Properties

 Label 8048.2.a.p.1.4 Level $8048$ Weight $2$ Character 8048.1 Self dual yes Analytic conductor $64.264$ Analytic rank $1$ Dimension $10$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8048 = 2^{4} \cdot 503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8048.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$64.2636035467$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 503) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$1.07636$$ of defining polynomial Character $$\chi$$ $$=$$ 8048.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-0.0763625 q^{3} +1.17276 q^{5} -0.469303 q^{7} -2.99417 q^{9} +O(q^{10})$$ $$q-0.0763625 q^{3} +1.17276 q^{5} -0.469303 q^{7} -2.99417 q^{9} +5.74596 q^{11} -1.85873 q^{13} -0.0895546 q^{15} +5.22916 q^{17} -2.12602 q^{19} +0.0358371 q^{21} -0.171951 q^{23} -3.62464 q^{25} +0.457730 q^{27} -6.19149 q^{29} +0.396234 q^{31} -0.438776 q^{33} -0.550377 q^{35} -8.17999 q^{37} +0.141937 q^{39} -12.4282 q^{41} +4.97920 q^{43} -3.51143 q^{45} +0.521599 q^{47} -6.77976 q^{49} -0.399312 q^{51} +8.76106 q^{53} +6.73861 q^{55} +0.162349 q^{57} -3.35297 q^{59} -5.38243 q^{61} +1.40517 q^{63} -2.17983 q^{65} -8.42823 q^{67} +0.0131306 q^{69} -7.47643 q^{71} -4.60009 q^{73} +0.276787 q^{75} -2.69660 q^{77} +17.1992 q^{79} +8.94755 q^{81} -5.97721 q^{83} +6.13253 q^{85} +0.472798 q^{87} -4.25595 q^{89} +0.872306 q^{91} -0.0302574 q^{93} -2.49331 q^{95} -2.64532 q^{97} -17.2044 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 8q^{3} - q^{5} + 5q^{7} - 2q^{9} + O(q^{10})$$ $$10q + 8q^{3} - q^{5} + 5q^{7} - 2q^{9} + 3q^{11} - 18q^{13} + 2q^{15} - 11q^{17} + q^{21} + 2q^{23} - 27q^{25} + 2q^{27} - 9q^{29} + 22q^{31} - 10q^{33} + 6q^{35} - 35q^{37} - 8q^{39} - 4q^{41} + 20q^{43} + 2q^{45} - 7q^{47} - 27q^{49} - 9q^{51} - 24q^{53} + 11q^{55} - 23q^{57} - 17q^{59} - 4q^{61} - 10q^{63} - 16q^{65} + 6q^{67} - 2q^{69} + q^{71} - 31q^{73} - 30q^{75} + 3q^{77} + 10q^{79} - 6q^{81} - 22q^{83} - 6q^{85} - 25q^{87} + q^{89} - 10q^{91} - 6q^{93} - 39q^{95} - 57q^{97} - 35q^{99} + O(q^{100})$$

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
$$3$$ −0.0763625 −0.0440879 −0.0220440 0.999757i $$-0.507017\pi$$
−0.0220440 + 0.999757i $$0.507017\pi$$
$$4$$ 0 0
$$5$$ 1.17276 0.524472 0.262236 0.965004i $$-0.415540\pi$$
0.262236 + 0.965004i $$0.415540\pi$$
$$6$$ 0 0
$$7$$ −0.469303 −0.177380 −0.0886899 0.996059i $$-0.528268\pi$$
−0.0886899 + 0.996059i $$0.528268\pi$$
$$8$$ 0 0
$$9$$ −2.99417 −0.998056
$$10$$ 0 0
$$11$$ 5.74596 1.73247 0.866237 0.499634i $$-0.166532\pi$$
0.866237 + 0.499634i $$0.166532\pi$$
$$12$$ 0 0
$$13$$ −1.85873 −0.515518 −0.257759 0.966209i $$-0.582984\pi$$
−0.257759 + 0.966209i $$0.582984\pi$$
$$14$$ 0 0
$$15$$ −0.0895546 −0.0231229
$$16$$ 0 0
$$17$$ 5.22916 1.26826 0.634129 0.773228i $$-0.281359\pi$$
0.634129 + 0.773228i $$0.281359\pi$$
$$18$$ 0 0
$$19$$ −2.12602 −0.487743 −0.243872 0.969808i $$-0.578418\pi$$
−0.243872 + 0.969808i $$0.578418\pi$$
$$20$$ 0 0
$$21$$ 0.0358371 0.00782030
$$22$$ 0 0
$$23$$ −0.171951 −0.0358543 −0.0179271 0.999839i $$-0.505707\pi$$
−0.0179271 + 0.999839i $$0.505707\pi$$
$$24$$ 0 0
$$25$$ −3.62464 −0.724929
$$26$$ 0 0
$$27$$ 0.457730 0.0880902
$$28$$ 0 0
$$29$$ −6.19149 −1.14973 −0.574865 0.818248i $$-0.694946\pi$$
−0.574865 + 0.818248i $$0.694946\pi$$
$$30$$ 0 0
$$31$$ 0.396234 0.0711658 0.0355829 0.999367i $$-0.488671\pi$$
0.0355829 + 0.999367i $$0.488671\pi$$
$$32$$ 0 0
$$33$$ −0.438776 −0.0763812
$$34$$ 0 0
$$35$$ −0.550377 −0.0930307
$$36$$ 0 0
$$37$$ −8.17999 −1.34478 −0.672391 0.740196i $$-0.734733\pi$$
−0.672391 + 0.740196i $$0.734733\pi$$
$$38$$ 0 0
$$39$$ 0.141937 0.0227281
$$40$$ 0 0
$$41$$ −12.4282 −1.94095 −0.970477 0.241193i $$-0.922461\pi$$
−0.970477 + 0.241193i $$0.922461\pi$$
$$42$$ 0 0
$$43$$ 4.97920 0.759321 0.379661 0.925126i $$-0.376041\pi$$
0.379661 + 0.925126i $$0.376041\pi$$
$$44$$ 0 0
$$45$$ −3.51143 −0.523453
$$46$$ 0 0
$$47$$ 0.521599 0.0760831 0.0380415 0.999276i $$-0.487888\pi$$
0.0380415 + 0.999276i $$0.487888\pi$$
$$48$$ 0 0
$$49$$ −6.77976 −0.968536
$$50$$ 0 0
$$51$$ −0.399312 −0.0559148
$$52$$ 0 0
$$53$$ 8.76106 1.20342 0.601712 0.798713i $$-0.294486\pi$$
0.601712 + 0.798713i $$0.294486\pi$$
$$54$$ 0 0
$$55$$ 6.73861 0.908634
