10.3923i − 1.03407i −0.855963 0.517036i $$-0.827035\pi$$
0.855963 0.517036i $$-0.172965\pi$$
$$102$$ 0 0
$$103$$ − 10.3923i − 1.02398i −0.858990 0.511992i $$-0.828908\pi$$
0.858990 0.511992i $$-0.171092\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 0 0
$$109$$ 3.46410i 0.331801i 0.986143 + 0.165900i $$0.0530530\pi$$
−0.986143 + 0.165900i $$0.946947\pi$$
$$110$$ 0 0
$$111$$ 10.3923 0.986394
$$112$$ 0 0
$$113$$ 15.0000i 1.41108i 0.708669 + 0.705541i $$0.249296\pi$$
−0.708669 + 0.705541i $$0.750704\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −6.92820 −0.640513
$$118$$ 0 0
$$119$$ −10.3923 −0.952661
$$120$$ 0 0
$$121$$ 2.00000 0.181818
$$122$$ 0 0
$$123$$ 9.00000 0.811503
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 3.46410 0.300376
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 9.00000i 0.768922i 0.923141 + 0.384461i $$0.125613\pi$$
−0.923141 + 0.384461i $$0.874387\pi$$
$$138$$ 0 0
$$139$$ 5.00000i 0.424094i 0.977259 + 0.212047i $$0.0680131\pi$$
−0.977259 + 0.212047i $$0.931987\pi$$
$$140$$ 0 0
$$141$$ − 10.3923i − 0.875190i
$$142$$ 0 0
$$143$$ 10.3923i 0.869048i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 5.00000 0.412393
$$148$$ 0 0
$$149$$ − 10.3923i − 0.851371i −0.904871 0.425685i $$-0.860033\pi$$
0.904871 0.425685i $$-0.139967\pi$$
$$150$$ 0 0
$$151$$ 3.46410 0.281905 0.140952 0.990016i $$-0.454984\pi$$
0.140952 + 0.990016i $$0.454984\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.92820 −0.552931 −0.276465 0.961024i $$-0.589163\pi$$
−0.276465 + 0.961024i $$0.589163\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −19.0000 −1.48819 −0.744097 0.668071i $$-0.767120\pi$$
−0.744097 + 0.668071i $$0.767120\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 10.3923i − 0.804181i −0.915600 0.402090i $$-0.868284\pi$$
0.915600 0.402090i $$-0.131716\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 2.00000i 0.152944i
$$172$$ 0 0
$$173$$ −20.7846 −1.58022 −0.790112 0.612962i $$-0.789978\pi$$
−0.790112 + 0.612962i $$0.789978\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ 0 0
$$179$$ 3.00000i 0.224231i 0.993695 + 0.112115i $$0.0357626\pi$$
−0.993695 + 0.112115i $$0.964237\pi$$
$$180$$ 0 0
$$181$$ 13.8564i 1.02994i 0.857209 + 0.514969i $$0.172197\pi$$
−0.857209 + 0.514969i $$0.827803\pi$$
$$182$$ 0 0
$$183$$ 3.46410i 0.256074i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −9.00000 −0.658145
$$188$$ 0 0
$$189$$ 17.3205i 1.25988i
$$190$$ 0 0
$$191$$ −20.7846 −1.50392 −0.751961 0.659208i $$-0.770892\pi$$
−0.751961 + 0.659208i $$0.770892\pi$$
$$192$$ 0 0
$$193$$ − 7.00000i − 0.503871i −0.967744 0.251936i $$-0.918933\pi$$
0.967744 0.251936i $$-0.0810671\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 20.7846 1.48084 0.740421 0.672143i $$-0.234626\pi$$
0.740421 + 0.672143i $$0.234626\pi$$
$$198$$ 0 0
$$199$$ 13.8564 0.982255 0.491127 0.871088i $$-0.336585\pi$$
0.491127 + 0.871088i $$0.336585\pi$$
$$200$$ 0 0
$$201$$ −11.0000 −0.775880
$$202$$ 0 0
$$203$$ 36.0000 2.52670
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ − 25.0000i − 1.72107i −0.509390 0.860535i $$-0.670129\pi$$
