# Properties

 Label 8470.2.a.ct.1.2 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5225.1 Defining polynomial: $$x^{4} - x^{3} - 8 x^{2} + x + 11$$ 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 $$-1.48718$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.919131 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.15520 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.919131 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.15520 q^{9} -1.00000 q^{10} -0.919131 q^{12} -3.72325 q^{13} +1.00000 q^{14} +0.919131 q^{15} +1.00000 q^{16} +1.31694 q^{17} -2.15520 q^{18} +2.70741 q^{19} -1.00000 q^{20} -0.919131 q^{21} -2.00000 q^{23} -0.919131 q^{24} +1.00000 q^{25} -3.72325 q^{26} +4.73830 q^{27} +1.00000 q^{28} +2.95002 q^{29} +0.919131 q^{30} +7.70820 q^{31} +1.00000 q^{32} +1.31694 q^{34} -1.00000 q^{35} -2.15520 q^{36} -8.04870 q^{37} +2.70741 q^{38} +3.42216 q^{39} -1.00000 q^{40} -3.24458 q^{41} -0.919131 q^{42} -6.48064 q^{43} +2.15520 q^{45} -2.00000 q^{46} -9.37090 q^{47} -0.919131 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.21044 q^{51} -3.72325 q^{52} +13.2848 q^{53} +4.73830 q^{54} +1.00000 q^{56} -2.48847 q^{57} +2.95002 q^{58} -2.83905 q^{59} +0.919131 q^{60} -1.44651 q^{61} +7.70820 q^{62} -2.15520 q^{63} +1.00000 q^{64} +3.72325 q^{65} -6.70741 q^{67} +1.31694 q^{68} +1.83826 q^{69} -1.00000 q^{70} +8.03365 q^{71} -2.15520 q^{72} +1.33071 q^{73} -8.04870 q^{74} -0.919131 q^{75} +2.70741 q^{76} +3.42216 q^{78} +7.72325 q^{79} -1.00000 q^{80} +2.11048 q^{81} -3.24458 q^{82} -0.285256 q^{83} -0.919131 q^{84} -1.31694 q^{85} -6.48064 q^{86} -2.71145 q^{87} -8.14015 q^{89} +2.15520 q^{90} -3.72325 q^{91} -2.00000 q^{92} -7.08485 q^{93} -9.37090 q^{94} -2.70741 q^{95} -0.919131 q^{96} +14.7984 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} + 2q^{6} + 4q^{7} + 4q^{8} + 6q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} + 2q^{6} + 4q^{7} + 4q^{8} + 6q^{9} - 4q^{10} + 2q^{12} + q^{13} + 4q^{14} - 2q^{15} + 4q^{16} + 2q^{17} + 6q^{18} - 3q^{19} - 4q^{20} + 2q^{21} - 8q^{23} + 2q^{24} + 4q^{25} + q^{26} + 14q^{27} + 4q^{28} + 15q^{29} - 2q^{30} + 4q^{31} + 4q^{32} + 2q^{34} - 4q^{35} + 6q^{36} + 2q^{37} - 3q^{38} - q^{39} - 4q^{40} + 11q^{41} + 2q^{42} + 7q^{43} - 6q^{45} - 8q^{46} + 5q^{47} + 2q^{48} + 4q^{49} + 4q^{50} + 18q^{51} + q^{52} + 10q^{53} + 14q^{54} + 4q^{56} - 34q^{57} + 15q^{58} + 13q^{59} - 2q^{60} + 26q^{61} + 4q^{62} + 6q^{63} + 4q^{64} - q^{65} - 13q^{67} + 2q^{68} - 4q^{69} - 4q^{70} - 13q^{71} + 6q^{72} - 18q^{73} + 2q^{74} + 2q^{75} - 3q^{76} - q^{78} + 15q^{79} - 4q^{80} - 8q^{81} + 11q^{82} - 2q^{83} + 2q^{84} - 2q^{85} + 7q^{86} + 26q^{87} - 7q^{89} - 6q^{90} + q^{91} - 8q^{92} + 2q^{93} + 5q^{94} + 3q^{95} + 2q^{96} + 10q^{97} + 4q^{98} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −0.919131 −0.530660 −0.265330 0.964158i $$-0.585481\pi$$
−0.265330 + 0.964158i $$0.585481\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −0.919131 −0.375234
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ −2.15520 −0.718400
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ −0.919131 −0.265330
$$13$$ −3.72325 −1.03264 −0.516322 0.856394i $$-0.672699\pi$$
−0.516322 + 0.856394i $$0.672699\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0.919131 0.237319
$$16$$ 1.00000 0.250000
$$17$$ 1.31694 0.319404 0.159702 0.987165i $$-0.448947\pi$$
0.159702 + 0.987165i $$0.448947\pi$$
$$18$$ −2.15520 −0.507985
$$19$$ 2.70741 0.621123 0.310561 0.950553i $$-0.399483\pi$$
0.310561 + 0.950553i $$0.399483\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −0.919131 −0.200571
$$22$$ 0 0
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ −0.919131 −0.187617
$$25$$ 1.00000 0.200000
$$26$$ −3.72325 −0.730190
$$27$$ 4.73830 0.911887
$$28$$ 1.00000 0.188982
$$29$$ 2.95002 0.547805 0.273902 0.961757i $$-0.411685\pi$$
0.273902 + 0.961757i $$0.411685\pi$$
$$30$$ 0.919131 0.167810
$$31$$ 7.70820 1.38443 0.692217 0.721689i $$-0.256634\pi$$
0.692217 + 0.721689i $$0.256634\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 1.31694 0.225853
$$35$$ −1.00000 −0.169031
$$36$$ −2.15520 −0.359200
$$37$$ −8.04870 −1.32320 −0.661599 0.749858i $$-0.730122\pi$$
−0.661599 + 0.749858i $$0.730122\pi$$
$$38$$ 2.70741 0.439200
$$39$$ 3.42216 0.547984
$$40$$ −1.00000 −0.158114
$$41$$ −3.24458 −0.506718 −0.253359 0.967372i $$-0.581535\pi$$
−0.253359 + 0.967372i $$0.581535\pi$$
$$42$$ −0.919131 −0.141825
$$43$$ −6.48064 −0.988289 −0.494145 0.869380i $$-0.664519\pi$$
−0.494145 + 0.869380i $$0.664519\pi$$
$$44$$ 0 0
$$45$$ 2.15520 0.321278
$$46$$ −2.00000 −0.294884
$$47$$ −9.37090 −1.36689 −0.683443 0.730004i $$-0.739518\pi$$
−0.683443 + 0.730004i $$0.739518\pi$$
$$48$$ −0.919131 −0.132665
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −1.21044 −0.169495
$$52$$ −3.72325 −0.516322
$$53$$ 13.2848 1.82480 0.912402 0.409296i $$-0.134226\pi$$
0.912402 + 0.409296i $$0.134226\pi$$
$$54$$ 4.73830 0.644801
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −2.48847 −0.329605
