# Properties

 Label 644.2.a.d.1.3 Level $644$ Weight $2$ Character 644.1 Self dual yes Analytic conductor $5.142$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$644 = 2^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 644.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.14236589017$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.6963152.1 Defining polynomial: $$x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10$$ x^5 - 2*x^4 - 10*x^3 + 10*x^2 + 29*x + 10 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.435854$$ of defining polynomial Character $$\chi$$ $$=$$ 644.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.43585 q^{3} +2.34236 q^{5} -1.00000 q^{7} -0.938323 q^{9} +O(q^{10})$$ $$q+1.43585 q^{3} +2.34236 q^{5} -1.00000 q^{7} -0.938323 q^{9} +5.38507 q^{11} -5.32339 q^{13} +3.36328 q^{15} +7.88754 q^{17} +6.06364 q^{19} -1.43585 q^{21} -1.00000 q^{23} +0.486637 q^{25} -5.65486 q^{27} +1.87468 q^{29} -5.83096 q^{31} +7.73218 q^{33} -2.34236 q^{35} +1.19193 q^{37} -7.64362 q^{39} +6.45169 q^{41} -6.93535 q^{43} -2.19789 q^{45} -10.8577 q^{47} +1.00000 q^{49} +11.3254 q^{51} +7.94029 q^{53} +12.6138 q^{55} +8.70650 q^{57} -2.69561 q^{59} -3.27771 q^{61} +0.938323 q^{63} -12.4693 q^{65} -1.82865 q^{67} -1.43585 q^{69} +7.30161 q^{71} -7.51039 q^{73} +0.698740 q^{75} -5.38507 q^{77} -2.49158 q^{79} -5.30458 q^{81} -16.8120 q^{83} +18.4754 q^{85} +2.69177 q^{87} +13.2665 q^{89} +5.32339 q^{91} -8.37240 q^{93} +14.2032 q^{95} +11.9624 q^{97} -5.05294 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10})$$ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 $$5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100})$$ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 + 2 * q^11 + 13 * q^13 + 4 * q^15 + 4 * q^17 + 12 * q^19 - 3 * q^21 - 5 * q^23 + 19 * q^25 + 15 * q^27 + 13 * q^29 - 3 * q^31 + 24 * q^33 - 2 * q^35 - 4 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - 16 * q^45 + 5 * q^47 + 5 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 12 * q^59 + 20 * q^61 - 10 * q^63 - 12 * q^65 - 12 * q^67 - 3 * q^69 + 9 * q^71 - 9 * q^73 + 35 * q^75 - 2 * q^77 - 8 * q^79 - 11 * q^81 - 28 * q^83 + 16 * q^85 - 15 * q^87 + 32 * q^89 - 13 * q^91 - 15 * q^93 - 36 * q^95 + 4 * q^97 + 28 * q^99

## 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
$$3$$ 1.43585 0.828991 0.414495 0.910051i $$-0.363958\pi$$
0.414495 + 0.910051i $$0.363958\pi$$
$$4$$ 0 0
$$5$$ 2.34236 1.04753 0.523767 0.851862i $$-0.324526\pi$$
0.523767 + 0.851862i $$0.324526\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −0.938323 −0.312774
$$10$$ 0 0
$$11$$ 5.38507 1.62366 0.811830 0.583894i $$-0.198472\pi$$
0.811830 + 0.583894i $$0.198472\pi$$
$$12$$ 0 0
$$13$$ −5.32339 −1.47644 −0.738222 0.674558i $$-0.764334\pi$$
−0.738222 + 0.674558i $$0.764334\pi$$
$$14$$ 0 0
$$15$$ 3.36328 0.868396
$$16$$ 0 0
$$17$$ 7.88754 1.91301 0.956505 0.291717i $$-0.0942265\pi$$
0.956505 + 0.291717i $$0.0942265\pi$$
$$18$$ 0 0
$$19$$ 6.06364 1.39109 0.695547 0.718480i $$-0.255162\pi$$
0.695547 + 0.718480i $$0.255162\pi$$
$$20$$ 0 0
$$21$$ −1.43585 −0.313329
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0.486637 0.0973275
$$26$$ 0 0
$$27$$ −5.65486 −1.08828
$$28$$ 0 0
$$29$$ 1.87468 0.348120 0.174060 0.984735i $$-0.444311\pi$$
0.174060 + 0.984735i $$0.444311\pi$$
$$30$$ 0 0
$$31$$ −5.83096 −1.04727 −0.523635 0.851942i $$-0.675425\pi$$
−0.523635 + 0.851942i $$0.675425\pi$$
$$32$$ 0 0
$$33$$ 7.73218 1.34600
$$34$$ 0 0
$$35$$ −2.34236 −0.395931
$$36$$ 0 0
$$37$$ 1.19193 0.195952 0.0979762 0.995189i $$-0.468763\pi$$
0.0979762 + 0.995189i $$0.468763\pi$$
$$38$$ 0 0
$$39$$ −7.64362 −1.22396
$$40$$ 0 0
$$41$$ 6.45169 1.00758 0.503792 0.863825i $$-0.331938\pi$$
0.503792 + 0.863825i $$0.331938\pi$$
$$42$$ 0 0
$$43$$ −6.93535 −1.05763 −0.528815 0.848737i $$-0.677364\pi$$
−0.528815 + 0.848737i $$0.677364\pi$$
$$44$$ 0 0
$$45$$ −2.19789 −0.327642
$$46$$ 0 0
$$47$$ −10.8577 −1.58376 −0.791878 0.610679i $$-0.790896\pi$$
−0.791878 + 0.610679i $$0.790896\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 11.3254 1.58587
$$52$$ 0 0
$$53$$ 7.94029 1.09068 0.545341 0.838214i $$-0.316400\pi$$
0.545341 + 0.838214i $$0.316400\pi$$
$$54$$ 0 0