0.509390 0.860535i $$-0.329871\pi$$
$$212$$ 0 0
$$213$$ 10.3923 0.712069
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 24.0000i − 1.62923i
$$218$$ 0 0
$$219$$ 7.00000i 0.473016i
$$220$$ 0 0
$$221$$ 10.3923i 0.699062i
$$222$$ 0 0
$$223$$ − 13.8564i − 0.927894i −0.885863 0.463947i $$-0.846433\pi$$
0.885863 0.463947i $$-0.153567\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ − 6.92820i − 0.457829i −0.973447 0.228914i $$-0.926482\pi$$
0.973447 0.228914i $$-0.0735176\pi$$
$$230$$ 0 0
$$231$$ 10.3923 0.683763
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 10.3923 0.675053
$$238$$ 0 0
$$239$$ 10.3923 0.672222 0.336111 0.941822i $$-0.390888\pi$$
0.336111 + 0.941822i $$0.390888\pi$$
$$240$$ 0 0
$$241$$ 5.00000 0.322078 0.161039 0.986948i $$-0.448515\pi$$
0.161039 + 0.986948i $$0.448515\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3.46410i − 0.220416i
$$248$$ 0 0
$$249$$ 15.0000 0.950586
$$250$$ 0 0
$$251$$ 9.00000i 0.568075i 0.958813 + 0.284037i $$0.0916740\pi$$
−0.958813 + 0.284037i $$0.908326\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ 0 0
$$259$$ − 36.0000i − 2.23693i
$$260$$ 0 0
$$261$$ 20.7846i 1.28654i
$$262$$ 0 0
$$263$$ 20.7846i 1.28163i 0.767694 + 0.640817i $$0.221404\pi$$
−0.767694 + 0.640817i $$0.778596\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −3.00000 −0.183597
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 20.7846 1.26258 0.631288 0.775549i $$-0.282527\pi$$
0.631288 + 0.775549i $$0.282527\pi$$
$$272$$ 0 0
$$273$$ − 12.0000i − 0.726273i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 17.3205 1.04069 0.520344 0.853957i $$-0.325804\pi$$
0.520344 + 0.853957i $$0.325804\pi$$
$$278$$ 0 0
$$279$$ 13.8564 0.829561
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 11.0000 0.653882 0.326941 0.945045i $$-0.393982\pi$$
0.326941 + 0.945045i $$0.393982\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 31.1769i − 1.84032i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ − 14.0000i − 0.820695i
$$292$$ 0 0
$$293$$ −10.3923 −0.607125 −0.303562 0.952812i $$-0.598176\pi$$
−0.303562 + 0.952812i $$0.598176\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 15.0000i 0.870388i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 13.8564i 0.798670i
$$302$$ 0 0
$$303$$ 10.3923i 0.597022i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −19.0000 −1.08439 −0.542194 0.840254i $$-0.682406\pi$$
−0.542194 + 0.840254i $$0.682406\pi$$
$$308$$ 0 0
$$309$$ 10.3923i 0.591198i
$$310$$ 0 0
$$311$$ 10.3923 0.589294 0.294647 0.955606i $$-0.404798\pi$$
0.294647 + 0.955606i $$0.404798\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 20.7846 1.16738 0.583690 0.811977i $$-0.301608\pi$$
0.583690 + 0.811977i $$0.301608\pi$$
$$318$$ 0 0
$$319$$ 31.1769 1.74557
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ 0 0
$$323$$ 3.00000 0.166924
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 3.46410i − 0.191565i
$$328$$ 0 0
$$329$$ −36.0000 −1.98474
$$330$$ 0 0
$$331$$ 19.0000i 1.04433i 0.852843 + 0.522167i $$0.174876\pi$$
−0.852843 + 0.522167i $$0.825124\pi$$
$$332$$ 0 0
$$333$$ 20.7846 1.13899