$$58$$ 2.95002 0.387357
$$59$$ −2.83905 −0.369613 −0.184807 0.982775i $$-0.559166\pi$$
−0.184807 + 0.982775i $$0.559166\pi$$
$$60$$ 0.919131 0.118659
$$61$$ −1.44651 −0.185206 −0.0926030 0.995703i $$-0.529519\pi$$
−0.0926030 + 0.995703i $$0.529519\pi$$
$$62$$ 7.70820 0.978943
$$63$$ −2.15520 −0.271530
$$64$$ 1.00000 0.125000
$$65$$ 3.72325 0.461813
$$66$$ 0 0
$$67$$ −6.70741 −0.819441 −0.409720 0.912211i $$-0.634374\pi$$
−0.409720 + 0.912211i $$0.634374\pi$$
$$68$$ 1.31694 0.159702
$$69$$ 1.83826 0.221301
$$70$$ −1.00000 −0.119523
$$71$$ 8.03365 0.953419 0.476709 0.879061i $$-0.341829\pi$$
0.476709 + 0.879061i $$0.341829\pi$$
$$72$$ −2.15520 −0.253993
$$73$$ 1.33071 0.155747 0.0778736 0.996963i $$-0.475187\pi$$
0.0778736 + 0.996963i $$0.475187\pi$$
$$74$$ −8.04870 −0.935642
$$75$$ −0.919131 −0.106132
$$76$$ 2.70741 0.310561
$$77$$ 0 0
$$78$$ 3.42216 0.387483
$$79$$ 7.72325 0.868934 0.434467 0.900688i $$-0.356937\pi$$
0.434467 + 0.900688i $$0.356937\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 2.11048 0.234498
$$82$$ −3.24458 −0.358304
$$83$$ −0.285256 −0.0313109 −0.0156555 0.999877i $$-0.504983\pi$$
−0.0156555 + 0.999877i $$0.504983\pi$$
$$84$$ −0.919131 −0.100285
$$85$$ −1.31694 −0.142842
$$86$$ −6.48064 −0.698826
$$87$$ −2.71145 −0.290698
$$88$$ 0 0
$$89$$ −8.14015 −0.862854 −0.431427 0.902148i $$-0.641990\pi$$
−0.431427 + 0.902148i $$0.641990\pi$$
$$90$$ 2.15520 0.227178
$$91$$ −3.72325 −0.390303
$$92$$ −2.00000 −0.208514
$$93$$ −7.08485 −0.734664
$$94$$ −9.37090 −0.966534
$$95$$ −2.70741 −0.277775
$$96$$ −0.919131 −0.0938084
$$97$$ 14.7984 1.50255 0.751274 0.659991i $$-0.229440\pi$$
0.751274 + 0.659991i $$0.229440\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 14.1000 1.40300 0.701499 0.712670i $$-0.252514\pi$$
0.701499 + 0.712670i $$0.252514\pi$$
$$102$$ −1.21044 −0.119851
$$103$$ 11.7627 1.15901 0.579504 0.814969i $$-0.303246\pi$$
0.579504 + 0.814969i $$0.303246\pi$$
$$104$$ −3.72325 −0.365095
$$105$$ 0.919131 0.0896980
$$106$$ 13.2848 1.29033
$$107$$ −19.8272 −1.91677 −0.958383 0.285484i $$-0.907846\pi$$
−0.958383 + 0.285484i $$0.907846\pi$$
$$108$$ 4.73830 0.455943
$$109$$ 0.995533 0.0953548 0.0476774 0.998863i $$-0.484818\pi$$
0.0476774 + 0.998863i $$0.484818\pi$$
$$110$$ 0 0
$$111$$ 7.39781 0.702169
$$112$$ 1.00000 0.0944911
$$113$$ 11.0691 1.04129 0.520645 0.853773i $$-0.325691\pi$$
0.520645 + 0.853773i $$0.325691\pi$$
$$114$$ −2.48847 −0.233066
$$115$$ 2.00000 0.186501
$$116$$ 2.95002 0.273902
$$117$$ 8.02435 0.741851
$$118$$ −2.83905 −0.261356
$$119$$ 1.31694 0.120723
$$120$$ 0.919131 0.0839048
$$121$$ 0 0
$$122$$ −1.44651 −0.130960
$$123$$ 2.98219 0.268895
$$124$$ 7.70820 0.692217
$$125$$ −1.00000 −0.0894427
$$126$$ −2.15520 −0.192000
$$127$$ −6.50223 −0.576980 −0.288490 0.957483i $$-0.593153\pi$$
−0.288490 + 0.957483i $$0.593153\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 5.95656 0.524446
$$130$$ 3.72325 0.326551
$$131$$ 21.7622 1.90137 0.950684 0.310160i $$-0.100383\pi$$
0.950684 + 0.310160i $$0.100383\pi$$
$$132$$ 0 0
$$133$$ 2.70741 0.234762
$$134$$ −6.70741 −0.579432
$$135$$ −4.73830 −0.407808
$$136$$ 1.31694 0.112926
$$137$$ −2.90408 −0.248112 −0.124056 0.992275i $$-0.539590\pi$$
−0.124056 + 0.992275i $$0.539590\pi$$
$$138$$ 1.83826 0.156483
$$139$$ 1.26775 0.107529 0.0537645 0.998554i $$-0.482878\pi$$
0.0537645 + 0.998554i $$0.482878\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 8.61308 0.725352
$$142$$ 8.03365 0.674169
$$143$$ 0 0
$$144$$ −2.15520 −0.179600
$$145$$ −2.95002 −0.244986
$$146$$ 1.33071 0.110130
$$147$$ −0.919131 −0.0758086
$$148$$ −8.04870 −0.661599
$$149$$ −5.85489 −0.479652 −0.239826 0.970816i $$-0.577090\pi$$
−0.239826 + 0.970816i $$0.577090\pi$$
$$150$$ −0.919131 −0.0750467
$$151$$ −10.2243 −0.832039 −0.416020 0.909356i $$-0.636575\pi$$
−0.416020 + 0.909356i $$0.636575\pi$$
$$152$$ 2.70741 0.219600
$$153$$ −2.83826 −0.229460
$$154$$ 0 0
$$155$$ −7.70820 −0.619138
$$156$$ 3.42216 0.273992
$$157$$ 1.18004 0.0941772 0.0470886 0.998891i $$-0.485006\pi$$
0.0470886 + 0.998891i $$0.485006\pi$$
$$158$$ 7.72325 0.614429
$$159$$ −12.2104 −0.968351
$$160$$ −1.00000 −0.0790569
$$161$$ −2.00000 −0.157622
$$162$$ 2.11048 0.165815
$$163$$ 13.5818 1.06381 0.531905 0.846804i $$-0.321476\pi$$
0.531905 + 0.846804i $$0.321476\pi$$
$$164$$ −3.24458 −0.253359
$$165$$ 0 0
$$166$$ −0.285256 −0.0221402
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −0.919131 −0.0709125
$$169$$ 0.862611 0.0663547
$$170$$ −1.31694 −0.101004
$$171$$ −5.83501 −0.446214
$$172$$ −6.48064 −0.494145
$$173$$ 12.4741 0.948389 0.474194 0.880420i $$-0.342739\pi$$
0.474194 + 0.880420i $$0.342739\pi$$
$$174$$ −2.71145 −0.205555
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 2.60946 0.196139
$$178$$ −8.14015 −0.610130
$$179$$ 12.2295 0.914078 0.457039 0.889447i $$-0.348910\pi$$
0.457039 + 0.889447i $$0.348910\pi$$
$$180$$ 2.15520 0.160639
$$181$$ 20.9718 1.55882 0.779411 0.626513i $$-0.215518\pi$$
0.779411 + 0.626513i $$0.215518\pi$$
$$182$$ −3.72325 −0.275986
$$183$$ 1.32953 0.0982815
