# Properties

 Label 8470.2.a.cy.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: 6.6.13298000.1 Defining polynomial: $$x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1$$ 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 $$2.79700$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.64424 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.64424 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.99201 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.64424 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.64424 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.99201 q^{9} -1.00000 q^{10} -2.64424 q^{12} -3.06037 q^{13} +1.00000 q^{14} -2.64424 q^{15} +1.00000 q^{16} +0.0683587 q^{17} -3.99201 q^{18} -2.77079 q^{19} +1.00000 q^{20} +2.64424 q^{21} +1.83120 q^{23} +2.64424 q^{24} +1.00000 q^{25} +3.06037 q^{26} -2.62310 q^{27} -1.00000 q^{28} -3.30937 q^{29} +2.64424 q^{30} +4.12571 q^{31} -1.00000 q^{32} -0.0683587 q^{34} -1.00000 q^{35} +3.99201 q^{36} -5.63448 q^{37} +2.77079 q^{38} +8.09234 q^{39} -1.00000 q^{40} +5.13257 q^{41} -2.64424 q^{42} -2.84053 q^{43} +3.99201 q^{45} -1.83120 q^{46} +11.9195 q^{47} -2.64424 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.180757 q^{51} -3.06037 q^{52} +0.123629 q^{53} +2.62310 q^{54} +1.00000 q^{56} +7.32664 q^{57} +3.30937 q^{58} -11.1929 q^{59} -2.64424 q^{60} -2.53653 q^{61} -4.12571 q^{62} -3.99201 q^{63} +1.00000 q^{64} -3.06037 q^{65} -2.20818 q^{67} +0.0683587 q^{68} -4.84213 q^{69} +1.00000 q^{70} +13.7247 q^{71} -3.99201 q^{72} -6.98905 q^{73} +5.63448 q^{74} -2.64424 q^{75} -2.77079 q^{76} -8.09234 q^{78} +12.4142 q^{79} +1.00000 q^{80} -5.03990 q^{81} -5.13257 q^{82} +3.05366 q^{83} +2.64424 q^{84} +0.0683587 q^{85} +2.84053 q^{86} +8.75076 q^{87} +13.5707 q^{89} -3.99201 q^{90} +3.06037 q^{91} +1.83120 q^{92} -10.9094 q^{93} -11.9195 q^{94} -2.77079 q^{95} +2.64424 q^{96} -2.74477 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} + O(q^{10})$$ $$6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} - 6 q^{10} + q^{12} - 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} - 7 q^{17} - 15 q^{18} - 11 q^{19} + 6 q^{20} - q^{21} - 6 q^{23} - q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} - 6 q^{28} - 2 q^{29} - q^{30} - 6 q^{32} + 7 q^{34} - 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} - 20 q^{39} - 6 q^{40} - 13 q^{41} + q^{42} - 19 q^{43} + 15 q^{45} + 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} - 6 q^{50} + 14 q^{51} - 2 q^{52} - 10 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} - 22 q^{61} - 15 q^{63} + 6 q^{64} - 2 q^{65} + 5 q^{67} - 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} - 15 q^{72} - 13 q^{73} + 14 q^{74} + q^{75} - 11 q^{76} + 20 q^{78} + 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} + 5 q^{83} - q^{84} - 7 q^{85} + 19 q^{86} - 14 q^{87} + q^{89} - 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} - 22 q^{94} - 11 q^{95} - q^{96} - 3 q^{97} - 6 q^{98} + O(q^{100})$$

## 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
$$3$$ −2.64424 −1.52665 −0.763326 0.646013i $$-0.776435\pi$$
−0.763326 + 0.646013i $$0.776435\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 2.64424 1.07951
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 3.99201 1.33067
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ −2.64424 −0.763326
$$13$$ −3.06037 −0.848793 −0.424396 0.905477i $$-0.639514\pi$$
−0.424396 + 0.905477i $$0.639514\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −2.64424 −0.682740
$$16$$ 1.00000 0.250000
$$17$$ 0.0683587 0.0165794 0.00828971 0.999966i $$-0.497361\pi$$
0.00828971 + 0.999966i $$0.497361\pi$$
$$18$$ −3.99201 −0.940925
$$19$$ −2.77079 −0.635664 −0.317832 0.948147i $$-0.602955\pi$$
−0.317832 + 0.948147i $$0.602955\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.64424 0.577021
$$22$$ 0 0
$$23$$ 1.83120 0.381831 0.190916 0.981606i $$-0.438854\pi$$
0.190916 + 0.981606i $$0.438854\pi$$
$$24$$ 2.64424 0.539753
$$25$$ 1.00000 0.200000
$$26$$ 3.06037 0.600187
$$27$$ −2.62310 −0.504817
$$28$$ −1.00000 −0.188982
