# Properties

 Label 8470.2.a.cw.1.2 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.19898000.1 Defining polynomial: $$x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5$$ x^6 - x^5 - 10*x^4 + 7*x^3 + 24*x^2 - 15*x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.68692$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.68692 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.68692 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.154300 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.68692 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.68692 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.154300 q^{9} -1.00000 q^{10} -1.68692 q^{12} +0.0425741 q^{13} -1.00000 q^{14} -1.68692 q^{15} +1.00000 q^{16} -4.72038 q^{17} +0.154300 q^{18} -1.71340 q^{19} +1.00000 q^{20} -1.68692 q^{21} -6.95966 q^{23} +1.68692 q^{24} +1.00000 q^{25} -0.0425741 q^{26} +5.32105 q^{27} +1.00000 q^{28} -1.34277 q^{29} +1.68692 q^{30} +6.01245 q^{31} -1.00000 q^{32} +4.72038 q^{34} +1.00000 q^{35} -0.154300 q^{36} +8.48459 q^{37} +1.71340 q^{38} -0.0718191 q^{39} -1.00000 q^{40} +3.91371 q^{41} +1.68692 q^{42} +6.42455 q^{43} -0.154300 q^{45} +6.95966 q^{46} +4.59129 q^{47} -1.68692 q^{48} +1.00000 q^{49} -1.00000 q^{50} +7.96291 q^{51} +0.0425741 q^{52} -14.1024 q^{53} -5.32105 q^{54} -1.00000 q^{56} +2.89036 q^{57} +1.34277 q^{58} -10.2177 q^{59} -1.68692 q^{60} +13.8661 q^{61} -6.01245 q^{62} -0.154300 q^{63} +1.00000 q^{64} +0.0425741 q^{65} +6.39600 q^{67} -4.72038 q^{68} +11.7404 q^{69} -1.00000 q^{70} +11.3041 q^{71} +0.154300 q^{72} -1.98728 q^{73} -8.48459 q^{74} -1.68692 q^{75} -1.71340 q^{76} +0.0718191 q^{78} -8.87065 q^{79} +1.00000 q^{80} -8.51329 q^{81} -3.91371 q^{82} -10.0807 q^{83} -1.68692 q^{84} -4.72038 q^{85} -6.42455 q^{86} +2.26515 q^{87} +2.93362 q^{89} +0.154300 q^{90} +0.0425741 q^{91} -6.95966 q^{92} -10.1425 q^{93} -4.59129 q^{94} -1.71340 q^{95} +1.68692 q^{96} -17.4812 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 + 3 * q^9 $$6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} - q^{12} - 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 9 q^{17} - 3 q^{18} - 12 q^{19} + 6 q^{20} - q^{21} + 4 q^{23} + q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} + 6 q^{28} - 15 q^{29} + q^{30} + 8 q^{31} - 6 q^{32} + 9 q^{34} + 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} + 19 q^{39} - 6 q^{40} - 4 q^{41} + q^{42} - 30 q^{43} + 3 q^{45} - 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 16 q^{51} - 9 q^{52} - 6 q^{53} + 4 q^{54} - 6 q^{56} + 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} + 14 q^{61} - 8 q^{62} + 3 q^{63} + 6 q^{64} - 9 q^{65} + 18 q^{67} - 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} - 3 q^{72} - 23 q^{73} + 4 q^{74} - q^{75} - 12 q^{76} - 19 q^{78} - 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} - 25 q^{83} - q^{84} - 9 q^{85} + 30 q^{86} - 14 q^{87} - 18 q^{89} - 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} + 7 q^{94} - 12 q^{95} + q^{96} + 7 q^{97} - 6 q^{98}+O(q^{100})$$ 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 + 3 * q^9 - 6 * q^10 - q^12 - 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 9 * q^17 - 3 * q^18 - 12 * q^19 + 6 * q^20 - q^21 + 4 * q^23 + q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 + 6 * q^28 - 15 * q^29 + q^30 + 8 * q^31 - 6 * q^32 + 9 * q^34 + 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 + 19 * q^39 - 6 * q^40 - 4 * q^41 + q^42 - 30 * q^43 + 3 * q^45 - 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 - 6 * q^50 - 16 * q^51 - 9 * q^52 - 6 * q^53 + 4 * q^54 - 6 * q^56 + 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 + 14 * q^61 - 8 * q^62 + 3 * q^63 + 6 * q^64 - 9 * q^65 + 18 * q^67 - 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 - 3 * q^72 - 23 * q^73 + 4 * q^74 - q^75 - 12 * q^76 - 19 * q^78 - 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 - 25 * q^83 - q^84 - 9 * q^85 + 30 * q^86 - 14 * q^87 - 18 * q^89 - 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 + 7 * q^94 - 12 * q^95 + q^96 + 7 * q^97 - 6 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.68692 −0.973944 −0.486972 0.873418i $$-0.661899\pi$$
−0.486972 + 0.873418i $$0.661899\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.68692 0.688682
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −0.154300 −0.0514332
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ −1.68692 −0.486972
$$13$$ 0.0425741 0.0118079 0.00590397 0.999983i $$-0.498121\pi$$
0.00590397 + 0.999983i $$0.498121\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −1.68692 −0.435561
$$16$$ 1.00000 0.250000
$$17$$ −4.72038 −1.14486 −0.572430 0.819953i $$-0.693999\pi$$
−0.572430 + 0.819953i $$0.693999\pi$$
$$18$$ 0.154300 0.0363688
$$19$$ −1.71340 −0.393080 −0.196540 0.980496i $$-0.562971\pi$$
−0.196540 + 0.980496i $$0.562971\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −1.68692 −0.368116
$$22$$ 0 0
$$23$$ −6.95966 −1.45119 −0.725595 0.688122i $$-0.758435\pi$$
−0.725595 + 0.688122i $$0.758435\pi$$
$$24$$ 1.68692 0.344341
$$25$$ 1.00000 0.200000
$$26$$ −0.0425741 −0.00834947
$$27$$ 5.32105 1.02404
$$28$$ 1.00000 0.188982
$$29$$ −1.34277 −0.249346 −0.124673 0.992198i $$-0.539788\pi$$