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 7.00000i − 0.381314i −0.981657 0.190657i $$-0.938938\pi$$
0.981657 0.190657i $$-0.0610619\pi$$
$$338$$ 0 0
$$339$$ − 15.0000i − 0.814688i
$$340$$ 0 0
$$341$$ − 20.7846i − 1.12555i
$$342$$ 0 0
$$343$$ 6.92820i 0.374088i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −33.0000 −1.77153 −0.885766 0.464131i $$-0.846367\pi$$
−0.885766 + 0.464131i $$0.846367\pi$$
$$348$$ 0 0
$$349$$ 10.3923i 0.556287i 0.960539 + 0.278144i $$0.0897191\pi$$
−0.960539 + 0.278144i $$0.910281\pi$$
$$350$$ 0 0
$$351$$ 17.3205 0.924500
$$352$$ 0 0
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 10.3923 0.550019
$$358$$ 0 0
$$359$$ −10.3923 −0.548485 −0.274242 0.961661i $$-0.588427\pi$$
−0.274242 + 0.961661i $$0.588427\pi$$
$$360$$ 0 0
$$361$$ 18.0000 0.947368
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 6.92820i − 0.361649i −0.983515 0.180825i $$-0.942123\pi$$
0.983515 0.180825i $$-0.0578766\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −13.8564 −0.717458 −0.358729 0.933442i $$-0.616790\pi$$
−0.358729 + 0.933442i $$0.616790\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ 19.0000i 0.975964i 0.872854 + 0.487982i $$0.162267\pi$$
−0.872854 + 0.487982i $$0.837733\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 31.1769i 1.59307i 0.604595 + 0.796533i $$0.293335\pi$$
−0.604595 + 0.796533i $$0.706665\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ 20.7846i 1.05382i 0.849921 + 0.526911i $$0.176650\pi$$
−0.849921 + 0.526911i $$0.823350\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 27.7128 1.39087 0.695433 0.718591i $$-0.255213\pi$$
0.695433 + 0.718591i $$0.255213\pi$$
$$398$$ 0 0
$$399$$ −3.46410 −0.173422
$$400$$ 0 0
$$401$$ −3.00000 −0.149813 −0.0749064 0.997191i $$-0.523866\pi$$
−0.0749064 + 0.997191i $$0.523866\pi$$
$$402$$ 0 0
$$403$$ −24.0000 −1.19553
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 31.1769i − 1.54538i
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ − 9.00000i − 0.443937i
$$412$$ 0 0
$$413$$ −41.5692 −2.04549
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 5.00000i − 0.244851i
$$418$$ 0 0
$$419$$ − 21.0000i − 1.02592i −0.858413 0.512959i $$-0.828549\pi$$
0.858413 0.512959i $$-0.171451\pi$$
$$420$$ 0 0
$$421$$ 20.7846i 1.01298i 0.862246 + 0.506490i $$0.169057\pi$$
−0.862246 + 0.506490i $$0.830943\pi$$
$$422$$ 0 0
$$423$$ − 20.7846i − 1.01058i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 12.0000 0.580721
$$428$$ 0 0
$$429$$ − 10.3923i − 0.501745i
$$430$$ 0 0
$$431$$ −10.3923 −0.500580 −0.250290 0.968171i $$-0.580526\pi$$
−0.250290 + 0.968171i $$0.580526\pi$$
$$432$$ 0 0
$$433$$ 1.00000i 0.0480569i 0.999711 + 0.0240285i $$0.00764923\pi$$
−0.999711 + 0.0240285i $$0.992351\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 17.3205 0.826663 0.413331 0.910581i $$-0.364365\pi$$
0.413331 + 0.910581i $$0.364365\pi$$
$$440$$ 0 0
$$441$$ 10.0000 0.476190
$$442$$ 0 0
$$443$$ 39.0000 1.85295 0.926473 0.376361i $$-0.122825\pi$$
0.926473 + 0.376361i $$0.122825\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 10.3923i 0.491539i
$$448$$ 0 0