$$56$$ 0 0
$$57$$ 0.162349 0.0215036
$$58$$ 0 0
$$59$$ −3.35297 −0.436519 −0.218260 0.975891i $$-0.570038\pi$$
−0.218260 + 0.975891i $$0.570038\pi$$
$$60$$ 0 0
$$61$$ −5.38243 −0.689150 −0.344575 0.938759i $$-0.611977\pi$$
−0.344575 + 0.938759i $$0.611977\pi$$
$$62$$ 0 0
$$63$$ 1.40517 0.177035
$$64$$ 0 0
$$65$$ −2.17983 −0.270375
$$66$$ 0 0
$$67$$ −8.42823 −1.02967 −0.514836 0.857289i $$-0.672147\pi$$
−0.514836 + 0.857289i $$0.672147\pi$$
$$68$$ 0 0
$$69$$ 0.0131306 0.00158074
$$70$$ 0 0
$$71$$ −7.47643 −0.887289 −0.443644 0.896203i $$-0.646315\pi$$
−0.443644 + 0.896203i $$0.646315\pi$$
$$72$$ 0 0
$$73$$ −4.60009 −0.538400 −0.269200 0.963084i $$-0.586759\pi$$
−0.269200 + 0.963084i $$0.586759\pi$$
$$74$$ 0 0
$$75$$ 0.276787 0.0319606
$$76$$ 0 0
$$77$$ −2.69660 −0.307306
$$78$$ 0 0
$$79$$ 17.1992 1.93506 0.967530 0.252758i $$-0.0813376\pi$$
0.967530 + 0.252758i $$0.0813376\pi$$
$$80$$ 0 0
$$81$$ 8.94755 0.994173
$$82$$ 0 0
$$83$$ −5.97721 −0.656084 −0.328042 0.944663i $$-0.606389\pi$$
−0.328042 + 0.944663i $$0.606389\pi$$
$$84$$ 0 0
$$85$$ 6.13253 0.665166
$$86$$ 0 0
$$87$$ 0.472798 0.0506892
$$88$$ 0 0
$$89$$ −4.25595 −0.451130 −0.225565 0.974228i $$-0.572423\pi$$
−0.225565 + 0.974228i $$0.572423\pi$$
$$90$$ 0 0
$$91$$ 0.872306 0.0914425
$$92$$ 0 0
$$93$$ −0.0302574 −0.00313755
$$94$$ 0 0
$$95$$ −2.49331 −0.255808
$$96$$ 0 0
$$97$$ −2.64532 −0.268592 −0.134296 0.990941i $$-0.542877\pi$$
−0.134296 + 0.990941i $$0.542877\pi$$
$$98$$ 0 0
$$99$$ −17.2044 −1.72911
$$100$$ 0 0
$$101$$ 6.39045 0.635873 0.317937 0.948112i $$-0.397010\pi$$
0.317937 + 0.948112i $$0.397010\pi$$
$$102$$ 0 0
$$103$$ 16.6014 1.63578 0.817890 0.575375i $$-0.195144\pi$$
0.817890 + 0.575375i $$0.195144\pi$$
$$104$$ 0 0
$$105$$ 0.0420282 0.00410153
$$106$$ 0 0
$$107$$ −9.15760 −0.885299 −0.442649 0.896695i $$-0.645961\pi$$
−0.442649 + 0.896695i $$0.645961\pi$$
$$108$$ 0 0
$$109$$ 6.51891 0.624398 0.312199 0.950017i $$-0.398934\pi$$
0.312199 + 0.950017i $$0.398934\pi$$
$$110$$ 0 0
$$111$$ 0.624645 0.0592887
$$112$$ 0 0
$$113$$ 1.36694 0.128591 0.0642954 0.997931i $$-0.479520\pi$$
0.0642954 + 0.997931i $$0.479520\pi$$
$$114$$ 0 0
$$115$$ −0.201656 −0.0188046
$$116$$ 0 0
$$117$$ 5.56535 0.514516
$$118$$ 0 0
$$119$$ −2.45406 −0.224963
$$120$$ 0 0
$$121$$ 22.0161 2.00146
$$122$$ 0 0
$$123$$ 0.949047 0.0855726
$$124$$ 0 0
$$125$$ −10.1146 −0.904677
$$126$$ 0 0
$$127$$ −6.77469 −0.601157 −0.300578 0.953757i $$-0.597180\pi$$
−0.300578 + 0.953757i $$0.597180\pi$$
$$128$$ 0 0
$$129$$ −0.380225 −0.0334769
$$130$$ 0 0
$$131$$ −16.6839 −1.45768 −0.728838 0.684687i $$-0.759939\pi$$
−0.728838 + 0.684687i $$0.759939\pi$$
$$132$$ 0 0
$$133$$ 0.997748 0.0865158
$$134$$ 0 0
$$135$$ 0.536805 0.0462008
$$136$$ 0 0
$$137$$ 13.0662 1.11632 0.558162 0.829732i $$-0.311507\pi$$
0.558162 + 0.829732i $$0.311507\pi$$
$$138$$ 0 0
$$139$$ 10.8468 0.920017 0.460008 0.887915i $$-0.347846\pi$$
0.460008 + 0.887915i $$0.347846\pi$$
$$140$$ 0 0
$$141$$ −0.0398306 −0.00335434
$$142$$ 0 0
$$143$$ −10.6802 −0.893122
$$144$$ 0 0
$$145$$ −7.26110 −0.603002
$$146$$ 0 0
$$147$$ 0.517719 0.0427008
$$148$$ 0 0
$$149$$ 0.0210826 0.00172716 0.000863578 1.00000i $$-0.499725\pi$$
0.000863578 1.00000i $$0.499725\pi$$
$$150$$ 0 0
$$151$$ −15.1704 −1.23455 −0.617276 0.786746i $$-0.711764\pi$$
−0.617276 + 0.786746i $$0.711764\pi$$
$$152$$ 0 0
$$153$$ −15.6570 −1.26579
$$154$$ 0 0
$$155$$ 0.464686 0.0373245
$$156$$ 0 0
$$157$$ 11.1437 0.889367 0.444684 0.895688i $$-0.353316\pi$$
0.444684 + 0.895688i $$0.353316\pi$$
$$158$$ 0 0
$$159$$ −0.669016 −0.0530565
$$160$$ 0 0
$$161$$ 0.0806970 0.00635982
$$162$$ 0 0
$$163$$ 10.3388 0.809797 0.404899 0.914362i $$-0.367307\pi$$
0.404899 + 0.914362i $$0.367307\pi$$
$$164$$ 0 0
$$165$$ −0.514578 −0.0400598
$$166$$ 0 0
$$167$$ 15.6444 1.21060 0.605300 0.795998i $$-0.293053\pi$$
0.605300 + 0.795998i $$0.293053\pi$$
$$168$$ 0 0
$$169$$ −9.54513 −0.734241
$$170$$ 0 0
$$171$$ 6.36567 0.486795
$$172$$ 0 0
$$173$$ −5.69316 −0.432843 −0.216422 0.976300i $$-0.569439\pi$$
−0.216422 + 0.976300i $$0.569439\pi$$
$$174$$ 0 0
$$175$$ 1.70105 0.128588
$$176$$ 0 0
$$177$$ 0.256041 0.0192452
$$178$$ 0 0
$$179$$ −10.8571 −0.811495 −0.405748 0.913985i $$-0.632989\pi$$
−0.405748 + 0.913985i $$0.632989\pi$$
$$180$$ 0 0