$$184$$ −2.00000 −0.147442
$$185$$ 8.04870 0.591752
$$186$$ −7.08485 −0.519486
$$187$$ 0 0
$$188$$ −9.37090 −0.683443
$$189$$ 4.73830 0.344661
$$190$$ −2.70741 −0.196416
$$191$$ 6.74632 0.488147 0.244073 0.969757i $$-0.421516\pi$$
0.244073 + 0.969757i $$0.421516\pi$$
$$192$$ −0.919131 −0.0663325
$$193$$ 19.9673 1.43728 0.718640 0.695382i $$-0.244765\pi$$
0.718640 + 0.695382i $$0.244765\pi$$
$$194$$ 14.7984 1.06246
$$195$$ −3.42216 −0.245066
$$196$$ 1.00000 0.0714286
$$197$$ 21.1717 1.50842 0.754212 0.656631i $$-0.228019\pi$$
0.754212 + 0.656631i $$0.228019\pi$$
$$198$$ 0 0
$$199$$ 8.27872 0.586863 0.293431 0.955980i $$-0.405203\pi$$
0.293431 + 0.955980i $$0.405203\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 6.16499 0.434845
$$202$$ 14.1000 0.992070
$$203$$ 2.95002 0.207051
$$204$$ −1.21044 −0.0847476
$$205$$ 3.24458 0.226611
$$206$$ 11.7627 0.819543
$$207$$ 4.31040 0.299593
$$208$$ −3.72325 −0.258161
$$209$$ 0 0
$$210$$ 0.919131 0.0634260
$$211$$ 15.5912 1.07334 0.536671 0.843792i $$-0.319682\pi$$
0.536671 + 0.843792i $$0.319682\pi$$
$$212$$ 13.2848 0.912402
$$213$$ −7.38397 −0.505942
$$214$$ −19.8272 −1.35536
$$215$$ 6.48064 0.441976
$$216$$ 4.73830 0.322401
$$217$$ 7.70820 0.523267
$$218$$ 0.995533 0.0674260
$$219$$ −1.22309 −0.0826489
$$220$$ 0 0
$$221$$ −4.90329 −0.329831
$$222$$ 7.39781 0.496508
$$223$$ −16.6313 −1.11372 −0.556858 0.830608i $$-0.687993\pi$$
−0.556858 + 0.830608i $$0.687993\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −2.15520 −0.143680
$$226$$ 11.0691 0.736304
$$227$$ 1.19716 0.0794582 0.0397291 0.999210i $$-0.487351\pi$$
0.0397291 + 0.999210i $$0.487351\pi$$
$$228$$ −2.48847 −0.164803
$$229$$ 14.8613 0.982064 0.491032 0.871141i $$-0.336620\pi$$
0.491032 + 0.871141i $$0.336620\pi$$
$$230$$ 2.00000 0.131876
$$231$$ 0 0
$$232$$ 2.95002 0.193678
$$233$$ −6.33199 −0.414822 −0.207411 0.978254i $$-0.566504\pi$$
−0.207411 + 0.978254i $$0.566504\pi$$
$$234$$ 8.02435 0.524568
$$235$$ 9.37090 0.611290
$$236$$ −2.83905 −0.184807
$$237$$ −7.09868 −0.461109
$$238$$ 1.31694 0.0853644
$$239$$ −11.6906 −0.756202 −0.378101 0.925764i $$-0.623423\pi$$
−0.378101 + 0.925764i $$0.623423\pi$$
$$240$$ 0.919131 0.0593296
$$241$$ 10.3945 0.669566 0.334783 0.942295i $$-0.391337\pi$$
0.334783 + 0.942295i $$0.391337\pi$$
$$242$$ 0 0
$$243$$ −16.1547 −1.03633
$$244$$ −1.44651 −0.0926030
$$245$$ −1.00000 −0.0638877
$$246$$ 2.98219 0.190138
$$247$$ −10.0804 −0.641399
$$248$$ 7.70820 0.489471
$$249$$ 0.262188 0.0166155
$$250$$ −1.00000 −0.0632456
$$251$$ −22.3395 −1.41006 −0.705029 0.709179i $$-0.749066\pi$$
−0.705029 + 0.709179i $$0.749066\pi$$
$$252$$ −2.15520 −0.135765
$$253$$ 0 0
$$254$$ −6.50223 −0.407986
$$255$$ 1.21044 0.0758005
$$256$$ 1.00000 0.0625000
$$257$$ −5.28201 −0.329482 −0.164741 0.986337i $$-0.552679\pi$$
−0.164741 + 0.986337i $$0.552679\pi$$
$$258$$ 5.95656 0.370839
$$259$$ −8.04870 −0.500122
$$260$$ 3.72325 0.230906
$$261$$ −6.35788 −0.393543
$$262$$ 21.7622 1.34447
$$263$$ 25.3847 1.56529 0.782645 0.622469i $$-0.213870\pi$$
0.782645 + 0.622469i $$0.213870\pi$$
$$264$$ 0 0
$$265$$ −13.2848 −0.816077
$$266$$ 2.70741 0.166002
$$267$$ 7.48186 0.457883
$$268$$ −6.70741 −0.409720
$$269$$ −15.3782 −0.937627 −0.468814 0.883297i $$-0.655318\pi$$
−0.468814 + 0.883297i $$0.655318\pi$$
$$270$$ −4.73830 −0.288364
$$271$$ 10.3274 0.627346 0.313673 0.949531i $$-0.398440\pi$$
0.313673 + 0.949531i $$0.398440\pi$$
$$272$$ 1.31694 0.0798510
$$273$$ 3.42216 0.207118
$$274$$ −2.90408 −0.175442
$$275$$ 0 0
$$276$$ 1.83826 0.110650
$$277$$ −22.7614 −1.36760 −0.683799 0.729670i $$-0.739674\pi$$
−0.683799 + 0.729670i $$0.739674\pi$$
$$278$$ 1.26775 0.0760345
$$279$$ −16.6127 −0.994577
$$280$$ −1.00000 −0.0597614
$$281$$ −3.07890 −0.183672 −0.0918359 0.995774i $$-0.529274\pi$$
−0.0918359 + 0.995774i $$0.529274\pi$$
$$282$$ 8.61308 0.512901
$$283$$ −10.9791 −0.652642 −0.326321 0.945259i $$-0.605809\pi$$
−0.326321 + 0.945259i $$0.605809\pi$$
$$284$$ 8.03365 0.476709
$$285$$ 2.48847 0.147404
$$286$$ 0 0
$$287$$ −3.24458 −0.191521
$$288$$ −2.15520 −0.126996
$$289$$ −15.2657 −0.897981
$$290$$ −2.95002 −0.173231
$$291$$ −13.6016 −0.797342
$$292$$ 1.33071 0.0778736
$$293$$ 8.05126 0.470360 0.235180 0.971952i $$-0.424432\pi$$
0.235180 + 0.971952i $$0.424432\pi$$
$$294$$ −0.919131 −0.0536048
$$295$$ 2.83905 0.165296
$$296$$ −8.04870 −0.467821
$$297$$ 0 0
$$298$$ −5.85489 −0.339165
$$299$$ 7.44651 0.430643
$$300$$ −0.919131 −0.0530660
$$301$$ −6.48064 −0.373538
$$302$$ −10.2243 −0.588341
$$303$$ −12.9597 −0.744516
$$304$$ 2.70741 0.155281
$$305$$ 1.44651 0.0828267
$$306$$ −2.83826 −0.162253
$$307$$ −27.5963 −1.57501 −0.787503 0.616311i $$-0.788627\pi$$
−0.787503 + 0.616311i $$0.788627\pi$$
$$308$$ 0 0
$$309$$ −10.8114 −0.615040
$$310$$ −7.70820 −0.437797
$$311$$ 29.7870 1.68907 0.844533 0.535504i $$-0.179878\pi$$
0.844533 + 0.535504i $$0.179878\pi$$
$$312$$ 3.42216 0.193741
$$313$$ 30.9153 1.74744 0.873718 0.486433i $$-0.161702\pi$$
0.873718 + 0.486433i $$0.161702\pi$$
$$314$$ 1.18004 0.0665934
$$315$$ 2.15520 0.121432