$$29$$ −3.30937 −0.614534 −0.307267 0.951623i $$-0.599414\pi$$
−0.307267 + 0.951623i $$0.599414\pi$$
$$30$$ 2.64424 0.482770
$$31$$ 4.12571 0.741000 0.370500 0.928833i $$-0.379186\pi$$
0.370500 + 0.928833i $$0.379186\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −0.0683587 −0.0117234
$$35$$ −1.00000 −0.169031
$$36$$ 3.99201 0.665334
$$37$$ −5.63448 −0.926303 −0.463151 0.886279i $$-0.653281\pi$$
−0.463151 + 0.886279i $$0.653281\pi$$
$$38$$ 2.77079 0.449482
$$39$$ 8.09234 1.29581
$$40$$ −1.00000 −0.158114
$$41$$ 5.13257 0.801573 0.400787 0.916171i $$-0.368737\pi$$
0.400787 + 0.916171i $$0.368737\pi$$
$$42$$ −2.64424 −0.408015
$$43$$ −2.84053 −0.433177 −0.216589 0.976263i $$-0.569493\pi$$
−0.216589 + 0.976263i $$0.569493\pi$$
$$44$$ 0 0
$$45$$ 3.99201 0.595093
$$46$$ −1.83120 −0.269996
$$47$$ 11.9195 1.73864 0.869322 0.494247i $$-0.164556\pi$$
0.869322 + 0.494247i $$0.164556\pi$$
$$48$$ −2.64424 −0.381663
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −0.180757 −0.0253110
$$52$$ −3.06037 −0.424396
$$53$$ 0.123629 0.0169818 0.00849088 0.999964i $$-0.497297\pi$$
0.00849088 + 0.999964i $$0.497297\pi$$
$$54$$ 2.62310 0.356959
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 7.32664 0.970438
$$58$$ 3.30937 0.434541
$$59$$ −11.1929 −1.45720 −0.728598 0.684941i $$-0.759828\pi$$
−0.728598 + 0.684941i $$0.759828\pi$$
$$60$$ −2.64424 −0.341370
$$61$$ −2.53653 −0.324770 −0.162385 0.986728i $$-0.551919\pi$$
−0.162385 + 0.986728i $$0.551919\pi$$
$$62$$ −4.12571 −0.523966
$$63$$ −3.99201 −0.502946
$$64$$ 1.00000 0.125000
$$65$$ −3.06037 −0.379592
$$66$$ 0 0
$$67$$ −2.20818 −0.269773 −0.134886 0.990861i $$-0.543067\pi$$
−0.134886 + 0.990861i $$0.543067\pi$$
$$68$$ 0.0683587 0.00828971
$$69$$ −4.84213 −0.582924
$$70$$ 1.00000 0.119523
$$71$$ 13.7247 1.62883 0.814413 0.580285i $$-0.197059\pi$$
0.814413 + 0.580285i $$0.197059\pi$$
$$72$$ −3.99201 −0.470462
$$73$$ −6.98905 −0.818007 −0.409003 0.912533i $$-0.634124\pi$$
−0.409003 + 0.912533i $$0.634124\pi$$
$$74$$ 5.63448 0.654995
$$75$$ −2.64424 −0.305331
$$76$$ −2.77079 −0.317832
$$77$$ 0 0
$$78$$ −8.09234 −0.916277
$$79$$ 12.4142 1.39671 0.698354 0.715752i $$-0.253916\pi$$
0.698354 + 0.715752i $$0.253916\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −5.03990 −0.559989
$$82$$ −5.13257 −0.566798
$$83$$ 3.05366 0.335183 0.167591 0.985857i $$-0.446401\pi$$
0.167591 + 0.985857i $$0.446401\pi$$
$$84$$ 2.64424 0.288510
$$85$$ 0.0683587 0.00741454
$$86$$ 2.84053 0.306303
$$87$$ 8.75076 0.938180
$$88$$ 0 0
$$89$$ 13.5707 1.43849 0.719246 0.694756i $$-0.244487\pi$$
0.719246 + 0.694756i $$0.244487\pi$$
$$90$$ −3.99201 −0.420794
$$91$$ 3.06037 0.320813
$$92$$ 1.83120 0.190916
$$93$$ −10.9094 −1.13125
$$94$$ −11.9195 −1.22941
$$95$$ −2.77079 −0.284277
$$96$$ 2.64424 0.269877
$$97$$ −2.74477 −0.278689 −0.139345 0.990244i $$-0.544500\pi$$
−0.139345 + 0.990244i $$0.544500\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 5.53961 0.551212 0.275606 0.961271i $$-0.411121\pi$$
0.275606 + 0.961271i $$0.411121\pi$$
$$102$$ 0.180757 0.0178976
$$103$$ −3.61697 −0.356390 −0.178195 0.983995i $$-0.557026\pi$$
−0.178195 + 0.983995i $$0.557026\pi$$
$$104$$ 3.06037 0.300094
$$105$$ 2.64424 0.258051
$$106$$ −0.123629 −0.0120079
$$107$$ −6.58182 −0.636288 −0.318144 0.948042i $$-0.603060\pi$$
−0.318144 + 0.948042i $$0.603060\pi$$
$$108$$ −2.62310 −0.252408
$$109$$ −1.14986 −0.110137 −0.0550683 0.998483i $$-0.517538\pi$$
−0.0550683 + 0.998483i $$0.517538\pi$$
$$110$$ 0 0
$$111$$ 14.8989 1.41414
$$112$$ −1.00000 −0.0944911
$$113$$ −12.1536 −1.14332 −0.571658 0.820492i $$-0.693700\pi$$
−0.571658 + 0.820492i $$0.693700\pi$$
$$114$$ −7.32664 −0.686203
$$115$$ 1.83120 0.170760
$$116$$ −3.30937 −0.307267
$$117$$ −12.2170 −1.12946
$$118$$ 11.1929 1.03039
$$119$$ −0.0683587 −0.00626643
$$120$$ 2.64424 0.241385
$$121$$ 0 0
$$122$$ 2.53653 0.229647
$$123$$ −13.5718 −1.22372
$$124$$ 4.12571 0.370500
$$125$$ 1.00000 0.0894427
$$126$$ 3.99201 0.355636
$$127$$ −13.7815 −1.22291 −0.611457 0.791278i $$-0.709416\pi$$
−0.611457 + 0.791278i $$0.709416\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 7.51106 0.661312
$$130$$ 3.06037 0.268412
