# Properties

 Label 3640.2.a.s.1.2 Level $3640$ Weight $2$ Character 3640.1 Self dual yes Analytic conductor $29.066$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.0655463357$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.13856$$ of defining polynomial Character $$\chi$$ $$=$$ 3640.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} +O(q^{10})$$ $$q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.13856 q^{15} +2.98080 q^{17} +1.93974 q^{19} +1.13856 q^{21} -2.17127 q^{23} +1.00000 q^{25} +5.35543 q^{27} +3.58697 q^{29} +7.29517 q^{31} +3.68447 q^{33} -1.00000 q^{35} +2.37463 q^{37} -1.13856 q^{39} +1.76846 q^{41} -5.51320 q^{43} -1.70367 q^{45} -7.53693 q^{47} +1.00000 q^{49} -3.39383 q^{51} -3.40734 q^{53} -3.23607 q^{55} -2.20852 q^{57} -11.6377 q^{59} -5.68447 q^{61} +1.70367 q^{63} +1.00000 q^{65} +2.04559 q^{67} +2.47214 q^{69} -10.9205 q^{71} -7.09181 q^{73} -1.13856 q^{75} +3.23607 q^{77} -14.7949 q^{79} -0.986489 q^{81} +2.61968 q^{83} +2.98080 q^{85} -4.08399 q^{87} +3.92952 q^{89} -1.00000 q^{91} -8.30602 q^{93} +1.93974 q^{95} -1.34192 q^{97} +5.51320 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10})$$ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 $$4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100})$$ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * 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
$$3$$ −1.13856 −0.657350 −0.328675 0.944443i $$-0.606602\pi$$
−0.328675 + 0.944443i $$0.606602\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −1.70367 −0.567890
$$10$$ 0 0
$$11$$ −3.23607 −0.975711 −0.487856 0.872924i $$-0.662221\pi$$
−0.487856 + 0.872924i $$0.662221\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.13856 −0.293976
$$16$$ 0 0
$$17$$ 2.98080 0.722950 0.361475 0.932382i $$-0.382273\pi$$
0.361475 + 0.932382i $$0.382273\pi$$
$$18$$ 0 0
$$19$$ 1.93974 0.445007 0.222503 0.974932i $$-0.428577\pi$$
0.222503 + 0.974932i $$0.428577\pi$$
$$20$$ 0 0
$$21$$ 1.13856 0.248455
$$22$$ 0 0
$$23$$ −2.17127 −0.452742 −0.226371 0.974041i $$-0.572686\pi$$
−0.226371 + 0.974041i $$0.572686\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.35543 1.03065
$$28$$ 0 0
$$29$$ 3.58697 0.666083 0.333042 0.942912i $$-0.391925\pi$$
0.333042 + 0.942912i $$0.391925\pi$$
$$30$$ 0 0
$$31$$ 7.29517 1.31025 0.655126 0.755520i $$-0.272616\pi$$
0.655126 + 0.755520i $$0.272616\pi$$
$$32$$ 0 0
$$33$$ 3.68447 0.641384
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 2.37463 0.390387 0.195194 0.980765i $$-0.437466\pi$$
0.195194 + 0.980765i $$0.437466\pi$$
$$38$$ 0 0
$$39$$ −1.13856 −0.182316
$$40$$ 0 0
$$41$$ 1.76846 0.276188 0.138094 0.990419i $$-0.455902\pi$$
0.138094 + 0.990419i $$0.455902\pi$$
$$42$$ 0 0
$$43$$ −5.51320 −0.840755 −0.420377 0.907349i $$-0.638102\pi$$
−0.420377 + 0.907349i $$0.638102\pi$$
$$44$$ 0 0
$$45$$ −1.70367 −0.253968
$$46$$ 0 0
$$47$$ −7.53693 −1.09937 −0.549687 0.835371i $$-0.685253\pi$$
−0.549687 + 0.835371i $$0.685253\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −3.39383 −0.475232
$$52$$ 0 0
$$53$$ −3.40734 −0.468035 −0.234017 0.972232i $$-0.575187\pi$$
−0.234017 + 0.972232i $$0.575187\pi$$
$$54$$ 0 0
$$55$$ −3.23607 −0.436351
$$56$$ 0 0
$$57$$ −2.20852 −0.292525
$$58$$ 0 0
$$59$$ −11.6377 −1.51510 −0.757551 0.652776i $$-0.773604\pi$$
−0.757551 + 0.652776i $$0.773604\pi$$
$$60$$ 0 0
$$61$$ −5.68447 −0.727822 −0.363911 0.931434i $$-0.618559\pi$$
−0.363911 + 0.931434i $$0.618559\pi$$
$$62$$ 0 0
$$63$$ 1.70367 0.214642
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 2.04559 0.249909 0.124954 0.992162i $$-0.460122\pi$$
0.124954 + 0.992162i $$0.460122\pi$$
$$68$$ 0 0
$$69$$ 2.47214 0.297610
$$70$$ 0 0
$$71$$ −10.9205 −1.29603 −0.648015 0.761628i $$-0.724401\pi$$
−0.648015 + 0.761628i $$0.724401\pi$$
$$72$$ 0 0
$$73$$ −7.09181 −0.830034 −0.415017 0.909814i $$-0.636224\pi$$
−0.415017 + 0.909814i $$0.636224\pi$$
$$74$$ 0 0
$$75$$ −1.13856 −0.131470
$$76$$ 0 0
$$77$$ 3.23607 0.368784
$$78$$ 0 0
$$79$$ −14.7949 −1.66455 −0.832276 0.554362i $$-0.812962\pi$$
−0.832276 + 0.554362i $$0.812962\pi$$
$$80$$ 0 0