$$316$$ 7.72325 0.434467
$$317$$ −2.32189 −0.130411 −0.0652053 0.997872i $$-0.520770\pi$$
−0.0652053 + 0.997872i $$0.520770\pi$$
$$318$$ −12.2104 −0.684727
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 18.2238 1.01715
$$322$$ −2.00000 −0.111456
$$323$$ 3.56549 0.198389
$$324$$ 2.11048 0.117249
$$325$$ −3.72325 −0.206529
$$326$$ 13.5818 0.752228
$$327$$ −0.915025 −0.0506010
$$328$$ −3.24458 −0.179152
$$329$$ −9.37090 −0.516634
$$330$$ 0 0
$$331$$ −9.95686 −0.547279 −0.273639 0.961832i $$-0.588227\pi$$
−0.273639 + 0.961832i $$0.588227\pi$$
$$332$$ −0.285256 −0.0156555
$$333$$ 17.3465 0.950585
$$334$$ 12.0000 0.656611
$$335$$ 6.70741 0.366465
$$336$$ −0.919131 −0.0501427
$$337$$ 20.7682 1.13132 0.565658 0.824640i $$-0.308622\pi$$
0.565658 + 0.824640i $$0.308622\pi$$
$$338$$ 0.862611 0.0469198
$$339$$ −10.1739 −0.552572
$$340$$ −1.31694 −0.0714210
$$341$$ 0 0
$$342$$ −5.83501 −0.315521
$$343$$ 1.00000 0.0539949
$$344$$ −6.48064 −0.349413
$$345$$ −1.83826 −0.0989687
$$346$$ 12.4741 0.670612
$$347$$ 15.7606 0.846072 0.423036 0.906113i $$-0.360964\pi$$
0.423036 + 0.906113i $$0.360964\pi$$
$$348$$ −2.71145 −0.145349
$$349$$ 23.2953 1.24697 0.623484 0.781836i $$-0.285717\pi$$
0.623484 + 0.781836i $$0.285717\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −17.6419 −0.941655
$$352$$ 0 0
$$353$$ 27.1303 1.44400 0.721999 0.691894i $$-0.243224\pi$$
0.721999 + 0.691894i $$0.243224\pi$$
$$354$$ 2.60946 0.138691
$$355$$ −8.03365 −0.426382
$$356$$ −8.14015 −0.431427
$$357$$ −1.21044 −0.0640631
$$358$$ 12.2295 0.646351
$$359$$ 32.0637 1.69226 0.846130 0.532977i $$-0.178927\pi$$
0.846130 + 0.532977i $$0.178927\pi$$
$$360$$ 2.15520 0.113589
$$361$$ −11.6699 −0.614206
$$362$$ 20.9718 1.10225
$$363$$ 0 0
$$364$$ −3.72325 −0.195151
$$365$$ −1.33071 −0.0696523
$$366$$ 1.32953 0.0694955
$$367$$ −11.6037 −0.605709 −0.302854 0.953037i $$-0.597940\pi$$
−0.302854 + 0.953037i $$0.597940\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 6.99271 0.364026
$$370$$ 8.04870 0.418432
$$371$$ 13.2848 0.689711
$$372$$ −7.08485 −0.367332
$$373$$ −17.7252 −0.917777 −0.458889 0.888494i $$-0.651752\pi$$
−0.458889 + 0.888494i $$0.651752\pi$$
$$374$$ 0 0
$$375$$ 0.919131 0.0474637
$$376$$ −9.37090 −0.483267
$$377$$ −10.9837 −0.565688
$$378$$ 4.73830 0.243712
$$379$$ −12.0540 −0.619174 −0.309587 0.950871i $$-0.600191\pi$$
−0.309587 + 0.950871i $$0.600191\pi$$
$$380$$ −2.70741 −0.138887
$$381$$ 5.97640 0.306180
$$382$$ 6.74632 0.345172
$$383$$ 6.75538 0.345184 0.172592 0.984993i $$-0.444786\pi$$
0.172592 + 0.984993i $$0.444786\pi$$
$$384$$ −0.919131 −0.0469042
$$385$$ 0 0
$$386$$ 19.9673 1.01631
$$387$$ 13.9671 0.709986
$$388$$ 14.7984 0.751274
$$389$$ −24.1441 −1.22416 −0.612078 0.790797i $$-0.709666\pi$$
−0.612078 + 0.790797i $$0.709666\pi$$
$$390$$ −3.42216 −0.173288
$$391$$ −2.63387 −0.133201
$$392$$ 1.00000 0.0505076
$$393$$ −20.0023 −1.00898
$$394$$ 21.1717 1.06662
$$395$$ −7.72325 −0.388599
$$396$$ 0 0
$$397$$ 10.5163 0.527798 0.263899 0.964550i $$-0.414991\pi$$
0.263899 + 0.964550i $$0.414991\pi$$
$$398$$ 8.27872 0.414975
$$399$$ −2.48847 −0.124579
$$400$$ 1.00000 0.0500000
$$401$$ −37.1997 −1.85766 −0.928831 0.370503i $$-0.879185\pi$$
−0.928831 + 0.370503i $$0.879185\pi$$
$$402$$ 6.16499 0.307482
$$403$$ −28.6996 −1.42963
$$404$$ 14.1000 0.701499
$$405$$ −2.11048 −0.104870
$$406$$ 2.95002 0.146407
$$407$$ 0 0
$$408$$ −1.21044 −0.0599256
$$409$$ 26.3908 1.30494 0.652470 0.757815i $$-0.273733\pi$$
0.652470 + 0.757815i $$0.273733\pi$$
$$410$$ 3.24458 0.160238
$$411$$ 2.66923 0.131663
$$412$$ 11.7627 0.579504
$$413$$ −2.83905 −0.139701
$$414$$ 4.31040 0.211844
$$415$$ 0.285256 0.0140027
$$416$$ −3.72325 −0.182547
$$417$$ −1.16523 −0.0570614
$$418$$ 0 0
$$419$$ 13.0447 0.637273 0.318637 0.947877i $$-0.396775\pi$$
0.318637 + 0.947877i $$0.396775\pi$$
$$420$$ 0.919131 0.0448490
$$421$$ −13.4510 −0.655563 −0.327782 0.944753i $$-0.606301\pi$$
−0.327782 + 0.944753i $$0.606301\pi$$
$$422$$ 15.5912 0.758967
$$423$$ 20.1961 0.981970
$$424$$ 13.2848 0.645165
$$425$$ 1.31694 0.0638808
$$426$$ −7.38397 −0.357755
$$427$$ −1.44651 −0.0700013
$$428$$ −19.8272 −0.958383
$$429$$ 0 0
$$430$$ 6.48064 0.312524
$$431$$ 32.7451 1.57728 0.788638 0.614858i $$-0.210787\pi$$
0.788638 + 0.614858i $$0.210787\pi$$
$$432$$ 4.73830 0.227972
$$433$$ 9.00434 0.432721 0.216361 0.976314i $$-0.430581\pi$$
0.216361 + 0.976314i $$0.430581\pi$$
$$434$$ 7.70820 0.370006
$$435$$ 2.71145 0.130004
$$436$$ 0.995533 0.0476774
$$437$$ −5.41482 −0.259026
$$438$$ −1.22309 −0.0584416
$$439$$ 3.88952 0.185637 0.0928184 0.995683i $$-0.470412\pi$$
0.0928184 + 0.995683i $$0.470412\pi$$
$$440$$ 0 0
$$441$$ −2.15520 −0.102629
$$442$$ −4.90329 −0.233226
$$443$$ 5.63377 0.267669 0.133834 0.991004i $$-0.457271\pi$$
0.133834 + 0.991004i $$0.457271\pi$$
$$444$$ 7.39781 0.351084
$$445$$ 8.14015 0.385880
$$446$$ −16.6313 −0.787515
$$447$$ 5.38141 0.254532
$$448$$ 1.00000 0.0472456
$$449$$ −5.62855 −0.265628 −0.132814 0.991141i $$-0.542401\pi$$
−0.132814 + 0.991141i $$0.542401\pi$$
$$450$$ −2.15520 −0.101597