$$131$$ −4.89161 −0.427382 −0.213691 0.976901i $$-0.568548\pi$$
−0.213691 + 0.976901i $$0.568548\pi$$
$$132$$ 0 0
$$133$$ 2.77079 0.240258
$$134$$ 2.20818 0.190758
$$135$$ −2.62310 −0.225761
$$136$$ −0.0683587 −0.00586171
$$137$$ −19.0559 −1.62805 −0.814026 0.580829i $$-0.802729\pi$$
−0.814026 + 0.580829i $$0.802729\pi$$
$$138$$ 4.84213 0.412190
$$139$$ −7.65318 −0.649135 −0.324567 0.945863i $$-0.605219\pi$$
−0.324567 + 0.945863i $$0.605219\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −31.5181 −2.65430
$$142$$ −13.7247 −1.15175
$$143$$ 0 0
$$144$$ 3.99201 0.332667
$$145$$ −3.30937 −0.274828
$$146$$ 6.98905 0.578418
$$147$$ −2.64424 −0.218093
$$148$$ −5.63448 −0.463151
$$149$$ 14.6774 1.20242 0.601208 0.799092i $$-0.294686\pi$$
0.601208 + 0.799092i $$0.294686\pi$$
$$150$$ 2.64424 0.215901
$$151$$ 11.0935 0.902774 0.451387 0.892328i $$-0.350929\pi$$
0.451387 + 0.892328i $$0.350929\pi$$
$$152$$ 2.77079 0.224741
$$153$$ 0.272888 0.0220617
$$154$$ 0 0
$$155$$ 4.12571 0.331385
$$156$$ 8.09234 0.647906
$$157$$ 13.1941 1.05300 0.526502 0.850174i $$-0.323503\pi$$
0.526502 + 0.850174i $$0.323503\pi$$
$$158$$ −12.4142 −0.987622
$$159$$ −0.326905 −0.0259253
$$160$$ −1.00000 −0.0790569
$$161$$ −1.83120 −0.144319
$$162$$ 5.03990 0.395972
$$163$$ −3.70882 −0.290497 −0.145248 0.989395i $$-0.546398\pi$$
−0.145248 + 0.989395i $$0.546398\pi$$
$$164$$ 5.13257 0.400787
$$165$$ 0 0
$$166$$ −3.05366 −0.237010
$$167$$ 8.75429 0.677428 0.338714 0.940889i $$-0.390008\pi$$
0.338714 + 0.940889i $$0.390008\pi$$
$$168$$ −2.64424 −0.204008
$$169$$ −3.63416 −0.279551
$$170$$ −0.0683587 −0.00524287
$$171$$ −11.0610 −0.845858
$$172$$ −2.84053 −0.216589
$$173$$ 3.03876 0.231033 0.115516 0.993306i $$-0.463148\pi$$
0.115516 + 0.993306i $$0.463148\pi$$
$$174$$ −8.75076 −0.663393
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 29.5968 2.22463
$$178$$ −13.5707 −1.01717
$$179$$ −7.29987 −0.545618 −0.272809 0.962068i $$-0.587953\pi$$
−0.272809 + 0.962068i $$0.587953\pi$$
$$180$$ 3.99201 0.297547
$$181$$ 10.9351 0.812800 0.406400 0.913695i $$-0.366784\pi$$
0.406400 + 0.913695i $$0.366784\pi$$
$$182$$ −3.06037 −0.226849
$$183$$ 6.70720 0.495811
$$184$$ −1.83120 −0.134998
$$185$$ −5.63448 −0.414255
$$186$$ 10.9094 0.799914
$$187$$ 0 0
$$188$$ 11.9195 0.869322
$$189$$ 2.62310 0.190803
$$190$$ 2.77079 0.201014
$$191$$ 22.6520 1.63904 0.819519 0.573051i $$-0.194241\pi$$
0.819519 + 0.573051i $$0.194241\pi$$
$$192$$ −2.64424 −0.190832
$$193$$ 16.9764 1.22199 0.610994 0.791635i $$-0.290770\pi$$
0.610994 + 0.791635i $$0.290770\pi$$
$$194$$ 2.74477 0.197063
$$195$$ 8.09234 0.579505
$$196$$ 1.00000 0.0714286
$$197$$ −26.3241 −1.87551 −0.937757 0.347293i $$-0.887101\pi$$
−0.937757 + 0.347293i $$0.887101\pi$$
$$198$$ 0 0
$$199$$ 16.9127 1.19891 0.599456 0.800408i $$-0.295384\pi$$
0.599456 + 0.800408i $$0.295384\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 5.83897 0.411849
$$202$$ −5.53961 −0.389766
$$203$$ 3.30937 0.232272
$$204$$ −0.180757 −0.0126555
$$205$$ 5.13257 0.358474
$$206$$ 3.61697 0.252006
$$207$$ 7.31016 0.508091
$$208$$ −3.06037 −0.212198
$$209$$ 0 0
$$210$$ −2.64424 −0.182470
$$211$$ 16.4483 1.13235 0.566173 0.824286i $$-0.308423\pi$$
0.566173 + 0.824286i $$0.308423\pi$$
$$212$$ 0.123629 0.00849088
$$213$$ −36.2915 −2.48665
$$214$$ 6.58182 0.449924
$$215$$ −2.84053 −0.193723
$$216$$ 2.62310 0.178480
$$217$$ −4.12571 −0.280072
$$218$$ 1.14986 0.0778784
$$219$$ 18.4807 1.24881
$$220$$ 0 0
$$221$$ −0.209203 −0.0140725
$$222$$ −14.8989 −0.999950
$$223$$ 20.4675 1.37060 0.685302 0.728259i $$-0.259670\pi$$
0.685302 + 0.728259i $$0.259670\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 3.99201 0.266134
$$226$$ 12.1536 0.808447
$$227$$ −23.7041 −1.57329 −0.786647 0.617403i $$-0.788185\pi$$
−0.786647 + 0.617403i $$0.788185\pi$$
$$228$$ 7.32664 0.485219
$$229$$ −26.6368 −1.76021 −0.880106 0.474778i $$-0.842528\pi$$
−0.880106 + 0.474778i $$0.842528\pi$$
$$230$$ −1.83120 −0.120746
$$231$$ 0 0
$$232$$ 3.30937 0.217271
$$233$$ −30.3368 −1.98743 −0.993714 0.111950i $$-0.964290\pi$$
−0.993714 + 0.111950i $$0.964290\pi$$
$$234$$ 12.2170 0.798650
$$235$$ 11.9195 0.777545