−0.124673 + 0.992198i $$0.539788\pi$$
$$30$$ 1.68692 0.307988
$$31$$ 6.01245 1.07987 0.539934 0.841707i $$-0.318449\pi$$
0.539934 + 0.841707i $$0.318449\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 4.72038 0.809539
$$35$$ 1.00000 0.169031
$$36$$ −0.154300 −0.0257166
$$37$$ 8.48459 1.39486 0.697429 0.716654i $$-0.254327\pi$$
0.697429 + 0.716654i $$0.254327\pi$$
$$38$$ 1.71340 0.277950
$$39$$ −0.0718191 −0.0115003
$$40$$ −1.00000 −0.158114
$$41$$ 3.91371 0.611219 0.305609 0.952157i $$-0.401140\pi$$
0.305609 + 0.952157i $$0.401140\pi$$
$$42$$ 1.68692 0.260297
$$43$$ 6.42455 0.979735 0.489867 0.871797i $$-0.337045\pi$$
0.489867 + 0.871797i $$0.337045\pi$$
$$44$$ 0 0
$$45$$ −0.154300 −0.0230016
$$46$$ 6.95966 1.02615
$$47$$ 4.59129 0.669709 0.334854 0.942270i $$-0.391313\pi$$
0.334854 + 0.942270i $$0.391313\pi$$
$$48$$ −1.68692 −0.243486
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 7.96291 1.11503
$$52$$ 0.0425741 0.00590397
$$53$$ −14.1024 −1.93711 −0.968554 0.248804i $$-0.919963\pi$$
−0.968554 + 0.248804i $$0.919963\pi$$
$$54$$ −5.32105 −0.724104
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 2.89036 0.382838
$$58$$ 1.34277 0.176315
$$59$$ −10.2177 −1.33023 −0.665117 0.746739i $$-0.731619\pi$$
−0.665117 + 0.746739i $$0.731619\pi$$
$$60$$ −1.68692 −0.217780
$$61$$ 13.8661 1.77538 0.887688 0.460446i $$-0.152310\pi$$
0.887688 + 0.460446i $$0.152310\pi$$
$$62$$ −6.01245 −0.763582
$$63$$ −0.154300 −0.0194399
$$64$$ 1.00000 0.125000
$$65$$ 0.0425741 0.00528067
$$66$$ 0 0
$$67$$ 6.39600 0.781396 0.390698 0.920519i $$-0.372234\pi$$
0.390698 + 0.920519i $$0.372234\pi$$
$$68$$ −4.72038 −0.572430
$$69$$ 11.7404 1.41338
$$70$$ −1.00000 −0.119523
$$71$$ 11.3041 1.34155 0.670774 0.741662i $$-0.265962\pi$$
0.670774 + 0.741662i $$0.265962\pi$$
$$72$$ 0.154300 0.0181844
$$73$$ −1.98728 −0.232594 −0.116297 0.993214i $$-0.537102\pi$$
−0.116297 + 0.993214i $$0.537102\pi$$
$$74$$ −8.48459 −0.986313
$$75$$ −1.68692 −0.194789
$$76$$ −1.71340 −0.196540
$$77$$ 0 0
$$78$$ 0.0718191 0.00813192
$$79$$ −8.87065 −0.998026 −0.499013 0.866595i $$-0.666304\pi$$
−0.499013 + 0.866595i $$0.666304\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −8.51329 −0.945921
$$82$$ −3.91371 −0.432197
$$83$$ −10.0807 −1.10650 −0.553249 0.833016i $$-0.686612\pi$$
−0.553249 + 0.833016i $$0.686612\pi$$
$$84$$ −1.68692 −0.184058
$$85$$ −4.72038 −0.511997
$$86$$ −6.42455 −0.692777
$$87$$ 2.26515 0.242849
$$88$$ 0 0
$$89$$ 2.93362 0.310963 0.155481 0.987839i $$-0.450307\pi$$
0.155481 + 0.987839i $$0.450307\pi$$
$$90$$ 0.154300 0.0162646
$$91$$ 0.0425741 0.00446298
$$92$$ −6.95966 −0.725595
$$93$$ −10.1425 −1.05173
$$94$$ −4.59129 −0.473556
$$95$$ −1.71340 −0.175791
$$96$$ 1.68692 0.172171
$$97$$ −17.4812 −1.77495 −0.887474 0.460857i $$-0.847542\pi$$
−0.887474 + 0.460857i $$0.847542\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −5.70914 −0.568081 −0.284040 0.958812i $$-0.591675\pi$$
−0.284040 + 0.958812i $$0.591675\pi$$
$$102$$ −7.96291 −0.788445
$$103$$ 13.4052 1.32085 0.660425 0.750892i $$-0.270376\pi$$
0.660425 + 0.750892i $$0.270376\pi$$
$$104$$ −0.0425741 −0.00417474
$$105$$ −1.68692 −0.164627
$$106$$ 14.1024 1.36974
$$107$$ −15.9669 −1.54358 −0.771790 0.635878i $$-0.780638\pi$$
−0.771790 + 0.635878i $$0.780638\pi$$
$$108$$ 5.32105 0.512019
$$109$$ 15.9501 1.52774 0.763871 0.645369i $$-0.223297\pi$$
0.763871 + 0.645369i $$0.223297\pi$$
$$110$$ 0 0
$$111$$ −14.3128 −1.35851
$$112$$ 1.00000 0.0944911
$$113$$ 8.65087 0.813805 0.406903 0.913472i $$-0.366609\pi$$
0.406903 + 0.913472i $$0.366609\pi$$
$$114$$ −2.89036 −0.270707
$$115$$ −6.95966 −0.648992
$$116$$ −1.34277 −0.124673
$$117$$ −0.00656917 −0.000607320 0
$$118$$ 10.2177 0.940618
$$119$$ −4.72038 −0.432717
$$120$$ 1.68692 0.153994
$$121$$ 0 0
$$122$$ −13.8661 −1.25538
$$123$$ −6.60212 −0.595293
$$124$$ 6.01245 0.539934
$$125$$ 1.00000 0.0894427
$$126$$ 0.154300 0.0137461
$$127$$ −1.82824 −0.162230 −0.0811149 0.996705i $$-0.525848\pi$$
−0.0811149 + 0.996705i $$0.525848\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −10.8377 −0.954207
$$130$$ −0.0425741 −0.00373400
$$131$$ −1.61678 −0.141259 −0.0706294 0.997503i $$-0.522501\pi$$
−0.0706294 + 0.997503i $$0.522501\pi$$
$$132$$ 0 0
$$133$$ −1.71340 −0.148570
$$134$$ −6.39600 −0.552530
$$135$$ 5.32105 0.457963
$$136$$ 4.72038 0.404769
$$137$$ 10.2667 0.877144 0.438572 0.898696i $$-0.355484\pi$$
0.438572 + 0.898696i $$0.355484\pi$$
$$138$$ −11.7404 −0.999409
$$139$$ −1.94758 −0.165191 −0.0825957 0.996583i $$-0.526321\pi$$
−0.0825957 + 0.996583i $$0.526321\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ −7.74515 −0.652259
$$142$$ −11.3041 −0.948617
$$143$$ 0 0
$$144$$ −0.154300 −0.0128583
$$145$$ −1.34277 −0.111511
$$146$$ 1.98728 0.164469
$$147$$ −1.68692 −0.139135
$$148$$ 8.48459 0.697429
$$149$$ −6.46920 −0.529978 −0.264989 0.964251i $$-0.585368\pi$$
−0.264989 + 0.964251i $$0.585368\pi$$
$$150$$ 1.68692 0.137736
$$151$$ −23.6110 −1.92143 −0.960717 0.277531i $$-0.910484\pi$$
−0.960717 + 0.277531i $$0.910484\pi$$
$$152$$ 1.71340 0.138975
$$153$$ 0.728353 0.0588839
$$154$$ 0 0
$$155$$ 6.01245 0.482932
$$156$$ −0.0718191 −0.00575013
$$157$$ 11.4891 0.916932 0.458466 0.888712i $$-0.348399\pi$$
0.458466 + 0.888712i $$0.348399\pi$$