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ − 27.0000i − 1.27138i
$$452$$ 0 0
$$453$$ −3.46410 −0.162758
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.0000i 1.73079i 0.501093 + 0.865393i $$0.332931\pi$$
−0.501093 + 0.865393i $$0.667069\pi$$
$$458$$ 0 0
$$459$$ 15.0000i 0.700140i
$$460$$ 0 0
$$461$$ 20.7846i 0.968036i 0.875058 + 0.484018i $$0.160823\pi$$
−0.875058 + 0.484018i $$0.839177\pi$$
$$462$$ 0 0
$$463$$ − 20.7846i − 0.965943i −0.875636 0.482971i $$-0.839558\pi$$
0.875636 0.482971i $$-0.160442\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 38.1051i 1.75953i
$$470$$ 0 0
$$471$$ 6.92820 0.319235
$$472$$ 0 0
$$473$$ 12.0000i 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −41.5692 −1.89935 −0.949673 0.313243i $$-0.898585\pi$$
−0.949673 + 0.313243i $$0.898585\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 13.8564i − 0.627894i −0.949441 0.313947i $$-0.898349\pi$$
0.949441 0.313947i $$-0.101651\pi$$
$$488$$ 0 0
$$489$$ 19.0000 0.859210
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ 31.1769 1.40414
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 36.0000i − 1.61482i
$$498$$ 0 0
$$499$$ 16.0000i 0.716258i 0.933672 + 0.358129i $$0.116585\pi$$
−0.933672 + 0.358129i $$0.883415\pi$$
$$500$$ 0 0
$$501$$ 10.3923i 0.464294i
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ − 20.7846i − 0.921262i −0.887592 0.460631i $$-0.847623\pi$$
0.887592 0.460631i $$-0.152377\pi$$
$$510$$ 0 0
$$511$$ 24.2487 1.07270
$$512$$ 0 0
$$513$$ − 5.00000i − 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −31.1769 −1.37116
$$518$$ 0 0
$$519$$ 20.7846 0.912343
$$520$$ 0 0
$$521$$ −21.0000 −0.920027 −0.460013 0.887912i $$-0.652155\pi$$
−0.460013 + 0.887912i $$0.652155\pi$$
$$522$$ 0 0
$$523$$ 1.00000 0.0437269 0.0218635 0.999761i $$-0.493040\pi$$
0.0218635 + 0.999761i $$0.493040\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 20.7846i − 0.905392i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ − 24.0000i − 1.04151i
$$532$$ 0 0
$$533$$ −31.1769 −1.35042
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 3.00000i − 0.129460i
$$538$$ 0 0
$$539$$ − 15.0000i − 0.646096i
$$540$$ 0 0
$$541$$ − 6.92820i − 0.297867i −0.988847 0.148933i $$-0.952416\pi$$
0.988847 0.148933i $$-0.0475840\pi$$
$$542$$ 0 0
$$543$$ − 13.8564i − 0.594635i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.00000 0.0427569 0.0213785 0.999771i $$-0.493195\pi$$
0.0213785 + 0.999771i $$0.493195\pi$$
$$548$$ 0 0
$$549$$ 6.92820i 0.295689i
$$550$$ 0 0
$$551$$ −10.3923 −0.442727
$$552$$ 0 0
$$553$$ − 36.0000i − 1.53088i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10.3923 −0.440336 −0.220168 0.975462i $$-0.570661\pi$$
−0.220168 + 0.975462i $$0.570661\pi$$
$$558$$ 0 0
$$559$$ 13.8564 0.586064
$$560$$ 0 0
$$561$$ 9.00000 0.379980
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3.46410i 0.145479i
$$568$$ 0 0
$$569$$ −3.00000 −0.125767 −0.0628833 0.998021i $$-0.520030\pi$$
−0.0628833 + 0.998021i $$0.520030\pi$$
$$570$$ 0 0
$$571$$ − 20.0000i − 0.836974i −0.908223 0.418487i $$-0.862561\pi$$
0.908223 0.418487i $$-0.137439\pi$$
$$572$$ 0 0