$$81$$ −0.986489 −0.109610
$$82$$ 0 0
$$83$$ 2.61968 0.287547 0.143774 0.989611i $$-0.454076\pi$$
0.143774 + 0.989611i $$0.454076\pi$$
$$84$$ 0 0
$$85$$ 2.98080 0.323313
$$86$$ 0 0
$$87$$ −4.08399 −0.437850
$$88$$ 0 0
$$89$$ 3.92952 0.416528 0.208264 0.978073i $$-0.433219\pi$$
0.208264 + 0.978073i $$0.433219\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −8.30602 −0.861294
$$94$$ 0 0
$$95$$ 1.93974 0.199013
$$96$$ 0 0
$$97$$ −1.34192 −0.136252 −0.0681258 0.997677i $$-0.521702\pi$$
−0.0681258 + 0.997677i $$0.521702\pi$$
$$98$$ 0 0
$$99$$ 5.51320 0.554097
$$100$$ 0 0
$$101$$ 7.49853 0.746132 0.373066 0.927805i $$-0.378307\pi$$
0.373066 + 0.927805i $$0.378307\pi$$
$$102$$ 0 0
$$103$$ 14.5685 1.43548 0.717738 0.696314i $$-0.245178\pi$$
0.717738 + 0.696314i $$0.245178\pi$$
$$104$$ 0 0
$$105$$ 1.13856 0.111112
$$106$$ 0 0
$$107$$ −11.7736 −1.13820 −0.569100 0.822269i $$-0.692708\pi$$
−0.569100 + 0.822269i $$0.692708\pi$$
$$108$$ 0 0
$$109$$ 0.787665 0.0754446 0.0377223 0.999288i $$-0.487990\pi$$
0.0377223 + 0.999288i $$0.487990\pi$$
$$110$$ 0 0
$$111$$ −2.70367 −0.256621
$$112$$ 0 0
$$113$$ −7.96160 −0.748964 −0.374482 0.927234i $$-0.622180\pi$$
−0.374482 + 0.927234i $$0.622180\pi$$
$$114$$ 0 0
$$115$$ −2.17127 −0.202472
$$116$$ 0 0
$$117$$ −1.70367 −0.157504
$$118$$ 0 0
$$119$$ −2.98080 −0.273249
$$120$$ 0 0
$$121$$ −0.527864 −0.0479876
$$122$$ 0 0
$$123$$ −2.01351 −0.181552
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −3.23607 −0.287155 −0.143577 0.989639i $$-0.545861\pi$$
−0.143577 + 0.989639i $$0.545861\pi$$
$$128$$ 0 0
$$129$$ 6.27713 0.552670
$$130$$ 0 0
$$131$$ −6.42467 −0.561326 −0.280663 0.959806i $$-0.590554\pi$$
−0.280663 + 0.959806i $$0.590554\pi$$
$$132$$ 0 0
$$133$$ −1.93974 −0.168197
$$134$$ 0 0
$$135$$ 5.35543 0.460922
$$136$$ 0 0
$$137$$ 1.83708 0.156952 0.0784760 0.996916i $$-0.474995\pi$$
0.0784760 + 0.996916i $$0.474995\pi$$
$$138$$ 0 0
$$139$$ 8.62874 0.731880 0.365940 0.930638i $$-0.380747\pi$$
0.365940 + 0.930638i $$0.380747\pi$$
$$140$$ 0 0
$$141$$ 8.58128 0.722674
$$142$$ 0 0
$$143$$ −3.23607 −0.270614
$$144$$ 0 0
$$145$$ 3.58697 0.297881
$$146$$ 0 0
$$147$$ −1.13856 −0.0939072
$$148$$ 0 0
$$149$$ −18.8405 −1.54347 −0.771735 0.635944i $$-0.780611\pi$$
−0.771735 + 0.635944i $$0.780611\pi$$
$$150$$ 0 0
$$151$$ −12.6704 −1.03111 −0.515553 0.856858i $$-0.672413\pi$$
−0.515553 + 0.856858i $$0.672413\pi$$
$$152$$ 0 0
$$153$$ −5.07830 −0.410557
$$154$$ 0 0
$$155$$ 7.29517 0.585962
$$156$$ 0 0
$$157$$ 2.32717 0.185728 0.0928641 0.995679i $$-0.470398\pi$$
0.0928641 + 0.995679i $$0.470398\pi$$
$$158$$ 0 0
$$159$$ 3.87948 0.307663
$$160$$ 0 0
$$161$$ 2.17127 0.171120
$$162$$ 0 0
$$163$$ −11.1623 −0.874299 −0.437149 0.899389i $$-0.644012\pi$$
−0.437149 + 0.899389i $$0.644012\pi$$
$$164$$ 0 0
$$165$$ 3.68447 0.286836
$$166$$ 0 0
$$167$$ −8.58128 −0.664039 −0.332020 0.943272i $$-0.607730\pi$$
−0.332020 + 0.943272i $$0.607730\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −3.30468 −0.252715
$$172$$ 0 0
$$173$$ −13.4010 −1.01886 −0.509431 0.860512i $$-0.670144\pi$$
−0.509431 + 0.860512i $$0.670144\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 13.2503 0.995953
$$178$$ 0 0
$$179$$ 8.45356 0.631849 0.315925 0.948784i $$-0.397685\pi$$
0.315925 + 0.948784i $$0.397685\pi$$
$$180$$ 0 0
$$181$$ −21.6989 −1.61286 −0.806432 0.591327i $$-0.798604\pi$$
−0.806432 + 0.591327i $$0.798604\pi$$
$$182$$ 0 0
$$183$$ 6.47214 0.478434
$$184$$ 0 0
$$185$$ 2.37463 0.174586
$$186$$ 0 0
$$187$$ −9.64607 −0.705391
$$188$$ 0 0
$$189$$ −5.35543 −0.389550
$$190$$ 0 0
$$191$$ −14.0975 −1.02006 −0.510030 0.860157i $$-0.670366\pi$$
−0.510030 + 0.860157i $$0.670366\pi$$
$$192$$ 0 0
$$193$$ 14.9077 1.07308 0.536541 0.843874i $$-0.319731\pi$$
0.536541 + 0.843874i $$0.319731\pi$$
$$194$$ 0 0
$$195$$ −1.13856 −0.0815343
$$196$$ 0 0
$$197$$ 18.1503 1.29315 0.646577 0.762848i $$-0.276200\pi$$
0.646577 + 0.762848i $$0.276200\pi$$
$$198$$ 0 0
$$199$$ −26.6070 −1.88612 −0.943062 0.332618i $$-0.892068\pi$$
−0.943062 + 0.332618i $$0.892068\pi$$
$$200$$ 0 0
$$201$$ −2.32904 −0.164278
$$202$$ 0 0