$$158$$ 8.87065 0.705711
$$159$$ 23.7896 1.88663
$$160$$ −1.00000 −0.0790569
$$161$$ −6.95966 −0.548498
$$162$$ 8.51329 0.668867
$$163$$ −14.1296 −1.10672 −0.553359 0.832943i $$-0.686654\pi$$
−0.553359 + 0.832943i $$0.686654\pi$$
$$164$$ 3.91371 0.305609
$$165$$ 0 0
$$166$$ 10.0807 0.782412
$$167$$ −15.6015 −1.20728 −0.603641 0.797256i $$-0.706284\pi$$
−0.603641 + 0.797256i $$0.706284\pi$$
$$168$$ 1.68692 0.130149
$$169$$ −12.9982 −0.999861
$$170$$ 4.72038 0.362037
$$171$$ 0.264377 0.0202174
$$172$$ 6.42455 0.489867
$$173$$ 6.21176 0.472271 0.236136 0.971720i $$-0.424119\pi$$
0.236136 + 0.971720i $$0.424119\pi$$
$$174$$ −2.26515 −0.171721
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 17.2365 1.29557
$$178$$ −2.93362 −0.219884
$$179$$ 15.9071 1.18895 0.594475 0.804114i $$-0.297360\pi$$
0.594475 + 0.804114i $$0.297360\pi$$
$$180$$ −0.154300 −0.0115008
$$181$$ −6.59938 −0.490528 −0.245264 0.969456i $$-0.578875\pi$$
−0.245264 + 0.969456i $$0.578875\pi$$
$$182$$ −0.0425741 −0.00315580
$$183$$ −23.3911 −1.72912
$$184$$ 6.95966 0.513073
$$185$$ 8.48459 0.623799
$$186$$ 10.1425 0.743686
$$187$$ 0 0
$$188$$ 4.59129 0.334854
$$189$$ 5.32105 0.387050
$$190$$ 1.71340 0.124303
$$191$$ 17.1690 1.24231 0.621153 0.783689i $$-0.286664\pi$$
0.621153 + 0.783689i $$0.286664\pi$$
$$192$$ −1.68692 −0.121743
$$193$$ −10.2185 −0.735541 −0.367771 0.929917i $$-0.619879\pi$$
−0.367771 + 0.929917i $$0.619879\pi$$
$$194$$ 17.4812 1.25508
$$195$$ −0.0718191 −0.00514308
$$196$$ 1.00000 0.0714286
$$197$$ −7.33268 −0.522432 −0.261216 0.965280i $$-0.584124\pi$$
−0.261216 + 0.965280i $$0.584124\pi$$
$$198$$ 0 0
$$199$$ −8.90352 −0.631154 −0.315577 0.948900i $$-0.602198\pi$$
−0.315577 + 0.948900i $$0.602198\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −10.7895 −0.761036
$$202$$ 5.70914 0.401694
$$203$$ −1.34277 −0.0942441
$$204$$ 7.96291 0.557515
$$205$$ 3.91371 0.273345
$$206$$ −13.4052 −0.933982
$$207$$ 1.07387 0.0746394
$$208$$ 0.0425741 0.00295198
$$209$$ 0 0
$$210$$ 1.68692 0.116409
$$211$$ 4.54767 0.313075 0.156537 0.987672i $$-0.449967\pi$$
0.156537 + 0.987672i $$0.449967\pi$$
$$212$$ −14.1024 −0.968554
$$213$$ −19.0691 −1.30659
$$214$$ 15.9669 1.09148
$$215$$ 6.42455 0.438151
$$216$$ −5.32105 −0.362052
$$217$$ 6.01245 0.408152
$$218$$ −15.9501 −1.08028
$$219$$ 3.35239 0.226533
$$220$$ 0 0
$$221$$ −0.200966 −0.0135184
$$222$$ 14.3128 0.960614
$$223$$ 18.0702 1.21007 0.605034 0.796200i $$-0.293160\pi$$
0.605034 + 0.796200i $$0.293160\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −0.154300 −0.0102866
$$226$$ −8.65087 −0.575447
$$227$$ 0.970501 0.0644144 0.0322072 0.999481i $$-0.489746\pi$$
0.0322072 + 0.999481i $$0.489746\pi$$
$$228$$ 2.89036 0.191419
$$229$$ −2.46292 −0.162755 −0.0813773 0.996683i $$-0.525932\pi$$
−0.0813773 + 0.996683i $$0.525932\pi$$
$$230$$ 6.95966 0.458907
$$231$$ 0 0
$$232$$ 1.34277 0.0881573
$$233$$ −6.65041 −0.435683 −0.217841 0.975984i $$-0.569902\pi$$
−0.217841 + 0.975984i $$0.569902\pi$$
$$234$$ 0.00656917 0.000429440 0
$$235$$ 4.59129 0.299503
$$236$$ −10.2177 −0.665117
$$237$$ 14.9641 0.972021
$$238$$ 4.72038 0.305977
$$239$$ 21.0529 1.36180 0.680900 0.732377i $$-0.261589\pi$$
0.680900 + 0.732377i $$0.261589\pi$$
$$240$$ −1.68692 −0.108890
$$241$$ −7.93244 −0.510973 −0.255487 0.966813i $$-0.582236\pi$$
−0.255487 + 0.966813i $$0.582236\pi$$
$$242$$ 0 0
$$243$$ −1.60191 −0.102763
$$244$$ 13.8661 0.887688
$$245$$ 1.00000 0.0638877
$$246$$ 6.60212 0.420936
$$247$$ −0.0729463 −0.00464146
$$248$$ −6.01245 −0.381791
$$249$$ 17.0053 1.07767
$$250$$ −1.00000 −0.0632456
$$251$$ 18.9431 1.19568 0.597839 0.801617i $$-0.296026\pi$$
0.597839 + 0.801617i $$0.296026\pi$$
$$252$$ −0.154300 −0.00971997
$$253$$ 0 0
$$254$$ 1.82824 0.114714
$$255$$ 7.96291 0.498657
$$256$$ 1.00000 0.0625000
$$257$$ −26.8598 −1.67547 −0.837736 0.546076i $$-0.816121\pi$$
−0.837736 + 0.546076i $$0.816121\pi$$
$$258$$ 10.8377 0.674726
$$259$$ 8.48459 0.527207
$$260$$ 0.0425741 0.00264033
$$261$$ 0.207189 0.0128247
$$262$$ 1.61678 0.0998850
$$263$$ −13.3750 −0.824740 −0.412370 0.911017i $$-0.635299\pi$$
−0.412370 + 0.911017i $$0.635299\pi$$
$$264$$ 0 0
$$265$$ −14.1024 −0.866301
$$266$$ 1.71340 0.105055
$$267$$ −4.94878 −0.302860
$$268$$ 6.39600 0.390698
$$269$$ −4.05979 −0.247530 −0.123765 0.992312i $$-0.539497\pi$$
−0.123765 + 0.992312i $$0.539497\pi$$
$$270$$ −5.32105 −0.323829
$$271$$ 23.0226 1.39852 0.699261 0.714866i $$-0.253512\pi$$
0.699261 + 0.714866i $$0.253512\pi$$
$$272$$ −4.72038 −0.286215
$$273$$ −0.0718191 −0.00434669
$$274$$ −10.2667 −0.620235
$$275$$ 0 0
$$276$$ 11.7404 0.706689
$$277$$ −22.2862 −1.33905 −0.669524 0.742790i $$-0.733502\pi$$
−0.669524 + 0.742790i $$0.733502\pi$$
$$278$$ 1.94758 0.116808
$$279$$ −0.927720 −0.0555411
$$280$$ −1.00000 −0.0597614
$$281$$ −2.24959 −0.134199 −0.0670996 0.997746i $$-0.521375\pi$$
−0.0670996 + 0.997746i $$0.521375\pi$$
$$282$$ 7.74515 0.461217
$$283$$ −3.71034 −0.220557 −0.110279 0.993901i $$-0.535174\pi$$
−0.110279 + 0.993901i $$0.535174\pi$$
$$284$$ 11.3041 0.670774
$$285$$ 2.89036 0.171210
$$286$$ 0 0
$$287$$ 3.91371 0.231019
$$288$$ 0.154300 0.00909220
$$289$$ 5.28199 0.310705
$$290$$ 1.34277 0.0788503
$$291$$ 29.4894 1.72870
$$292$$ −1.98728 −0.116297