$$236$$ −11.1929 −0.728598
$$237$$ −32.8262 −2.13229
$$238$$ 0.0683587 0.00443104
$$239$$ 9.95180 0.643728 0.321864 0.946786i $$-0.395691\pi$$
0.321864 + 0.946786i $$0.395691\pi$$
$$240$$ −2.64424 −0.170685
$$241$$ 30.1849 1.94438 0.972191 0.234189i $$-0.0752433\pi$$
0.972191 + 0.234189i $$0.0752433\pi$$
$$242$$ 0 0
$$243$$ 21.1960 1.35973
$$244$$ −2.53653 −0.162385
$$245$$ 1.00000 0.0638877
$$246$$ 13.5718 0.865303
$$247$$ 8.47964 0.539547
$$248$$ −4.12571 −0.261983
$$249$$ −8.07461 −0.511707
$$250$$ −1.00000 −0.0632456
$$251$$ −1.16998 −0.0738485 −0.0369242 0.999318i $$-0.511756\pi$$
−0.0369242 + 0.999318i $$0.511756\pi$$
$$252$$ −3.99201 −0.251473
$$253$$ 0 0
$$254$$ 13.7815 0.864730
$$255$$ −0.180757 −0.0113194
$$256$$ 1.00000 0.0625000
$$257$$ −0.980916 −0.0611879 −0.0305939 0.999532i $$-0.509740\pi$$
−0.0305939 + 0.999532i $$0.509740\pi$$
$$258$$ −7.51106 −0.467618
$$259$$ 5.63448 0.350110
$$260$$ −3.06037 −0.189796
$$261$$ −13.2110 −0.817741
$$262$$ 4.89161 0.302204
$$263$$ 15.9923 0.986125 0.493063 0.869994i $$-0.335877\pi$$
0.493063 + 0.869994i $$0.335877\pi$$
$$264$$ 0 0
$$265$$ 0.123629 0.00759448
$$266$$ −2.77079 −0.169888
$$267$$ −35.8842 −2.19608
$$268$$ −2.20818 −0.134886
$$269$$ 15.1048 0.920953 0.460476 0.887672i $$-0.347679\pi$$
0.460476 + 0.887672i $$0.347679\pi$$
$$270$$ 2.62310 0.159637
$$271$$ 0.499548 0.0303454 0.0151727 0.999885i $$-0.495170\pi$$
0.0151727 + 0.999885i $$0.495170\pi$$
$$272$$ 0.0683587 0.00414486
$$273$$ −8.09234 −0.489771
$$274$$ 19.0559 1.15121
$$275$$ 0 0
$$276$$ −4.84213 −0.291462
$$277$$ 11.1061 0.667302 0.333651 0.942697i $$-0.391719\pi$$
0.333651 + 0.942697i $$0.391719\pi$$
$$278$$ 7.65318 0.459007
$$279$$ 16.4699 0.986025
$$280$$ 1.00000 0.0597614
$$281$$ −15.4512 −0.921742 −0.460871 0.887467i $$-0.652463\pi$$
−0.460871 + 0.887467i $$0.652463\pi$$
$$282$$ 31.5181 1.87688
$$283$$ −14.3139 −0.850876 −0.425438 0.904988i $$-0.639880\pi$$
−0.425438 + 0.904988i $$0.639880\pi$$
$$284$$ 13.7247 0.814413
$$285$$ 7.32664 0.433993
$$286$$ 0 0
$$287$$ −5.13257 −0.302966
$$288$$ −3.99201 −0.235231
$$289$$ −16.9953 −0.999725
$$290$$ 3.30937 0.194333
$$291$$ 7.25783 0.425462
$$292$$ −6.98905 −0.409003
$$293$$ 17.5749 1.02673 0.513367 0.858169i $$-0.328398\pi$$
0.513367 + 0.858169i $$0.328398\pi$$
$$294$$ 2.64424 0.154215
$$295$$ −11.1929 −0.651678
$$296$$ 5.63448 0.327498
$$297$$ 0 0
$$298$$ −14.6774 −0.850237
$$299$$ −5.60414 −0.324096
$$300$$ −2.64424 −0.152665
$$301$$ 2.84053 0.163726
$$302$$ −11.0935 −0.638358
$$303$$ −14.6481 −0.841509
$$304$$ −2.77079 −0.158916
$$305$$ −2.53653 −0.145241
$$306$$ −0.272888 −0.0156000
$$307$$ −5.38032 −0.307071 −0.153536 0.988143i $$-0.549066\pi$$
−0.153536 + 0.988143i $$0.549066\pi$$
$$308$$ 0 0
$$309$$ 9.56413 0.544084
$$310$$ −4.12571 −0.234325
$$311$$ 16.9028 0.958470 0.479235 0.877687i $$-0.340914\pi$$
0.479235 + 0.877687i $$0.340914\pi$$
$$312$$ −8.09234 −0.458139
$$313$$ −1.35523 −0.0766020 −0.0383010 0.999266i $$-0.512195\pi$$
−0.0383010 + 0.999266i $$0.512195\pi$$
$$314$$ −13.1941 −0.744587
$$315$$ −3.99201 −0.224924
$$316$$ 12.4142 0.698354
$$317$$ 29.3217 1.64687 0.823434 0.567412i $$-0.192055\pi$$
0.823434 + 0.567412i $$0.192055\pi$$
$$318$$ 0.326905 0.0183319
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 17.4039 0.971391
$$322$$ 1.83120 0.102049
$$323$$ −0.189408 −0.0105389
$$324$$ −5.03990 −0.279995
$$325$$ −3.06037 −0.169759
$$326$$ 3.70882 0.205412
$$327$$ 3.04051 0.168140
$$328$$ −5.13257 −0.283399
$$329$$ −11.9195 −0.657145
$$330$$ 0 0
$$331$$ 7.10546 0.390551 0.195276 0.980748i $$-0.437440\pi$$
0.195276 + 0.980748i $$0.437440\pi$$
$$332$$ 3.05366 0.167591
$$333$$ −22.4929 −1.23260
$$334$$ −8.75429 −0.479014
$$335$$ −2.20818 −0.120646
$$336$$ 2.64424 0.144255
$$337$$ −28.7650 −1.56693 −0.783463 0.621438i $$-0.786549\pi$$
−0.783463 + 0.621438i $$0.786549\pi$$
$$338$$ 3.63416 0.197672
$$339$$ 32.1371 1.74545
$$340$$ 0.0683587 0.00370727
$$341$$ 0 0
$$342$$ 11.0610 0.598112
$$343$$ −1.00000 −0.0539949
$$344$$ 2.84053 0.153151
$$345$$ −4.84213 −0.260692
$$346$$ −3.03876 −0.163365
$$347$$ 25.7893 1.38444 0.692222 0.721685i $$-0.256632\pi$$