$$573$$ 20.7846 0.868290
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 17.0000i 0.707719i 0.935299 + 0.353860i $$0.115131\pi$$
−0.935299 + 0.353860i $$0.884869\pi$$
$$578$$ 0 0
$$579$$ 7.00000i 0.290910i
$$580$$ 0 0
$$581$$ − 51.9615i − 2.15573i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −15.0000 −0.619116 −0.309558 0.950881i $$-0.600181\pi$$
−0.309558 + 0.950881i $$0.600181\pi$$
$$588$$ 0 0
$$589$$ 6.92820i 0.285472i
$$590$$ 0 0
$$591$$ −20.7846 −0.854965
$$592$$ 0 0
$$593$$ 3.00000i 0.123195i 0.998101 + 0.0615976i $$0.0196196\pi$$
−0.998101 + 0.0615976i $$0.980380\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −13.8564 −0.567105
$$598$$ 0 0
$$599$$ −31.1769 −1.27385 −0.636927 0.770924i $$-0.719795\pi$$
−0.636927 + 0.770924i $$0.719795\pi$$
$$600$$ 0 0
$$601$$ 41.0000 1.67242 0.836212 0.548406i $$-0.184765\pi$$
0.836212 + 0.548406i $$0.184765\pi$$
$$602$$ 0 0
$$603$$ −22.0000 −0.895909
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 20.7846i 0.843621i 0.906684 + 0.421811i $$0.138605\pi$$
−0.906684 + 0.421811i $$0.861395\pi$$
$$608$$ 0 0
$$609$$ −36.0000 −1.45879
$$610$$ 0 0
$$611$$ 36.0000i 1.45640i
$$612$$ 0 0
$$613$$ 20.7846 0.839482 0.419741 0.907644i $$-0.362121\pi$$
0.419741 + 0.907644i $$0.362121\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 10.3923i 0.416359i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −3.00000 −0.119808
$$628$$ 0 0
$$629$$ − 31.1769i − 1.24310i
$$630$$ 0 0
$$631$$ −13.8564 −0.551615 −0.275807 0.961213i $$-0.588945\pi$$
−0.275807 + 0.961213i $$0.588945\pi$$
$$632$$ 0 0
$$633$$ 25.0000i 0.993661i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −17.3205 −0.686264
$$638$$ 0 0
$$639$$ 20.7846 0.822226
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 41.5692i 1.63425i 0.576457 + 0.817127i $$0.304435\pi$$
−0.576457 + 0.817127i $$0.695565\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 24.0000i 0.940634i
$$652$$ 0 0
$$653$$ −31.1769 −1.22005 −0.610023 0.792383i $$-0.708840\pi$$
−0.610023 + 0.792383i $$0.708840\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000i 0.546192i
$$658$$ 0 0
$$659$$ 33.0000i 1.28550i 0.766077 + 0.642749i $$0.222206\pi$$
−0.766077 + 0.642749i $$0.777794\pi$$
$$660$$ 0 0
$$661$$ 10.3923i 0.404214i 0.979363 + 0.202107i $$0.0647788\pi$$
−0.979363 + 0.202107i $$0.935221\pi$$
$$662$$ 0 0
$$663$$ − 10.3923i − 0.403604i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 13.8564i 0.535720i
$$670$$ 0 0
$$671$$ 10.3923 0.401190
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 20.7846 0.798817 0.399409 0.916773i $$-0.369215\pi$$
0.399409 + 0.916773i $$0.369215\pi$$
$$678$$ 0 0
$$679$$ −48.4974 −1.86116
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.92820i 0.264327i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 37.0000i 1.40755i 0.710425 + 0.703773i $$0.248503\pi$$
−0.710425 + 0.703773i $$0.751497\pi$$
$$692$$ 0 0
$$693$$ 20.7846 0.789542
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 27.0000i − 1.02270i
$$698$$ 0 0
$$699$$ − 6.00000i − 0.226941i
$$700$$ 0 0
$$701$$ − 31.1769i − 1.17754i −0.808302 0.588768i $$-0.799613\pi$$