$$55$$ 12.6138 1.70084
$$56$$ 0 0
$$57$$ 8.70650 1.15320
$$58$$ 0 0
$$59$$ −2.69561 −0.350938 −0.175469 0.984485i $$-0.556144\pi$$
−0.175469 + 0.984485i $$0.556144\pi$$
$$60$$ 0 0
$$61$$ −3.27771 −0.419667 −0.209834 0.977737i $$-0.567292\pi$$
−0.209834 + 0.977737i $$0.567292\pi$$
$$62$$ 0 0
$$63$$ 0.938323 0.118218
$$64$$ 0 0
$$65$$ −12.4693 −1.54663
$$66$$ 0 0
$$67$$ −1.82865 −0.223405 −0.111702 0.993742i $$-0.535630\pi$$
−0.111702 + 0.993742i $$0.535630\pi$$
$$68$$ 0 0
$$69$$ −1.43585 −0.172857
$$70$$ 0 0
$$71$$ 7.30161 0.866541 0.433271 0.901264i $$-0.357359\pi$$
0.433271 + 0.901264i $$0.357359\pi$$
$$72$$ 0 0
$$73$$ −7.51039 −0.879024 −0.439512 0.898237i $$-0.644849\pi$$
−0.439512 + 0.898237i $$0.644849\pi$$
$$74$$ 0 0
$$75$$ 0.698740 0.0806836
$$76$$ 0 0
$$77$$ −5.38507 −0.613686
$$78$$ 0 0
$$79$$ −2.49158 −0.280324 −0.140162 0.990129i $$-0.544762\pi$$
−0.140162 + 0.990129i $$0.544762\pi$$
$$80$$ 0 0
$$81$$ −5.30458 −0.589398
$$82$$ 0 0
$$83$$ −16.8120 −1.84536 −0.922678 0.385571i $$-0.874004\pi$$
−0.922678 + 0.385571i $$0.874004\pi$$
$$84$$ 0 0
$$85$$ 18.4754 2.00394
$$86$$ 0 0
$$87$$ 2.69177 0.288588
$$88$$ 0 0
$$89$$ 13.2665 1.40624 0.703121 0.711070i $$-0.251789\pi$$
0.703121 + 0.711070i $$0.251789\pi$$
$$90$$ 0 0
$$91$$ 5.32339 0.558043
$$92$$ 0 0
$$93$$ −8.37240 −0.868178
$$94$$ 0 0
$$95$$ 14.2032 1.45722
$$96$$ 0 0
$$97$$ 11.9624 1.21460 0.607300 0.794473i $$-0.292253\pi$$
0.607300 + 0.794473i $$0.292253\pi$$
$$98$$ 0 0
$$99$$ −5.05294 −0.507839
$$100$$ 0 0
$$101$$ −14.4918 −1.44199 −0.720993 0.692943i $$-0.756314\pi$$
−0.720993 + 0.692943i $$0.756314\pi$$
$$102$$ 0 0
$$103$$ 7.55642 0.744556 0.372278 0.928121i $$-0.378577\pi$$
0.372278 + 0.928121i $$0.378577\pi$$
$$104$$ 0 0
$$105$$ −3.36328 −0.328223
$$106$$ 0 0
$$107$$ −10.9353 −1.05716 −0.528580 0.848883i $$-0.677275\pi$$
−0.528580 + 0.848883i $$0.677275\pi$$
$$108$$ 0 0
$$109$$ −12.6250 −1.20926 −0.604628 0.796508i $$-0.706678\pi$$
−0.604628 + 0.796508i $$0.706678\pi$$
$$110$$ 0 0
$$111$$ 1.71144 0.162443
$$112$$ 0 0
$$113$$ −3.15008 −0.296335 −0.148167 0.988962i $$-0.547337\pi$$
−0.148167 + 0.988962i $$0.547337\pi$$
$$114$$ 0 0
$$115$$ −2.34236 −0.218426
$$116$$ 0 0
$$117$$ 4.99506 0.461794
$$118$$ 0 0
$$119$$ −7.88754 −0.723050
$$120$$ 0 0
$$121$$ 17.9990 1.63627
$$122$$ 0 0
$$123$$ 9.26368 0.835278
$$124$$ 0 0
$$125$$ −10.5719 −0.945580
$$126$$ 0 0
$$127$$ −9.25975 −0.821670 −0.410835 0.911710i $$-0.634763\pi$$
−0.410835 + 0.911710i $$0.634763\pi$$
$$128$$ 0 0
$$129$$ −9.95815 −0.876766
$$130$$ 0 0
$$131$$ 5.02289 0.438852 0.219426 0.975629i $$-0.429582\pi$$
0.219426 + 0.975629i $$0.429582\pi$$
$$132$$ 0 0
$$133$$ −6.06364 −0.525784
$$134$$ 0 0
$$135$$ −13.2457 −1.14001
$$136$$ 0 0
$$137$$ −15.7114 −1.34232 −0.671159 0.741313i $$-0.734203\pi$$
−0.671159 + 0.741313i $$0.734203\pi$$
$$138$$ 0 0
$$139$$ −0.0569290 −0.00482865 −0.00241433 0.999997i $$-0.500769\pi$$
−0.00241433 + 0.999997i $$0.500769\pi$$
$$140$$ 0 0
$$141$$ −15.5900 −1.31292
$$142$$ 0 0
$$143$$ −28.6669 −2.39724
$$144$$ 0 0
$$145$$ 4.39118 0.364667
$$146$$ 0 0
$$147$$ 1.43585 0.118427
$$148$$ 0 0
$$149$$ 1.55642 0.127507 0.0637536 0.997966i $$-0.479693\pi$$
0.0637536 + 0.997966i $$0.479693\pi$$
$$150$$ 0 0
$$151$$ −3.44675 −0.280492 −0.140246 0.990117i $$-0.544789\pi$$
−0.140246 + 0.990117i $$0.544789\pi$$
$$152$$ 0 0
$$153$$ −7.40106 −0.598340
$$154$$ 0 0
$$155$$ −13.6582 −1.09705
$$156$$ 0 0
$$157$$ 20.0366 1.59910 0.799548 0.600603i $$-0.205073\pi$$
0.799548 + 0.600603i $$0.205073\pi$$
$$158$$ 0 0
$$159$$ 11.4011 0.904165
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ −20.4556 −1.60221 −0.801104 0.598526i $$-0.795753\pi$$
−0.801104 + 0.598526i $$0.795753\pi$$
$$164$$ 0 0
$$165$$ 18.1115 1.40998
$$166$$ 0 0
$$167$$ 18.3044 1.41644 0.708220 0.705992i $$-0.249499\pi$$
0.708220 + 0.705992i $$0.249499\pi$$
$$168$$ 0 0
$$169$$ 15.3385 1.17989
$$170$$ 0 0
$$171$$ −5.68965 −0.435099
$$172$$ 0 0
$$173$$ 0.0636396 0.00483843 0.00241922 0.999997i $$-0.499230\pi$$
0.00241922 + 0.999997i $$0.499230\pi$$
$$174$$ 0 0
$$175$$ −0.486637 −0.0367863
$$176$$ 0 0
$$177$$ −3.87050 −0.290925
$$178$$ 0 0
$$179$$ −10.0500 −0.751169 −0.375585 0.926788i $$-0.622558\pi$$
−0.375585 + 0.926788i $$0.622558\pi$$
$$180$$ 0 0