0.692222 + 0.721685i $$0.256632\pi$$
$$348$$ 8.75076 0.469090
$$349$$ −36.6973 −1.96436 −0.982180 0.187942i $$-0.939818\pi$$
−0.982180 + 0.187942i $$0.939818\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 8.02766 0.428485
$$352$$ 0 0
$$353$$ 16.9704 0.903241 0.451620 0.892210i $$-0.350846\pi$$
0.451620 + 0.892210i $$0.350846\pi$$
$$354$$ −29.5968 −1.57305
$$355$$ 13.7247 0.728434
$$356$$ 13.5707 0.719246
$$357$$ 0.180757 0.00956667
$$358$$ 7.29987 0.385810
$$359$$ −21.1819 −1.11794 −0.558969 0.829189i $$-0.688803\pi$$
−0.558969 + 0.829189i $$0.688803\pi$$
$$360$$ −3.99201 −0.210397
$$361$$ −11.3227 −0.595932
$$362$$ −10.9351 −0.574736
$$363$$ 0 0
$$364$$ 3.06037 0.160407
$$365$$ −6.98905 −0.365824
$$366$$ −6.70720 −0.350591
$$367$$ 36.4966 1.90511 0.952554 0.304371i $$-0.0984461\pi$$
0.952554 + 0.304371i $$0.0984461\pi$$
$$368$$ 1.83120 0.0954579
$$369$$ 20.4893 1.06663
$$370$$ 5.63448 0.292923
$$371$$ −0.123629 −0.00641850
$$372$$ −10.9094 −0.565625
$$373$$ 6.23055 0.322606 0.161303 0.986905i $$-0.448430\pi$$
0.161303 + 0.986905i $$0.448430\pi$$
$$374$$ 0 0
$$375$$ −2.64424 −0.136548
$$376$$ −11.9195 −0.614703
$$377$$ 10.1279 0.521612
$$378$$ −2.62310 −0.134918
$$379$$ 0.463665 0.0238169 0.0119084 0.999929i $$-0.496209\pi$$
0.0119084 + 0.999929i $$0.496209\pi$$
$$380$$ −2.77079 −0.142139
$$381$$ 36.4417 1.86696
$$382$$ −22.6520 −1.15898
$$383$$ 30.4947 1.55821 0.779103 0.626896i $$-0.215675\pi$$
0.779103 + 0.626896i $$0.215675\pi$$
$$384$$ 2.64424 0.134938
$$385$$ 0 0
$$386$$ −16.9764 −0.864076
$$387$$ −11.3394 −0.576416
$$388$$ −2.74477 −0.139345
$$389$$ −13.9789 −0.708758 −0.354379 0.935102i $$-0.615308\pi$$
−0.354379 + 0.935102i $$0.615308\pi$$
$$390$$ −8.09234 −0.409772
$$391$$ 0.125178 0.00633054
$$392$$ −1.00000 −0.0505076
$$393$$ 12.9346 0.652463
$$394$$ 26.3241 1.32619
$$395$$ 12.4142 0.624627
$$396$$ 0 0
$$397$$ −11.9510 −0.599801 −0.299901 0.953970i $$-0.596953\pi$$
−0.299901 + 0.953970i $$0.596953\pi$$
$$398$$ −16.9127 −0.847759
$$399$$ −7.32664 −0.366791
$$400$$ 1.00000 0.0500000
$$401$$ −14.4578 −0.721990 −0.360995 0.932568i $$-0.617563\pi$$
−0.360995 + 0.932568i $$0.617563\pi$$
$$402$$ −5.83897 −0.291221
$$403$$ −12.6262 −0.628955
$$404$$ 5.53961 0.275606
$$405$$ −5.03990 −0.250435
$$406$$ −3.30937 −0.164241
$$407$$ 0 0
$$408$$ 0.180757 0.00894880
$$409$$ 3.54419 0.175249 0.0876246 0.996154i $$-0.472072\pi$$
0.0876246 + 0.996154i $$0.472072\pi$$
$$410$$ −5.13257 −0.253480
$$411$$ 50.3882 2.48547
$$412$$ −3.61697 −0.178195
$$413$$ 11.1929 0.550768
$$414$$ −7.31016 −0.359275
$$415$$ 3.05366 0.149898
$$416$$ 3.06037 0.150047
$$417$$ 20.2369 0.991003
$$418$$ 0 0
$$419$$ −35.5085 −1.73471 −0.867353 0.497694i $$-0.834180\pi$$
−0.867353 + 0.497694i $$0.834180\pi$$
$$420$$ 2.64424 0.129026
$$421$$ −26.6190 −1.29733 −0.648665 0.761074i $$-0.724672\pi$$
−0.648665 + 0.761074i $$0.724672\pi$$
$$422$$ −16.4483 −0.800690
$$423$$ 47.5829 2.31356
$$424$$ −0.123629 −0.00600396
$$425$$ 0.0683587 0.00331588
$$426$$ 36.2915 1.75833
$$427$$ 2.53653 0.122751
$$428$$ −6.58182 −0.318144
$$429$$ 0 0
$$430$$ 2.84053 0.136983
$$431$$ −26.0439 −1.25449 −0.627244 0.778823i $$-0.715817\pi$$
−0.627244 + 0.778823i $$0.715817\pi$$
$$432$$ −2.62310 −0.126204
$$433$$ 1.69218 0.0813208 0.0406604 0.999173i $$-0.487054\pi$$
0.0406604 + 0.999173i $$0.487054\pi$$
$$434$$ 4.12571 0.198040
$$435$$ 8.75076 0.419567
$$436$$ −1.14986 −0.0550683
$$437$$ −5.07387 −0.242716
$$438$$ −18.4807 −0.883044
$$439$$ −12.8065 −0.611223 −0.305611 0.952156i $$-0.598861\pi$$
−0.305611 + 0.952156i $$0.598861\pi$$
$$440$$ 0 0
$$441$$ 3.99201 0.190096
$$442$$ 0.209203 0.00995075
$$443$$ 0.704398 0.0334670 0.0167335 0.999860i $$-0.494673\pi$$
0.0167335 + 0.999860i $$0.494673\pi$$
$$444$$ 14.8989 0.707071
$$445$$ 13.5707 0.643313
$$446$$ −20.4675 −0.969163
$$447$$ −38.8105 −1.83567
$$448$$ −1.00000 −0.0472456
$$449$$ −18.4066 −0.868660 −0.434330 0.900754i $$-0.643015\pi$$
−0.434330 + 0.900754i $$0.643015\pi$$
$$450$$ −3.99201 −0.188185
$$451$$ 0 0
$$452$$ −12.1536 −0.571658
$$453$$ −29.3338 −1.37822
$$454$$ 23.7041 1.11249
$$455$$ 3.06037 0.143472
$$456$$ −7.32664 −0.343101