$$203$$ −3.58697 −0.251756
$$204$$ 0 0
$$205$$ 1.76846 0.123515
$$206$$ 0 0
$$207$$ 3.69914 0.257108
$$208$$ 0 0
$$209$$ −6.27713 −0.434198
$$210$$ 0 0
$$211$$ 6.28432 0.432631 0.216315 0.976324i $$-0.430596\pi$$
0.216315 + 0.976324i $$0.430596\pi$$
$$212$$ 0 0
$$213$$ 12.4337 0.851946
$$214$$ 0 0
$$215$$ −5.51320 −0.375997
$$216$$ 0 0
$$217$$ −7.29517 −0.495229
$$218$$ 0 0
$$219$$ 8.07449 0.545623
$$220$$ 0 0
$$221$$ 2.98080 0.200510
$$222$$ 0 0
$$223$$ 17.8885 1.19791 0.598953 0.800784i $$-0.295584\pi$$
0.598953 + 0.800784i $$0.295584\pi$$
$$224$$ 0 0
$$225$$ −1.70367 −0.113578
$$226$$ 0 0
$$227$$ 7.77566 0.516089 0.258044 0.966133i $$-0.416922\pi$$
0.258044 + 0.966133i $$0.416922\pi$$
$$228$$ 0 0
$$229$$ −5.60048 −0.370090 −0.185045 0.982730i $$-0.559243\pi$$
−0.185045 + 0.982730i $$0.559243\pi$$
$$230$$ 0 0
$$231$$ −3.68447 −0.242420
$$232$$ 0 0
$$233$$ 6.88011 0.450731 0.225365 0.974274i $$-0.427642\pi$$
0.225365 + 0.974274i $$0.427642\pi$$
$$234$$ 0 0
$$235$$ −7.53693 −0.491655
$$236$$ 0 0
$$237$$ 16.8449 1.09419
$$238$$ 0 0
$$239$$ 21.2722 1.37598 0.687991 0.725720i $$-0.258493\pi$$
0.687991 + 0.725720i $$0.258493\pi$$
$$240$$ 0 0
$$241$$ −7.89805 −0.508758 −0.254379 0.967105i $$-0.581871\pi$$
−0.254379 + 0.967105i $$0.581871\pi$$
$$242$$ 0 0
$$243$$ −14.9431 −0.958601
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 1.93974 0.123423
$$248$$ 0 0
$$249$$ −2.98267 −0.189019
$$250$$ 0 0
$$251$$ −29.0792 −1.83546 −0.917731 0.397203i $$-0.869981\pi$$
−0.917731 + 0.397203i $$0.869981\pi$$
$$252$$ 0 0
$$253$$ 7.02639 0.441746
$$254$$ 0 0
$$255$$ −3.39383 −0.212530
$$256$$ 0 0
$$257$$ 8.12959 0.507110 0.253555 0.967321i $$-0.418400\pi$$
0.253555 + 0.967321i $$0.418400\pi$$
$$258$$ 0 0
$$259$$ −2.37463 −0.147552
$$260$$ 0 0
$$261$$ −6.11101 −0.378262
$$262$$ 0 0
$$263$$ 0.0416886 0.00257063 0.00128531 0.999999i $$-0.499591\pi$$
0.00128531 + 0.999999i $$0.499591\pi$$
$$264$$ 0 0
$$265$$ −3.40734 −0.209311
$$266$$ 0 0
$$267$$ −4.47401 −0.273805
$$268$$ 0 0
$$269$$ 24.0068 1.46372 0.731859 0.681456i $$-0.238653\pi$$
0.731859 + 0.681456i $$0.238653\pi$$
$$270$$ 0 0
$$271$$ −23.8815 −1.45070 −0.725349 0.688381i $$-0.758322\pi$$
−0.725349 + 0.688381i $$0.758322\pi$$
$$272$$ 0 0
$$273$$ 1.13856 0.0689090
$$274$$ 0 0
$$275$$ −3.23607 −0.195142
$$276$$ 0 0
$$277$$ 23.5256 1.41351 0.706757 0.707457i $$-0.250158\pi$$
0.706757 + 0.707457i $$0.250158\pi$$
$$278$$ 0 0
$$279$$ −12.4286 −0.744079
$$280$$ 0 0
$$281$$ −18.8405 −1.12393 −0.561964 0.827162i $$-0.689954\pi$$
−0.561964 + 0.827162i $$0.689954\pi$$
$$282$$ 0 0
$$283$$ −9.11742 −0.541974 −0.270987 0.962583i $$-0.587350\pi$$
−0.270987 + 0.962583i $$0.587350\pi$$
$$284$$ 0 0
$$285$$ −2.20852 −0.130821
$$286$$ 0 0
$$287$$ −1.76846 −0.104389
$$288$$ 0 0
$$289$$ −8.11483 −0.477343
$$290$$ 0 0
$$291$$ 1.52786 0.0895650
$$292$$ 0 0
$$293$$ −10.2387 −0.598153 −0.299076 0.954229i $$-0.596679\pi$$
−0.299076 + 0.954229i $$0.596679\pi$$
$$294$$ 0 0
$$295$$ −11.6377 −0.677574
$$296$$ 0 0
$$297$$ −17.3305 −1.00562
$$298$$ 0 0
$$299$$ −2.17127 −0.125568
$$300$$ 0 0
$$301$$ 5.51320 0.317775
$$302$$ 0 0
$$303$$ −8.53756 −0.490470
$$304$$ 0 0
$$305$$ −5.68447 −0.325492
$$306$$ 0 0
$$307$$ 18.9059 1.07902 0.539508 0.841981i $$-0.318610\pi$$
0.539508 + 0.841981i $$0.318610\pi$$
$$308$$ 0 0
$$309$$ −16.5872 −0.943610
$$310$$ 0 0
$$311$$ −17.7499 −1.00650 −0.503252 0.864140i $$-0.667863\pi$$
−0.503252 + 0.864140i $$0.667863\pi$$
$$312$$ 0 0
$$313$$ 5.53187 0.312680 0.156340 0.987703i $$-0.450031\pi$$
0.156340 + 0.987703i $$0.450031\pi$$
$$314$$ 0 0
$$315$$ 1.70367 0.0959910
$$316$$ 0 0
$$317$$ −2.13491 −0.119908 −0.0599542 0.998201i $$-0.519095\pi$$
−0.0599542 + 0.998201i $$0.519095\pi$$
$$318$$ 0 0
$$319$$ −11.6077 −0.649905
$$320$$ 0 0
$$321$$ 13.4050 0.748196
$$322$$ 0 0
$$323$$ 5.78198 0.321718
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ −0.896807 −0.0495935
$$328$$ 0 0
$$329$$ 7.53693 0.415524
$$330$$ 0 0
$$331$$ 12.8528 0.706454 0.353227 0.935538i $$-0.385084\pi$$
0.353227 + 0.935538i $$0.385084\pi$$
$$332$$ 0 0
$$333$$ −4.04559 −0.221697
$$334$$ 0 0