$$181$$ −0.138173 −0.0102703 −0.00513516 0.999987i $$-0.501635\pi$$
−0.00513516 + 0.999987i $$0.501635\pi$$
$$182$$ 0 0
$$183$$ −4.70631 −0.347900
$$184$$ 0 0
$$185$$ 2.79193 0.205267
$$186$$ 0 0
$$187$$ 42.4750 3.10608
$$188$$ 0 0
$$189$$ 5.65486 0.411330
$$190$$ 0 0
$$191$$ −8.55149 −0.618764 −0.309382 0.950938i $$-0.600122\pi$$
−0.309382 + 0.950938i $$0.600122\pi$$
$$192$$ 0 0
$$193$$ −24.3765 −1.75466 −0.877330 0.479887i $$-0.840678\pi$$
−0.877330 + 0.479887i $$0.840678\pi$$
$$194$$ 0 0
$$195$$ −17.9041 −1.28214
$$196$$ 0 0
$$197$$ 21.5421 1.53481 0.767404 0.641164i $$-0.221548\pi$$
0.767404 + 0.641164i $$0.221548\pi$$
$$198$$ 0 0
$$199$$ 6.51951 0.462156 0.231078 0.972935i $$-0.425775\pi$$
0.231078 + 0.972935i $$0.425775\pi$$
$$200$$ 0 0
$$201$$ −2.62567 −0.185201
$$202$$ 0 0
$$203$$ −1.87468 −0.131577
$$204$$ 0 0
$$205$$ 15.1122 1.05548
$$206$$ 0 0
$$207$$ 0.938323 0.0652180
$$208$$ 0 0
$$209$$ 32.6531 2.25866
$$210$$ 0 0
$$211$$ −18.2352 −1.25536 −0.627681 0.778471i $$-0.715996\pi$$
−0.627681 + 0.778471i $$0.715996\pi$$
$$212$$ 0 0
$$213$$ 10.4840 0.718355
$$214$$ 0 0
$$215$$ −16.2451 −1.10790
$$216$$ 0 0
$$217$$ 5.83096 0.395831
$$218$$ 0 0
$$219$$ −10.7838 −0.728703
$$220$$ 0 0
$$221$$ −41.9885 −2.82445
$$222$$ 0 0
$$223$$ −19.8496 −1.32923 −0.664614 0.747187i $$-0.731404\pi$$
−0.664614 + 0.747187i $$0.731404\pi$$
$$224$$ 0 0
$$225$$ −0.456623 −0.0304415
$$226$$ 0 0
$$227$$ −0.579223 −0.0384444 −0.0192222 0.999815i $$-0.506119\pi$$
−0.0192222 + 0.999815i $$0.506119\pi$$
$$228$$ 0 0
$$229$$ 14.6538 0.968347 0.484174 0.874972i $$-0.339120\pi$$
0.484174 + 0.874972i $$0.339120\pi$$
$$230$$ 0 0
$$231$$ −7.73218 −0.508740
$$232$$ 0 0
$$233$$ 14.5953 0.956170 0.478085 0.878314i $$-0.341331\pi$$
0.478085 + 0.878314i $$0.341331\pi$$
$$234$$ 0 0
$$235$$ −25.4326 −1.65904
$$236$$ 0 0
$$237$$ −3.57754 −0.232386
$$238$$ 0 0
$$239$$ −12.9435 −0.837243 −0.418621 0.908161i $$-0.637487\pi$$
−0.418621 + 0.908161i $$0.637487\pi$$
$$240$$ 0 0
$$241$$ 1.24075 0.0799239 0.0399619 0.999201i $$-0.487276\pi$$
0.0399619 + 0.999201i $$0.487276\pi$$
$$242$$ 0 0
$$243$$ 9.34797 0.599672
$$244$$ 0 0
$$245$$ 2.34236 0.149648
$$246$$ 0 0
$$247$$ −32.2791 −2.05387
$$248$$ 0 0
$$249$$ −24.1396 −1.52978
$$250$$ 0 0
$$251$$ 19.8766 1.25460 0.627301 0.778777i $$-0.284159\pi$$
0.627301 + 0.778777i $$0.284159\pi$$
$$252$$ 0 0
$$253$$ −5.38507 −0.338557
$$254$$ 0 0
$$255$$ 26.5280 1.66125
$$256$$ 0 0
$$257$$ 4.85803 0.303036 0.151518 0.988455i $$-0.451584\pi$$
0.151518 + 0.988455i $$0.451584\pi$$
$$258$$ 0 0
$$259$$ −1.19193 −0.0740630
$$260$$ 0 0
$$261$$ −1.75906 −0.108883
$$262$$ 0 0
$$263$$ 11.5760 0.713806 0.356903 0.934141i $$-0.383833\pi$$
0.356903 + 0.934141i $$0.383833\pi$$
$$264$$ 0 0
$$265$$ 18.5990 1.14253
$$266$$ 0 0
$$267$$ 19.0487 1.16576
$$268$$ 0 0
$$269$$ −4.38805 −0.267544 −0.133772 0.991012i $$-0.542709\pi$$
−0.133772 + 0.991012i $$0.542709\pi$$
$$270$$ 0 0
$$271$$ −1.44396 −0.0877145 −0.0438572 0.999038i $$-0.513965\pi$$
−0.0438572 + 0.999038i $$0.513965\pi$$
$$272$$ 0 0
$$273$$ 7.64362 0.462613
$$274$$ 0 0
$$275$$ 2.62058 0.158027
$$276$$ 0 0
$$277$$ 0.281027 0.0168853 0.00844263 0.999964i $$-0.497313\pi$$
0.00844263 + 0.999964i $$0.497313\pi$$
$$278$$ 0 0
$$279$$ 5.47132 0.327559
$$280$$ 0 0
$$281$$ −19.5185 −1.16438 −0.582188 0.813054i $$-0.697803\pi$$
−0.582188 + 0.813054i $$0.697803\pi$$
$$282$$ 0 0
$$283$$ 27.3097 1.62339 0.811697 0.584079i $$-0.198544\pi$$
0.811697 + 0.584079i $$0.198544\pi$$
$$284$$ 0 0
$$285$$ 20.3937 1.20802
$$286$$ 0 0
$$287$$ −6.45169 −0.380831
$$288$$ 0 0
$$289$$ 45.2133 2.65960
$$290$$ 0 0
$$291$$ 17.1763 1.00689
$$292$$ 0 0
$$293$$ 20.8180 1.21620 0.608100 0.793860i $$-0.291932\pi$$
0.608100 + 0.793860i $$0.291932\pi$$
$$294$$ 0 0
$$295$$ −6.31408 −0.367620
$$296$$ 0 0
$$297$$ −30.4518 −1.76699
$$298$$ 0 0
$$299$$ 5.32339 0.307860
$$300$$ 0 0
$$301$$ 6.93535 0.399747
$$302$$ 0 0
$$303$$ −20.8081 −1.19539
$$304$$ 0 0
$$305$$ −7.67756 −0.439616
$$306$$ 0 0
$$307$$ 18.1772 1.03742 0.518712 0.854949i $$-0.326412\pi$$
0.518712 + 0.854949i $$0.326412\pi$$
$$308$$ 0 0
$$309$$ 10.8499 0.617230
$$310$$ 0 0
$$311$$ −12.6176 −0.715478 −0.357739 0.933822i $$-0.616452\pi$$
−0.357739 + 0.933822i $$0.616452\pi$$
$$312$$ 0 0