$$457$$ −16.9955 −0.795016 −0.397508 0.917599i $$-0.630125\pi$$
−0.397508 + 0.917599i $$0.630125\pi$$
$$458$$ 26.6368 1.24466
$$459$$ −0.179312 −0.00836957
$$460$$ 1.83120 0.0853801
$$461$$ −13.3276 −0.620729 −0.310364 0.950618i $$-0.600451\pi$$
−0.310364 + 0.950618i $$0.600451\pi$$
$$462$$ 0 0
$$463$$ −34.9814 −1.62572 −0.812861 0.582457i $$-0.802091\pi$$
−0.812861 + 0.582457i $$0.802091\pi$$
$$464$$ −3.30937 −0.153634
$$465$$ −10.9094 −0.505910
$$466$$ 30.3368 1.40532
$$467$$ −3.57861 −0.165599 −0.0827993 0.996566i $$-0.526386\pi$$
−0.0827993 + 0.996566i $$0.526386\pi$$
$$468$$ −12.2170 −0.564731
$$469$$ 2.20818 0.101964
$$470$$ −11.9195 −0.549807
$$471$$ −34.8884 −1.60757
$$472$$ 11.1929 0.515197
$$473$$ 0 0
$$474$$ 32.8262 1.50776
$$475$$ −2.77079 −0.127133
$$476$$ −0.0683587 −0.00313322
$$477$$ 0.493528 0.0225971
$$478$$ −9.95180 −0.455185
$$479$$ −18.3300 −0.837518 −0.418759 0.908097i $$-0.637535\pi$$
−0.418759 + 0.908097i $$0.637535\pi$$
$$480$$ 2.64424 0.120693
$$481$$ 17.2436 0.786239
$$482$$ −30.1849 −1.37489
$$483$$ 4.84213 0.220325
$$484$$ 0 0
$$485$$ −2.74477 −0.124634
$$486$$ −21.1960 −0.961471
$$487$$ 0.487041 0.0220699 0.0110350 0.999939i $$-0.496487\pi$$
0.0110350 + 0.999939i $$0.496487\pi$$
$$488$$ 2.53653 0.114823
$$489$$ 9.80700 0.443488
$$490$$ −1.00000 −0.0451754
$$491$$ −6.09167 −0.274913 −0.137457 0.990508i $$-0.543893\pi$$
−0.137457 + 0.990508i $$0.543893\pi$$
$$492$$ −13.5718 −0.611862
$$493$$ −0.226224 −0.0101886
$$494$$ −8.47964 −0.381517
$$495$$ 0 0
$$496$$ 4.12571 0.185250
$$497$$ −13.7247 −0.615639
$$498$$ 8.07461 0.361832
$$499$$ 42.3141 1.89424 0.947120 0.320879i $$-0.103978\pi$$
0.947120 + 0.320879i $$0.103978\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −23.1485 −1.03420
$$502$$ 1.16998 0.0522188
$$503$$ −32.3677 −1.44320 −0.721602 0.692308i $$-0.756594\pi$$
−0.721602 + 0.692308i $$0.756594\pi$$
$$504$$ 3.99201 0.177818
$$505$$ 5.53961 0.246510
$$506$$ 0 0
$$507$$ 9.60960 0.426777
$$508$$ −13.7815 −0.611457
$$509$$ 23.7319 1.05190 0.525950 0.850516i $$-0.323710\pi$$
0.525950 + 0.850516i $$0.323710\pi$$
$$510$$ 0.180757 0.00800405
$$511$$ 6.98905 0.309178
$$512$$ −1.00000 −0.0441942
$$513$$ 7.26808 0.320893
$$514$$ 0.980916 0.0432664
$$515$$ −3.61697 −0.159383
$$516$$ 7.51106 0.330656
$$517$$ 0 0
$$518$$ −5.63448 −0.247565
$$519$$ −8.03522 −0.352707
$$520$$ 3.06037 0.134206
$$521$$ −30.8219 −1.35033 −0.675167 0.737665i $$-0.735928\pi$$
−0.675167 + 0.737665i $$0.735928\pi$$
$$522$$ 13.2110 0.578230
$$523$$ −37.6079 −1.64448 −0.822240 0.569141i $$-0.807276\pi$$
−0.822240 + 0.569141i $$0.807276\pi$$
$$524$$ −4.89161 −0.213691
$$525$$ 2.64424 0.115404
$$526$$ −15.9923 −0.697296
$$527$$ 0.282028 0.0122853
$$528$$ 0 0
$$529$$ −19.6467 −0.854205
$$530$$ −0.123629 −0.00537011
$$531$$ −44.6823 −1.93905
$$532$$ 2.77079 0.120129
$$533$$ −15.7075 −0.680369
$$534$$ 35.8842 1.55286
$$535$$ −6.58182 −0.284557
$$536$$ 2.20818 0.0953790
$$537$$ 19.3026 0.832969
$$538$$ −15.1048 −0.651212
$$539$$ 0 0
$$540$$ −2.62310 −0.112880
$$541$$ 7.18833 0.309050 0.154525 0.987989i $$-0.450615\pi$$
0.154525 + 0.987989i $$0.450615\pi$$
$$542$$ −0.499548 −0.0214574
$$543$$ −28.9150 −1.24086
$$544$$ −0.0683587 −0.00293086
$$545$$ −1.14986 −0.0492546
$$546$$ 8.09234 0.346320
$$547$$ −17.0757 −0.730103 −0.365052 0.930987i $$-0.618949\pi$$
−0.365052 + 0.930987i $$0.618949\pi$$
$$548$$ −19.0559 −0.814026
$$549$$ −10.1259 −0.432161
$$550$$ 0 0
$$551$$ 9.16957 0.390637
$$552$$ 4.84213 0.206095
$$553$$ −12.4142 −0.527906
$$554$$ −11.1061 −0.471854
$$555$$ 14.8989 0.632424
$$556$$ −7.65318 −0.324567
$$557$$ −35.7390 −1.51431 −0.757156 0.653234i $$-0.773412\pi$$
−0.757156 + 0.653234i $$0.773412\pi$$
$$558$$ −16.4699 −0.697225
$$559$$ 8.69307 0.367678
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ 15.4512 0.651770
$$563$$ 18.6595 0.786404 0.393202 0.919452i $$-0.371367\pi$$
0.393202 + 0.919452i $$0.371367\pi$$
$$564$$ −31.5181 −1.32715
$$565$$ −12.1536 −0.511307
$$566$$ 14.3139 0.601660
$$567$$ 5.03990 0.211656
$$568$$ −13.7247 −0.575877
$$569$$ −39.7394 −1.66596 −0.832981 0.553301i $$-0.813368\pi$$
−0.832981 + 0.553301i $$0.813368\pi$$