$$335$$ 2.04559 0.111763
$$336$$ 0 0
$$337$$ 20.8147 1.13385 0.566924 0.823770i $$-0.308133\pi$$
0.566924 + 0.823770i $$0.308133\pi$$
$$338$$ 0 0
$$339$$ 9.06479 0.492332
$$340$$ 0 0
$$341$$ −23.6077 −1.27843
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 2.47214 0.133095
$$346$$ 0 0
$$347$$ 17.2889 0.928114 0.464057 0.885805i $$-0.346393\pi$$
0.464057 + 0.885805i $$0.346393\pi$$
$$348$$ 0 0
$$349$$ −28.9920 −1.55191 −0.775953 0.630791i $$-0.782731\pi$$
−0.775953 + 0.630791i $$0.782731\pi$$
$$350$$ 0 0
$$351$$ 5.35543 0.285852
$$352$$ 0 0
$$353$$ −30.9149 −1.64544 −0.822718 0.568450i $$-0.807543\pi$$
−0.822718 + 0.568450i $$0.807543\pi$$
$$354$$ 0 0
$$355$$ −10.9205 −0.579602
$$356$$ 0 0
$$357$$ 3.39383 0.179621
$$358$$ 0 0
$$359$$ −25.6584 −1.35420 −0.677100 0.735891i $$-0.736764\pi$$
−0.677100 + 0.735891i $$0.736764\pi$$
$$360$$ 0 0
$$361$$ −15.2374 −0.801969
$$362$$ 0 0
$$363$$ 0.601007 0.0315447
$$364$$ 0 0
$$365$$ −7.09181 −0.371203
$$366$$ 0 0
$$367$$ −12.4317 −0.648930 −0.324465 0.945898i $$-0.605184\pi$$
−0.324465 + 0.945898i $$0.605184\pi$$
$$368$$ 0 0
$$369$$ −3.01288 −0.156844
$$370$$ 0 0
$$371$$ 3.40734 0.176900
$$372$$ 0 0
$$373$$ −29.1613 −1.50991 −0.754957 0.655774i $$-0.772343\pi$$
−0.754957 + 0.655774i $$0.772343\pi$$
$$374$$ 0 0
$$375$$ −1.13856 −0.0587952
$$376$$ 0 0
$$377$$ 3.58697 0.184738
$$378$$ 0 0
$$379$$ 12.4921 0.641677 0.320839 0.947134i $$-0.396035\pi$$
0.320839 + 0.947134i $$0.396035\pi$$
$$380$$ 0 0
$$381$$ 3.68447 0.188761
$$382$$ 0 0
$$383$$ −11.7012 −0.597902 −0.298951 0.954268i $$-0.596637\pi$$
−0.298951 + 0.954268i $$0.596637\pi$$
$$384$$ 0 0
$$385$$ 3.23607 0.164925
$$386$$ 0 0
$$387$$ 9.39268 0.477457
$$388$$ 0 0
$$389$$ 15.1758 0.769444 0.384722 0.923032i $$-0.374297\pi$$
0.384722 + 0.923032i $$0.374297\pi$$
$$390$$ 0 0
$$391$$ −6.47214 −0.327310
$$392$$ 0 0
$$393$$ 7.31490 0.368988
$$394$$ 0 0
$$395$$ −14.7949 −0.744410
$$396$$ 0 0
$$397$$ 35.0150 1.75735 0.878677 0.477417i $$-0.158427\pi$$
0.878677 + 0.477417i $$0.158427\pi$$
$$398$$ 0 0
$$399$$ 2.20852 0.110564
$$400$$ 0 0
$$401$$ −17.0059 −0.849237 −0.424618 0.905372i $$-0.639592\pi$$
−0.424618 + 0.905372i $$0.639592\pi$$
$$402$$ 0 0
$$403$$ 7.29517 0.363398
$$404$$ 0 0
$$405$$ −0.986489 −0.0490191
$$406$$ 0 0
$$407$$ −7.68447 −0.380905
$$408$$ 0 0
$$409$$ 4.05279 0.200397 0.100199 0.994967i $$-0.468052\pi$$
0.100199 + 0.994967i $$0.468052\pi$$
$$410$$ 0 0
$$411$$ −2.09163 −0.103172
$$412$$ 0 0
$$413$$ 11.6377 0.572655
$$414$$ 0 0
$$415$$ 2.61968 0.128595
$$416$$ 0 0
$$417$$ −9.82438 −0.481102
$$418$$ 0 0
$$419$$ −8.73256 −0.426614 −0.213307 0.976985i $$-0.568423\pi$$
−0.213307 + 0.976985i $$0.568423\pi$$
$$420$$ 0 0
$$421$$ 18.0000 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$422$$ 0 0
$$423$$ 12.8405 0.624324
$$424$$ 0 0
$$425$$ 2.98080 0.144590
$$426$$ 0 0
$$427$$ 5.68447 0.275091
$$428$$ 0 0
$$429$$ 3.68447 0.177888
$$430$$ 0 0
$$431$$ 3.20530 0.154394 0.0771970 0.997016i $$-0.475403\pi$$
0.0771970 + 0.997016i $$0.475403\pi$$
$$432$$ 0 0
$$433$$ −33.4199 −1.60606 −0.803028 0.595941i $$-0.796779\pi$$
−0.803028 + 0.595941i $$0.796779\pi$$
$$434$$ 0 0
$$435$$ −4.08399 −0.195812
$$436$$ 0 0
$$437$$ −4.21171 −0.201473
$$438$$ 0 0
$$439$$ 23.6551 1.12900 0.564499 0.825434i $$-0.309069\pi$$
0.564499 + 0.825434i $$0.309069\pi$$
$$440$$ 0 0
$$441$$ −1.70367 −0.0811272
$$442$$ 0 0
$$443$$ 6.15864 0.292606 0.146303 0.989240i $$-0.453263\pi$$
0.146303 + 0.989240i $$0.453263\pi$$
$$444$$ 0 0
$$445$$ 3.92952 0.186277
$$446$$ 0 0
$$447$$ 21.4511 1.01460
$$448$$ 0 0
$$449$$ 15.4073 0.727117 0.363559 0.931571i $$-0.381562\pi$$
0.363559 + 0.931571i $$0.381562\pi$$
$$450$$ 0 0
$$451$$ −5.72287 −0.269479
$$452$$ 0 0
$$453$$ 14.4261 0.677797
$$454$$ 0 0
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ 12.6518 0.591824 0.295912 0.955215i $$-0.404376\pi$$
0.295912 + 0.955215i $$0.404376\pi$$
$$458$$ 0 0
$$459$$ 15.9635 0.745111
$$460$$ 0 0
$$461$$ 7.19838 0.335262 0.167631 0.985850i $$-0.446388\pi$$
0.167631 + 0.985850i $$0.446388\pi$$
$$462$$ 0 0
$$463$$ 19.2849 0.896248 0.448124 0.893972i $$-0.352092\pi$$