$$451$$ 0 0
$$452$$ 11.0691 0.520645
$$453$$ 9.39744 0.441530
$$454$$ 1.19716 0.0561855
$$455$$ 3.72325 0.174549
$$456$$ −2.48847 −0.116533
$$457$$ 1.15770 0.0541548 0.0270774 0.999633i $$-0.491380\pi$$
0.0270774 + 0.999633i $$0.491380\pi$$
$$458$$ 14.8613 0.694424
$$459$$ 6.24005 0.291260
$$460$$ 2.00000 0.0932505
$$461$$ 10.3088 0.480129 0.240065 0.970757i $$-0.422831\pi$$
0.240065 + 0.970757i $$0.422831\pi$$
$$462$$ 0 0
$$463$$ −29.4740 −1.36977 −0.684887 0.728649i $$-0.740149\pi$$
−0.684887 + 0.728649i $$0.740149\pi$$
$$464$$ 2.95002 0.136951
$$465$$ 7.08485 0.328552
$$466$$ −6.33199 −0.293324
$$467$$ 20.5977 0.953146 0.476573 0.879135i $$-0.341879\pi$$
0.476573 + 0.879135i $$0.341879\pi$$
$$468$$ 8.02435 0.370926
$$469$$ −6.70741 −0.309720
$$470$$ 9.37090 0.432247
$$471$$ −1.08461 −0.0499761
$$472$$ −2.83905 −0.130678
$$473$$ 0 0
$$474$$ −7.09868 −0.326053
$$475$$ 2.70741 0.124225
$$476$$ 1.31694 0.0603617
$$477$$ −28.6313 −1.31094
$$478$$ −11.6906 −0.534715
$$479$$ −4.18526 −0.191229 −0.0956146 0.995418i $$-0.530482\pi$$
−0.0956146 + 0.995418i $$0.530482\pi$$
$$480$$ 0.919131 0.0419524
$$481$$ 29.9673 1.36639
$$482$$ 10.3945 0.473454
$$483$$ 1.83826 0.0836438
$$484$$ 0 0
$$485$$ −14.7984 −0.671960
$$486$$ −16.1547 −0.732792
$$487$$ −8.06730 −0.365564 −0.182782 0.983153i $$-0.558510\pi$$
−0.182782 + 0.983153i $$0.558510\pi$$
$$488$$ −1.44651 −0.0654802
$$489$$ −12.4835 −0.564522
$$490$$ −1.00000 −0.0451754
$$491$$ −23.5882 −1.06452 −0.532261 0.846580i $$-0.678658\pi$$
−0.532261 + 0.846580i $$0.678658\pi$$
$$492$$ 2.98219 0.134448
$$493$$ 3.88499 0.174971
$$494$$ −10.0804 −0.453538
$$495$$ 0 0
$$496$$ 7.70820 0.346109
$$497$$ 8.03365 0.360358
$$498$$ 0.262188 0.0117489
$$499$$ 7.22181 0.323293 0.161646 0.986849i $$-0.448320\pi$$
0.161646 + 0.986849i $$0.448320\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −11.0296 −0.492765
$$502$$ −22.3395 −0.997061
$$503$$ −11.3828 −0.507532 −0.253766 0.967266i $$-0.581669\pi$$
−0.253766 + 0.967266i $$0.581669\pi$$
$$504$$ −2.15520 −0.0960002
$$505$$ −14.1000 −0.627440
$$506$$ 0 0
$$507$$ −0.792852 −0.0352118
$$508$$ −6.50223 −0.288490
$$509$$ 12.1422 0.538192 0.269096 0.963113i $$-0.413275\pi$$
0.269096 + 0.963113i $$0.413275\pi$$
$$510$$ 1.21044 0.0535991
$$511$$ 1.33071 0.0588669
$$512$$ 1.00000 0.0441942
$$513$$ 12.8285 0.566394
$$514$$ −5.28201 −0.232979
$$515$$ −11.7627 −0.518324
$$516$$ 5.95656 0.262223
$$517$$ 0 0
$$518$$ −8.04870 −0.353640
$$519$$ −11.4653 −0.503272
$$520$$ 3.72325 0.163275
$$521$$ 12.1565 0.532585 0.266293 0.963892i $$-0.414201\pi$$
0.266293 + 0.963892i $$0.414201\pi$$
$$522$$ −6.35788 −0.278277
$$523$$ 29.3199 1.28207 0.641035 0.767512i $$-0.278505\pi$$
0.641035 + 0.767512i $$0.278505\pi$$
$$524$$ 21.7622 0.950684
$$525$$ −0.919131 −0.0401142
$$526$$ 25.3847 1.10683
$$527$$ 10.1512 0.442194
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ −13.2848 −0.577053
$$531$$ 6.11872 0.265530
$$532$$ 2.70741 0.117381
$$533$$ 12.0804 0.523259
$$534$$ 7.48186 0.323772
$$535$$ 19.8272 0.857204
$$536$$ −6.70741 −0.289716
$$537$$ −11.2405 −0.485065
$$538$$ −15.3782 −0.663002
$$539$$ 0 0
$$540$$ −4.73830 −0.203904
$$541$$ −2.54640 −0.109478 −0.0547392 0.998501i $$-0.517433\pi$$
−0.0547392 + 0.998501i $$0.517433\pi$$
$$542$$ 10.3274 0.443600
$$543$$ −19.2758 −0.827205
$$544$$ 1.31694 0.0564632
$$545$$ −0.995533 −0.0426439
$$546$$ 3.42216 0.146455
$$547$$ −32.1359 −1.37403 −0.687016 0.726642i $$-0.741080\pi$$
−0.687016 + 0.726642i $$0.741080\pi$$
$$548$$ −2.90408 −0.124056
$$549$$ 3.11751 0.133052
$$550$$ 0 0
$$551$$ 7.98692 0.340254
$$552$$ 1.83826 0.0782416
$$553$$ 7.72325 0.328426
$$554$$ −22.7614 −0.967038
$$555$$ −7.39781 −0.314019
$$556$$ 1.26775 0.0537645
$$557$$ −5.98042 −0.253399 −0.126699 0.991941i $$-0.540438\pi$$
−0.126699 + 0.991941i $$0.540438\pi$$
$$558$$ −16.6127 −0.703272
$$559$$ 24.1291 1.02055
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −3.07890 −0.129876
$$563$$ 1.80284 0.0759807 0.0379903 0.999278i $$-0.487904\pi$$
0.0379903 + 0.999278i $$0.487904\pi$$
$$564$$ 8.61308 0.362676
$$565$$ −11.0691 −0.465679
$$566$$ −10.9791 −0.461488
$$567$$ 2.11048 0.0886317
$$568$$ 8.03365 0.337084
$$569$$ 30.1377 1.26344 0.631718 0.775198i $$-0.282350\pi$$
0.631718 + 0.775198i $$0.282350\pi$$
$$570$$ 2.48847 0.104230
$$571$$ 30.3322 1.26936 0.634681 0.772774i $$-0.281131\pi$$
0.634681 + 0.772774i $$0.281131\pi$$
$$572$$ 0 0
$$573$$ −6.20075 −0.259040
$$574$$ −3.24458 −0.135426
$$575$$ −2.00000 −0.0834058
$$576$$ −2.15520 −0.0897999
$$577$$ −15.7849 −0.657135 −0.328568 0.944480i $$-0.606566\pi$$
−0.328568 + 0.944480i $$0.606566\pi$$
$$578$$ −15.2657 −0.634968
$$579$$ −18.3526 −0.762708
$$580$$ −2.95002 −0.122493
$$581$$ −0.285256 −0.0118344
$$582$$ −13.6016 −0.563806
$$583$$ 0 0
$$584$$ 1.33071 0.0550650
$$585$$ −8.02435 −0.331766
$$586$$ 8.05126 0.332595
$$587$$ 25.9219 1.06991 0.534956 0.844880i $$-0.320328\pi$$
0.534956 + 0.844880i $$0.320328\pi$$
$$588$$ −0.919131 −0.0379043
$$589$$ 20.8693 0.859904
$$590$$ 2.83905 0.116882
$$591$$ −19.4596 −0.800460