$$570$$ −7.32664 −0.306879
$$571$$ −13.3763 −0.559779 −0.279890 0.960032i $$-0.590298\pi$$
−0.279890 + 0.960032i $$0.590298\pi$$
$$572$$ 0 0
$$573$$ −59.8972 −2.50224
$$574$$ 5.13257 0.214229
$$575$$ 1.83120 0.0763663
$$576$$ 3.99201 0.166334
$$577$$ 19.8754 0.827422 0.413711 0.910408i $$-0.364232\pi$$
0.413711 + 0.910408i $$0.364232\pi$$
$$578$$ 16.9953 0.706912
$$579$$ −44.8897 −1.86555
$$580$$ −3.30937 −0.137414
$$581$$ −3.05366 −0.126687
$$582$$ −7.25783 −0.300847
$$583$$ 0 0
$$584$$ 6.98905 0.289209
$$585$$ −12.2170 −0.505111
$$586$$ −17.5749 −0.726010
$$587$$ 5.68609 0.234690 0.117345 0.993091i $$-0.462562\pi$$
0.117345 + 0.993091i $$0.462562\pi$$
$$588$$ −2.64424 −0.109047
$$589$$ −11.4315 −0.471026
$$590$$ 11.1929 0.460806
$$591$$ 69.6072 2.86326
$$592$$ −5.63448 −0.231576
$$593$$ −28.1824 −1.15731 −0.578656 0.815571i $$-0.696423\pi$$
−0.578656 + 0.815571i $$0.696423\pi$$
$$594$$ 0 0
$$595$$ −0.0683587 −0.00280243
$$596$$ 14.6774 0.601208
$$597$$ −44.7214 −1.83032
$$598$$ 5.60414 0.229170
$$599$$ 36.2189 1.47987 0.739933 0.672681i $$-0.234857\pi$$
0.739933 + 0.672681i $$0.234857\pi$$
$$600$$ 2.64424 0.107951
$$601$$ 14.3423 0.585034 0.292517 0.956260i $$-0.405507\pi$$
0.292517 + 0.956260i $$0.405507\pi$$
$$602$$ −2.84053 −0.115772
$$603$$ −8.81509 −0.358978
$$604$$ 11.0935 0.451387
$$605$$ 0 0
$$606$$ 14.6481 0.595037
$$607$$ 3.54637 0.143943 0.0719714 0.997407i $$-0.477071\pi$$
0.0719714 + 0.997407i $$0.477071\pi$$
$$608$$ 2.77079 0.112370
$$609$$ −8.75076 −0.354599
$$610$$ 2.53653 0.102701
$$611$$ −36.4781 −1.47575
$$612$$ 0.272888 0.0110309
$$613$$ 20.4324 0.825259 0.412629 0.910899i $$-0.364610\pi$$
0.412629 + 0.910899i $$0.364610\pi$$
$$614$$ 5.38032 0.217132
$$615$$ −13.5718 −0.547266
$$616$$ 0 0
$$617$$ 19.3755 0.780028 0.390014 0.920809i $$-0.372470\pi$$
0.390014 + 0.920809i $$0.372470\pi$$
$$618$$ −9.56413 −0.384726
$$619$$ −6.00523 −0.241371 −0.120685 0.992691i $$-0.538509\pi$$
−0.120685 + 0.992691i $$0.538509\pi$$
$$620$$ 4.12571 0.165693
$$621$$ −4.80343 −0.192755
$$622$$ −16.9028 −0.677741
$$623$$ −13.5707 −0.543699
$$624$$ 8.09234 0.323953
$$625$$ 1.00000 0.0400000
$$626$$ 1.35523 0.0541658
$$627$$ 0 0
$$628$$ 13.1941 0.526502
$$629$$ −0.385166 −0.0153576
$$630$$ 3.99201 0.159045
$$631$$ 21.7929 0.867562 0.433781 0.901018i $$-0.357179\pi$$
0.433781 + 0.901018i $$0.357179\pi$$
$$632$$ −12.4142 −0.493811
$$633$$ −43.4932 −1.72870
$$634$$ −29.3217 −1.16451
$$635$$ −13.7815 −0.546903
$$636$$ −0.326905 −0.0129626
$$637$$ −3.06037 −0.121256
$$638$$ 0 0
$$639$$ 54.7892 2.16743
$$640$$ −1.00000 −0.0395285
$$641$$ −41.0344 −1.62076 −0.810380 0.585904i $$-0.800739\pi$$
−0.810380 + 0.585904i $$0.800739\pi$$
$$642$$ −17.4039 −0.686877
$$643$$ −24.7624 −0.976534 −0.488267 0.872694i $$-0.662371\pi$$
−0.488267 + 0.872694i $$0.662371\pi$$
$$644$$ −1.83120 −0.0721594
$$645$$ 7.51106 0.295748
$$646$$ 0.189408 0.00745215
$$647$$ 44.1102 1.73415 0.867076 0.498175i $$-0.165996\pi$$
0.867076 + 0.498175i $$0.165996\pi$$
$$648$$ 5.03990 0.197986
$$649$$ 0 0
$$650$$ 3.06037 0.120037
$$651$$ 10.9094 0.427572
$$652$$ −3.70882 −0.145248
$$653$$ −22.0286 −0.862046 −0.431023 0.902341i $$-0.641847\pi$$
−0.431023 + 0.902341i $$0.641847\pi$$
$$654$$ −3.04051 −0.118893
$$655$$ −4.89161 −0.191131
$$656$$ 5.13257 0.200393
$$657$$ −27.9003 −1.08850
$$658$$ 11.9195 0.464672
$$659$$ −19.5066 −0.759868 −0.379934 0.925014i $$-0.624053\pi$$
−0.379934 + 0.925014i $$0.624053\pi$$
$$660$$ 0 0
$$661$$ 8.75637 0.340583 0.170292 0.985394i $$-0.445529\pi$$
0.170292 + 0.985394i $$0.445529\pi$$
$$662$$ −7.10546 −0.276161
$$663$$ 0.553182 0.0214838
$$664$$ −3.05366 −0.118505
$$665$$ 2.77079 0.107447
$$666$$ 22.4929 0.871582
$$667$$ −6.06011 −0.234648
$$668$$ 8.75429 0.338714
$$669$$ −54.1209 −2.09244
$$670$$ 2.20818 0.0853096
$$671$$ 0 0
$$672$$ −2.64424 −0.102004
$$673$$ 2.41083 0.0929306 0.0464653 0.998920i $$-0.485204\pi$$
0.0464653 + 0.998920i $$0.485204\pi$$
$$674$$ 28.7650 1.10798
$$675$$ −2.62310 −0.100963
$$676$$ −3.63416 −0.139776
$$677$$ −2.32293 −0.0892776 −0.0446388 0.999003i $$-0.514214\pi$$
−0.0446388 + 0.999003i $$0.514214\pi$$