0.808302 0.588768i $$-0.200387\pi$$
$$702$$ 0 0
$$703$$ 10.3923i 0.391953i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 36.0000 1.35392
$$708$$ 0 0
$$709$$ − 34.6410i − 1.30097i −0.759519 0.650485i $$-0.774566\pi$$
0.759519 0.650485i $$-0.225434\pi$$
$$710$$ 0 0
$$711$$ 20.7846 0.779484
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −10.3923 −0.388108
$$718$$ 0 0
$$719$$ −10.3923 −0.387568 −0.193784 0.981044i $$-0.562076\pi$$
−0.193784 + 0.981044i $$0.562076\pi$$
$$720$$ 0 0
$$721$$ 36.0000 1.34071
$$722$$ 0 0
$$723$$ −5.00000 −0.185952
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 45.0333i 1.67019i 0.550103 + 0.835097i $$0.314588\pi$$
−0.550103 + 0.835097i $$0.685412\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 12.0000i 0.443836i
$$732$$ 0 0
$$733$$ 31.1769 1.15155 0.575773 0.817610i $$-0.304701\pi$$
0.575773 + 0.817610i $$0.304701\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 33.0000i 1.21557i
$$738$$ 0 0
$$739$$ − 20.0000i − 0.735712i −0.929883 0.367856i $$-0.880092\pi$$
0.929883 0.367856i $$-0.119908\pi$$
$$740$$ 0 0
$$741$$ 3.46410i 0.127257i
$$742$$ 0 0
$$743$$ 41.5692i 1.52503i 0.646972 + 0.762513i $$0.276035\pi$$
−0.646972 + 0.762513i $$0.723965\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 30.0000 1.09764
$$748$$ 0 0
$$749$$ − 31.1769i − 1.13918i
$$750$$ 0 0
$$751$$ −51.9615 −1.89610 −0.948051 0.318117i $$-0.896950\pi$$
−0.948051 + 0.318117i $$0.896950\pi$$
$$752$$ 0 0
$$753$$ − 9.00000i − 0.327978i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6.92820 −0.251810 −0.125905 0.992042i $$-0.540183\pi$$
−0.125905 + 0.992042i $$0.540183\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3.00000 0.108750 0.0543750 0.998521i $$-0.482683\pi$$
0.0543750 + 0.998521i $$0.482683\pi$$
$$762$$ 0 0
$$763$$ −12.0000 −0.434429
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 41.5692i 1.50098i
$$768$$ 0 0
$$769$$ 7.00000 0.252426 0.126213 0.992003i $$-0.459718\pi$$
0.126213 + 0.992003i $$0.459718\pi$$
$$770$$ 0 0
$$771$$ 6.00000i 0.216085i
$$772$$ 0 0
$$773$$ 10.3923 0.373785 0.186893 0.982380i $$-0.440158\pi$$
0.186893 + 0.982380i $$0.440158\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 36.0000i 1.29149i
$$778$$ 0 0
$$779$$ 9.00000i 0.322458i
$$780$$ 0 0
$$781$$ − 31.1769i − 1.11560i
$$782$$ 0 0
$$783$$ − 51.9615i − 1.85695i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 16.0000 0.570338 0.285169 0.958477i $$-0.407950\pi$$
0.285169 + 0.958477i $$0.407950\pi$$
$$788$$ 0 0
$$789$$ − 20.7846i − 0.739952i
$$790$$ 0 0
$$791$$ −51.9615 −1.84754
$$792$$ 0 0
$$793$$ − 12.0000i − 0.426132i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −20.7846 −0.736229 −0.368114 0.929781i $$-0.619996\pi$$
−0.368114 + 0.929781i $$0.619996\pi$$
$$798$$ 0 0
$$799$$ −31.1769 −1.10296
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 21.0000 0.741074
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ − 28.0000i − 0.983213i −0.870817 0.491606i $$-0.836410\pi$$
0.870817 0.491606i $$-0.163590\pi$$
$$812$$ 0 0
$$813$$ −20.7846 −0.728948
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 4.00000i − 0.139942i