0.448124 + 0.893972i $$0.352092\pi$$
$$464$$ 0 0
$$465$$ −8.30602 −0.385183
$$466$$ 0 0
$$467$$ 2.59407 0.120039 0.0600197 0.998197i $$-0.480884\pi$$
0.0600197 + 0.998197i $$0.480884\pi$$
$$468$$ 0 0
$$469$$ −2.04559 −0.0944567
$$470$$ 0 0
$$471$$ −2.64963 −0.122088
$$472$$ 0 0
$$473$$ 17.8411 0.820334
$$474$$ 0 0
$$475$$ 1.93974 0.0890013
$$476$$ 0 0
$$477$$ 5.80499 0.265792
$$478$$ 0 0
$$479$$ −18.6808 −0.853548 −0.426774 0.904358i $$-0.640350\pi$$
−0.426774 + 0.904358i $$0.640350\pi$$
$$480$$ 0 0
$$481$$ 2.37463 0.108274
$$482$$ 0 0
$$483$$ −2.47214 −0.112486
$$484$$ 0 0
$$485$$ −1.34192 −0.0609335
$$486$$ 0 0
$$487$$ −8.27081 −0.374786 −0.187393 0.982285i $$-0.560004\pi$$
−0.187393 + 0.982285i $$0.560004\pi$$
$$488$$ 0 0
$$489$$ 12.7090 0.574720
$$490$$ 0 0
$$491$$ 1.87448 0.0845940 0.0422970 0.999105i $$-0.486532\pi$$
0.0422970 + 0.999105i $$0.486532\pi$$
$$492$$ 0 0
$$493$$ 10.6920 0.481545
$$494$$ 0 0
$$495$$ 5.51320 0.247800
$$496$$ 0 0
$$497$$ 10.9205 0.489853
$$498$$ 0 0
$$499$$ −36.2217 −1.62151 −0.810754 0.585387i $$-0.800943\pi$$
−0.810754 + 0.585387i $$0.800943\pi$$
$$500$$ 0 0
$$501$$ 9.77034 0.436506
$$502$$ 0 0
$$503$$ −1.53364 −0.0683817 −0.0341908 0.999415i $$-0.510885\pi$$
−0.0341908 + 0.999415i $$0.510885\pi$$
$$504$$ 0 0
$$505$$ 7.49853 0.333680
$$506$$ 0 0
$$507$$ −1.13856 −0.0505654
$$508$$ 0 0
$$509$$ 8.06229 0.357355 0.178677 0.983908i $$-0.442818\pi$$
0.178677 + 0.983908i $$0.442818\pi$$
$$510$$ 0 0
$$511$$ 7.09181 0.313723
$$512$$ 0 0
$$513$$ 10.3881 0.458648
$$514$$ 0 0
$$515$$ 14.5685 0.641964
$$516$$ 0 0
$$517$$ 24.3900 1.07267
$$518$$ 0 0
$$519$$ 15.2579 0.669749
$$520$$ 0 0
$$521$$ −10.4835 −0.459291 −0.229646 0.973274i $$-0.573757\pi$$
−0.229646 + 0.973274i $$0.573757\pi$$
$$522$$ 0 0
$$523$$ −25.2517 −1.10418 −0.552090 0.833784i $$-0.686170\pi$$
−0.552090 + 0.833784i $$0.686170\pi$$
$$524$$ 0 0
$$525$$ 1.13856 0.0496910
$$526$$ 0 0
$$527$$ 21.7454 0.947247
$$528$$ 0 0
$$529$$ −18.2856 −0.795025
$$530$$ 0 0
$$531$$ 19.8269 0.860412
$$532$$ 0 0
$$533$$ 1.76846 0.0766007
$$534$$ 0 0
$$535$$ −11.7736 −0.509018
$$536$$ 0 0
$$537$$ −9.62493 −0.415346
$$538$$ 0 0
$$539$$ −3.23607 −0.139387
$$540$$ 0 0
$$541$$ −15.3947 −0.661870 −0.330935 0.943654i $$-0.607364\pi$$
−0.330935 + 0.943654i $$0.607364\pi$$
$$542$$ 0 0
$$543$$ 24.7055 1.06022
$$544$$ 0 0
$$545$$ 0.787665 0.0337398
$$546$$ 0 0
$$547$$ 37.9613 1.62311 0.811554 0.584277i $$-0.198622\pi$$
0.811554 + 0.584277i $$0.198622\pi$$
$$548$$ 0 0
$$549$$ 9.68447 0.413323
$$550$$ 0 0
$$551$$ 6.95778 0.296411
$$552$$ 0 0
$$553$$ 14.7949 0.629141
$$554$$ 0 0
$$555$$ −2.70367 −0.114764
$$556$$ 0 0
$$557$$ −43.5779 −1.84645 −0.923227 0.384254i $$-0.874459\pi$$
−0.923227 + 0.384254i $$0.874459\pi$$
$$558$$ 0 0
$$559$$ −5.51320 −0.233183
$$560$$ 0 0
$$561$$ 10.9827 0.463689
$$562$$ 0 0
$$563$$ 21.6423 0.912116 0.456058 0.889950i $$-0.349261\pi$$
0.456058 + 0.889950i $$0.349261\pi$$
$$564$$ 0 0
$$565$$ −7.96160 −0.334947
$$566$$ 0 0
$$567$$ 0.986489 0.0414287
$$568$$ 0 0
$$569$$ −2.47845 −0.103902 −0.0519511 0.998650i $$-0.516544\pi$$
−0.0519511 + 0.998650i $$0.516544\pi$$
$$570$$ 0 0
$$571$$ −10.0040 −0.418655 −0.209327 0.977846i $$-0.567127\pi$$
−0.209327 + 0.977846i $$0.567127\pi$$
$$572$$ 0 0
$$573$$ 16.0509 0.670537
$$574$$ 0 0
$$575$$ −2.17127 −0.0905484
$$576$$ 0 0
$$577$$ 28.2953 1.17795 0.588974 0.808152i $$-0.299532\pi$$
0.588974 + 0.808152i $$0.299532\pi$$
$$578$$ 0 0
$$579$$ −16.9734 −0.705391
$$580$$ 0 0
$$581$$ −2.61968 −0.108683
$$582$$ 0 0
$$583$$ 11.0264 0.456667
$$584$$ 0 0
$$585$$ −1.70367 −0.0704381
$$586$$ 0 0
$$587$$ 0.614357 0.0253572 0.0126786 0.999920i $$-0.495964\pi$$
0.0126786 + 0.999920i $$0.495964\pi$$
$$588$$ 0 0
$$589$$ 14.1507 0.583071
$$590$$ 0 0
$$591$$ −20.6653 −0.850056
$$592$$ 0 0
$$593$$ −34.6227 −1.42178 −0.710892 0.703302i $$-0.751708\pi$$
−0.710892 + 0.703302i $$0.751708\pi$$
$$594$$ 0 0
$$595$$ −2.98080 −0.122201
$$596$$ 0 0
$$597$$ 30.2938 1.23984
$$598$$ 0 0
$$599$$ 38.2311 1.56208 0.781040 0.624481i $$-0.214689\pi$$
0.781040 + 0.624481i $$0.214689\pi$$
$$600$$ 0 0