$$181$$ 6.26595 0.465744 0.232872 0.972507i $$-0.425188\pi$$
0.232872 + 0.972507i $$0.425188\pi$$
$$182$$ 0 0
$$183$$ 0.411016 0.0303832
$$184$$ 0 0
$$185$$ −9.59313 −0.705301
$$186$$ 0 0
$$187$$ 30.0466 2.19722
$$188$$ 0 0
$$189$$ −0.214814 −0.0156254
$$190$$ 0 0
$$191$$ −8.26296 −0.597887 −0.298943 0.954271i $$-0.596634\pi$$
−0.298943 + 0.954271i $$0.596634\pi$$
$$192$$ 0 0
$$193$$ −19.8469 −1.42861 −0.714306 0.699833i $$-0.753258\pi$$
−0.714306 + 0.699833i $$0.753258\pi$$
$$194$$ 0 0
$$195$$ 0.166458 0.0119203
$$196$$ 0 0
$$197$$ −4.24866 −0.302704 −0.151352 0.988480i $$-0.548363\pi$$
−0.151352 + 0.988480i $$0.548363\pi$$
$$198$$ 0 0
$$199$$ −4.63286 −0.328415 −0.164207 0.986426i $$-0.552507\pi$$
−0.164207 + 0.986426i $$0.552507\pi$$
$$200$$ 0 0
$$201$$ 0.643601 0.0453961
$$202$$ 0 0
$$203$$ 2.90568 0.203939
$$204$$ 0 0
$$205$$ −14.5752 −1.01798
$$206$$ 0 0
$$207$$ 0.514850 0.0357846
$$208$$ 0 0
$$209$$ −12.2161 −0.845002
$$210$$ 0 0
$$211$$ −10.2699 −0.707007 −0.353503 0.935433i $$-0.615010\pi$$
−0.353503 + 0.935433i $$0.615010\pi$$
$$212$$ 0 0
$$213$$ 0.570919 0.0391187
$$214$$ 0 0
$$215$$ 5.83939 0.398243
$$216$$ 0 0
$$217$$ −0.185954 −0.0126234
$$218$$ 0 0
$$219$$ 0.351275 0.0237369
$$220$$ 0 0
$$221$$ −9.71958 −0.653810
$$222$$ 0 0
$$223$$ −14.7887 −0.990326 −0.495163 0.868800i $$-0.664892\pi$$
−0.495163 + 0.868800i $$0.664892\pi$$
$$224$$ 0 0
$$225$$ 10.8528 0.723520
$$226$$ 0 0
$$227$$ −14.9154 −0.989969 −0.494984 0.868902i $$-0.664826\pi$$
−0.494984 + 0.868902i $$0.664826\pi$$
$$228$$ 0 0
$$229$$ −18.2488 −1.20592 −0.602959 0.797772i $$-0.706012\pi$$
−0.602959 + 0.797772i $$0.706012\pi$$
$$230$$ 0 0
$$231$$ 0.205919 0.0135485
$$232$$ 0 0
$$233$$ −16.6741 −1.09235 −0.546177 0.837670i $$-0.683918\pi$$
−0.546177 + 0.837670i $$0.683918\pi$$
$$234$$ 0 0
$$235$$ 0.611708 0.0399035
$$236$$ 0 0
$$237$$ −1.31337 −0.0853127
$$238$$ 0 0
$$239$$ −24.3691 −1.57631 −0.788153 0.615479i $$-0.788963\pi$$
−0.788153 + 0.615479i $$0.788963\pi$$
$$240$$ 0 0
$$241$$ 18.1071 1.16638 0.583190 0.812336i $$-0.301804\pi$$
0.583190 + 0.812336i $$0.301804\pi$$
$$242$$ 0 0
$$243$$ −2.05645 −0.131921
$$244$$ 0 0
$$245$$ −7.95100 −0.507971
$$246$$ 0 0
$$247$$ 3.95170 0.251441
$$248$$ 0 0
$$249$$ 0.456435 0.0289254
$$250$$ 0 0
$$251$$ −26.9687 −1.70225 −0.851124 0.524965i $$-0.824078\pi$$
−0.851124 + 0.524965i $$0.824078\pi$$
$$252$$ 0 0
$$253$$ −0.988024 −0.0621165
$$254$$ 0 0
$$255$$ −0.468295 −0.0293258
$$256$$ 0 0
$$257$$ 20.3022 1.26642 0.633208 0.773982i $$-0.281738\pi$$
0.633208 + 0.773982i $$0.281738\pi$$
$$258$$ 0 0
$$259$$ 3.83889 0.238537
$$260$$ 0 0
$$261$$ 18.5384 1.14750
$$262$$ 0 0
$$263$$ 12.0595 0.743622 0.371811 0.928309i $$-0.378737\pi$$
0.371811 + 0.928309i $$0.378737\pi$$
$$264$$ 0 0
$$265$$ 10.2746 0.631162
$$266$$ 0 0
$$267$$ 0.324995 0.0198894
$$268$$ 0 0
$$269$$ −25.0120 −1.52501 −0.762506 0.646982i $$-0.776031\pi$$
−0.762506 + 0.646982i $$0.776031\pi$$
$$270$$ 0 0
$$271$$ 18.0354 1.09557 0.547787 0.836618i $$-0.315470\pi$$
0.547787 + 0.836618i $$0.315470\pi$$
$$272$$ 0 0
$$273$$ −0.0666115 −0.00403151
$$274$$ 0 0
$$275$$ −20.8271 −1.25592
$$276$$ 0 0
$$277$$ −29.3688 −1.76460 −0.882301 0.470686i $$-0.844006\pi$$
−0.882301 + 0.470686i $$0.844006\pi$$
$$278$$ 0 0
$$279$$ −1.18639 −0.0710274
$$280$$ 0 0
$$281$$ 12.0977 0.721686 0.360843 0.932626i $$-0.382489\pi$$
0.360843 + 0.932626i $$0.382489\pi$$
$$282$$ 0 0
$$283$$ 8.04149 0.478017 0.239009 0.971017i $$-0.423178\pi$$
0.239009 + 0.971017i $$0.423178\pi$$
$$284$$ 0 0
$$285$$ 0.190395 0.0112780
$$286$$ 0 0
$$287$$ 5.83257 0.344286
$$288$$ 0 0
$$289$$ 10.3441 0.608477
$$290$$ 0 0
$$291$$ 0.202003 0.0118416
$$292$$ 0 0
$$293$$ −13.5236 −0.790056 −0.395028 0.918669i $$-0.629265\pi$$
−0.395028 + 0.918669i $$0.629265\pi$$
$$294$$ 0 0
$$295$$ −3.93221 −0.228942
$$296$$ 0 0
$$297$$ 2.63010 0.152614
$$298$$ 0 0
$$299$$ 0.319610 0.0184835
$$300$$ 0 0
$$301$$ −2.33675 −0.134688
$$302$$ 0 0
$$303$$ −0.487991 −0.0280343
$$304$$ 0 0
$$305$$ −6.31228 −0.361440
$$306$$ 0 0
$$307$$ −3.60632 −0.205824 −0.102912 0.994690i $$-0.532816\pi$$
−0.102912 + 0.994690i $$0.532816\pi$$
$$308$$ 0 0
$$309$$ −1.26772 −0.0721181
$$310$$ 0 0
$$311$$ 27.5438 1.56187 0.780934 0.624613i $$-0.214743\pi$$
0.780934 + 0.624613i $$0.214743\pi$$
$$312$$ 0 0