$$592$$ −8.04870 −0.330799
$$593$$ −2.58469 −0.106140 −0.0530702 0.998591i $$-0.516901\pi$$
−0.0530702 + 0.998591i $$0.516901\pi$$
$$594$$ 0 0
$$595$$ −1.31694 −0.0539892
$$596$$ −5.85489 −0.239826
$$597$$ −7.60922 −0.311425
$$598$$ 7.44651 0.304510
$$599$$ 8.01529 0.327496 0.163748 0.986502i $$-0.447642\pi$$
0.163748 + 0.986502i $$0.447642\pi$$
$$600$$ −0.919131 −0.0375234
$$601$$ 1.07488 0.0438454 0.0219227 0.999760i $$-0.493021\pi$$
0.0219227 + 0.999760i $$0.493021\pi$$
$$602$$ −6.48064 −0.264131
$$603$$ 14.4558 0.588686
$$604$$ −10.2243 −0.416020
$$605$$ 0 0
$$606$$ −12.9597 −0.526452
$$607$$ 12.6752 0.514472 0.257236 0.966349i $$-0.417188\pi$$
0.257236 + 0.966349i $$0.417188\pi$$
$$608$$ 2.70741 0.109800
$$609$$ −2.71145 −0.109874
$$610$$ 1.44651 0.0585673
$$611$$ 34.8902 1.41151
$$612$$ −2.83826 −0.114730
$$613$$ −21.0929 −0.851935 −0.425968 0.904738i $$-0.640066\pi$$
−0.425968 + 0.904738i $$0.640066\pi$$
$$614$$ −27.5963 −1.11370
$$615$$ −2.98219 −0.120254
$$616$$ 0 0
$$617$$ −15.4070 −0.620263 −0.310131 0.950694i $$-0.600373\pi$$
−0.310131 + 0.950694i $$0.600373\pi$$
$$618$$ −10.8114 −0.434899
$$619$$ 18.6431 0.749328 0.374664 0.927161i $$-0.377758\pi$$
0.374664 + 0.927161i $$0.377758\pi$$
$$620$$ −7.70820 −0.309569
$$621$$ −9.47660 −0.380283
$$622$$ 29.7870 1.19435
$$623$$ −8.14015 −0.326128
$$624$$ 3.42216 0.136996
$$625$$ 1.00000 0.0400000
$$626$$ 30.9153 1.23562
$$627$$ 0 0
$$628$$ 1.18004 0.0470886
$$629$$ −10.5996 −0.422635
$$630$$ 2.15520 0.0858652
$$631$$ 27.4709 1.09360 0.546799 0.837264i $$-0.315846\pi$$
0.546799 + 0.837264i $$0.315846\pi$$
$$632$$ 7.72325 0.307214
$$633$$ −14.3303 −0.569580
$$634$$ −2.32189 −0.0922142
$$635$$ 6.50223 0.258033
$$636$$ −12.2104 −0.484175
$$637$$ −3.72325 −0.147521
$$638$$ 0 0
$$639$$ −17.3141 −0.684936
$$640$$ −1.00000 −0.0395285
$$641$$ −32.4224 −1.28061 −0.640304 0.768121i $$-0.721192\pi$$
−0.640304 + 0.768121i $$0.721192\pi$$
$$642$$ 18.2238 0.719235
$$643$$ 1.62945 0.0642591 0.0321296 0.999484i $$-0.489771\pi$$
0.0321296 + 0.999484i $$0.489771\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ −5.95656 −0.234539
$$646$$ 3.56549 0.140282
$$647$$ 36.2914 1.42676 0.713382 0.700775i $$-0.247163\pi$$
0.713382 + 0.700775i $$0.247163\pi$$
$$648$$ 2.11048 0.0829074
$$649$$ 0 0
$$650$$ −3.72325 −0.146038
$$651$$ −7.08485 −0.277677
$$652$$ 13.5818 0.531905
$$653$$ −32.1275 −1.25725 −0.628623 0.777710i $$-0.716381\pi$$
−0.628623 + 0.777710i $$0.716381\pi$$
$$654$$ −0.915025 −0.0357803
$$655$$ −21.7622 −0.850318
$$656$$ −3.24458 −0.126679
$$657$$ −2.86793 −0.111889
$$658$$ −9.37090 −0.365316
$$659$$ −24.7215 −0.963012 −0.481506 0.876443i $$-0.659910\pi$$
−0.481506 + 0.876443i $$0.659910\pi$$
$$660$$ 0 0
$$661$$ −24.3370 −0.946598 −0.473299 0.880902i $$-0.656937\pi$$
−0.473299 + 0.880902i $$0.656937\pi$$
$$662$$ −9.95686 −0.386984
$$663$$ 4.50676 0.175028
$$664$$ −0.285256 −0.0110701
$$665$$ −2.70741 −0.104989
$$666$$ 17.3465 0.672165
$$667$$ −5.90004 −0.228450
$$668$$ 12.0000 0.464294
$$669$$ 15.2864 0.591004
$$670$$ 6.70741 0.259130
$$671$$ 0 0
$$672$$ −0.919131 −0.0354562
$$673$$ 43.7873 1.68787 0.843937 0.536442i $$-0.180232\pi$$
0.843937 + 0.536442i $$0.180232\pi$$
$$674$$ 20.7682 0.799962
$$675$$ 4.73830 0.182377
$$676$$ 0.862611 0.0331773
$$677$$ −20.9199 −0.804018 −0.402009 0.915636i $$-0.631688\pi$$
−0.402009 + 0.915636i $$0.631688\pi$$
$$678$$ −10.1739 −0.390727
$$679$$ 14.7984 0.567909
$$680$$ −1.31694 −0.0505022
$$681$$ −1.10035 −0.0421653
$$682$$ 0 0
$$683$$ −6.79955 −0.260178 −0.130089 0.991502i $$-0.541526\pi$$
−0.130089 + 0.991502i $$0.541526\pi$$
$$684$$ −5.83501 −0.223107
$$685$$ 2.90408 0.110959
$$686$$ 1.00000 0.0381802
$$687$$ −13.6595 −0.521143
$$688$$ −6.48064 −0.247072
$$689$$ −49.4625 −1.88437
$$690$$ −1.83826 −0.0699814
$$691$$ 33.0127 1.25586 0.627931 0.778269i $$-0.283902\pi$$
0.627931 + 0.778269i $$0.283902\pi$$
$$692$$ 12.4741 0.474194
$$693$$ 0 0
$$694$$ 15.7606 0.598263
$$695$$ −1.26775 −0.0480885
$$696$$ −2.71145 −0.102777
$$697$$ −4.27290 −0.161848
$$698$$ 23.2953 0.881740
$$699$$ 5.81992 0.220130
$$700$$ 1.00000 0.0377964
$$701$$ 34.2118 1.29216 0.646081 0.763269i $$-0.276407\pi$$
0.646081 + 0.763269i $$0.276407\pi$$
$$702$$ −17.6419 −0.665850
$$703$$ −21.7911 −0.821869
$$704$$ 0 0
$$705$$ −8.61308 −0.324387
$$706$$ 27.1303 1.02106
$$707$$ 14.1000 0.530284
$$708$$ 2.60946 0.0980695
$$709$$ −10.7815 −0.404909 −0.202455 0.979292i $$-0.564892\pi$$
−0.202455 + 0.979292i $$0.564892\pi$$
$$710$$ −8.03365 −0.301498
$$711$$ −16.6451 −0.624242
$$712$$ −8.14015 −0.305065
$$713$$ −15.4164 −0.577349
$$714$$ −1.21044 −0.0452995
$$715$$ 0 0
$$716$$ 12.2295 0.457039
$$717$$ 10.7452 0.401286
$$718$$ 32.0637 1.19661
$$719$$ −48.7392 −1.81767 −0.908833 0.417160i $$-0.863026\pi$$
−0.908833 + 0.417160i $$0.863026\pi$$
$$720$$ 2.15520 0.0803195
$$721$$ 11.7627 0.438064
$$722$$ −11.6699 −0.434309
$$723$$ −9.55386 −0.355312
$$724$$ 20.9718 0.779411
$$725$$ 2.95002 0.109561
$$726$$ 0 0
$$727$$ 11.9416 0.442891 0.221446 0.975173i $$-0.428922\pi$$