$$313$$ −19.2510 −1.08813 −0.544066 0.839042i $$-0.683116\pi$$
−0.544066 + 0.839042i $$0.683116\pi$$
$$314$$ 0 0
$$315$$ 2.19789 0.123837
$$316$$ 0 0
$$317$$ 10.9197 0.613312 0.306656 0.951820i $$-0.400790\pi$$
0.306656 + 0.951820i $$0.400790\pi$$
$$318$$ 0 0
$$319$$ 10.0953 0.565228
$$320$$ 0 0
$$321$$ −15.7016 −0.876376
$$322$$ 0 0
$$323$$ 47.8272 2.66118
$$324$$ 0 0
$$325$$ −2.59056 −0.143699
$$326$$ 0 0
$$327$$ −18.1277 −1.00246
$$328$$ 0 0
$$329$$ 10.8577 0.598603
$$330$$ 0 0
$$331$$ 33.1364 1.82134 0.910671 0.413133i $$-0.135566\pi$$
0.910671 + 0.413133i $$0.135566\pi$$
$$332$$ 0 0
$$333$$ −1.11842 −0.0612888
$$334$$ 0 0
$$335$$ −4.28335 −0.234024
$$336$$ 0 0
$$337$$ 35.6458 1.94175 0.970875 0.239588i $$-0.0770125\pi$$
0.970875 + 0.239588i $$0.0770125\pi$$
$$338$$ 0 0
$$339$$ −4.52305 −0.245659
$$340$$ 0 0
$$341$$ −31.4001 −1.70041
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −3.36328 −0.181073
$$346$$ 0 0
$$347$$ −0.117783 −0.00632293 −0.00316146 0.999995i $$-0.501006\pi$$
−0.00316146 + 0.999995i $$0.501006\pi$$
$$348$$ 0 0
$$349$$ −12.7083 −0.680258 −0.340129 0.940379i $$-0.610471\pi$$
−0.340129 + 0.940379i $$0.610471\pi$$
$$350$$ 0 0
$$351$$ 30.1030 1.60678
$$352$$ 0 0
$$353$$ −15.4791 −0.823870 −0.411935 0.911213i $$-0.635147\pi$$
−0.411935 + 0.911213i $$0.635147\pi$$
$$354$$ 0 0
$$355$$ 17.1030 0.907731
$$356$$ 0 0
$$357$$ −11.3254 −0.599401
$$358$$ 0 0
$$359$$ 7.66806 0.404705 0.202352 0.979313i $$-0.435141\pi$$
0.202352 + 0.979313i $$0.435141\pi$$
$$360$$ 0 0
$$361$$ 17.7677 0.935143
$$362$$ 0 0
$$363$$ 25.8439 1.35645
$$364$$ 0 0
$$365$$ −17.5920 −0.920808
$$366$$ 0 0
$$367$$ 10.1079 0.527628 0.263814 0.964574i $$-0.415019\pi$$
0.263814 + 0.964574i $$0.415019\pi$$
$$368$$ 0 0
$$369$$ −6.05376 −0.315146
$$370$$ 0 0
$$371$$ −7.94029 −0.412239
$$372$$ 0 0
$$373$$ −28.8437 −1.49347 −0.746734 0.665123i $$-0.768379\pi$$
−0.746734 + 0.665123i $$0.768379\pi$$
$$374$$ 0 0
$$375$$ −15.1797 −0.783877
$$376$$ 0 0
$$377$$ −9.97968 −0.513980
$$378$$ 0 0
$$379$$ 27.2020 1.39727 0.698636 0.715477i $$-0.253790\pi$$
0.698636 + 0.715477i $$0.253790\pi$$
$$380$$ 0 0
$$381$$ −13.2957 −0.681157
$$382$$ 0 0
$$383$$ 19.6517 1.00416 0.502078 0.864822i $$-0.332569\pi$$
0.502078 + 0.864822i $$0.332569\pi$$
$$384$$ 0 0
$$385$$ −12.6138 −0.642857
$$386$$ 0 0
$$387$$ 6.50760 0.330800
$$388$$ 0 0
$$389$$ 12.9990 0.659075 0.329537 0.944143i $$-0.393107\pi$$
0.329537 + 0.944143i $$0.393107\pi$$
$$390$$ 0 0
$$391$$ −7.88754 −0.398890
$$392$$ 0 0
$$393$$ 7.21214 0.363804
$$394$$ 0 0
$$395$$ −5.83616 −0.293649
$$396$$ 0 0
$$397$$ 27.2637 1.36832 0.684162 0.729330i $$-0.260168\pi$$
0.684162 + 0.729330i $$0.260168\pi$$
$$398$$ 0 0
$$399$$ −8.70650 −0.435870
$$400$$ 0 0
$$401$$ 10.3543 0.517069 0.258535 0.966002i $$-0.416760\pi$$
0.258535 + 0.966002i $$0.416760\pi$$
$$402$$ 0 0
$$403$$ 31.0405 1.54624
$$404$$ 0 0
$$405$$ −12.4252 −0.617414
$$406$$ 0 0
$$407$$ 6.41863 0.318160
$$408$$ 0 0
$$409$$ 14.4338 0.713707 0.356853 0.934160i $$-0.383850\pi$$
0.356853 + 0.934160i $$0.383850\pi$$
$$410$$ 0 0
$$411$$ −22.5593 −1.11277
$$412$$ 0 0
$$413$$ 2.69561 0.132642
$$414$$ 0 0
$$415$$ −39.3797 −1.93307
$$416$$ 0 0
$$417$$ −0.0817417 −0.00400291
$$418$$ 0 0
$$419$$ 38.7780 1.89443 0.947214 0.320601i $$-0.103885\pi$$
0.947214 + 0.320601i $$0.103885\pi$$
$$420$$ 0 0
$$421$$ 33.3097 1.62342 0.811708 0.584063i $$-0.198538\pi$$
0.811708 + 0.584063i $$0.198538\pi$$
$$422$$ 0 0
$$423$$ 10.1880 0.495358
$$424$$ 0 0
$$425$$ 3.83837 0.186188
$$426$$ 0 0
$$427$$ 3.27771 0.158619
$$428$$ 0 0
$$429$$ −41.1614 −1.98729
$$430$$ 0 0
$$431$$ −0.967325 −0.0465944 −0.0232972 0.999729i $$-0.507416\pi$$
−0.0232972 + 0.999729i $$0.507416\pi$$
$$432$$ 0 0
$$433$$ 11.4173 0.548679 0.274340 0.961633i $$-0.411541\pi$$
0.274340 + 0.961633i $$0.411541\pi$$
$$434$$ 0 0
$$435$$ 6.30509 0.302306
$$436$$ 0 0
$$437$$ −6.06364 −0.290063
$$438$$ 0 0
$$439$$ −10.3789 −0.495357 −0.247678 0.968842i $$-0.579668\pi$$
−0.247678 + 0.968842i $$0.579668\pi$$
$$440$$ 0 0
$$441$$ −0.938323 −0.0446820
$$442$$ 0 0
$$443$$ −23.2218 −1.10330 −0.551651 0.834075i $$-0.686002\pi$$
−0.551651 + 0.834075i $$0.686002\pi$$
$$444$$ 0 0
$$445$$ 31.0748 1.47309
$$446$$ 0 0
$$447$$ 2.23480 0.105702
$$448$$ 0 0