$$313$$ −30.2190 −1.70808 −0.854039 0.520209i $$-0.825854\pi$$
−0.854039 + 0.520209i $$0.825854\pi$$
$$314$$ 0 0
$$315$$ 1.64792 0.0928499
$$316$$ 0 0
$$317$$ −5.23152 −0.293831 −0.146916 0.989149i $$-0.546935\pi$$
−0.146916 + 0.989149i $$0.546935\pi$$
$$318$$ 0 0
$$319$$ −35.5761 −1.99188
$$320$$ 0 0
$$321$$ 0.699298 0.0390310
$$322$$ 0 0
$$323$$ −11.1173 −0.618584
$$324$$ 0 0
$$325$$ 6.73723 0.373714
$$326$$ 0 0
$$327$$ −0.497800 −0.0275284
$$328$$ 0 0
$$329$$ −0.244788 −0.0134956
$$330$$ 0 0
$$331$$ 5.56111 0.305666 0.152833 0.988252i $$-0.451160\pi$$
0.152833 + 0.988252i $$0.451160\pi$$
$$332$$ 0 0
$$333$$ 24.4923 1.34217
$$334$$ 0 0
$$335$$ −9.88426 −0.540035
$$336$$ 0 0
$$337$$ 1.20314 0.0655392 0.0327696 0.999463i $$-0.489567\pi$$
0.0327696 + 0.999463i $$0.489567\pi$$
$$338$$ 0 0
$$339$$ −0.104383 −0.00566930
$$340$$ 0 0
$$341$$ 2.27675 0.123293
$$342$$ 0 0
$$343$$ 6.46688 0.349178
$$344$$ 0 0
$$345$$ 0.0153990 0.000829054 0
$$346$$ 0 0
$$347$$ −18.8760 −1.01332 −0.506659 0.862147i $$-0.669120\pi$$
−0.506659 + 0.862147i $$0.669120\pi$$
$$348$$ 0 0
$$349$$ 18.3633 0.982966 0.491483 0.870887i $$-0.336455\pi$$
0.491483 + 0.870887i $$0.336455\pi$$
$$350$$ 0 0
$$351$$ −0.850795 −0.0454121
$$352$$ 0 0
$$353$$ −2.06519 −0.109919 −0.0549596 0.998489i $$-0.517503\pi$$
−0.0549596 + 0.998489i $$0.517503\pi$$
$$354$$ 0 0
$$355$$ −8.76802 −0.465358
$$356$$ 0 0
$$357$$ 0.187398 0.00991816
$$358$$ 0 0
$$359$$ 7.83605 0.413571 0.206786 0.978386i $$-0.433700\pi$$
0.206786 + 0.978386i $$0.433700\pi$$
$$360$$ 0 0
$$361$$ −14.4800 −0.762107
$$362$$ 0 0
$$363$$ −1.68121 −0.0882404
$$364$$ 0 0
$$365$$ −5.39478 −0.282376
$$366$$ 0 0
$$367$$ −27.3479 −1.42755 −0.713775 0.700375i $$-0.753016\pi$$
−0.713775 + 0.700375i $$0.753016\pi$$
$$368$$ 0 0
$$369$$ 37.2120 1.93718
$$370$$ 0 0
$$371$$ −4.11159 −0.213463
$$372$$ 0 0
$$373$$ 31.3381 1.62263 0.811313 0.584612i $$-0.198753\pi$$
0.811313 + 0.584612i $$0.198753\pi$$
$$374$$ 0 0
$$375$$ 0.772376 0.0398853
$$376$$ 0 0
$$377$$ 11.5083 0.592707
$$378$$ 0 0
$$379$$ 1.97192 0.101291 0.0506454 0.998717i $$-0.483872\pi$$
0.0506454 + 0.998717i $$0.483872\pi$$
$$380$$ 0 0
$$381$$ 0.517333 0.0265038
$$382$$ 0 0
$$383$$ 24.9455 1.27466 0.637328 0.770593i $$-0.280040\pi$$
0.637328 + 0.770593i $$0.280040\pi$$
$$384$$ 0 0
$$385$$ −3.16245 −0.161173
$$386$$ 0 0
$$387$$ −14.9086 −0.757845
$$388$$ 0 0
$$389$$ −8.98152 −0.455381 −0.227691 0.973734i $$-0.573117\pi$$
−0.227691 + 0.973734i $$0.573117\pi$$
$$390$$ 0 0
$$391$$ −0.899159 −0.0454724
$$392$$ 0 0
$$393$$ 1.27402 0.0642659
$$394$$ 0 0
$$395$$ 20.1704 1.01488
$$396$$ 0 0
$$397$$ −10.3125 −0.517569 −0.258784 0.965935i $$-0.583322\pi$$
−0.258784 + 0.965935i $$0.583322\pi$$
$$398$$ 0 0
$$399$$ −0.0761906 −0.00381430
$$400$$ 0 0
$$401$$ −22.2691 −1.11207 −0.556033 0.831160i $$-0.687677\pi$$
−0.556033 + 0.831160i $$0.687677\pi$$
$$402$$ 0 0
$$403$$ −0.736492 −0.0366873
$$404$$ 0 0
$$405$$ 10.4933 0.521416
$$406$$ 0 0
$$407$$ −47.0019 −2.32980
$$408$$ 0 0
$$409$$ −9.36108 −0.462876 −0.231438 0.972850i $$-0.574343\pi$$
−0.231438 + 0.972850i $$0.574343\pi$$
$$410$$ 0 0
$$411$$ −0.997771 −0.0492164
$$412$$ 0 0
$$413$$ 1.57356 0.0774297
$$414$$ 0 0
$$415$$ −7.00980 −0.344098
$$416$$ 0 0
$$417$$ −0.828292 −0.0405616
$$418$$ 0 0
$$419$$ 34.9797 1.70887 0.854436 0.519557i $$-0.173903\pi$$
0.854436 + 0.519557i $$0.173903\pi$$
$$420$$ 0 0
$$421$$ 15.9244 0.776107 0.388054 0.921637i $$-0.373147\pi$$
0.388054 + 0.921637i $$0.373147\pi$$
$$422$$ 0 0
$$423$$ −1.56176 −0.0759352
$$424$$ 0 0
$$425$$ −18.9538 −0.919396
$$426$$ 0 0
$$427$$ 2.52599 0.122241
$$428$$ 0 0
$$429$$ 0.815566 0.0393759
$$430$$ 0 0
$$431$$ 37.5257 1.80755 0.903775 0.428008i $$-0.140785\pi$$
0.903775 + 0.428008i $$0.140785\pi$$
$$432$$ 0 0
$$433$$ 27.7776 1.33491 0.667454 0.744651i $$-0.267384\pi$$
0.667454 + 0.744651i $$0.267384\pi$$
$$434$$ 0 0
$$435$$ 0.554476 0.0265851
$$436$$ 0 0
$$437$$ 0.365572 0.0174877
$$438$$ 0 0
$$439$$ −37.3507 −1.78265 −0.891327 0.453362i $$-0.850225\pi$$
−0.891327 + 0.453362i $$0.850225\pi$$
$$440$$ 0 0
$$441$$ 20.2997 0.966654
$$442$$ 0 0
$$443$$ −15.8245 −0.751844 −0.375922 0.926651i $$-0.622674\pi$$
−0.375922 + 0.926651i $$0.622674\pi$$
$$444$$ 0 0
$$445$$ −4.99119 −0.236605
$$446$$ 0 0