$$818$$ 0 0
$$819$$ − 24.0000i − 0.838628i
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ − 34.6410i − 1.20751i −0.797170 0.603755i $$-0.793671\pi$$
0.797170 0.603755i $$-0.206329\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −3.00000 −0.104320 −0.0521601 0.998639i $$-0.516611\pi$$
−0.0521601 + 0.998639i $$0.516611\pi$$
$$828$$ 0 0
$$829$$ − 10.3923i − 0.360940i −0.983581 0.180470i $$-0.942238\pi$$
0.983581 0.180470i $$-0.0577618\pi$$
$$830$$ 0 0
$$831$$ −17.3205 −0.600842
$$832$$ 0 0
$$833$$ − 15.0000i − 0.519719i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −34.6410 −1.19737
$$838$$ 0 0
$$839$$ 41.5692 1.43513 0.717564 0.696492i $$-0.245257\pi$$
0.717564 + 0.696492i $$0.245257\pi$$
$$840$$ 0 0
$$841$$ −79.0000 −2.72414
$$842$$ 0 0
$$843$$ −18.0000 −0.619953
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.92820i 0.238056i
$$848$$ 0 0
$$849$$ −11.0000 −0.377519
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 55.4256 1.89774 0.948869 0.315671i $$-0.102230\pi$$
0.948869 + 0.315671i $$0.102230\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 39.0000i − 1.33221i −0.745856 0.666107i $$-0.767959\pi$$
0.745856 0.666107i $$-0.232041\pi$$
$$858$$ 0 0
$$859$$ 13.0000i 0.443554i 0.975097 + 0.221777i $$0.0711857\pi$$
−0.975097 + 0.221777i $$0.928814\pi$$
$$860$$ 0 0
$$861$$ 31.1769i 1.06251i
$$862$$ 0 0
$$863$$ 10.3923i 0.353758i 0.984233 + 0.176879i $$0.0566002\pi$$
−0.984233 + 0.176879i $$0.943400\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ − 31.1769i − 1.05760i
$$870$$ 0 0
$$871$$ 38.1051 1.29114
$$872$$ 0 0
$$873$$ − 28.0000i − 0.947656i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −48.4974 −1.63764 −0.818821 0.574049i $$-0.805372\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 0 0
$$879$$ 10.3923 0.350524
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ 35.0000 1.17784 0.588922 0.808190i $$-0.299553\pi$$
0.588922 + 0.808190i $$0.299553\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.00000i 0.100504i
$$892$$ 0 0
$$893$$ 10.3923 0.347765
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 72.0000i 2.40133i
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 13.8564i − 0.461112i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 0 0
$$909$$ 20.7846i 0.689382i
$$910$$ 0 0
$$911$$ −20.7846 −0.688625 −0.344312 0.938855i $$-0.611888\pi$$
−0.344312 + 0.938855i $$0.611888\pi$$
$$912$$ 0 0
$$913$$ − 45.0000i − 1.48928i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 27.7128 0.914161 0.457081 0.889425i $$-0.348895\pi$$
0.457081 + 0.889425i $$0.348895\pi$$
$$920$$ 0 0
$$921$$ 19.0000 0.626071
$$922$$ 0 0
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 20.7846i 0.682656i
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 5.00000i 0.163868i
$$932$$ 0 0
$$933$$ −10.3923 −0.340229
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 35.0000i 1.14340i 0.820463 + 0.571700i $$0.193716\pi$$
−0.820463 + 0.571700i $$0.806284\pi$$
$$938$$ 0 0
$$939$$ − 22.0000i − 0.717943i
$$940$$ 0 0
$$941$$ − 51.9615i − 1.69390i −0.531675 0.846949i $$-0.678437\pi$$