0.221446 + 0.975173i $$0.428922\pi$$
$$728$$ −3.72325 −0.137993
$$729$$ 8.51686 0.315439
$$730$$ −1.33071 −0.0492516
$$731$$ −8.53460 −0.315664
$$732$$ 1.32953 0.0491408
$$733$$ 38.9332 1.43803 0.719015 0.694995i $$-0.244593\pi$$
0.719015 + 0.694995i $$0.244593\pi$$
$$734$$ −11.6037 −0.428301
$$735$$ 0.919131 0.0339026
$$736$$ −2.00000 −0.0737210
$$737$$ 0 0
$$738$$ 6.99271 0.257405
$$739$$ 29.3583 1.07996 0.539981 0.841677i $$-0.318431\pi$$
0.539981 + 0.841677i $$0.318431\pi$$
$$740$$ 8.04870 0.295876
$$741$$ 9.26519 0.340365
$$742$$ 13.2848 0.487699
$$743$$ 27.1351 0.995491 0.497746 0.867323i $$-0.334161\pi$$
0.497746 + 0.867323i $$0.334161\pi$$
$$744$$ −7.08485 −0.259743
$$745$$ 5.85489 0.214507
$$746$$ −17.7252 −0.648966
$$747$$ 0.614784 0.0224938
$$748$$ 0 0
$$749$$ −19.8272 −0.724470
$$750$$ 0.919131 0.0335619
$$751$$ −36.8507 −1.34470 −0.672350 0.740233i $$-0.734715\pi$$
−0.672350 + 0.740233i $$0.734715\pi$$
$$752$$ −9.37090 −0.341721
$$753$$ 20.5329 0.748262
$$754$$ −10.9837 −0.400002
$$755$$ 10.2243 0.372099
$$756$$ 4.73830 0.172330
$$757$$ 1.04667 0.0380417 0.0190209 0.999819i $$-0.493945\pi$$
0.0190209 + 0.999819i $$0.493945\pi$$
$$758$$ −12.0540 −0.437822
$$759$$ 0 0
$$760$$ −2.70741 −0.0982082
$$761$$ 2.62959 0.0953227 0.0476614 0.998864i $$-0.484823\pi$$
0.0476614 + 0.998864i $$0.484823\pi$$
$$762$$ 5.97640 0.216502
$$763$$ 0.995533 0.0360407
$$764$$ 6.74632 0.244073
$$765$$ 2.83826 0.102618
$$766$$ 6.75538 0.244082
$$767$$ 10.5705 0.381679
$$768$$ −0.919131 −0.0331663
$$769$$ 31.0270 1.11886 0.559431 0.828877i $$-0.311020\pi$$
0.559431 + 0.828877i $$0.311020\pi$$
$$770$$ 0 0
$$771$$ 4.85485 0.174843
$$772$$ 19.9673 0.718640
$$773$$ −8.84786 −0.318236 −0.159118 0.987260i $$-0.550865\pi$$
−0.159118 + 0.987260i $$0.550865\pi$$
$$774$$ 13.9671 0.502036
$$775$$ 7.70820 0.276887
$$776$$ 14.7984 0.531231
$$777$$ 7.39781 0.265395
$$778$$ −24.1441 −0.865609
$$779$$ −8.78441 −0.314734
$$780$$ −3.42216 −0.122533
$$781$$ 0 0
$$782$$ −2.63387 −0.0941872
$$783$$ 13.9781 0.499536
$$784$$ 1.00000 0.0357143
$$785$$ −1.18004 −0.0421173
$$786$$ −20.0023 −0.713457
$$787$$ 6.15195 0.219293 0.109647 0.993971i $$-0.465028\pi$$
0.109647 + 0.993971i $$0.465028\pi$$
$$788$$ 21.1717 0.754212
$$789$$ −23.3319 −0.830637
$$790$$ −7.72325 −0.274781
$$791$$ 11.0691 0.393571
$$792$$ 0 0
$$793$$ 5.38571 0.191252
$$794$$ 10.5163 0.373210
$$795$$ 12.2104 0.433060
$$796$$ 8.27872 0.293431
$$797$$ −47.3736 −1.67806 −0.839030 0.544085i $$-0.816877\pi$$
−0.839030 + 0.544085i $$0.816877\pi$$
$$798$$ −2.48847 −0.0880907
$$799$$ −12.3409 −0.436589
$$800$$ 1.00000 0.0353553
$$801$$ 17.5436 0.619874
$$802$$ −37.1997 −1.31357
$$803$$ 0 0
$$804$$ 6.16499 0.217422
$$805$$ 2.00000 0.0704907
$$806$$ −28.6996 −1.01090
$$807$$ 14.1346 0.497562
$$808$$ 14.1000 0.496035
$$809$$ 9.79837 0.344492 0.172246 0.985054i $$-0.444898\pi$$
0.172246 + 0.985054i $$0.444898\pi$$
$$810$$ −2.11048 −0.0741546
$$811$$ 20.9999 0.737406 0.368703 0.929547i $$-0.379802\pi$$
0.368703 + 0.929547i $$0.379802\pi$$
$$812$$ 2.95002 0.103525
$$813$$ −9.49224 −0.332908
$$814$$ 0 0
$$815$$ −13.5818 −0.475750
$$816$$ −1.21044 −0.0423738
$$817$$ −17.5458 −0.613849
$$818$$ 26.3908 0.922732
$$819$$ 8.02435 0.280393
$$820$$ 3.24458 0.113306
$$821$$ 48.1483 1.68039 0.840194 0.542286i $$-0.182441\pi$$
0.840194 + 0.542286i $$0.182441\pi$$
$$822$$ 2.66923 0.0931001
$$823$$ −35.5348 −1.23867 −0.619333 0.785128i $$-0.712597\pi$$
−0.619333 + 0.785128i $$0.712597\pi$$
$$824$$ 11.7627 0.409771
$$825$$ 0 0
$$826$$ −2.83905 −0.0987833
$$827$$ 44.0977 1.53343 0.766714 0.641988i $$-0.221890\pi$$
0.766714 + 0.641988i $$0.221890\pi$$
$$828$$ 4.31040 0.149797
$$829$$ −6.77860 −0.235430 −0.117715 0.993047i $$-0.537557\pi$$
−0.117715 + 0.993047i $$0.537557\pi$$
$$830$$ 0.285256 0.00990139
$$831$$ 20.9207 0.725730
$$832$$ −3.72325 −0.129081
$$833$$ 1.31694 0.0456292
$$834$$ −1.16523 −0.0403485
$$835$$ −12.0000 −0.415277
$$836$$ 0 0
$$837$$ 36.5238 1.26245
$$838$$ 13.0447 0.450620
$$839$$ −53.6528 −1.85230 −0.926150 0.377155i $$-0.876902\pi$$
−0.926150 + 0.377155i $$0.876902\pi$$
$$840$$ 0.919131 0.0317130
$$841$$ −20.2974 −0.699910
$$842$$ −13.4510 −0.463553
$$843$$ 2.82991 0.0974673
$$844$$ 15.5912 0.536671
$$845$$ −0.862611 −0.0296747
$$846$$ 20.1961 0.694358
$$847$$ 0 0
$$848$$ 13.2848 0.456201
$$849$$ 10.0913 0.346331
$$850$$ 1.31694 0.0451706
$$851$$ 16.0974 0.551812
$$852$$ −7.38397 −0.252971
$$853$$ 8.65701 0.296410 0.148205 0.988957i $$-0.452650\pi$$
0.148205 + 0.988957i $$0.452650\pi$$
$$854$$ −1.44651 −0.0494984
$$855$$ 5.83501 0.199553
$$856$$ −19.8272 −0.677679
$$857$$ 46.7535 1.59707 0.798534 0.601950i $$-0.205609\pi$$
0.798534 + 0.601950i $$0.205609\pi$$
$$858$$ 0 0
$$859$$ −35.5377 −1.21253 −0.606265 0.795263i $$-0.707333\pi$$
−0.606265 + 0.795263i $$0.707333\pi$$
$$860$$ 6.48064 0.220988
$$861$$ 2.98219 0.101633
$$862$$ 32.7451 1.11530
$$863$$ 40.0753 1.36418 0.682089 0.731269i $$-0.261072\pi$$
0.682089 + 0.731269i $$0.261072\pi$$
$$864$$ 4.73830 0.161200
$$865$$ −12.4741 −0.424132