$$447$$ −0.00160992 −7.61467e−5 0
$$448$$ 0 0
$$449$$ −33.0395 −1.55923 −0.779616 0.626258i $$-0.784586\pi$$
−0.779616 + 0.626258i $$0.784586\pi$$
$$450$$ 0 0
$$451$$ −71.4118 −3.36265
$$452$$ 0 0
$$453$$ 1.15845 0.0544289
$$454$$ 0 0
$$455$$ 1.02300 0.0479591
$$456$$ 0 0
$$457$$ −6.38585 −0.298717 −0.149359 0.988783i $$-0.547721\pi$$
−0.149359 + 0.988783i $$0.547721\pi$$
$$458$$ 0 0
$$459$$ 2.39354 0.111721
$$460$$ 0 0
$$461$$ 4.10510 0.191193 0.0955967 0.995420i $$-0.469524\pi$$
0.0955967 + 0.995420i $$0.469524\pi$$
$$462$$ 0 0
$$463$$ 21.9482 1.02002 0.510011 0.860168i $$-0.329641\pi$$
0.510011 + 0.860168i $$0.329641\pi$$
$$464$$ 0 0
$$465$$ −0.0354846 −0.00164556
$$466$$ 0 0
$$467$$ 20.9539 0.969633 0.484816 0.874616i $$-0.338886\pi$$
0.484816 + 0.874616i $$0.338886\pi$$
$$468$$ 0 0
$$469$$ 3.95539 0.182643
$$470$$ 0 0
$$471$$ −0.850964 −0.0392104
$$472$$ 0 0
$$473$$ 28.6103 1.31550
$$474$$ 0 0
$$475$$ 7.70608 0.353579
$$476$$ 0 0
$$477$$ −26.2321 −1.20108
$$478$$ 0 0
$$479$$ −40.1617 −1.83503 −0.917517 0.397698i $$-0.869809\pi$$
−0.917517 + 0.397698i $$0.869809\pi$$
$$480$$ 0 0
$$481$$ 15.2044 0.693260
$$482$$ 0 0
$$483$$ −0.00616223 −0.000280391 0
$$484$$ 0 0
$$485$$ −3.10232 −0.140869
$$486$$ 0 0
$$487$$ 1.56804 0.0710547 0.0355273 0.999369i $$-0.488689\pi$$
0.0355273 + 0.999369i $$0.488689\pi$$
$$488$$ 0 0
$$489$$ −0.789497 −0.0357023
$$490$$ 0 0
$$491$$ −9.66243 −0.436059 −0.218030 0.975942i $$-0.569963\pi$$
−0.218030 + 0.975942i $$0.569963\pi$$
$$492$$ 0 0
$$493$$ −32.3763 −1.45815
$$494$$ 0 0
$$495$$ −20.1765 −0.906868
$$496$$ 0 0
$$497$$ 3.50871 0.157387
$$498$$ 0 0
$$499$$ 20.6363 0.923810 0.461905 0.886930i $$-0.347166\pi$$
0.461905 + 0.886930i $$0.347166\pi$$
$$500$$ 0 0
$$501$$ −1.19465 −0.0533728
$$502$$ 0 0
$$503$$ 1.00000 0.0445878
$$504$$ 0 0
$$505$$ 7.49444 0.333498
$$506$$ 0 0
$$507$$ 0.728890 0.0323712
$$508$$ 0 0
$$509$$ −23.3284 −1.03401 −0.517006 0.855982i $$-0.672953\pi$$
−0.517006 + 0.855982i $$0.672953\pi$$
$$510$$ 0 0
$$511$$ 2.15883 0.0955012
$$512$$ 0 0
$$513$$ −0.973144 −0.0429654
$$514$$ 0 0
$$515$$ 19.4693 0.857921
$$516$$ 0 0
$$517$$ 2.99709 0.131812
$$518$$ 0 0
$$519$$ 0.434744 0.0190832
$$520$$ 0 0
$$521$$ −43.2647 −1.89546 −0.947730 0.319073i $$-0.896629\pi$$
−0.947730 + 0.319073i $$0.896629\pi$$
$$522$$ 0 0
$$523$$ 43.4082 1.89811 0.949054 0.315112i $$-0.102042\pi$$
0.949054 + 0.315112i $$0.102042\pi$$
$$524$$ 0 0
$$525$$ −0.129897 −0.00566916
$$526$$ 0 0
$$527$$ 2.07197 0.0902565
$$528$$ 0 0
$$529$$ −22.9704 −0.998714
$$530$$ 0 0
$$531$$ 10.0394 0.435671
$$532$$ 0 0
$$533$$ 23.1006 1.00060
$$534$$ 0 0
$$535$$ −10.7396 −0.464315
$$536$$ 0 0
$$537$$ 0.829073 0.0357771
$$538$$ 0 0
$$539$$ −38.9562 −1.67796
$$540$$ 0 0
$$541$$ 32.5657 1.40011 0.700054 0.714090i $$-0.253159\pi$$
0.700054 + 0.714090i $$0.253159\pi$$
$$542$$ 0 0
$$543$$ −0.478484 −0.0205337
$$544$$ 0 0
$$545$$ 7.64509 0.327480
$$546$$ 0 0
$$547$$ −27.8377 −1.19025 −0.595127 0.803631i $$-0.702898\pi$$
−0.595127 + 0.803631i $$0.702898\pi$$
$$548$$ 0 0
$$549$$ 16.1159 0.687811
$$550$$ 0 0
$$551$$ 13.1633 0.560773
$$552$$ 0 0
$$553$$ −8.07162 −0.343240
$$554$$ 0 0
$$555$$ 0.732556 0.0310953
$$556$$ 0 0
$$557$$ 15.7225 0.666182 0.333091 0.942895i $$-0.391908\pi$$
0.333091 + 0.942895i $$0.391908\pi$$
$$558$$ 0 0
$$559$$ −9.25498 −0.391444
$$560$$ 0 0
$$561$$ −2.29443 −0.0968710
$$562$$ 0 0
$$563$$ 24.7817 1.04442 0.522212 0.852816i $$-0.325107\pi$$
0.522212 + 0.852816i $$0.325107\pi$$
$$564$$ 0 0
$$565$$ 1.60309 0.0674423
$$566$$ 0 0
$$567$$ −4.19911 −0.176346
$$568$$ 0 0
$$569$$ 2.95346 0.123816 0.0619078 0.998082i $$-0.480282\pi$$
0.0619078 + 0.998082i $$0.480282\pi$$
$$570$$ 0 0
$$571$$ 10.7926 0.451656 0.225828 0.974167i $$-0.427491\pi$$
0.225828 + 0.974167i $$0.427491\pi$$
$$572$$ 0 0
$$573$$ 0.630981 0.0263596
$$574$$ 0 0
$$575$$ 0.623261 0.0259918
$$576$$ 0 0
$$577$$ −5.92593 −0.246700 −0.123350 0.992363i $$-0.539364\pi$$
−0.123350 + 0.992363i $$0.539364\pi$$
$$578$$ 0 0
$$579$$ 1.51556 0.0629845
$$580$$ 0 0
$$581$$ 2.80512 0.116376
$$582$$ 0 0
$$583$$ 50.3407 2.08490
$$584$$ 0 0
$$585$$ 6.52679 0.269850
$$586$$ 0 0
$$587$$ 14.8090 0.611234 0.305617 0.952155i $$-0.401137\pi$$
0.305617 + 0.952155i $$0.401137\pi$$
$$588$$ 0 0
$$589$$ −0.842403 −0.0347106
$$590$$ 0 0