0.531675 0.846949i $$-0.321563\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ − 24.2487i − 0.787146i
$$950$$ 0 0
$$951$$ −20.7846 −0.673987
$$952$$ 0 0
$$953$$ 39.0000i 1.26333i 0.775240 + 0.631667i $$0.217629\pi$$
−0.775240 + 0.631667i $$0.782371\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −31.1769 −1.00781
$$958$$ 0 0
$$959$$ −31.1769 −1.00676
$$960$$ 0 0
$$961$$ 17.0000 0.548387
$$962$$ 0 0
$$963$$ 18.0000 0.580042
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 20.7846i 0.668388i 0.942504 + 0.334194i $$0.108464\pi$$
−0.942504 + 0.334194i $$0.891536\pi$$
$$968$$ 0 0
$$969$$ −3.00000 −0.0963739
$$970$$ 0 0
$$971$$ − 57.0000i − 1.82922i −0.404341 0.914609i $$-0.632499\pi$$
0.404341 0.914609i $$-0.367501\pi$$
$$972$$ 0 0
$$973$$ −17.3205 −0.555270
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 27.0000i 0.863807i 0.901920 + 0.431903i $$0.142158\pi$$
−0.901920 + 0.431903i $$0.857842\pi$$
$$978$$ 0 0
$$979$$ 9.00000i 0.287641i
$$980$$ 0 0
$$981$$ − 6.92820i − 0.221201i
$$982$$ 0 0
$$983$$ 10.3923i 0.331463i 0.986171 + 0.165732i $$0.0529985\pi$$
−0.986171 + 0.165732i $$0.947001\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 36.0000 1.14589
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 24.2487 0.770286 0.385143 0.922857i $$-0.374152\pi$$
0.385143 + 0.922857i $$0.374152\pi$$
$$992$$ 0 0
$$993$$ − 19.0000i − 0.602947i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 0 0
$$999$$ −51.9615 −1.64399
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 1600.2.f.c.1249.4 4
4.3 odd 2 1600.2.f.g.1249.1 4
5.2 odd 4 1600.2.d.e.801.3 yes 4
5.3 odd 4 1600.2.d.f.801.2 yes 4
5.4 even 2 1600.2.f.g.1249.2 4
8.3 odd 2 inner 1600.2.f.c.1249.2 4
8.5 even 2 1600.2.f.g.1249.3 4
20.3 even 4 1600.2.d.f.801.3 yes 4
20.7 even 4 1600.2.d.e.801.2 yes 4
20.19 odd 2 inner 1600.2.f.c.1249.3 4
40.3 even 4 1600.2.d.f.801.1 yes 4
40.13 odd 4 1600.2.d.f.801.4 yes 4
40.19 odd 2 1600.2.f.g.1249.4 4
40.27 even 4 1600.2.d.e.801.4 yes 4
40.29 even 2 inner 1600.2.f.c.1249.1 4
40.37 odd 4 1600.2.d.e.801.1 4
80.3 even 4 6400.2.a.ba.1.2 2
80.13 odd 4 6400.2.a.cg.1.1 2
80.27 even 4 6400.2.a.bb.1.1 2
80.37 odd 4 6400.2.a.cf.1.2 2
80.43 even 4 6400.2.a.cg.1.2 2
80.53 odd 4 6400.2.a.ba.1.1 2
80.67 even 4 6400.2.a.cf.1.1 2
80.77 odd 4 6400.2.a.bb.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.e.801.1 4 40.37 odd 4
1600.2.d.e.801.2 yes 4 20.7 even 4
1600.2.d.e.801.3 yes 4 5.2 odd 4
1600.2.d.e.801.4 yes 4 40.27 even 4
1600.2.d.f.801.1 yes 4 40.3 even 4
1600.2.d.f.801.2 yes 4 5.3 odd 4
1600.2.d.f.801.3 yes 4 20.3 even 4
1600.2.d.f.801.4 yes 4 40.13 odd 4
1600.2.f.c.1249.1 4 40.29 even 2 inner
1600.2.f.c.1249.2 4 8.3 odd 2 inner
1600.2.f.c.1249.3 4 20.19 odd 2 inner
1600.2.f.c.1249.4 4 1.1 even 1 trivial
1600.2.f.g.1249.1 4 4.3 odd 2
1600.2.f.g.1249.2 4 5.4 even 2
1600.2.f.g.1249.3 4 8.5 even 2
1600.2.f.g.1249.4 4 40.19 odd 2
6400.2.a.ba.1.1 2 80.53 odd 4
6400.2.a.ba.1.2 2 80.3 even 4
6400.2.a.bb.1.1 2 80.27 even 4
6400.2.a.bb.1.2 2 80.77 odd 4
6400.2.a.cf.1.1 2 80.67 even 4
6400.2.a.cf.1.2 2 80.37 odd 4
6400.2.a.cg.1.1 2 80.13 odd 4
6400.2.a.cg.1.2 2 80.43 even 4