$$601$$ −36.3390 −1.48230 −0.741149 0.671341i $$-0.765719\pi$$
−0.741149 + 0.671341i $$0.765719\pi$$
$$602$$ 0 0
$$603$$ −3.48502 −0.141921
$$604$$ 0 0
$$605$$ −0.527864 −0.0214607
$$606$$ 0 0
$$607$$ −2.03537 −0.0826132 −0.0413066 0.999147i $$-0.513152\pi$$
−0.0413066 + 0.999147i $$0.513152\pi$$
$$608$$ 0 0
$$609$$ 4.08399 0.165492
$$610$$ 0 0
$$611$$ −7.53693 −0.304912
$$612$$ 0 0
$$613$$ 29.8464 1.20548 0.602742 0.797936i $$-0.294075\pi$$
0.602742 + 0.797936i $$0.294075\pi$$
$$614$$ 0 0
$$615$$ −2.01351 −0.0811926
$$616$$ 0 0
$$617$$ 2.72163 0.109569 0.0547843 0.998498i $$-0.482553\pi$$
0.0547843 + 0.998498i $$0.482553\pi$$
$$618$$ 0 0
$$619$$ −8.52795 −0.342767 −0.171384 0.985204i $$-0.554824\pi$$
−0.171384 + 0.985204i $$0.554824\pi$$
$$620$$ 0 0
$$621$$ −11.6281 −0.466620
$$622$$ 0 0
$$623$$ −3.92952 −0.157433
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 7.14691 0.285420
$$628$$ 0 0
$$629$$ 7.07830 0.282230
$$630$$ 0 0
$$631$$ −1.39268 −0.0554415 −0.0277208 0.999616i $$-0.508825\pi$$
−0.0277208 + 0.999616i $$0.508825\pi$$
$$632$$ 0 0
$$633$$ −7.15510 −0.284390
$$634$$ 0 0
$$635$$ −3.23607 −0.128419
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 18.6050 0.736003
$$640$$ 0 0
$$641$$ 3.19838 0.126329 0.0631643 0.998003i $$-0.479881\pi$$
0.0631643 + 0.998003i $$0.479881\pi$$
$$642$$ 0 0
$$643$$ −25.2845 −0.997124 −0.498562 0.866854i $$-0.666138\pi$$
−0.498562 + 0.866854i $$0.666138\pi$$
$$644$$ 0 0
$$645$$ 6.27713 0.247162
$$646$$ 0 0
$$647$$ 0.414007 0.0162763 0.00813814 0.999967i $$-0.497410\pi$$
0.00813814 + 0.999967i $$0.497410\pi$$
$$648$$ 0 0
$$649$$ 37.6605 1.47830
$$650$$ 0 0
$$651$$ 8.30602 0.325539
$$652$$ 0 0
$$653$$ 22.4170 0.877246 0.438623 0.898671i $$-0.355466\pi$$
0.438623 + 0.898671i $$0.355466\pi$$
$$654$$ 0 0
$$655$$ −6.42467 −0.251033
$$656$$ 0 0
$$657$$ 12.0821 0.471368
$$658$$ 0 0
$$659$$ 18.8679 0.734990 0.367495 0.930026i $$-0.380216\pi$$
0.367495 + 0.930026i $$0.380216\pi$$
$$660$$ 0 0
$$661$$ 14.8398 0.577203 0.288601 0.957449i $$-0.406810\pi$$
0.288601 + 0.957449i $$0.406810\pi$$
$$662$$ 0 0
$$663$$ −3.39383 −0.131806
$$664$$ 0 0
$$665$$ −1.93974 −0.0752199
$$666$$ 0 0
$$667$$ −7.78829 −0.301564
$$668$$ 0 0
$$669$$ −20.3673 −0.787444
$$670$$ 0 0
$$671$$ 18.3953 0.710144
$$672$$ 0 0
$$673$$ −0.488836 −0.0188432 −0.00942162 0.999956i $$-0.502999\pi$$
−0.00942162 + 0.999956i $$0.502999\pi$$
$$674$$ 0 0
$$675$$ 5.35543 0.206131
$$676$$ 0 0
$$677$$ 24.5736 0.944442 0.472221 0.881480i $$-0.343452\pi$$
0.472221 + 0.881480i $$0.343452\pi$$
$$678$$ 0 0
$$679$$ 1.34192 0.0514982
$$680$$ 0 0
$$681$$ −8.85309 −0.339251
$$682$$ 0 0
$$683$$ 19.8392 0.759126 0.379563 0.925166i $$-0.376074\pi$$
0.379563 + 0.925166i $$0.376074\pi$$
$$684$$ 0 0
$$685$$ 1.83708 0.0701910
$$686$$ 0 0
$$687$$ 6.37650 0.243279
$$688$$ 0 0
$$689$$ −3.40734 −0.129809
$$690$$ 0 0
$$691$$ 25.4788 0.969259 0.484630 0.874719i $$-0.338954\pi$$
0.484630 + 0.874719i $$0.338954\pi$$
$$692$$ 0 0
$$693$$ −5.51320 −0.209429
$$694$$ 0 0
$$695$$ 8.62874 0.327307
$$696$$ 0 0
$$697$$ 5.27144 0.199670
$$698$$ 0 0
$$699$$ −7.83344 −0.296288
$$700$$ 0 0
$$701$$ 27.8981 1.05369 0.526847 0.849960i $$-0.323374\pi$$
0.526847 + 0.849960i $$0.323374\pi$$
$$702$$ 0 0
$$703$$ 4.60617 0.173725
$$704$$ 0 0
$$705$$ 8.58128 0.323190
$$706$$ 0 0
$$707$$ −7.49853 −0.282011
$$708$$ 0 0
$$709$$ 21.7860 0.818190 0.409095 0.912492i $$-0.365845\pi$$
0.409095 + 0.912492i $$0.365845\pi$$
$$710$$ 0 0
$$711$$ 25.2056 0.945283
$$712$$ 0 0
$$713$$ −15.8398 −0.593206
$$714$$ 0 0
$$715$$ −3.23607 −0.121022
$$716$$ 0 0
$$717$$ −24.2197 −0.904502
$$718$$ 0 0
$$719$$ −2.76064 −0.102955 −0.0514773 0.998674i $$-0.516393\pi$$
−0.0514773 + 0.998674i $$0.516393\pi$$
$$720$$ 0 0
$$721$$ −14.5685 −0.542559
$$722$$ 0 0
$$723$$ 8.99244 0.334432
$$724$$ 0 0
$$725$$ 3.58697 0.133217
$$726$$ 0 0
$$727$$ 5.00366 0.185575 0.0927877 0.995686i $$-0.470422\pi$$
0.0927877 + 0.995686i $$0.470422\pi$$
$$728$$ 0 0
$$729$$ 19.9732 0.739747
$$730$$ 0 0
$$731$$ −16.4337 −0.607824
$$732$$ 0 0
$$733$$ −1.26512 −0.0467283 −0.0233642 0.999727i $$-0.507438\pi$$