$$591$$ 0.324438 0.0133456
$$592$$ 0 0
$$593$$ 33.4465 1.37348 0.686741 0.726902i $$-0.259041\pi$$
0.686741 + 0.726902i $$0.259041\pi$$
$$594$$ 0 0
$$595$$ −2.87801 −0.117987
$$596$$ 0 0
$$597$$ 0.353777 0.0144791
$$598$$ 0 0
$$599$$ −9.35823 −0.382367 −0.191183 0.981554i $$-0.561233\pi$$
−0.191183 + 0.981554i $$0.561233\pi$$
$$600$$ 0 0
$$601$$ −6.16182 −0.251346 −0.125673 0.992072i $$-0.540109\pi$$
−0.125673 + 0.992072i $$0.540109\pi$$
$$602$$ 0 0
$$603$$ 25.2355 1.02767
$$604$$ 0 0
$$605$$ 25.8195 1.04971
$$606$$ 0 0
$$607$$ −35.7361 −1.45048 −0.725242 0.688494i $$-0.758272\pi$$
−0.725242 + 0.688494i $$0.758272\pi$$
$$608$$ 0 0
$$609$$ −0.221885 −0.00899124
$$610$$ 0 0
$$611$$ −0.969511 −0.0392222
$$612$$ 0 0
$$613$$ −1.96528 −0.0793771 −0.0396885 0.999212i $$-0.512637\pi$$
−0.0396885 + 0.999212i $$0.512637\pi$$
$$614$$ 0 0
$$615$$ 1.11300 0.0448805
$$616$$ 0 0
$$617$$ −35.4461 −1.42701 −0.713503 0.700652i $$-0.752893\pi$$
−0.713503 + 0.700652i $$0.752893\pi$$
$$618$$ 0 0
$$619$$ 34.4643 1.38524 0.692618 0.721304i $$-0.256457\pi$$
0.692618 + 0.721304i $$0.256457\pi$$
$$620$$ 0 0
$$621$$ −0.0787071 −0.00315841
$$622$$ 0 0
$$623$$ 1.99733 0.0800213
$$624$$ 0 0
$$625$$ 6.26126 0.250451
$$626$$ 0 0
$$627$$ 0.932849 0.0372544
$$628$$ 0 0
$$629$$ −42.7745 −1.70553
$$630$$ 0 0
$$631$$ 29.7560 1.18457 0.592284 0.805729i $$-0.298226\pi$$
0.592284 + 0.805729i $$0.298226\pi$$
$$632$$ 0 0
$$633$$ 0.784233 0.0311705
$$634$$ 0 0
$$635$$ −7.94506 −0.315290
$$636$$ 0 0
$$637$$ 12.6017 0.499298
$$638$$ 0 0
$$639$$ 22.3857 0.885564
$$640$$ 0 0
$$641$$ −32.9327 −1.30077 −0.650383 0.759607i $$-0.725391\pi$$
−0.650383 + 0.759607i $$0.725391\pi$$
$$642$$ 0 0
$$643$$ 0.492253 0.0194126 0.00970628 0.999953i $$-0.496910\pi$$
0.00970628 + 0.999953i $$0.496910\pi$$
$$644$$ 0 0
$$645$$ −0.445911 −0.0175577
$$646$$ 0 0
$$647$$ −3.76376 −0.147969 −0.0739844 0.997259i $$-0.523571\pi$$
−0.0739844 + 0.997259i $$0.523571\pi$$
$$648$$ 0 0
$$649$$ −19.2660 −0.756258
$$650$$ 0 0
$$651$$ 0.0141999 0.000556538 0
$$652$$ 0 0
$$653$$ 18.5728 0.726809 0.363404 0.931631i $$-0.381614\pi$$
0.363404 + 0.931631i $$0.381614\pi$$
$$654$$ 0 0
$$655$$ −19.5661 −0.764510
$$656$$ 0 0
$$657$$ 13.7734 0.537353
$$658$$ 0 0
$$659$$ 34.1930 1.33197 0.665985 0.745965i $$-0.268012\pi$$
0.665985 + 0.745965i $$0.268012\pi$$
$$660$$ 0 0
$$661$$ 29.5439 1.14912 0.574562 0.818461i $$-0.305173\pi$$
0.574562 + 0.818461i $$0.305173\pi$$
$$662$$ 0 0
$$663$$ 0.742212 0.0288251
$$664$$ 0 0
$$665$$ 1.17012 0.0453751
$$666$$ 0 0
$$667$$ 1.06463 0.0412227
$$668$$ 0 0
$$669$$ 1.12930 0.0436614
$$670$$ 0 0
$$671$$ −30.9273 −1.19393
$$672$$ 0 0
$$673$$ −9.15185 −0.352778 −0.176389 0.984321i $$-0.556442\pi$$
−0.176389 + 0.984321i $$0.556442\pi$$
$$674$$ 0 0
$$675$$ −1.65911 −0.0638591
$$676$$ 0 0
$$677$$ −33.1256 −1.27312 −0.636561 0.771227i $$-0.719644\pi$$
−0.636561 + 0.771227i $$0.719644\pi$$
$$678$$ 0 0
$$679$$ 1.24146 0.0476427
$$680$$ 0 0
$$681$$ 1.13898 0.0436457
$$682$$ 0 0
$$683$$ −49.1452 −1.88049 −0.940245 0.340498i $$-0.889404\pi$$
−0.940245 + 0.340498i $$0.889404\pi$$
$$684$$ 0 0
$$685$$ 15.3235 0.585481
$$686$$ 0 0
$$687$$ 1.39353 0.0531664
$$688$$ 0 0
$$689$$ −16.2844 −0.620387
$$690$$ 0 0
$$691$$ −34.1013 −1.29727 −0.648637 0.761098i $$-0.724661\pi$$
−0.648637 + 0.761098i $$0.724661\pi$$
$$692$$ 0 0
$$693$$ 8.07406 0.306708
$$694$$ 0 0
$$695$$ 12.7207 0.482523
$$696$$ 0 0
$$697$$ −64.9889 −2.46163
$$698$$ 0 0
$$699$$ 1.27327 0.0481596
$$700$$ 0 0
$$701$$ 26.2590 0.991787 0.495894 0.868383i $$-0.334841\pi$$
0.495894 + 0.868383i $$0.334841\pi$$
$$702$$ 0 0
$$703$$ 17.3909 0.655909
$$704$$ 0 0
$$705$$ −0.0467116 −0.00175926
$$706$$ 0 0
$$707$$ −2.99905 −0.112791
$$708$$ 0 0
$$709$$ 28.4811 1.06963 0.534815 0.844969i $$-0.320381\pi$$
0.534815 + 0.844969i $$0.320381\pi$$
$$710$$ 0 0
$$711$$ −51.4973 −1.93130
$$712$$ 0 0
$$713$$ −0.0681329 −0.00255160
$$714$$ 0 0
$$715$$ −12.5252 −0.468418
$$716$$ 0 0
$$717$$ 1.86089 0.0694961
$$718$$ 0 0
$$719$$ −13.8368 −0.516024 −0.258012 0.966142i $$-0.583067\pi$$
−0.258012 + 0.966142i $$0.583067\pi$$
$$720$$ 0 0
$$721$$ −7.79106 −0.290154
$$722$$ 0 0
$$723$$ −1.38270 −0.0514233
$$724$$ 0 0
$$725$$ 22.4419 0.833473
$$726$$ 0 0
$$727$$ −9.61759 −0.356697 −0.178348 0.983967i $$-0.557075\pi$$