$$866$$ 9.00434 0.305980
$$867$$ 14.0312 0.476523
$$868$$ 7.70820 0.261633
$$869$$ 0 0
$$870$$ 2.71145 0.0919269
$$871$$ 24.9734 0.846191
$$872$$ 0.995533 0.0337130
$$873$$ −31.8934 −1.07943
$$874$$ −5.41482 −0.183159
$$875$$ −1.00000 −0.0338062
$$876$$ −1.22309 −0.0413244
$$877$$ −37.0839 −1.25223 −0.626117 0.779729i $$-0.715357\pi$$
−0.626117 + 0.779729i $$0.715357\pi$$
$$878$$ 3.88952 0.131265
$$879$$ −7.40016 −0.249601
$$880$$ 0 0
$$881$$ −24.7764 −0.834737 −0.417369 0.908737i $$-0.637048\pi$$
−0.417369 + 0.908737i $$0.637048\pi$$
$$882$$ −2.15520 −0.0725693
$$883$$ −5.49721 −0.184996 −0.0924980 0.995713i $$-0.529485\pi$$
−0.0924980 + 0.995713i $$0.529485\pi$$
$$884$$ −4.90329 −0.164915
$$885$$ −2.60946 −0.0877161
$$886$$ 5.63377 0.189270
$$887$$ −22.5044 −0.755625 −0.377813 0.925882i $$-0.623324\pi$$
−0.377813 + 0.925882i $$0.623324\pi$$
$$888$$ 7.39781 0.248254
$$889$$ −6.50223 −0.218078
$$890$$ 8.14015 0.272858
$$891$$ 0 0
$$892$$ −16.6313 −0.556858
$$893$$ −25.3709 −0.849004
$$894$$ 5.38141 0.179981
$$895$$ −12.2295 −0.408788
$$896$$ 1.00000 0.0334077
$$897$$ −6.84431 −0.228525
$$898$$ −5.62855 −0.187827
$$899$$ 22.7394 0.758400
$$900$$ −2.15520 −0.0718400
$$901$$ 17.4952 0.582850
$$902$$ 0 0
$$903$$ 5.95656 0.198222
$$904$$ 11.0691 0.368152
$$905$$ −20.9718 −0.697126
$$906$$ 9.39744 0.312209
$$907$$ −10.0800 −0.334702 −0.167351 0.985897i $$-0.553521\pi$$
−0.167351 + 0.985897i $$0.553521\pi$$
$$908$$ 1.19716 0.0397291
$$909$$ −30.3882 −1.00791
$$910$$ 3.72325 0.123425
$$911$$ 16.3242 0.540846 0.270423 0.962742i $$-0.412836\pi$$
0.270423 + 0.962742i $$0.412836\pi$$
$$912$$ −2.48847 −0.0824013
$$913$$ 0 0
$$914$$ 1.15770 0.0382932
$$915$$ −1.32953 −0.0439528
$$916$$ 14.8613 0.491032
$$917$$ 21.7622 0.718650
$$918$$ 6.24005 0.205952
$$919$$ −18.9580 −0.625366 −0.312683 0.949858i $$-0.601228\pi$$
−0.312683 + 0.949858i $$0.601228\pi$$
$$920$$ 2.00000 0.0659380
$$921$$ 25.3646 0.835793
$$922$$ 10.3088 0.339503
$$923$$ −29.9113 −0.984543
$$924$$ 0 0
$$925$$ −8.04870 −0.264640
$$926$$ −29.4740 −0.968577
$$927$$ −25.3509 −0.832631
$$928$$ 2.95002 0.0968392
$$929$$ −15.5873 −0.511404 −0.255702 0.966756i $$-0.582307\pi$$
−0.255702 + 0.966756i $$0.582307\pi$$
$$930$$ 7.08485 0.232321
$$931$$ 2.70741 0.0887319
$$932$$ −6.33199 −0.207411
$$933$$ −27.3781 −0.896320
$$934$$ 20.5977 0.673976
$$935$$ 0 0
$$936$$ 8.02435 0.262284
$$937$$ −32.7215 −1.06896 −0.534482 0.845180i $$-0.679493\pi$$
−0.534482 + 0.845180i $$0.679493\pi$$
$$938$$ −6.70741 −0.219005
$$939$$ −28.4152 −0.927295
$$940$$ 9.37090 0.305645
$$941$$ 14.5193 0.473314 0.236657 0.971593i $$-0.423948\pi$$
0.236657 + 0.971593i $$0.423948\pi$$
$$942$$ −1.08461 −0.0353385
$$943$$ 6.48915 0.211316
$$944$$ −2.83905 −0.0924033
$$945$$ −4.73830 −0.154137
$$946$$ 0 0
$$947$$ 16.5854 0.538954 0.269477 0.963007i $$-0.413149\pi$$
0.269477 + 0.963007i $$0.413149\pi$$
$$948$$ −7.09868 −0.230554
$$949$$ −4.95455 −0.160832
$$950$$ 2.70741 0.0878401
$$951$$ 2.13412 0.0692037
$$952$$ 1.31694 0.0426822
$$953$$ −29.5354 −0.956745 −0.478373 0.878157i $$-0.658773\pi$$
−0.478373 + 0.878157i $$0.658773\pi$$
$$954$$ −28.6313 −0.926973
$$955$$ −6.74632 −0.218306
$$956$$ −11.6906 −0.378101
$$957$$ 0 0
$$958$$ −4.18526 −0.135220
$$959$$ −2.90408 −0.0937777
$$960$$ 0.919131 0.0296648
$$961$$ 28.4164 0.916658
$$962$$ 29.9673 0.966186
$$963$$ 42.7315 1.37700
$$964$$ 10.3945 0.334783
$$965$$ −19.9673 −0.642771
$$966$$ 1.83826 0.0591451
$$967$$ −36.7434 −1.18159 −0.590794 0.806823i $$-0.701185\pi$$
−0.590794 + 0.806823i $$0.701185\pi$$
$$968$$ 0 0
$$969$$ −3.27715 −0.105277
$$970$$ −14.7984 −0.475147
$$971$$ −22.2877 −0.715245 −0.357622 0.933866i $$-0.616412\pi$$
−0.357622 + 0.933866i $$0.616412\pi$$
$$972$$ −16.1547 −0.518163
$$973$$ 1.26775 0.0406422
$$974$$ −8.06730 −0.258493
$$975$$ 3.42216 0.109597
$$976$$ −1.44651 −0.0463015
$$977$$ −40.5549 −1.29747 −0.648733 0.761016i $$-0.724701\pi$$
−0.648733 + 0.761016i $$0.724701\pi$$
$$978$$ −12.4835 −0.399177
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −2.14557 −0.0685028
$$982$$ −23.5882 −0.752731
$$983$$ −39.8706 −1.27167 −0.635837 0.771824i $$-0.719345\pi$$
−0.635837 + 0.771824i $$0.719345\pi$$
$$984$$ 2.98219 0.0950688
$$985$$ −21.1717 −0.674587
$$986$$ 3.88499 0.123723
$$987$$ 8.61308 0.274157
$$988$$ −10.0804 −0.320700
$$989$$ 12.9613 0.412145
$$990$$ 0 0
$$991$$ 35.5410 1.12900 0.564498 0.825435i $$-0.309070\pi$$
0.564498 + 0.825435i $$0.309070\pi$$
$$992$$ 7.70820 0.244736
$$993$$ 9.15166 0.290419
$$994$$ 8.03365 0.254812
$$995$$ −8.27872 −0.262453
$$996$$ 0.262188 0.00830774
$$997$$ −47.4408 −1.50246 −0.751232 0.660038i $$-0.770540\pi$$
−0.751232 + 0.660038i $$0.770540\pi$$
$$998$$ 7.22181 0.228602
$$999$$ −38.1372 −1.20661
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.ct.1.2 4
11.7 odd 10 770.2.n.e.71.2 8
11.8 odd 10 770.2.n.e.141.2 yes 8
11.10 odd 2 8470.2.a.cq.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.e.71.2 8 11.7 odd 10
770.2.n.e.141.2 yes 8 11.8 odd 10
8470.2.a.cq.1.2 4 11.10 odd 2
8470.2.a.ct.1.2 4 1.1 even 1 trivial