−0.0233642 + 0.999727i $$0.507438\pi$$
$$734$$ 0 0
$$735$$ −1.13856 −0.0419966
$$736$$ 0 0
$$737$$ −6.61968 −0.243839
$$738$$ 0 0
$$739$$ 28.1432 1.03526 0.517632 0.855603i $$-0.326814\pi$$
0.517632 + 0.855603i $$0.326814\pi$$
$$740$$ 0 0
$$741$$ −2.20852 −0.0811319
$$742$$ 0 0
$$743$$ −13.1298 −0.481685 −0.240842 0.970564i $$-0.577424\pi$$
−0.240842 + 0.970564i $$0.577424\pi$$
$$744$$ 0 0
$$745$$ −18.8405 −0.690261
$$746$$ 0 0
$$747$$ −4.46307 −0.163295
$$748$$ 0 0
$$749$$ 11.7736 0.430199
$$750$$ 0 0
$$751$$ 27.3409 0.997685 0.498842 0.866693i $$-0.333759\pi$$
0.498842 + 0.866693i $$0.333759\pi$$
$$752$$ 0 0
$$753$$ 33.1085 1.20654
$$754$$ 0 0
$$755$$ −12.6704 −0.461124
$$756$$ 0 0
$$757$$ 29.5640 1.07452 0.537260 0.843417i $$-0.319459\pi$$
0.537260 + 0.843417i $$0.319459\pi$$
$$758$$ 0 0
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −17.3639 −0.629440 −0.314720 0.949185i $$-0.601911\pi$$
−0.314720 + 0.949185i $$0.601911\pi$$
$$762$$ 0 0
$$763$$ −0.787665 −0.0285154
$$764$$ 0 0
$$765$$ −5.07830 −0.183606
$$766$$ 0 0
$$767$$ −11.6377 −0.420214
$$768$$ 0 0
$$769$$ 10.9443 0.394661 0.197330 0.980337i $$-0.436773\pi$$
0.197330 + 0.980337i $$0.436773\pi$$
$$770$$ 0 0
$$771$$ −9.25606 −0.333349
$$772$$ 0 0
$$773$$ −38.4003 −1.38116 −0.690582 0.723254i $$-0.742646\pi$$
−0.690582 + 0.723254i $$0.742646\pi$$
$$774$$ 0 0
$$775$$ 7.29517 0.262050
$$776$$ 0 0
$$777$$ 2.70367 0.0969937
$$778$$ 0 0
$$779$$ 3.43036 0.122905
$$780$$ 0 0
$$781$$ 35.3396 1.26455
$$782$$ 0 0
$$783$$ 19.2098 0.686501
$$784$$ 0 0
$$785$$ 2.32717 0.0830602
$$786$$ 0 0
$$787$$ 22.1836 0.790761 0.395380 0.918517i $$-0.370613\pi$$
0.395380 + 0.918517i $$0.370613\pi$$
$$788$$ 0 0
$$789$$ −0.0474651 −0.00168980
$$790$$ 0 0
$$791$$ 7.96160 0.283082
$$792$$ 0 0
$$793$$ −5.68447 −0.201861
$$794$$ 0 0
$$795$$ 3.87948 0.137591
$$796$$ 0 0
$$797$$ 21.2368 0.752245 0.376123 0.926570i $$-0.377257\pi$$
0.376123 + 0.926570i $$0.377257\pi$$
$$798$$ 0 0
$$799$$ −22.4661 −0.794793
$$800$$ 0 0
$$801$$ −6.69461 −0.236542
$$802$$ 0 0
$$803$$ 22.9496 0.809874
$$804$$ 0 0
$$805$$ 2.17127 0.0765274
$$806$$ 0 0
$$807$$ −27.3332 −0.962175
$$808$$ 0 0
$$809$$ −20.3677 −0.716090 −0.358045 0.933704i $$-0.616557\pi$$
−0.358045 + 0.933704i $$0.616557\pi$$
$$810$$ 0 0
$$811$$ 41.3800 1.45305 0.726525 0.687140i $$-0.241134\pi$$
0.726525 + 0.687140i $$0.241134\pi$$
$$812$$ 0 0
$$813$$ 27.1906 0.953617
$$814$$ 0 0
$$815$$ −11.1623 −0.390998
$$816$$ 0 0
$$817$$ −10.6942 −0.374141
$$818$$ 0 0
$$819$$ 1.70367 0.0595311
$$820$$ 0 0
$$821$$ −28.8328 −1.00627 −0.503136 0.864207i $$-0.667821\pi$$
−0.503136 + 0.864207i $$0.667821\pi$$
$$822$$ 0 0
$$823$$ −11.1527 −0.388758 −0.194379 0.980926i $$-0.562269\pi$$
−0.194379 + 0.980926i $$0.562269\pi$$
$$824$$ 0 0
$$825$$ 3.68447 0.128277
$$826$$ 0 0
$$827$$ −27.0330 −0.940028 −0.470014 0.882659i $$-0.655751\pi$$
−0.470014 + 0.882659i $$0.655751\pi$$
$$828$$ 0 0
$$829$$ 22.2928 0.774260 0.387130 0.922025i $$-0.373466\pi$$
0.387130 + 0.922025i $$0.373466\pi$$
$$830$$ 0 0
$$831$$ −26.7854 −0.929174
$$832$$ 0 0
$$833$$ 2.98080 0.103279
$$834$$ 0 0
$$835$$ −8.58128 −0.296967
$$836$$ 0 0
$$837$$ 39.0688 1.35042
$$838$$ 0 0
$$839$$ 50.3602 1.73863 0.869314 0.494260i $$-0.164561\pi$$
0.869314 + 0.494260i $$0.164561\pi$$
$$840$$ 0 0
$$841$$ −16.1337 −0.556333
$$842$$ 0 0
$$843$$ 21.4511 0.738814
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 0.527864 0.0181376
$$848$$ 0 0
$$849$$ 10.3808 0.356267
$$850$$ 0 0
$$851$$ −5.15598 −0.176745
$$852$$ 0 0
$$853$$ 35.8989 1.22915 0.614577 0.788857i $$-0.289327\pi$$
0.614577 + 0.788857i $$0.289327\pi$$
$$854$$ 0 0
$$855$$ −3.30468 −0.113018
$$856$$ 0 0
$$857$$ −8.48752 −0.289928 −0.144964 0.989437i $$-0.546307\pi$$
−0.144964 + 0.989437i $$0.546307\pi$$
$$858$$ 0 0
$$859$$ 16.8200 0.573891 0.286946 0.957947i $$-0.407360\pi$$
0.286946 + 0.957947i $$0.407360\pi$$
$$860$$ 0 0
$$861$$ 2.01351 0.0686203
$$862$$ 0 0
$$863$$ −28.5718 −0.972594 −0.486297 0.873793i $$-0.661653\pi$$
−0.486297 + 0.873793i $$0.661653\pi$$
$$864$$ 0 0
$$865$$ −13.4010 −0.455649
$$866$$ 0 0
$$867$$ 9.23926 0.313782
$$868$$ 0 0