−0.178348 + 0.983967i $$0.557075\pi$$
$$728$$ 0 0
$$729$$ −26.6856 −0.988356
$$730$$ 0 0
$$731$$ 26.0370 0.963015
$$732$$ 0 0
$$733$$ −16.9530 −0.626175 −0.313087 0.949724i $$-0.601363\pi$$
−0.313087 + 0.949724i $$0.601363\pi$$
$$734$$ 0 0
$$735$$ 0.607158 0.0223954
$$736$$ 0 0
$$737$$ −48.4283 −1.78388
$$738$$ 0 0
$$739$$ −9.82452 −0.361401 −0.180700 0.983538i $$-0.557836\pi$$
−0.180700 + 0.983538i $$0.557836\pi$$
$$740$$ 0 0
$$741$$ −0.301762 −0.0110855
$$742$$ 0 0
$$743$$ −28.8650 −1.05896 −0.529478 0.848324i $$-0.677612\pi$$
−0.529478 + 0.848324i $$0.677612\pi$$
$$744$$ 0 0
$$745$$ 0.0247248 0.000905845 0
$$746$$ 0 0
$$747$$ 17.8968 0.654808
$$748$$ 0 0
$$749$$ 4.29769 0.157034
$$750$$ 0 0
$$751$$ 9.70834 0.354262 0.177131 0.984187i $$-0.443318\pi$$
0.177131 + 0.984187i $$0.443318\pi$$
$$752$$ 0 0
$$753$$ 2.05940 0.0750486
$$754$$ 0 0
$$755$$ −17.7912 −0.647489
$$756$$ 0 0
$$757$$ −5.99476 −0.217883 −0.108942 0.994048i $$-0.534746\pi$$
−0.108942 + 0.994048i $$0.534746\pi$$
$$758$$ 0 0
$$759$$ 0.0754480 0.00273859
$$760$$ 0 0
$$761$$ 15.4793 0.561123 0.280562 0.959836i $$-0.409479\pi$$
0.280562 + 0.959836i $$0.409479\pi$$
$$762$$ 0 0
$$763$$ −3.05934 −0.110756
$$764$$ 0 0
$$765$$ −18.3618 −0.663873
$$766$$ 0 0
$$767$$ 6.23226 0.225034
$$768$$ 0 0
$$769$$ −27.9740 −1.00877 −0.504384 0.863480i $$-0.668280\pi$$
−0.504384 + 0.863480i $$0.668280\pi$$
$$770$$ 0 0
$$771$$ −1.55033 −0.0558336
$$772$$ 0 0
$$773$$ 31.7330 1.14136 0.570679 0.821173i $$-0.306680\pi$$
0.570679 + 0.821173i $$0.306680\pi$$
$$774$$ 0 0
$$775$$ −1.43621 −0.0515901
$$776$$ 0 0
$$777$$ −0.293147 −0.0105166
$$778$$ 0 0
$$779$$ 26.4226 0.946687
$$780$$ 0 0
$$781$$ −42.9593 −1.53720
$$782$$ 0 0
$$783$$ −2.83403 −0.101280
$$784$$ 0 0
$$785$$ 13.0689 0.466448
$$786$$ 0 0
$$787$$ −19.1339 −0.682051 −0.341025 0.940054i $$-0.610774\pi$$
−0.341025 + 0.940054i $$0.610774\pi$$
$$788$$ 0 0
$$789$$ −0.920895 −0.0327847
$$790$$ 0 0
$$791$$ −0.641508 −0.0228094
$$792$$ 0 0
$$793$$ 10.0045 0.355270
$$794$$ 0 0
$$795$$ −0.784593 −0.0278266
$$796$$ 0 0
$$797$$ −2.45993 −0.0871352 −0.0435676 0.999050i $$-0.513872\pi$$
−0.0435676 + 0.999050i $$0.513872\pi$$
$$798$$ 0 0
$$799$$ 2.72752 0.0964929
$$800$$ 0 0
$$801$$ 12.7430 0.450253
$$802$$ 0 0
$$803$$ −26.4320 −0.932764
$$804$$ 0 0
$$805$$ 0.0946379 0.00333555
$$806$$ 0 0
$$807$$ 1.90998 0.0672346
$$808$$ 0 0
$$809$$ 16.9350 0.595404 0.297702 0.954659i $$-0.403780\pi$$
0.297702 + 0.954659i $$0.403780\pi$$
$$810$$ 0 0
$$811$$ 23.7584 0.834269 0.417135 0.908845i $$-0.363034\pi$$
0.417135 + 0.908845i $$0.363034\pi$$
$$812$$ 0 0
$$813$$ −1.37723 −0.0483016
$$814$$ 0 0
$$815$$ 12.1249 0.424716
$$816$$ 0 0
$$817$$ −10.5859 −0.370354
$$818$$ 0 0
$$819$$ −2.61183 −0.0912648
$$820$$ 0 0
$$821$$ 19.2765 0.672755 0.336378 0.941727i $$-0.390798\pi$$
0.336378 + 0.941727i $$0.390798\pi$$
$$822$$ 0 0
$$823$$ −52.8192 −1.84116 −0.920580 0.390553i $$-0.872284\pi$$
−0.920580 + 0.390553i $$0.872284\pi$$
$$824$$ 0 0
$$825$$ 1.59041 0.0553709
$$826$$ 0 0
$$827$$ 25.7144 0.894176 0.447088 0.894490i $$-0.352461\pi$$
0.447088 + 0.894490i $$0.352461\pi$$
$$828$$ 0 0
$$829$$ 23.3380 0.810562 0.405281 0.914192i $$-0.367174\pi$$
0.405281 + 0.914192i $$0.367174\pi$$
$$830$$ 0 0
$$831$$ 2.24268 0.0777976
$$832$$ 0 0
$$833$$ −35.4524 −1.22835
$$834$$ 0 0
$$835$$ 18.3471 0.634926
$$836$$ 0 0
$$837$$ 0.181368 0.00626900
$$838$$ 0 0
$$839$$ −18.8028 −0.649144 −0.324572 0.945861i $$-0.605220\pi$$
−0.324572 + 0.945861i $$0.605220\pi$$
$$840$$ 0 0
$$841$$ 9.33453 0.321880
$$842$$ 0 0
$$843$$ −0.923809 −0.0318177
$$844$$ 0 0
$$845$$ −11.1941 −0.385089
$$846$$ 0 0
$$847$$ −10.3322 −0.355019
$$848$$ 0 0
$$849$$ −0.614069 −0.0210748
$$850$$ 0 0
$$851$$ 1.40656 0.0482162
$$852$$ 0 0
$$853$$ −45.0553 −1.54266 −0.771332 0.636433i $$-0.780409\pi$$
−0.771332 + 0.636433i $$0.780409\pi$$
$$854$$ 0 0
$$855$$ 7.46538 0.255311
$$856$$ 0 0
$$857$$ −33.4362 −1.14216 −0.571079 0.820895i $$-0.693475\pi$$
−0.571079 + 0.820895i $$0.693475\pi$$
$$858$$ 0 0
$$859$$ −19.4483 −0.663567 −0.331784 0.943355i $$-0.607650\pi$$
−0.331784 + 0.943355i $$0.607650\pi$$
$$860$$ 0 0
$$861$$ −0.445390 −0.0151789
$$862$$ 0 0
$$863$$ 16.1628 0.550186 0.275093 0.961418i $$-0.411291\pi$$
0.275093 + 0.961418i $$0.411291\pi$$