$$449$$ −10.6505 −0.502629 −0.251314 0.967905i $$-0.580863\pi$$
−0.251314 + 0.967905i $$0.580863\pi$$
$$450$$ 0 0
$$451$$ 34.7428 1.63597
$$452$$ 0 0
$$453$$ −4.94903 −0.232526
$$454$$ 0 0
$$455$$ 12.4693 0.584569
$$456$$ 0 0
$$457$$ 0.379938 0.0177727 0.00888637 0.999961i $$-0.497171\pi$$
0.00888637 + 0.999961i $$0.497171\pi$$
$$458$$ 0 0
$$459$$ −44.6029 −2.08189
$$460$$ 0 0
$$461$$ −16.3880 −0.763267 −0.381634 0.924314i $$-0.624638\pi$$
−0.381634 + 0.924314i $$0.624638\pi$$
$$462$$ 0 0
$$463$$ −36.2349 −1.68398 −0.841989 0.539495i $$-0.818615\pi$$
−0.841989 + 0.539495i $$0.818615\pi$$
$$464$$ 0 0
$$465$$ −19.6112 −0.909446
$$466$$ 0 0
$$467$$ 17.1603 0.794083 0.397041 0.917801i $$-0.370037\pi$$
0.397041 + 0.917801i $$0.370037\pi$$
$$468$$ 0 0
$$469$$ 1.82865 0.0844391
$$470$$ 0 0
$$471$$ 28.7696 1.32564
$$472$$ 0 0
$$473$$ −37.3473 −1.71723
$$474$$ 0 0
$$475$$ 2.95079 0.135392
$$476$$ 0 0
$$477$$ −7.45055 −0.341137
$$478$$ 0 0
$$479$$ −16.8299 −0.768976 −0.384488 0.923130i $$-0.625622\pi$$
−0.384488 + 0.923130i $$0.625622\pi$$
$$480$$ 0 0
$$481$$ −6.34512 −0.289313
$$482$$ 0 0
$$483$$ 1.43585 0.0653336
$$484$$ 0 0
$$485$$ 28.0203 1.27233
$$486$$ 0 0
$$487$$ −26.9276 −1.22021 −0.610103 0.792322i $$-0.708872\pi$$
−0.610103 + 0.792322i $$0.708872\pi$$
$$488$$ 0 0
$$489$$ −29.3713 −1.32821
$$490$$ 0 0
$$491$$ −21.3432 −0.963203 −0.481602 0.876390i $$-0.659945\pi$$
−0.481602 + 0.876390i $$0.659945\pi$$
$$492$$ 0 0
$$493$$ 14.7866 0.665957
$$494$$ 0 0
$$495$$ −11.8358 −0.531979
$$496$$ 0 0
$$497$$ −7.30161 −0.327522
$$498$$ 0 0
$$499$$ −19.5203 −0.873847 −0.436923 0.899499i $$-0.643932\pi$$
−0.436923 + 0.899499i $$0.643932\pi$$
$$500$$ 0 0
$$501$$ 26.2825 1.17422
$$502$$ 0 0
$$503$$ 5.02673 0.224131 0.112065 0.993701i $$-0.464253\pi$$
0.112065 + 0.993701i $$0.464253\pi$$
$$504$$ 0 0
$$505$$ −33.9449 −1.51053
$$506$$ 0 0
$$507$$ 22.0239 0.978115
$$508$$ 0 0
$$509$$ −10.5354 −0.466973 −0.233486 0.972360i $$-0.575013\pi$$
−0.233486 + 0.972360i $$0.575013\pi$$
$$510$$ 0 0
$$511$$ 7.51039 0.332240
$$512$$ 0 0
$$513$$ −34.2890 −1.51390
$$514$$ 0 0
$$515$$ 17.6998 0.779948
$$516$$ 0 0
$$517$$ −58.4694 −2.57148
$$518$$ 0 0
$$519$$ 0.0913772 0.00401101
$$520$$ 0 0
$$521$$ 10.8559 0.475606 0.237803 0.971313i $$-0.423573\pi$$
0.237803 + 0.971313i $$0.423573\pi$$
$$522$$ 0 0
$$523$$ 16.5175 0.722259 0.361130 0.932516i $$-0.382391\pi$$
0.361130 + 0.932516i $$0.382391\pi$$
$$524$$ 0 0
$$525$$ −0.698740 −0.0304955
$$526$$ 0 0
$$527$$ −45.9919 −2.00344
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 2.52935 0.109764
$$532$$ 0 0
$$533$$ −34.3449 −1.48764
$$534$$ 0 0
$$535$$ −25.6145 −1.10741
$$536$$ 0 0
$$537$$ −14.4303 −0.622712
$$538$$ 0 0
$$539$$ 5.38507 0.231951
$$540$$ 0 0
$$541$$ −11.0983 −0.477152 −0.238576 0.971124i $$-0.576681\pi$$
−0.238576 + 0.971124i $$0.576681\pi$$
$$542$$ 0 0
$$543$$ −0.198396 −0.00851400
$$544$$ 0 0
$$545$$ −29.5723 −1.26674
$$546$$ 0 0
$$547$$ −3.70757 −0.158524 −0.0792621 0.996854i $$-0.525256\pi$$
−0.0792621 + 0.996854i $$0.525256\pi$$
$$548$$ 0 0
$$549$$ 3.07555 0.131261
$$550$$ 0 0
$$551$$ 11.3674 0.484268
$$552$$ 0 0
$$553$$ 2.49158 0.105953
$$554$$ 0 0
$$555$$ 4.00880 0.170164
$$556$$ 0 0
$$557$$ −14.5792 −0.617741 −0.308871 0.951104i $$-0.599951\pi$$
−0.308871 + 0.951104i $$0.599951\pi$$
$$558$$ 0 0
$$559$$ 36.9196 1.56153
$$560$$ 0 0
$$561$$ 60.9878 2.57491
$$562$$ 0 0
$$563$$ −15.2872 −0.644280 −0.322140 0.946692i $$-0.604402\pi$$
−0.322140 + 0.946692i $$0.604402\pi$$
$$564$$ 0 0
$$565$$ −7.37861 −0.310420
$$566$$ 0 0
$$567$$ 5.30458 0.222771
$$568$$ 0 0
$$569$$ 17.3853 0.728828 0.364414 0.931237i $$-0.381269\pi$$
0.364414 + 0.931237i $$0.381269\pi$$
$$570$$ 0 0
$$571$$ −5.96802 −0.249754 −0.124877 0.992172i $$-0.539854\pi$$
−0.124877 + 0.992172i $$0.539854\pi$$
$$572$$ 0 0
$$573$$ −12.2787 −0.512949
$$574$$ 0 0
$$575$$ −0.486637 −0.0202942
$$576$$ 0 0
$$577$$ −22.1611 −0.922579 −0.461289 0.887250i $$-0.652613\pi$$
−0.461289 + 0.887250i $$0.652613\pi$$
$$578$$ 0 0
$$579$$ −35.0011 −1.45460
$$580$$ 0 0
$$581$$ 16.8120 0.697479
$$582$$ 0 0
$$583$$ 42.7590 1.77090
$$584$$ 0 0
$$585$$ 11.7002 0.483745
$$586$$ 0 0
$$587$$ 9.96131 0.411147 0.205574 0.978642i $$-0.434094\pi$$
0.205574 + 0.978642i $$0.434094\pi$$