$$293$$ 1.85170 0.108177 0.0540886 0.998536i $$-0.482775\pi$$
0.0540886 + 0.998536i $$0.482775\pi$$
$$294$$ 1.68692 0.0983832
$$295$$ −10.2177 −0.594899
$$296$$ −8.48459 −0.493157
$$297$$ 0 0
$$298$$ 6.46920 0.374751
$$299$$ −0.296302 −0.0171356
$$300$$ −1.68692 −0.0973944
$$301$$ 6.42455 0.370305
$$302$$ 23.6110 1.35866
$$303$$ 9.63087 0.553279
$$304$$ −1.71340 −0.0982700
$$305$$ 13.8661 0.793972
$$306$$ −0.728353 −0.0416372
$$307$$ −6.66980 −0.380666 −0.190333 0.981720i $$-0.560957\pi$$
−0.190333 + 0.981720i $$0.560957\pi$$
$$308$$ 0 0
$$309$$ −22.6134 −1.28643
$$310$$ −6.01245 −0.341484
$$311$$ 15.0969 0.856067 0.428034 0.903763i $$-0.359207\pi$$
0.428034 + 0.903763i $$0.359207\pi$$
$$312$$ 0.0718191 0.00406596
$$313$$ 7.33744 0.414736 0.207368 0.978263i $$-0.433510\pi$$
0.207368 + 0.978263i $$0.433510\pi$$
$$314$$ −11.4891 −0.648369
$$315$$ −0.154300 −0.00869380
$$316$$ −8.87065 −0.499013
$$317$$ 22.4900 1.26316 0.631581 0.775310i $$-0.282406\pi$$
0.631581 + 0.775310i $$0.282406\pi$$
$$318$$ −23.7896 −1.33405
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 26.9349 1.50336
$$322$$ 6.95966 0.387847
$$323$$ 8.08788 0.450022
$$324$$ −8.51329 −0.472961
$$325$$ 0.0425741 0.00236159
$$326$$ 14.1296 0.782568
$$327$$ −26.9065 −1.48793
$$328$$ −3.91371 −0.216099
$$329$$ 4.59129 0.253126
$$330$$ 0 0
$$331$$ 10.3154 0.566987 0.283494 0.958974i $$-0.408507\pi$$
0.283494 + 0.958974i $$0.408507\pi$$
$$332$$ −10.0807 −0.553249
$$333$$ −1.30917 −0.0717420
$$334$$ 15.6015 0.853677
$$335$$ 6.39600 0.349451
$$336$$ −1.68692 −0.0920291
$$337$$ −27.2073 −1.48208 −0.741038 0.671463i $$-0.765666\pi$$
−0.741038 + 0.671463i $$0.765666\pi$$
$$338$$ 12.9982 0.707008
$$339$$ −14.5933 −0.792600
$$340$$ −4.72038 −0.255999
$$341$$ 0 0
$$342$$ −0.264377 −0.0142958
$$343$$ 1.00000 0.0539949
$$344$$ −6.42455 −0.346389
$$345$$ 11.7404 0.632082
$$346$$ −6.21176 −0.333946
$$347$$ −6.65299 −0.357151 −0.178576 0.983926i $$-0.557149\pi$$
−0.178576 + 0.983926i $$0.557149\pi$$
$$348$$ 2.26515 0.121425
$$349$$ −25.4581 −1.36274 −0.681371 0.731938i $$-0.738616\pi$$
−0.681371 + 0.731938i $$0.738616\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 0.226539 0.0120918
$$352$$ 0 0
$$353$$ −29.0584 −1.54662 −0.773311 0.634026i $$-0.781401\pi$$
−0.773311 + 0.634026i $$0.781401\pi$$
$$354$$ −17.2365 −0.916109
$$355$$ 11.3041 0.599958
$$356$$ 2.93362 0.155481
$$357$$ 7.96291 0.421442
$$358$$ −15.9071 −0.840715
$$359$$ −28.3224 −1.49480 −0.747400 0.664375i $$-0.768698\pi$$
−0.747400 + 0.664375i $$0.768698\pi$$
$$360$$ 0.154300 0.00813231
$$361$$ −16.0643 −0.845488
$$362$$ 6.59938 0.346856
$$363$$ 0 0
$$364$$ 0.0425741 0.00223149
$$365$$ −1.98728 −0.104019
$$366$$ 23.3911 1.22267
$$367$$ −27.5184 −1.43645 −0.718224 0.695812i $$-0.755044\pi$$
−0.718224 + 0.695812i $$0.755044\pi$$
$$368$$ −6.95966 −0.362798
$$369$$ −0.603884 −0.0314370
$$370$$ −8.48459 −0.441093
$$371$$ −14.1024 −0.732158
$$372$$ −10.1425 −0.525866
$$373$$ −22.9306 −1.18730 −0.593650 0.804724i $$-0.702313\pi$$
−0.593650 + 0.804724i $$0.702313\pi$$
$$374$$ 0 0
$$375$$ −1.68692 −0.0871122
$$376$$ −4.59129 −0.236778
$$377$$ −0.0571673 −0.00294427
$$378$$ −5.32105 −0.273685
$$379$$ 15.4140 0.791764 0.395882 0.918301i $$-0.370439\pi$$
0.395882 + 0.918301i $$0.370439\pi$$
$$380$$ −1.71340 −0.0878954
$$381$$ 3.08409 0.158003
$$382$$ −17.1690 −0.878443
$$383$$ −30.8927 −1.57854 −0.789270 0.614046i $$-0.789541\pi$$
−0.789270 + 0.614046i $$0.789541\pi$$
$$384$$ 1.68692 0.0860853
$$385$$ 0 0
$$386$$ 10.2185 0.520106
$$387$$ −0.991306 −0.0503909
$$388$$ −17.4812 −0.887474
$$389$$ 0.452651 0.0229503 0.0114752 0.999934i $$-0.496347\pi$$
0.0114752 + 0.999934i $$0.496347\pi$$
$$390$$ 0.0718191 0.00363670
$$391$$ 32.8523 1.66141
$$392$$ −1.00000 −0.0505076
$$393$$ 2.72738 0.137578
$$394$$ 7.33268 0.369415
$$395$$ −8.87065 −0.446331
$$396$$ 0 0
$$397$$ 24.1263 1.21086 0.605431 0.795897i $$-0.293001\pi$$
0.605431 + 0.795897i $$0.293001\pi$$
$$398$$ 8.90352 0.446293
$$399$$ 2.89036 0.144699
$$400$$ 1.00000 0.0500000
$$401$$ −11.2261 −0.560603 −0.280301 0.959912i $$-0.590434\pi$$
−0.280301 + 0.959912i $$0.590434\pi$$
$$402$$ 10.7895 0.538134
$$403$$ 0.255975 0.0127510
$$404$$ −5.70914 −0.284040
$$405$$ −8.51329 −0.423029
$$406$$ 1.34277 0.0666406
$$407$$ 0 0
$$408$$ −7.96291 −0.394223
$$409$$ −30.4772 −1.50700 −0.753500 0.657448i $$-0.771636\pi$$
−0.753500 + 0.657448i $$0.771636\pi$$
$$410$$ −3.91371 −0.193284
$$411$$ −17.3191 −0.854289
$$412$$ 13.4052 0.660425
$$413$$ −10.2177 −0.502782
$$414$$ −1.07387 −0.0527780
$$415$$ −10.0807 −0.494841
$$416$$ −0.0425741 −0.00208737
$$417$$ 3.28541 0.160887
$$418$$ 0 0
$$419$$ 11.8710 0.579937 0.289968 0.957036i $$-0.406355\pi$$
0.289968 + 0.957036i $$0.406355\pi$$
$$420$$ −1.68692 −0.0823133
$$421$$ 23.7676 1.15836 0.579181 0.815199i $$-0.303373\pi$$
0.579181 + 0.815199i $$0.303373\pi$$
$$422$$ −4.54767 −0.221377
$$423$$ −0.708435 −0.0344453
$$424$$ 14.1024 0.684871
$$425$$ −4.72038 −0.228972
$$426$$ 19.0691 0.923900
$$427$$ 13.8661 0.671029
$$428$$ −15.9669 −0.771790
$$429$$ 0 0
$$430$$ −6.42455 −0.309819
$$431$$ 27.1939 1.30989 0.654943 0.755678i $$-0.272693\pi$$
0.654943 + 0.755678i $$0.272693\pi$$
$$432$$ 5.32105 0.256009
$$433$$ −2.21065 −0.106237 −0.0531185 0.998588i $$-0.516916\pi$$