$$869$$ 47.8772 1.62412
$$870$$ 0 0
$$871$$ 2.04559 0.0693123
$$872$$ 0 0
$$873$$ 2.28619 0.0773759
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ 16.7638 0.566072 0.283036 0.959109i $$-0.408658\pi$$
0.283036 + 0.959109i $$0.408658\pi$$
$$878$$ 0 0
$$879$$ 11.6575 0.393196
$$880$$ 0 0
$$881$$ −29.5933 −0.997023 −0.498512 0.866883i $$-0.666120\pi$$
−0.498512 + 0.866883i $$0.666120\pi$$
$$882$$ 0 0
$$883$$ 3.18203 0.107084 0.0535418 0.998566i $$-0.482949\pi$$
0.0535418 + 0.998566i $$0.482949\pi$$
$$884$$ 0 0
$$885$$ 13.2503 0.445404
$$886$$ 0 0
$$887$$ −4.31856 −0.145003 −0.0725015 0.997368i $$-0.523098\pi$$
−0.0725015 + 0.997368i $$0.523098\pi$$
$$888$$ 0 0
$$889$$ 3.23607 0.108534
$$890$$ 0 0
$$891$$ 3.19235 0.106948
$$892$$ 0 0
$$893$$ −14.6197 −0.489229
$$894$$ 0 0
$$895$$ 8.45356 0.282571
$$896$$ 0 0
$$897$$ 2.47214 0.0825422
$$898$$ 0 0
$$899$$ 26.1675 0.872736
$$900$$ 0 0
$$901$$ −10.1566 −0.338366
$$902$$ 0 0
$$903$$ −6.27713 −0.208890
$$904$$ 0 0
$$905$$ −21.6989 −0.721294
$$906$$ 0 0
$$907$$ −8.46653 −0.281127 −0.140563 0.990072i $$-0.544891\pi$$
−0.140563 + 0.990072i $$0.544891\pi$$
$$908$$ 0 0
$$909$$ −12.7750 −0.423721
$$910$$ 0 0
$$911$$ 4.45356 0.147553 0.0737766 0.997275i $$-0.476495\pi$$
0.0737766 + 0.997275i $$0.476495\pi$$
$$912$$ 0 0
$$913$$ −8.47746 −0.280563
$$914$$ 0 0
$$915$$ 6.47214 0.213962
$$916$$ 0 0
$$917$$ 6.42467 0.212161
$$918$$ 0 0
$$919$$ −0.310467 −0.0102414 −0.00512068 0.999987i $$-0.501630\pi$$
−0.00512068 + 0.999987i $$0.501630\pi$$
$$920$$ 0 0
$$921$$ −21.5256 −0.709291
$$922$$ 0 0
$$923$$ −10.9205 −0.359454
$$924$$ 0 0
$$925$$ 2.37463 0.0780774
$$926$$ 0 0
$$927$$ −24.8199 −0.815193
$$928$$ 0 0
$$929$$ −0.886424 −0.0290826 −0.0145413 0.999894i $$-0.504629\pi$$
−0.0145413 + 0.999894i $$0.504629\pi$$
$$930$$ 0 0
$$931$$ 1.93974 0.0635724
$$932$$ 0 0
$$933$$ 20.2094 0.661626
$$934$$ 0 0
$$935$$ −9.64607 −0.315460
$$936$$ 0 0
$$937$$ −22.3307 −0.729513 −0.364757 0.931103i $$-0.618848\pi$$
−0.364757 + 0.931103i $$0.618848\pi$$
$$938$$ 0 0
$$939$$ −6.29839 −0.205540
$$940$$ 0 0
$$941$$ −10.0673 −0.328184 −0.164092 0.986445i $$-0.552469\pi$$
−0.164092 + 0.986445i $$0.552469\pi$$
$$942$$ 0 0
$$943$$ −3.83982 −0.125042
$$944$$ 0 0
$$945$$ −5.35543 −0.174212
$$946$$ 0 0
$$947$$ 26.0559 0.846703 0.423352 0.905965i $$-0.360853\pi$$
0.423352 + 0.905965i $$0.360853\pi$$
$$948$$ 0 0
$$949$$ −7.09181 −0.230210
$$950$$ 0 0
$$951$$ 2.43073 0.0788218
$$952$$ 0 0
$$953$$ 11.7206 0.379666 0.189833 0.981816i $$-0.439205\pi$$
0.189833 + 0.981816i $$0.439205\pi$$
$$954$$ 0 0
$$955$$ −14.0975 −0.456185
$$956$$ 0 0
$$957$$ 13.2161 0.427215
$$958$$ 0 0
$$959$$ −1.83708 −0.0593222
$$960$$ 0 0
$$961$$ 22.2195 0.716759
$$962$$ 0 0
$$963$$ 20.0584 0.646373
$$964$$ 0 0
$$965$$ 14.9077 0.479897
$$966$$ 0 0
$$967$$ −12.0758 −0.388332 −0.194166 0.980969i $$-0.562200\pi$$
−0.194166 + 0.980969i $$0.562200\pi$$
$$968$$ 0 0
$$969$$ −6.58315 −0.211481
$$970$$ 0 0
$$971$$ −60.2174 −1.93247 −0.966234 0.257666i $$-0.917047\pi$$
−0.966234 + 0.257666i $$0.917047\pi$$
$$972$$ 0 0
$$973$$ −8.62874 −0.276625
$$974$$ 0 0
$$975$$ −1.13856 −0.0364632
$$976$$ 0 0
$$977$$ −27.1157 −0.867508 −0.433754 0.901031i $$-0.642811\pi$$
−0.433754 + 0.901031i $$0.642811\pi$$
$$978$$ 0 0
$$979$$ −12.7162 −0.406411
$$980$$ 0 0
$$981$$ −1.34192 −0.0428443
$$982$$ 0 0
$$983$$ −28.4274 −0.906692 −0.453346 0.891335i $$-0.649770\pi$$
−0.453346 + 0.891335i $$0.649770\pi$$
$$984$$ 0 0
$$985$$ 18.1503 0.578316
$$986$$ 0 0
$$987$$ −8.58128 −0.273145
$$988$$ 0 0
$$989$$ 11.9707 0.380645
$$990$$ 0 0
$$991$$ 25.3582 0.805529 0.402765 0.915304i $$-0.368049\pi$$
0.402765 + 0.915304i $$0.368049\pi$$
$$992$$ 0 0
$$993$$ −14.6337 −0.464388
$$994$$ 0 0
$$995$$ −26.6070 −0.843500
$$996$$ 0 0
$$997$$ −40.7554 −1.29074 −0.645368 0.763872i $$-0.723296\pi$$
−0.645368 + 0.763872i $$0.723296\pi$$
$$998$$ 0 0
$$999$$ 12.7172 0.402354
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 3640.2.a.s.1.2 4
4.3 odd 2 7280.2.a.ca.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.s.1.2 4 1.1 even 1 trivial
7280.2.a.ca.1.3 4 4.3 odd 2