$$588$$ 0 0
$$589$$ −35.3568 −1.45685
$$590$$ 0 0
$$591$$ 30.9312 1.27234
$$592$$ 0 0
$$593$$ 8.77014 0.360147 0.180073 0.983653i $$-0.442367\pi$$
0.180073 + 0.983653i $$0.442367\pi$$
$$594$$ 0 0
$$595$$ −18.4754 −0.757419
$$596$$ 0 0
$$597$$ 9.36106 0.383123
$$598$$ 0 0
$$599$$ 11.3276 0.462832 0.231416 0.972855i $$-0.425664\pi$$
0.231416 + 0.972855i $$0.425664\pi$$
$$600$$ 0 0
$$601$$ 5.77090 0.235400 0.117700 0.993049i $$-0.462448\pi$$
0.117700 + 0.993049i $$0.462448\pi$$
$$602$$ 0 0
$$603$$ 1.71586 0.0698753
$$604$$ 0 0
$$605$$ 42.1601 1.71405
$$606$$ 0 0
$$607$$ 1.19400 0.0484630 0.0242315 0.999706i $$-0.492286\pi$$
0.0242315 + 0.999706i $$0.492286\pi$$
$$608$$ 0 0
$$609$$ −2.69177 −0.109076
$$610$$ 0 0
$$611$$ 57.7997 2.33833
$$612$$ 0 0
$$613$$ −42.3622 −1.71099 −0.855496 0.517810i $$-0.826747\pi$$
−0.855496 + 0.517810i $$0.826747\pi$$
$$614$$ 0 0
$$615$$ 21.6988 0.874982
$$616$$ 0 0
$$617$$ 33.9063 1.36502 0.682508 0.730878i $$-0.260889\pi$$
0.682508 + 0.730878i $$0.260889\pi$$
$$618$$ 0 0
$$619$$ 27.6830 1.11267 0.556337 0.830957i $$-0.312206\pi$$
0.556337 + 0.830957i $$0.312206\pi$$
$$620$$ 0 0
$$621$$ 5.65486 0.226922
$$622$$ 0 0
$$623$$ −13.2665 −0.531510
$$624$$ 0 0
$$625$$ −27.1964 −1.08785
$$626$$ 0 0
$$627$$ 46.8851 1.87241
$$628$$ 0 0
$$629$$ 9.40141 0.374859
$$630$$ 0 0
$$631$$ −0.801923 −0.0319240 −0.0159620 0.999873i $$-0.505081\pi$$
−0.0159620 + 0.999873i $$0.505081\pi$$
$$632$$ 0 0
$$633$$ −26.1831 −1.04068
$$634$$ 0 0
$$635$$ −21.6897 −0.860728
$$636$$ 0 0
$$637$$ −5.32339 −0.210921
$$638$$ 0 0
$$639$$ −6.85126 −0.271032
$$640$$ 0 0
$$641$$ 19.1287 0.755538 0.377769 0.925900i $$-0.376691\pi$$
0.377769 + 0.925900i $$0.376691\pi$$
$$642$$ 0 0
$$643$$ −43.8827 −1.73056 −0.865282 0.501286i $$-0.832861\pi$$
−0.865282 + 0.501286i $$0.832861\pi$$
$$644$$ 0 0
$$645$$ −23.3255 −0.918442
$$646$$ 0 0
$$647$$ 41.0032 1.61200 0.806001 0.591914i $$-0.201627\pi$$
0.806001 + 0.591914i $$0.201627\pi$$
$$648$$ 0 0
$$649$$ −14.5160 −0.569804
$$650$$ 0 0
$$651$$ 8.37240 0.328140
$$652$$ 0 0
$$653$$ 35.1744 1.37648 0.688240 0.725483i $$-0.258383\pi$$
0.688240 + 0.725483i $$0.258383\pi$$
$$654$$ 0 0
$$655$$ 11.7654 0.459712
$$656$$ 0 0
$$657$$ 7.04717 0.274936
$$658$$ 0 0
$$659$$ −24.7839 −0.965445 −0.482723 0.875773i $$-0.660352\pi$$
−0.482723 + 0.875773i $$0.660352\pi$$
$$660$$ 0 0
$$661$$ 41.4545 1.61239 0.806197 0.591648i $$-0.201522\pi$$
0.806197 + 0.591648i $$0.201522\pi$$
$$662$$ 0 0
$$663$$ −60.2893 −2.34144
$$664$$ 0 0
$$665$$ −14.2032 −0.550777
$$666$$ 0 0
$$667$$ −1.87468 −0.0725880
$$668$$ 0 0
$$669$$ −28.5011 −1.10192
$$670$$ 0 0
$$671$$ −17.6507 −0.681397
$$672$$ 0 0
$$673$$ −36.0660 −1.39024 −0.695121 0.718893i $$-0.744649\pi$$
−0.695121 + 0.718893i $$0.744649\pi$$
$$674$$ 0 0
$$675$$ −2.75187 −0.105919
$$676$$ 0 0
$$677$$ 3.41266 0.131159 0.0655795 0.997847i $$-0.479110\pi$$
0.0655795 + 0.997847i $$0.479110\pi$$
$$678$$ 0 0
$$679$$ −11.9624 −0.459076
$$680$$ 0 0
$$681$$ −0.831679 −0.0318700
$$682$$ 0 0
$$683$$ −24.2314 −0.927188 −0.463594 0.886048i $$-0.653440\pi$$
−0.463594 + 0.886048i $$0.653440\pi$$
$$684$$ 0 0
$$685$$ −36.8018 −1.40612
$$686$$ 0 0
$$687$$ 21.0407 0.802751
$$688$$ 0 0
$$689$$ −42.2693 −1.61033
$$690$$ 0 0
$$691$$ 7.37573 0.280586 0.140293 0.990110i $$-0.455196\pi$$
0.140293 + 0.990110i $$0.455196\pi$$
$$692$$ 0 0
$$693$$ 5.05294 0.191945
$$694$$ 0 0
$$695$$ −0.133348 −0.00505818
$$696$$ 0 0
$$697$$ 50.8879 1.92752
$$698$$ 0 0
$$699$$ 20.9567 0.792656
$$700$$ 0 0
$$701$$ 3.25950 0.123109 0.0615547 0.998104i $$-0.480394\pi$$
0.0615547 + 0.998104i $$0.480394\pi$$
$$702$$ 0 0
$$703$$ 7.22744 0.272588
$$704$$ 0 0
$$705$$ −36.5175 −1.37533
$$706$$ 0 0
$$707$$ 14.4918 0.545019
$$708$$ 0 0
$$709$$ 43.0749 1.61771 0.808856 0.588006i $$-0.200087\pi$$
0.808856 + 0.588006i $$0.200087\pi$$
$$710$$ 0 0
$$711$$ 2.33790 0.0876782
$$712$$ 0 0
$$713$$ 5.83096 0.218371
$$714$$ 0 0
$$715$$ −67.1480 −2.51119
$$716$$ 0 0
$$717$$ −18.5849 −0.694066
$$718$$ 0 0
$$719$$ 0.219003 0.00816744 0.00408372 0.999992i $$-0.498700\pi$$
0.00408372 + 0.999992i $$0.498700\pi$$
$$720$$ 0 0
$$721$$ −7.55642 −0.281416
$$722$$ 0 0
$$723$$ 1.78154 0.0662562
$$724$$ 0 0
$$725$$ 0.912291 0.0338816
$$726$$ 0 0