−0.0531185 + 0.998588i $$0.516916\pi$$
$$434$$ −6.01245 −0.288607
$$435$$ 2.26515 0.108606
$$436$$ 15.9501 0.763871
$$437$$ 11.9247 0.570434
$$438$$ −3.35239 −0.160183
$$439$$ −6.47985 −0.309267 −0.154633 0.987972i $$-0.549420\pi$$
−0.154633 + 0.987972i $$0.549420\pi$$
$$440$$ 0 0
$$441$$ −0.154300 −0.00734760
$$442$$ 0.200966 0.00955898
$$443$$ 17.0792 0.811456 0.405728 0.913994i $$-0.367018\pi$$
0.405728 + 0.913994i $$0.367018\pi$$
$$444$$ −14.3128 −0.679257
$$445$$ 2.93362 0.139067
$$446$$ −18.0702 −0.855647
$$447$$ 10.9130 0.516169
$$448$$ 1.00000 0.0472456
$$449$$ 13.1328 0.619774 0.309887 0.950773i $$-0.399709\pi$$
0.309887 + 0.950773i $$0.399709\pi$$
$$450$$ 0.154300 0.00727376
$$451$$ 0 0
$$452$$ 8.65087 0.406903
$$453$$ 39.8298 1.87137
$$454$$ −0.970501 −0.0455479
$$455$$ 0.0425741 0.00199591
$$456$$ −2.89036 −0.135354
$$457$$ 15.4508 0.722758 0.361379 0.932419i $$-0.382306\pi$$
0.361379 + 0.932419i $$0.382306\pi$$
$$458$$ 2.46292 0.115085
$$459$$ −25.1174 −1.17238
$$460$$ −6.95966 −0.324496
$$461$$ −33.8352 −1.57586 −0.787932 0.615763i $$-0.788848\pi$$
−0.787932 + 0.615763i $$0.788848\pi$$
$$462$$ 0 0
$$463$$ 31.7915 1.47748 0.738738 0.673993i $$-0.235422\pi$$
0.738738 + 0.673993i $$0.235422\pi$$
$$464$$ −1.34277 −0.0623366
$$465$$ −10.1425 −0.470349
$$466$$ 6.65041 0.308074
$$467$$ −17.0295 −0.788032 −0.394016 0.919104i $$-0.628915\pi$$
−0.394016 + 0.919104i $$0.628915\pi$$
$$468$$ −0.00656917 −0.000303660 0
$$469$$ 6.39600 0.295340
$$470$$ −4.59129 −0.211781
$$471$$ −19.3812 −0.893040
$$472$$ 10.2177 0.470309
$$473$$ 0 0
$$474$$ −14.9641 −0.687323
$$475$$ −1.71340 −0.0786160
$$476$$ −4.72038 −0.216358
$$477$$ 2.17599 0.0996317
$$478$$ −21.0529 −0.962938
$$479$$ 29.9718 1.36945 0.684724 0.728802i $$-0.259923\pi$$
0.684724 + 0.728802i $$0.259923\pi$$
$$480$$ 1.68692 0.0769970
$$481$$ 0.361224 0.0164704
$$482$$ 7.93244 0.361313
$$483$$ 11.7404 0.534207
$$484$$ 0 0
$$485$$ −17.4812 −0.793781
$$486$$ 1.60191 0.0726641
$$487$$ 0.733047 0.0332175 0.0166088 0.999862i $$-0.494713\pi$$
0.0166088 + 0.999862i $$0.494713\pi$$
$$488$$ −13.8661 −0.627690
$$489$$ 23.8356 1.07788
$$490$$ −1.00000 −0.0451754
$$491$$ 10.1494 0.458035 0.229018 0.973422i $$-0.426449\pi$$
0.229018 + 0.973422i $$0.426449\pi$$
$$492$$ −6.60212 −0.297646
$$493$$ 6.33839 0.285467
$$494$$ 0.0729463 0.00328201
$$495$$ 0 0
$$496$$ 6.01245 0.269967
$$497$$ 11.3041 0.507057
$$498$$ −17.0053 −0.762025
$$499$$ −11.1673 −0.499919 −0.249959 0.968256i $$-0.580417\pi$$
−0.249959 + 0.968256i $$0.580417\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 26.3185 1.17582
$$502$$ −18.9431 −0.845471
$$503$$ −7.46249 −0.332736 −0.166368 0.986064i $$-0.553204\pi$$
−0.166368 + 0.986064i $$0.553204\pi$$
$$504$$ 0.154300 0.00687305
$$505$$ −5.70914 −0.254053
$$506$$ 0 0
$$507$$ 21.9269 0.973808
$$508$$ −1.82824 −0.0811149
$$509$$ −30.8240 −1.36625 −0.683125 0.730302i $$-0.739380\pi$$
−0.683125 + 0.730302i $$0.739380\pi$$
$$510$$ −7.96291 −0.352603
$$511$$ −1.98728 −0.0879122
$$512$$ −1.00000 −0.0441942
$$513$$ −9.11707 −0.402529
$$514$$ 26.8598 1.18474
$$515$$ 13.4052 0.590702
$$516$$ −10.8377 −0.477103
$$517$$ 0 0
$$518$$ −8.48459 −0.372791
$$519$$ −10.4787 −0.459966
$$520$$ −0.0425741 −0.00186700
$$521$$ −14.5848 −0.638970 −0.319485 0.947591i $$-0.603510\pi$$
−0.319485 + 0.947591i $$0.603510\pi$$
$$522$$ −0.207189 −0.00906843
$$523$$ −9.58569 −0.419153 −0.209576 0.977792i $$-0.567208\pi$$
−0.209576 + 0.977792i $$0.567208\pi$$
$$524$$ −1.61678 −0.0706294
$$525$$ −1.68692 −0.0736232
$$526$$ 13.3750 0.583179
$$527$$ −28.3811 −1.23630
$$528$$ 0 0
$$529$$ 25.4369 1.10595
$$530$$ 14.1024 0.612567
$$531$$ 1.57659 0.0684183
$$532$$ −1.71340 −0.0742852
$$533$$ 0.166623 0.00721723
$$534$$ 4.94878 0.214155
$$535$$ −15.9669 −0.690310
$$536$$ −6.39600 −0.276265
$$537$$ −26.8340 −1.15797
$$538$$ 4.05979 0.175030
$$539$$ 0 0
$$540$$ 5.32105 0.228982
$$541$$ 1.71177 0.0735948 0.0367974 0.999323i $$-0.488284\pi$$
0.0367974 + 0.999323i $$0.488284\pi$$
$$542$$ −23.0226 −0.988905
$$543$$ 11.1326 0.477747
$$544$$ 4.72038 0.202385
$$545$$ 15.9501 0.683227
$$546$$ 0.0718191 0.00307358
$$547$$ 17.4770 0.747262 0.373631 0.927577i $$-0.378113\pi$$
0.373631 + 0.927577i $$0.378113\pi$$
$$548$$ 10.2667 0.438572
$$549$$ −2.13954 −0.0913133
$$550$$ 0 0
$$551$$ 2.30070 0.0980131
$$552$$ −11.7404 −0.499705
$$553$$ −8.87065 −0.377218
$$554$$ 22.2862 0.946850
$$555$$ −14.3128 −0.607546
$$556$$ −1.94758 −0.0825957
$$557$$ −25.6895 −1.08850 −0.544250 0.838923i $$-0.683186\pi$$
−0.544250 + 0.838923i $$0.683186\pi$$
$$558$$ 0.927720 0.0392735
$$559$$ 0.273520 0.0115686
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 2.24959 0.0948932
$$563$$ −19.3434 −0.815225 −0.407613 0.913155i $$-0.633639\pi$$
−0.407613 + 0.913155i $$0.633639\pi$$
$$564$$ −7.74515 −0.326129
$$565$$ 8.65087 0.363945
$$566$$ 3.71034 0.155957
$$567$$ −8.51329 −0.357525
$$568$$ −11.3041 −0.474309
$$569$$ 1.51921 0.0636886 0.0318443 0.999493i $$-0.489862\pi$$
0.0318443 + 0.999493i $$0.489862\pi$$
$$570$$ −2.89036 −0.121064
$$571$$ 4.97687 0.208276 0.104138 0.994563i $$-0.466792\pi$$
0.104138 + 0.994563i $$0.466792\pi$$
$$572$$ 0 0
$$573$$ −28.9628 −1.20994
$$574$$ −3.91371 −0.163355
$$575$$ −6.95966 −0.290238
$$576$$ −0.154300 −0.00642915