$$864$$ 0 0
$$865$$ −6.67669 −0.227014
$$866$$ 0 0
$$867$$ −0.789902 −0.0268265
$$868$$ 0 0
$$869$$ 98.8259 3.35244
$$870$$ 0 0
$$871$$ 15.6658 0.530815
$$872$$ 0 0
$$873$$ 7.92054 0.268070
$$874$$ 0 0
$$875$$ 4.74681 0.160471
$$876$$ 0 0
$$877$$ −22.3959 −0.756255 −0.378127 0.925754i $$-0.623432\pi$$
−0.378127 + 0.925754i $$0.623432\pi$$
$$878$$ 0 0
$$879$$ 1.03269 0.0348319
$$880$$ 0 0
$$881$$ 10.4074 0.350633 0.175316 0.984512i $$-0.443905\pi$$
0.175316 + 0.984512i $$0.443905\pi$$
$$882$$ 0 0
$$883$$ 18.3582 0.617803 0.308902 0.951094i $$-0.400039\pi$$
0.308902 + 0.951094i $$0.400039\pi$$
$$884$$ 0 0
$$885$$ 0.300274 0.0100936
$$886$$ 0 0
$$887$$ 22.9758 0.771453 0.385726 0.922613i $$-0.373951\pi$$
0.385726 + 0.922613i $$0.373951\pi$$
$$888$$ 0 0
$$889$$ 3.17938 0.106633
$$890$$ 0 0
$$891$$ 51.4123 1.72238
$$892$$ 0 0
$$893$$ −1.10893 −0.0371090
$$894$$ 0 0
$$895$$ −12.7327 −0.425607
$$896$$ 0 0
$$897$$ −0.0244062 −0.000814900 0
$$898$$ 0 0
$$899$$ −2.45328 −0.0818215
$$900$$ 0 0
$$901$$ 45.8130 1.52625
$$902$$ 0 0
$$903$$ 0.178440 0.00593812
$$904$$ 0 0
$$905$$ 7.34843 0.244270
$$906$$ 0 0
$$907$$ −0.415835 −0.0138076 −0.00690378 0.999976i $$-0.502198\pi$$
−0.00690378 + 0.999976i $$0.502198\pi$$
$$908$$ 0 0
$$909$$ −19.1341 −0.634638
$$910$$ 0 0
$$911$$ 13.6089 0.450881 0.225441 0.974257i $$-0.427618\pi$$
0.225441 + 0.974257i $$0.427618\pi$$
$$912$$ 0 0
$$913$$ −34.3448 −1.13665
$$914$$ 0 0
$$915$$ 0.482022 0.0159351
$$916$$ 0 0
$$917$$ 7.82978 0.258562
$$918$$ 0 0
$$919$$ 14.8913 0.491218 0.245609 0.969369i $$-0.421012\pi$$
0.245609 + 0.969369i $$0.421012\pi$$
$$920$$ 0 0
$$921$$ 0.275388 0.00907433
$$922$$ 0 0
$$923$$ 13.8966 0.457414
$$924$$ 0 0
$$925$$ 29.6496 0.974871
$$926$$ 0 0
$$927$$ −49.7072 −1.63260
$$928$$ 0 0
$$929$$ −19.4977 −0.639699 −0.319850 0.947468i $$-0.603632\pi$$
−0.319850 + 0.947468i $$0.603632\pi$$
$$930$$ 0 0
$$931$$ 14.4139 0.472397
$$932$$ 0 0
$$933$$ −2.10332 −0.0688595
$$934$$ 0 0
$$935$$ 35.2373 1.15238
$$936$$ 0 0
$$937$$ 12.5125 0.408764 0.204382 0.978891i $$-0.434481\pi$$
0.204382 + 0.978891i $$0.434481\pi$$
$$938$$ 0 0
$$939$$ 2.30760 0.0753056
$$940$$ 0 0
$$941$$ 4.08256 0.133088 0.0665439 0.997783i $$-0.478803\pi$$
0.0665439 + 0.997783i $$0.478803\pi$$
$$942$$ 0 0
$$943$$ 2.13704 0.0695915
$$944$$ 0 0
$$945$$ −0.251924 −0.00819509
$$946$$ 0 0
$$947$$ 0.698574 0.0227006 0.0113503 0.999936i $$-0.496387\pi$$
0.0113503 + 0.999936i $$0.496387\pi$$
$$948$$ 0 0
$$949$$ 8.55032 0.277555
$$950$$ 0 0
$$951$$ 0.399492 0.0129544
$$952$$ 0 0
$$953$$ 7.31223 0.236866 0.118433 0.992962i $$-0.462213\pi$$
0.118433 + 0.992962i $$0.462213\pi$$
$$954$$ 0 0
$$955$$ −9.69044 −0.313575
$$956$$ 0 0
$$957$$ 2.71668 0.0878178
$$958$$ 0 0
$$959$$ −6.13202 −0.198013
$$960$$ 0 0
$$961$$ −30.8430 −0.994935
$$962$$ 0 0
$$963$$ 27.4194 0.883578
$$964$$ 0 0
$$965$$ −23.2756 −0.749267
$$966$$ 0 0
$$967$$ −38.1107 −1.22556 −0.612778 0.790255i $$-0.709948\pi$$
−0.612778 + 0.790255i $$0.709948\pi$$
$$968$$ 0 0
$$969$$ 0.848946 0.0272721
$$970$$ 0 0
$$971$$ 51.0424 1.63803 0.819014 0.573773i $$-0.194521\pi$$
0.819014 + 0.573773i $$0.194521\pi$$
$$972$$ 0 0
$$973$$ −5.09045 −0.163192
$$974$$ 0 0
$$975$$ −0.514472 −0.0164763
$$976$$ 0 0
$$977$$ 53.4134 1.70885 0.854424 0.519577i $$-0.173910\pi$$
0.854424 + 0.519577i $$0.173910\pi$$
$$978$$ 0 0
$$979$$ −24.4546 −0.781571
$$980$$ 0 0
$$981$$ −19.5187 −0.623185
$$982$$ 0 0
$$983$$ −1.20932 −0.0385714 −0.0192857 0.999814i $$-0.506139\pi$$
−0.0192857 + 0.999814i $$0.506139\pi$$
$$984$$ 0 0
$$985$$ −4.98264 −0.158760
$$986$$ 0 0
$$987$$ 0.0186926 0.000594993 0
$$988$$ 0 0
$$989$$ −0.856179 −0.0272249
$$990$$ 0 0
$$991$$ 13.3298 0.423433 0.211717 0.977331i $$-0.432095\pi$$
0.211717 + 0.977331i $$0.432095\pi$$
$$992$$ 0 0
$$993$$ −0.424660 −0.0134762
$$994$$ 0 0
$$995$$ −5.43321 −0.172244
$$996$$ 0 0
$$997$$ −39.2475 −1.24298 −0.621490 0.783422i $$-0.713472\pi$$
−0.621490 + 0.783422i $$0.713472\pi$$
$$998$$ 0 0
$$999$$ −3.74423 −0.118462
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 8048.2.a.p.1.4 10
4.3 odd 2 503.2.a.e.1.4 10
12.11 even 2 4527.2.a.k.1.7 10

By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.4 10 4.3 odd 2
4527.2.a.k.1.7 10 12.11 even 2
8048.2.a.p.1.4 10 1.1 even 1 trivial