$$727$$ −22.7045 −0.842062 −0.421031 0.907046i $$-0.638332\pi$$
−0.421031 + 0.907046i $$0.638332\pi$$
$$728$$ 0 0
$$729$$ 29.3361 1.08652
$$730$$ 0 0
$$731$$ −54.7028 −2.02326
$$732$$ 0 0
$$733$$ 17.9129 0.661628 0.330814 0.943696i $$-0.392677\pi$$
0.330814 + 0.943696i $$0.392677\pi$$
$$734$$ 0 0
$$735$$ 3.36328 0.124057
$$736$$ 0 0
$$737$$ −9.84740 −0.362734
$$738$$ 0 0
$$739$$ −11.3452 −0.417339 −0.208670 0.977986i $$-0.566913\pi$$
−0.208670 + 0.977986i $$0.566913\pi$$
$$740$$ 0 0
$$741$$ −46.3481 −1.70264
$$742$$ 0 0
$$743$$ 9.31548 0.341752 0.170876 0.985293i $$-0.445340\pi$$
0.170876 + 0.985293i $$0.445340\pi$$
$$744$$ 0 0
$$745$$ 3.64570 0.133568
$$746$$ 0 0
$$747$$ 15.7751 0.577180
$$748$$ 0 0
$$749$$ 10.9353 0.399569
$$750$$ 0 0
$$751$$ 5.43047 0.198161 0.0990803 0.995079i $$-0.468410\pi$$
0.0990803 + 0.995079i $$0.468410\pi$$
$$752$$ 0 0
$$753$$ 28.5400 1.04005
$$754$$ 0 0
$$755$$ −8.07351 −0.293825
$$756$$ 0 0
$$757$$ −9.50823 −0.345582 −0.172791 0.984958i $$-0.555279\pi$$
−0.172791 + 0.984958i $$0.555279\pi$$
$$758$$ 0 0
$$759$$ −7.73218 −0.280660
$$760$$ 0 0
$$761$$ 21.6974 0.786529 0.393265 0.919425i $$-0.371346\pi$$
0.393265 + 0.919425i $$0.371346\pi$$
$$762$$ 0 0
$$763$$ 12.6250 0.457056
$$764$$ 0 0
$$765$$ −17.3359 −0.626782
$$766$$ 0 0
$$767$$ 14.3498 0.518141
$$768$$ 0 0
$$769$$ −10.7850 −0.388916 −0.194458 0.980911i $$-0.562295\pi$$
−0.194458 + 0.980911i $$0.562295\pi$$
$$770$$ 0 0
$$771$$ 6.97542 0.251214
$$772$$ 0 0
$$773$$ −25.1118 −0.903210 −0.451605 0.892218i $$-0.649148\pi$$
−0.451605 + 0.892218i $$0.649148\pi$$
$$774$$ 0 0
$$775$$ −2.83756 −0.101928
$$776$$ 0 0
$$777$$ −1.71144 −0.0613976
$$778$$ 0 0
$$779$$ 39.1207 1.40164
$$780$$ 0 0
$$781$$ 39.3197 1.40697
$$782$$ 0 0
$$783$$ −10.6011 −0.378851
$$784$$ 0 0
$$785$$ 46.9329 1.67511
$$786$$ 0 0
$$787$$ 1.64538 0.0586516 0.0293258 0.999570i $$-0.490664\pi$$
0.0293258 + 0.999570i $$0.490664\pi$$
$$788$$ 0 0
$$789$$ 16.6214 0.591739
$$790$$ 0 0
$$791$$ 3.15008 0.112004
$$792$$ 0 0
$$793$$ 17.4485 0.619615
$$794$$ 0 0
$$795$$ 26.7054 0.947144
$$796$$ 0 0
$$797$$ 12.0243 0.425923 0.212962 0.977061i $$-0.431689\pi$$
0.212962 + 0.977061i $$0.431689\pi$$
$$798$$ 0 0
$$799$$ −85.6404 −3.02974
$$800$$ 0 0
$$801$$ −12.4482 −0.439837
$$802$$ 0 0
$$803$$ −40.4440 −1.42724
$$804$$ 0 0
$$805$$ 2.34236 0.0825572
$$806$$ 0 0
$$807$$ −6.30059 −0.221791
$$808$$ 0 0
$$809$$ 11.4285 0.401803 0.200901 0.979611i $$-0.435613\pi$$
0.200901 + 0.979611i $$0.435613\pi$$
$$810$$ 0 0
$$811$$ 24.2552 0.851714 0.425857 0.904790i $$-0.359973\pi$$
0.425857 + 0.904790i $$0.359973\pi$$
$$812$$ 0 0
$$813$$ −2.07332 −0.0727145
$$814$$ 0 0
$$815$$ −47.9143 −1.67837
$$816$$ 0 0
$$817$$ −42.0535 −1.47126
$$818$$ 0 0
$$819$$ −4.99506 −0.174542
$$820$$ 0 0
$$821$$ 16.1394 0.563268 0.281634 0.959522i $$-0.409124\pi$$
0.281634 + 0.959522i $$0.409124\pi$$
$$822$$ 0 0
$$823$$ −5.72512 −0.199565 −0.0997825 0.995009i $$-0.531815\pi$$
−0.0997825 + 0.995009i $$0.531815\pi$$
$$824$$ 0 0
$$825$$ 3.76277 0.131003
$$826$$ 0 0
$$827$$ −27.6141 −0.960237 −0.480118 0.877204i $$-0.659406\pi$$
−0.480118 + 0.877204i $$0.659406\pi$$
$$828$$ 0 0
$$829$$ 27.3694 0.950580 0.475290 0.879829i $$-0.342343\pi$$
0.475290 + 0.879829i $$0.342343\pi$$
$$830$$ 0 0
$$831$$ 0.403514 0.0139977
$$832$$ 0 0
$$833$$ 7.88754 0.273287
$$834$$ 0 0
$$835$$ 42.8755 1.48377
$$836$$ 0 0
$$837$$ 32.9732 1.13972
$$838$$ 0 0
$$839$$ 9.24538 0.319186 0.159593 0.987183i $$-0.448982\pi$$
0.159593 + 0.987183i $$0.448982\pi$$
$$840$$ 0 0
$$841$$ −25.4856 −0.878813
$$842$$ 0 0
$$843$$ −28.0257 −0.965257
$$844$$ 0 0
$$845$$ 35.9283 1.23597
$$846$$ 0 0
$$847$$ −17.9990 −0.618453
$$848$$ 0 0
$$849$$ 39.2128 1.34578
$$850$$ 0 0
$$851$$ −1.19193 −0.0408589
$$852$$ 0 0
$$853$$ 48.3601 1.65582 0.827910 0.560862i $$-0.189530\pi$$
0.827910 + 0.560862i $$0.189530\pi$$
$$854$$ 0 0
$$855$$ −13.3272 −0.455781
$$856$$ 0 0
$$857$$ −1.42104 −0.0485416 −0.0242708 0.999705i $$-0.507726\pi$$
−0.0242708 + 0.999705i $$0.507726\pi$$
$$858$$ 0 0
$$859$$ −7.68890 −0.262342 −0.131171 0.991360i $$-0.541874\pi$$
−0.131171 + 0.991360i $$0.541874\pi$$
$$860$$ 0 0
$$861$$ −9.26368 −0.315705
$$862$$ 0 0
$$863$$ 1.91806 0.0652916 0.0326458 0.999467i $$-0.489607\pi$$
0.0326458 + 0.999467i $$0.489607\pi$$