$$577$$ −31.6102 −1.31595 −0.657976 0.753039i $$-0.728587\pi$$
−0.657976 + 0.753039i $$0.728587\pi$$
$$578$$ −5.28199 −0.219702
$$579$$ 17.2377 0.716376
$$580$$ −1.34277 −0.0557556
$$581$$ −10.0807 −0.418217
$$582$$ −29.4894 −1.22238
$$583$$ 0 0
$$584$$ 1.98728 0.0822343
$$585$$ −0.00656917 −0.000271602 0
$$586$$ −1.85170 −0.0764928
$$587$$ 7.11885 0.293826 0.146913 0.989149i $$-0.453066\pi$$
0.146913 + 0.989149i $$0.453066\pi$$
$$588$$ −1.68692 −0.0695674
$$589$$ −10.3017 −0.424475
$$590$$ 10.2177 0.420657
$$591$$ 12.3697 0.508820
$$592$$ 8.48459 0.348714
$$593$$ −18.4209 −0.756456 −0.378228 0.925713i $$-0.623466\pi$$
−0.378228 + 0.925713i $$0.623466\pi$$
$$594$$ 0 0
$$595$$ −4.72038 −0.193517
$$596$$ −6.46920 −0.264989
$$597$$ 15.0195 0.614709
$$598$$ 0.296302 0.0121167
$$599$$ −1.22319 −0.0499783 −0.0249892 0.999688i $$-0.507955\pi$$
−0.0249892 + 0.999688i $$0.507955\pi$$
$$600$$ 1.68692 0.0688682
$$601$$ −20.6888 −0.843916 −0.421958 0.906615i $$-0.638657\pi$$
−0.421958 + 0.906615i $$0.638657\pi$$
$$602$$ −6.42455 −0.261845
$$603$$ −0.986901 −0.0401897
$$604$$ −23.6110 −0.960717
$$605$$ 0 0
$$606$$ −9.63087 −0.391227
$$607$$ −10.3033 −0.418199 −0.209100 0.977894i $$-0.567053\pi$$
−0.209100 + 0.977894i $$0.567053\pi$$
$$608$$ 1.71340 0.0694874
$$609$$ 2.26515 0.0917885
$$610$$ −13.8661 −0.561423
$$611$$ 0.195470 0.00790788
$$612$$ 0.728353 0.0294419
$$613$$ −42.7503 −1.72667 −0.863333 0.504634i $$-0.831627\pi$$
−0.863333 + 0.504634i $$0.831627\pi$$
$$614$$ 6.66980 0.269171
$$615$$ −6.60212 −0.266223
$$616$$ 0 0
$$617$$ 10.3582 0.417004 0.208502 0.978022i $$-0.433141\pi$$
0.208502 + 0.978022i $$0.433141\pi$$
$$618$$ 22.6134 0.909646
$$619$$ 14.2901 0.574368 0.287184 0.957875i $$-0.407281\pi$$
0.287184 + 0.957875i $$0.407281\pi$$
$$620$$ 6.01245 0.241466
$$621$$ −37.0327 −1.48607
$$622$$ −15.0969 −0.605331
$$623$$ 2.93362 0.117533
$$624$$ −0.0718191 −0.00287507
$$625$$ 1.00000 0.0400000
$$626$$ −7.33744 −0.293263
$$627$$ 0 0
$$628$$ 11.4891 0.458466
$$629$$ −40.0505 −1.59692
$$630$$ 0.154300 0.00614745
$$631$$ −11.6481 −0.463703 −0.231851 0.972751i $$-0.574478\pi$$
−0.231851 + 0.972751i $$0.574478\pi$$
$$632$$ 8.87065 0.352855
$$633$$ −7.67156 −0.304917
$$634$$ −22.4900 −0.893190
$$635$$ −1.82824 −0.0725514
$$636$$ 23.7896 0.943317
$$637$$ 0.0425741 0.00168685
$$638$$ 0 0
$$639$$ −1.74422 −0.0690001
$$640$$ −1.00000 −0.0395285
$$641$$ 18.0247 0.711931 0.355965 0.934499i $$-0.384152\pi$$
0.355965 + 0.934499i $$0.384152\pi$$
$$642$$ −26.9349 −1.06304
$$643$$ −10.6620 −0.420469 −0.210235 0.977651i $$-0.567423\pi$$
−0.210235 + 0.977651i $$0.567423\pi$$
$$644$$ −6.95966 −0.274249
$$645$$ −10.8377 −0.426734
$$646$$ −8.08788 −0.318214
$$647$$ −21.7314 −0.854348 −0.427174 0.904170i $$-0.640491\pi$$
−0.427174 + 0.904170i $$0.640491\pi$$
$$648$$ 8.51329 0.334434
$$649$$ 0 0
$$650$$ −0.0425741 −0.00166989
$$651$$ −10.1425 −0.397517
$$652$$ −14.1296 −0.553359
$$653$$ 1.06200 0.0415592 0.0207796 0.999784i $$-0.493385\pi$$
0.0207796 + 0.999784i $$0.493385\pi$$
$$654$$ 26.9065 1.05213
$$655$$ −1.61678 −0.0631728
$$656$$ 3.91371 0.152805
$$657$$ 0.306637 0.0119630
$$658$$ −4.59129 −0.178987
$$659$$ 33.2641 1.29578 0.647892 0.761732i $$-0.275651\pi$$
0.647892 + 0.761732i $$0.275651\pi$$
$$660$$ 0 0
$$661$$ −8.99126 −0.349719 −0.174860 0.984593i $$-0.555947\pi$$
−0.174860 + 0.984593i $$0.555947\pi$$
$$662$$ −10.3154 −0.400920
$$663$$ 0.339014 0.0131662
$$664$$ 10.0807 0.391206
$$665$$ −1.71340 −0.0664427
$$666$$ 1.30917 0.0507293
$$667$$ 9.34524 0.361849
$$668$$ −15.6015 −0.603641
$$669$$ −30.4829 −1.17854
$$670$$ −6.39600 −0.247099
$$671$$ 0 0
$$672$$ 1.68692 0.0650744
$$673$$ −9.38976 −0.361949 −0.180974 0.983488i $$-0.557925\pi$$
−0.180974 + 0.983488i $$0.557925\pi$$
$$674$$ 27.2073 1.04799
$$675$$ 5.32105 0.204807
$$676$$ −12.9982 −0.499930
$$677$$ 37.7060 1.44916 0.724579 0.689191i $$-0.242034\pi$$
0.724579 + 0.689191i $$0.242034\pi$$
$$678$$ 14.5933 0.560453
$$679$$ −17.4812 −0.670868
$$680$$ 4.72038 0.181018
$$681$$ −1.63716 −0.0627360
$$682$$ 0 0
$$683$$ 11.5317 0.441248 0.220624 0.975359i $$-0.429191\pi$$
0.220624 + 0.975359i $$0.429191\pi$$
$$684$$ 0.264377 0.0101087
$$685$$ 10.2667 0.392271
$$686$$ −1.00000 −0.0381802
$$687$$ 4.15476 0.158514
$$688$$ 6.42455 0.244934
$$689$$ −0.600395 −0.0228732
$$690$$ −11.7404 −0.446949
$$691$$ −12.7280 −0.484196 −0.242098 0.970252i $$-0.577835\pi$$
−0.242098 + 0.970252i $$0.577835\pi$$
$$692$$ 6.21176 0.236136
$$693$$ 0 0
$$694$$ 6.65299 0.252544
$$695$$ −1.94758 −0.0738758
$$696$$ −2.26515 −0.0858603
$$697$$ −18.4742 −0.699760
$$698$$ 25.4581 0.963604
$$699$$ 11.2187 0.424331
$$700$$ 1.00000 0.0377964
$$701$$ 11.1988 0.422972 0.211486 0.977381i $$-0.432170\pi$$
0.211486 + 0.977381i $$0.432170\pi$$
$$702$$ −0.226539 −0.00855017
$$703$$ −14.5375 −0.548291
$$704$$ 0 0
$$705$$ −7.74515 −0.291699
$$706$$ 29.0584 1.09363
$$707$$ −5.70914 −0.214714
$$708$$ 17.2365 0.647787
$$709$$ −28.4744 −1.06938 −0.534688 0.845049i $$-0.679571\pi$$
−0.534688 + 0.845049i $$0.679571\pi$$
$$710$$ −11.3041 −0.424234
$$711$$ 1.36874 0.0513317
$$712$$ −2.93362 −0.109942
$$713$$ −41.8447 −1.56709
$$714$$ −7.96291 −0.298004
$$715$$ 0 0
$$716$$ 15.9071 0.594475
$$717$$ −35.5146 −1.32632
$$718$$ 28.3224 1.05698