$$864$$ 0 0
$$865$$ 0.149067 0.00506842
$$866$$ 0 0
$$867$$ 64.9197 2.20479
$$868$$ 0 0
$$869$$ −13.4173 −0.455151
$$870$$ 0 0
$$871$$ 9.73461 0.329845
$$872$$ 0 0
$$873$$ −11.2246 −0.379896
$$874$$ 0 0
$$875$$ 10.5719 0.357396
$$876$$ 0 0
$$877$$ −18.2136 −0.615029 −0.307515 0.951543i $$-0.599497\pi$$
−0.307515 + 0.951543i $$0.599497\pi$$
$$878$$ 0 0
$$879$$ 29.8916 1.00822
$$880$$ 0 0
$$881$$ −42.2662 −1.42399 −0.711993 0.702187i $$-0.752207\pi$$
−0.711993 + 0.702187i $$0.752207\pi$$
$$882$$ 0 0
$$883$$ 17.1758 0.578011 0.289006 0.957327i $$-0.406675\pi$$
0.289006 + 0.957327i $$0.406675\pi$$
$$884$$ 0 0
$$885$$ −9.06609 −0.304753
$$886$$ 0 0
$$887$$ 11.8742 0.398696 0.199348 0.979929i $$-0.436118\pi$$
0.199348 + 0.979929i $$0.436118\pi$$
$$888$$ 0 0
$$889$$ 9.25975 0.310562
$$890$$ 0 0
$$891$$ −28.5655 −0.956982
$$892$$ 0 0
$$893$$ −65.8371 −2.20315
$$894$$ 0 0
$$895$$ −23.5406 −0.786875
$$896$$ 0 0
$$897$$ 7.64362 0.255213
$$898$$ 0 0
$$899$$ −10.9312 −0.364576
$$900$$ 0 0
$$901$$ 62.6293 2.08649
$$902$$ 0 0
$$903$$ 9.95815 0.331386
$$904$$ 0 0
$$905$$ −0.323651 −0.0107585
$$906$$ 0 0
$$907$$ 16.7923 0.557579 0.278790 0.960352i $$-0.410067\pi$$
0.278790 + 0.960352i $$0.410067\pi$$
$$908$$ 0 0
$$909$$ 13.5980 0.451016
$$910$$ 0 0
$$911$$ 42.6355 1.41258 0.706288 0.707925i $$-0.250368\pi$$
0.706288 + 0.707925i $$0.250368\pi$$
$$912$$ 0 0
$$913$$ −90.5338 −2.99623
$$914$$ 0 0
$$915$$ −11.0239 −0.364437
$$916$$ 0 0
$$917$$ −5.02289 −0.165870
$$918$$ 0 0
$$919$$ −19.9742 −0.658887 −0.329444 0.944175i $$-0.606861\pi$$
−0.329444 + 0.944175i $$0.606861\pi$$
$$920$$ 0 0
$$921$$ 26.0997 0.860016
$$922$$ 0 0
$$923$$ −38.8693 −1.27940
$$924$$ 0 0
$$925$$ 0.580038 0.0190715
$$926$$ 0 0
$$927$$ −7.09036 −0.232878
$$928$$ 0 0
$$929$$ 6.02222 0.197583 0.0987914 0.995108i $$-0.468502\pi$$
0.0987914 + 0.995108i $$0.468502\pi$$
$$930$$ 0 0
$$931$$ 6.06364 0.198728
$$932$$ 0 0
$$933$$ −18.1170 −0.593125
$$934$$ 0 0
$$935$$ 99.4915 3.25372
$$936$$ 0 0
$$937$$ −24.6553 −0.805452 −0.402726 0.915320i $$-0.631937\pi$$
−0.402726 + 0.915320i $$0.631937\pi$$
$$938$$ 0 0
$$939$$ −27.6417 −0.902051
$$940$$ 0 0
$$941$$ 50.1049 1.63337 0.816686 0.577082i $$-0.195809\pi$$
0.816686 + 0.577082i $$0.195809\pi$$
$$942$$ 0 0
$$943$$ −6.45169 −0.210096
$$944$$ 0 0
$$945$$ 13.2457 0.430883
$$946$$ 0 0
$$947$$ 19.9519 0.648351 0.324175 0.945997i $$-0.394913\pi$$
0.324175 + 0.945997i $$0.394913\pi$$
$$948$$ 0 0
$$949$$ 39.9807 1.29783
$$950$$ 0 0
$$951$$ 15.6791 0.508430
$$952$$ 0 0
$$953$$ −40.2873 −1.30503 −0.652517 0.757774i $$-0.726287\pi$$
−0.652517 + 0.757774i $$0.726287\pi$$
$$954$$ 0 0
$$955$$ −20.0306 −0.648176
$$956$$ 0 0
$$957$$ 14.4954 0.468569
$$958$$ 0 0
$$959$$ 15.7114 0.507349
$$960$$ 0 0
$$961$$ 3.00006 0.0967762
$$962$$ 0 0
$$963$$ 10.2609 0.330652
$$964$$ 0 0
$$965$$ −57.0985 −1.83807
$$966$$ 0 0
$$967$$ −31.2689 −1.00554 −0.502771 0.864420i $$-0.667686\pi$$
−0.502771 + 0.864420i $$0.667686\pi$$
$$968$$ 0 0
$$969$$ 68.6729 2.20609
$$970$$ 0 0
$$971$$ −42.5037 −1.36401 −0.682005 0.731347i $$-0.738892\pi$$
−0.682005 + 0.731347i $$0.738892\pi$$
$$972$$ 0 0
$$973$$ 0.0569290 0.00182506
$$974$$ 0 0
$$975$$ −3.71967 −0.119125
$$976$$ 0 0
$$977$$ −33.2556 −1.06394 −0.531971 0.846762i $$-0.678549\pi$$
−0.531971 + 0.846762i $$0.678549\pi$$
$$978$$ 0 0
$$979$$ 71.4409 2.28326
$$980$$ 0 0
$$981$$ 11.8463 0.378224
$$982$$ 0 0
$$983$$ 46.5016 1.48317 0.741586 0.670858i $$-0.234074\pi$$
0.741586 + 0.670858i $$0.234074\pi$$
$$984$$ 0 0
$$985$$ 50.4592 1.60776
$$986$$ 0 0
$$987$$ 15.5900 0.496237
$$988$$ 0 0
$$989$$ 6.93535 0.220531
$$990$$ 0 0
$$991$$ −31.1965 −0.990991 −0.495496 0.868610i $$-0.665014\pi$$
−0.495496 + 0.868610i $$0.665014\pi$$
$$992$$ 0 0
$$993$$ 47.5790 1.50988
$$994$$ 0 0
$$995$$ 15.2710 0.484124
$$996$$ 0 0
$$997$$ 7.37962 0.233715 0.116857 0.993149i $$-0.462718\pi$$
0.116857 + 0.993149i $$0.462718\pi$$
$$998$$ 0 0
$$999$$ −6.74020 −0.213251
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 644.2.a.d.1.3 5
3.2 odd 2 5796.2.a.t.1.2 5
4.3 odd 2 2576.2.a.bb.1.3 5
7.6 odd 2 4508.2.a.f.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.d.1.3 5 1.1 even 1 trivial
2576.2.a.bb.1.3 5 4.3 odd 2
4508.2.a.f.1.3 5 7.6 odd 2
5796.2.a.t.1.2 5 3.2 odd 2