$$719$$ −44.5968 −1.66318 −0.831589 0.555391i $$-0.812569\pi$$
−0.831589 + 0.555391i $$0.812569\pi$$
$$720$$ −0.154300 −0.00575041
$$721$$ 13.4052 0.499234
$$722$$ 16.0643 0.597850
$$723$$ 13.3814 0.497659
$$724$$ −6.59938 −0.245264
$$725$$ −1.34277 −0.0498693
$$726$$ 0 0
$$727$$ 50.4565 1.87133 0.935664 0.352892i $$-0.114802\pi$$
0.935664 + 0.352892i $$0.114802\pi$$
$$728$$ −0.0425741 −0.00157790
$$729$$ 28.2422 1.04601
$$730$$ 1.98728 0.0735526
$$731$$ −30.3263 −1.12166
$$732$$ −23.3911 −0.864558
$$733$$ 38.0995 1.40724 0.703618 0.710578i $$-0.251566\pi$$
0.703618 + 0.710578i $$0.251566\pi$$
$$734$$ 27.5184 1.01572
$$735$$ −1.68692 −0.0622230
$$736$$ 6.95966 0.256537
$$737$$ 0 0
$$738$$ 0.603884 0.0222293
$$739$$ 0.758286 0.0278940 0.0139470 0.999903i $$-0.495560\pi$$
0.0139470 + 0.999903i $$0.495560\pi$$
$$740$$ 8.48459 0.311900
$$741$$ 0.123055 0.00452053
$$742$$ 14.1024 0.517714
$$743$$ −34.7125 −1.27348 −0.636739 0.771079i $$-0.719717\pi$$
−0.636739 + 0.771079i $$0.719717\pi$$
$$744$$ 10.1425 0.371843
$$745$$ −6.46920 −0.237013
$$746$$ 22.9306 0.839547
$$747$$ 1.55544 0.0569107
$$748$$ 0 0
$$749$$ −15.9669 −0.583418
$$750$$ 1.68692 0.0615976
$$751$$ −46.0088 −1.67889 −0.839443 0.543448i $$-0.817119\pi$$
−0.839443 + 0.543448i $$0.817119\pi$$
$$752$$ 4.59129 0.167427
$$753$$ −31.9555 −1.16452
$$754$$ 0.0571673 0.00208191
$$755$$ −23.6110 −0.859291
$$756$$ 5.32105 0.193525
$$757$$ −23.1505 −0.841420 −0.420710 0.907195i $$-0.638219\pi$$
−0.420710 + 0.907195i $$0.638219\pi$$
$$758$$ −15.4140 −0.559862
$$759$$ 0 0
$$760$$ 1.71340 0.0621514
$$761$$ 7.19624 0.260864 0.130432 0.991457i $$-0.458364\pi$$
0.130432 + 0.991457i $$0.458364\pi$$
$$762$$ −3.08409 −0.111725
$$763$$ 15.9501 0.577432
$$764$$ 17.1690 0.621153
$$765$$ 0.728353 0.0263337
$$766$$ 30.8927 1.11620
$$767$$ −0.435011 −0.0157073
$$768$$ −1.68692 −0.0608715
$$769$$ 8.64296 0.311673 0.155837 0.987783i $$-0.450193\pi$$
0.155837 + 0.987783i $$0.450193\pi$$
$$770$$ 0 0
$$771$$ 45.3104 1.63181
$$772$$ −10.2185 −0.367771
$$773$$ −39.4190 −1.41780 −0.708901 0.705308i $$-0.750809\pi$$
−0.708901 + 0.705308i $$0.750809\pi$$
$$774$$ 0.991306 0.0356318
$$775$$ 6.01245 0.215974
$$776$$ 17.4812 0.627539
$$777$$ −14.3128 −0.513470
$$778$$ −0.452651 −0.0162283
$$779$$ −6.70574 −0.240258
$$780$$ −0.0718191 −0.00257154
$$781$$ 0 0
$$782$$ −32.8523 −1.17479
$$783$$ −7.14496 −0.255340
$$784$$ 1.00000 0.0357143
$$785$$ 11.4891 0.410064
$$786$$ −2.72738 −0.0972824
$$787$$ −44.2814 −1.57846 −0.789230 0.614098i $$-0.789520\pi$$
−0.789230 + 0.614098i $$0.789520\pi$$
$$788$$ −7.33268 −0.261216
$$789$$ 22.5626 0.803250
$$790$$ 8.87065 0.315603
$$791$$ 8.65087 0.307589
$$792$$ 0 0
$$793$$ 0.590338 0.0209635
$$794$$ −24.1263 −0.856209
$$795$$ 23.7896 0.843729
$$796$$ −8.90352 −0.315577
$$797$$ −20.9781 −0.743081 −0.371540 0.928417i $$-0.621170\pi$$
−0.371540 + 0.928417i $$0.621170\pi$$
$$798$$ −2.89036 −0.102318
$$799$$ −21.6726 −0.766723
$$800$$ −1.00000 −0.0353553
$$801$$ −0.452656 −0.0159938
$$802$$ 11.2261 0.396406
$$803$$ 0 0
$$804$$ −10.7895 −0.380518
$$805$$ −6.95966 −0.245296
$$806$$ −0.255975 −0.00901633
$$807$$ 6.84854 0.241080
$$808$$ 5.70914 0.200847
$$809$$ −24.2672 −0.853190 −0.426595 0.904443i $$-0.640287\pi$$
−0.426595 + 0.904443i $$0.640287\pi$$
$$810$$ 8.51329 0.299127
$$811$$ −37.0315 −1.30035 −0.650175 0.759784i $$-0.725305\pi$$
−0.650175 + 0.759784i $$0.725305\pi$$
$$812$$ −1.34277 −0.0471221
$$813$$ −38.8373 −1.36208
$$814$$ 0 0
$$815$$ −14.1296 −0.494940
$$816$$ 7.96291 0.278757
$$817$$ −11.0078 −0.385114
$$818$$ 30.4772 1.06561
$$819$$ −0.00656917 −0.000229546 0
$$820$$ 3.91371 0.136673
$$821$$ 22.0348 0.769019 0.384509 0.923121i $$-0.374371\pi$$
0.384509 + 0.923121i $$0.374371\pi$$
$$822$$ 17.3191 0.604074
$$823$$ 25.6954 0.895684 0.447842 0.894113i $$-0.352193\pi$$
0.447842 + 0.894113i $$0.352193\pi$$
$$824$$ −13.4052 −0.466991
$$825$$ 0 0
$$826$$ 10.2177 0.355520
$$827$$ −16.8897 −0.587313 −0.293657 0.955911i $$-0.594872\pi$$
−0.293657 + 0.955911i $$0.594872\pi$$
$$828$$ 1.07387 0.0373197
$$829$$ −20.4637 −0.710732 −0.355366 0.934727i $$-0.615644\pi$$
−0.355366 + 0.934727i $$0.615644\pi$$
$$830$$ 10.0807 0.349905
$$831$$ 37.5951 1.30416
$$832$$ 0.0425741 0.00147599
$$833$$ −4.72038 −0.163551
$$834$$ −3.28541 −0.113764
$$835$$ −15.6015 −0.539913
$$836$$ 0 0
$$837$$ 31.9926 1.10583
$$838$$ −11.8710 −0.410077
$$839$$ 34.0393 1.17517 0.587584 0.809163i $$-0.300079\pi$$
0.587584 + 0.809163i $$0.300079\pi$$
$$840$$ 1.68692 0.0582043
$$841$$ −27.1970 −0.937826
$$842$$ −23.7676 −0.819085
$$843$$ 3.79488 0.130703
$$844$$ 4.54767 0.156537
$$845$$ −12.9982 −0.447151
$$846$$ 0.708435 0.0243565
$$847$$ 0 0
$$848$$ −14.1024 −0.484277
$$849$$ 6.25906 0.214810
$$850$$ 4.72038 0.161908
$$851$$ −59.0499 −2.02420
$$852$$ −19.0691 −0.653296
$$853$$ −20.6791 −0.708040 −0.354020 0.935238i $$-0.615186\pi$$
−0.354020 + 0.935238i $$0.615186\pi$$
$$854$$ −13.8661 −0.474489
$$855$$ 0.264377 0.00904149
$$856$$ 15.9669 0.545738
$$857$$ −44.5639 −1.52227 −0.761136 0.648592i $$-0.775358\pi$$
−0.761136 + 0.648592i $$0.775358\pi$$
$$858$$ 0 0
$$859$$ 0.992443 0.0338617 0.0169309 0.999857i $$-0.494610\pi$$
0.0169309 + 0.999857i $$0.494610\pi$$
$$860$$ 6.42455 0.219075
$$861$$ −6.60212 −0.225000
$$862$$ −27.1939 −0.926229
$$863$$ 34.0157 1.15791 0.578954 0.815360i $$-0.303461\pi$$
0.578954 + 0.815360i $$0.303461\pi$$
$$864$$ −5.32105 −0.181026
$$865$$ 6.21176 0.211206
$$866$$ 2.21065 0.0751209
$$867$$ −8.91030 −0.302610
$$868$$ 6.01245 0.204076
$$869$$ 0 0
$$870$$ −2.26515 −0.0767957
$$871$$ 0.272304 0.00922667
$$872$$ −15.9501 −0.540138
$$873$$ 2.69735 0.0912914
$$874$$ −11.9247 −0.403358
$$875$$ 1.00000 0.0338062
$$876$$ 3.35239 0.113267
$$877$$ 37.7370 1.27429 0.637144 0.770745i $$-0.280116\pi$$
0.637144 + 0.770745i $$0.280116\pi$$
$$878$$ 6.47985 0.218685
$$879$$ −3.12366 −0.105359
$$880$$ 0 0
$$881$$ 12.1850 0.410522 0.205261 0.978707i $$-0.434196\pi$$
0.205261 + 0.978707i $$0.434196\pi$$
$$882$$ 0.154300 0.00519554
$$883$$ −23.9895 −0.807311 −0.403656 0.914911i $$-0.632261\pi$$
−0.403656 + 0.914911i $$0.632261\pi$$
$$884$$ −0.200966 −0.00675922
$$885$$ 17.2365 0.579398
$$886$$ −17.0792 −0.573786
$$887$$ −42.9265 −1.44133 −0.720666 0.693282i $$-0.756164\pi$$
−0.720666 + 0.693282i $$0.756164\pi$$
$$888$$ 14.3128 0.480307
$$889$$ −1.82824 −0.0613171
$$890$$ −2.93362 −0.0983351
$$891$$ 0 0
$$892$$ 18.0702 0.605034
$$893$$ −7.86670 −0.263249
$$894$$ −10.9130 −0.364986
$$895$$ 15.9071 0.531715
$$896$$ −1.00000 −0.0334077
$$897$$ 0.499837 0.0166891
$$898$$ −13.1328 −0.438246
$$899$$ −8.07335 −0.269261
$$900$$ −0.154300 −0.00514332
$$901$$ 66.5685 2.21772
$$902$$ 0 0
$$903$$ −10.8377 −0.360656
$$904$$ −8.65087 −0.287724
$$905$$ −6.59938 −0.219371
$$906$$ −39.8298 −1.32326
$$907$$ 17.8773 0.593606 0.296803 0.954939i $$-0.404079\pi$$
0.296803 + 0.954939i $$0.404079\pi$$
$$908$$ 0.970501 0.0322072
$$909$$ 0.880919 0.0292182
$$910$$ −0.0425741 −0.00141132
$$911$$ −26.8557 −0.889769 −0.444885 0.895588i $$-0.646755\pi$$
−0.444885 + 0.895588i $$0.646755\pi$$
$$912$$ 2.89036 0.0957095
$$913$$ 0 0
$$914$$ −15.4508 −0.511067
$$915$$ −23.3911 −0.773284
$$916$$ −2.46292 −0.0813773
$$917$$ −1.61678 −0.0533908
$$918$$ 25.1174 0.828997
$$919$$ 16.2819 0.537091 0.268546 0.963267i $$-0.413457\pi$$
0.268546 + 0.963267i $$0.413457\pi$$
$$920$$ 6.95966 0.229453
$$921$$ 11.2514 0.370747
$$922$$ 33.8352 1.11430
$$923$$ 0.481261 0.0158409
$$924$$ 0 0
$$925$$ 8.48459 0.278972
$$926$$ −31.7915 −1.04473
$$927$$ −2.06841 −0.0679356
$$928$$ 1.34277 0.0440786
$$929$$ −36.6780 −1.20337 −0.601684 0.798734i $$-0.705503\pi$$
−0.601684 + 0.798734i $$0.705503\pi$$
$$930$$ 10.1425 0.332587
$$931$$ −1.71340 −0.0561543
$$932$$ −6.65041 −0.217841
$$933$$ −25.4673 −0.833761
$$934$$ 17.0295 0.557223
$$935$$ 0 0
$$936$$ 0.00656917 0.000214720 0
$$937$$ −29.0963 −0.950536 −0.475268 0.879841i $$-0.657649\pi$$
−0.475268 + 0.879841i $$0.657649\pi$$
$$938$$ −6.39600 −0.208837
$$939$$ −12.3777 −0.403930
$$940$$ 4.59129 0.149751
$$941$$ −5.63123 −0.183573 −0.0917865 0.995779i $$-0.529258\pi$$
−0.0917865 + 0.995779i $$0.529258\pi$$
$$942$$ 19.3812 0.631475
$$943$$ −27.2381 −0.886995
$$944$$ −10.2177 −0.332559
$$945$$ 5.32105 0.173094
$$946$$ 0 0
$$947$$ −25.4396 −0.826677 −0.413338 0.910577i $$-0.635637\pi$$
−0.413338 + 0.910577i $$0.635637\pi$$
$$948$$ 14.9641 0.486011
$$949$$ −0.0846068 −0.00274645
$$950$$ 1.71340 0.0555899
$$951$$ −37.9388 −1.23025
$$952$$ 4.72038 0.152988
$$953$$ −36.6858 −1.18837 −0.594184 0.804329i $$-0.702525\pi$$
−0.594184 + 0.804329i $$0.702525\pi$$
$$954$$ −2.17599 −0.0704503
$$955$$ 17.1690 0.555576
$$956$$ 21.0529 0.680900
$$957$$ 0 0
$$958$$ −29.9718 −0.968346
$$959$$ 10.2667 0.331529
$$960$$ −1.68692 −0.0544451
$$961$$ 5.14959 0.166116
$$962$$ −0.361224 −0.0116463
$$963$$ 2.46369 0.0793913
$$964$$ −7.93244 −0.255487
$$965$$ −10.2185 −0.328944
$$966$$ −11.7404 −0.377741
$$967$$ 23.5537 0.757438 0.378719 0.925512i $$-0.376365\pi$$
0.378719 + 0.925512i $$0.376365\pi$$
$$968$$ 0 0
$$969$$ −13.6436 −0.438296
$$970$$ 17.4812 0.561288
$$971$$ 7.95989 0.255445 0.127722 0.991810i $$-0.459233\pi$$
0.127722 + 0.991810i $$0.459233\pi$$
$$972$$ −1.60191 −0.0513813
$$973$$ −1.94758 −0.0624365
$$974$$ −0.733047 −0.0234884
$$975$$ −0.0718191 −0.00230005
$$976$$ 13.8661 0.443844
$$977$$ 16.8029 0.537574 0.268787 0.963200i $$-0.413377\pi$$
0.268787 + 0.963200i $$0.413377\pi$$
$$978$$ −23.8356 −0.762178
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ −2.46109 −0.0785767
$$982$$ −10.1494 −0.323880
$$983$$ −6.64006 −0.211785 −0.105893 0.994378i $$-0.533770\pi$$
−0.105893 + 0.994378i $$0.533770\pi$$
$$984$$ 6.60212 0.210468
$$985$$ −7.33268 −0.233639
$$986$$ −6.33839 −0.201856
$$987$$ −7.74515 −0.246531
$$988$$ −0.0729463 −0.00232073
$$989$$ −44.7127 −1.42178
$$990$$ 0 0
$$991$$ −49.2695 −1.56510 −0.782548 0.622590i $$-0.786080\pi$$
−0.782548 + 0.622590i $$0.786080\pi$$
$$992$$ −6.01245 −0.190896
$$993$$ −17.4013 −0.552214
$$994$$ −11.3041 −0.358544
$$995$$ −8.90352 −0.282261
$$996$$ 17.0053 0.538833
$$997$$ 32.5523 1.03094 0.515471 0.856907i $$-0.327617\pi$$
0.515471 + 0.856907i $$0.327617\pi$$
$$998$$ 11.1673 0.353496
$$999$$ 45.1469 1.42839
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 8470.2.a.cw.1.2 6
11.5 even 5 770.2.n.j.421.1 12
11.9 even 5 770.2.n.j.631.1 yes 12
11.10 odd 2 8470.2.a.dc.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.421.1 12 11.5 even 5
770.2.n.j.631.1 yes 12 11.9 even 5
8470.2.a.cw.1.2 6 1.1 even 1 trivial
8470.2.a.dc.1.2 6 11.10 odd 2