$$678$$ −32.1371 −1.23422
$$679$$ 2.74477 0.105335
$$680$$ −0.0683587 −0.00262144
$$681$$ 62.6792 2.40187
$$682$$ 0 0
$$683$$ −4.35856 −0.166776 −0.0833879 0.996517i $$-0.526574\pi$$
−0.0833879 + 0.996517i $$0.526574\pi$$
$$684$$ −11.0610 −0.422929
$$685$$ −19.0559 −0.728087
$$686$$ 1.00000 0.0381802
$$687$$ 70.4342 2.68723
$$688$$ −2.84053 −0.108294
$$689$$ −0.378350 −0.0144140
$$690$$ 4.84213 0.184337
$$691$$ −20.4837 −0.779236 −0.389618 0.920977i $$-0.627393\pi$$
−0.389618 + 0.920977i $$0.627393\pi$$
$$692$$ 3.03876 0.115516
$$693$$ 0 0
$$694$$ −25.7893 −0.978950
$$695$$ −7.65318 −0.290302
$$696$$ −8.75076 −0.331697
$$697$$ 0.350856 0.0132896
$$698$$ 36.6973 1.38901
$$699$$ 80.2177 3.03411
$$700$$ −1.00000 −0.0377964
$$701$$ 25.5850 0.966333 0.483166 0.875529i $$-0.339487\pi$$
0.483166 + 0.875529i $$0.339487\pi$$
$$702$$ −8.02766 −0.302984
$$703$$ 15.6120 0.588817
$$704$$ 0 0
$$705$$ −31.5181 −1.18704
$$706$$ −16.9704 −0.638688
$$707$$ −5.53961 −0.208339
$$708$$ 29.5968 1.11232
$$709$$ −20.1461 −0.756604 −0.378302 0.925682i $$-0.623492\pi$$
−0.378302 + 0.925682i $$0.623492\pi$$
$$710$$ −13.7247 −0.515080
$$711$$ 49.5576 1.85856
$$712$$ −13.5707 −0.508584
$$713$$ 7.55500 0.282937
$$714$$ −0.180757 −0.00676465
$$715$$ 0 0
$$716$$ −7.29987 −0.272809
$$717$$ −26.3150 −0.982750
$$718$$ 21.1819 0.790501
$$719$$ −41.3909 −1.54362 −0.771811 0.635852i $$-0.780649\pi$$
−0.771811 + 0.635852i $$0.780649\pi$$
$$720$$ 3.99201 0.148773
$$721$$ 3.61697 0.134703
$$722$$ 11.3227 0.421387
$$723$$ −79.8162 −2.96840
$$724$$ 10.9351 0.406400
$$725$$ −3.30937 −0.122907
$$726$$ 0 0
$$727$$ −18.3813 −0.681723 −0.340862 0.940113i $$-0.610719\pi$$
−0.340862 + 0.940113i $$0.610719\pi$$
$$728$$ −3.06037 −0.113425
$$729$$ −40.9277 −1.51584
$$730$$ 6.98905 0.258676
$$731$$ −0.194175 −0.00718183
$$732$$ 6.70720 0.247905
$$733$$ −47.3817 −1.75008 −0.875042 0.484048i $$-0.839166\pi$$
−0.875042 + 0.484048i $$0.839166\pi$$
$$734$$ −36.4966 −1.34711
$$735$$ −2.64424 −0.0975343
$$736$$ −1.83120 −0.0674989
$$737$$ 0 0
$$738$$ −20.4893 −0.754220
$$739$$ 36.4944 1.34247 0.671234 0.741246i $$-0.265765\pi$$
0.671234 + 0.741246i $$0.265765\pi$$
$$740$$ −5.63448 −0.207128
$$741$$ −22.4222 −0.823700
$$742$$ 0.123629 0.00453857
$$743$$ −19.4544 −0.713714 −0.356857 0.934159i $$-0.616152\pi$$
−0.356857 + 0.934159i $$0.616152\pi$$
$$744$$ 10.9094 0.399957
$$745$$ 14.6774 0.537737
$$746$$ −6.23055 −0.228117
$$747$$ 12.1902 0.446017
$$748$$ 0 0
$$749$$ 6.58182 0.240494
$$750$$ 2.64424 0.0965540
$$751$$ −34.0903 −1.24397 −0.621986 0.783029i $$-0.713674\pi$$
−0.621986 + 0.783029i $$0.713674\pi$$
$$752$$ 11.9195 0.434661
$$753$$ 3.09371 0.112741
$$754$$ −10.1279 −0.368835
$$755$$ 11.0935 0.403733
$$756$$ 2.62310 0.0954014
$$757$$ −8.47015 −0.307853 −0.153926 0.988082i $$-0.549192\pi$$
−0.153926 + 0.988082i $$0.549192\pi$$
$$758$$ −0.463665 −0.0168411
$$759$$ 0 0
$$760$$ 2.77079 0.100507
$$761$$ −14.8964 −0.539993 −0.269997 0.962861i $$-0.587023\pi$$
−0.269997 + 0.962861i $$0.587023\pi$$
$$762$$ −36.4417 −1.32014
$$763$$ 1.14986 0.0416277
$$764$$ 22.6520 0.819519
$$765$$ 0.272888 0.00986630
$$766$$ −30.4947 −1.10182
$$767$$ 34.2545 1.23686
$$768$$ −2.64424 −0.0954158
$$769$$ −26.5771 −0.958396 −0.479198 0.877707i $$-0.659072\pi$$
−0.479198 + 0.877707i $$0.659072\pi$$
$$770$$ 0 0
$$771$$ 2.59378 0.0934126
$$772$$ 16.9764 0.610994
$$773$$ 23.5243 0.846112 0.423056 0.906104i $$-0.360957\pi$$
0.423056 + 0.906104i $$0.360957\pi$$
$$774$$ 11.3394 0.407587
$$775$$ 4.12571 0.148200
$$776$$ 2.74477 0.0985315
$$777$$ −14.8989 −0.534496
$$778$$ 13.9789 0.501167
$$779$$ −14.2213 −0.509531
$$780$$ 8.09234 0.289752
$$781$$ 0 0
$$782$$ −0.125178 −0.00447637
$$783$$ 8.68081 0.310227
$$784$$ 1.00000 0.0357143
$$785$$ 13.1941 0.470918
$$786$$ −12.9346 −0.461361
$$787$$ −9.85816 −0.351405 −0.175703 0.984443i $$-0.556220\pi$$
−0.175703 + 0.984443i $$0.556220\pi$$
$$788$$ −26.3241 −0.937757
$$789$$ −42.2874 −1.50547
$$790$$ −12.4142 −0.441678
$$791$$ 12.1536 0.432133
$$792$$ 0 0
$$793$$ 7.76271 0.275662
$$794$$ 11.9510 0.424123
$$795$$ −0.326905 −0.0115941
$$796$$ 16.9127 0.599456
$$797$$ −38.6360 −1.36856 −0.684279 0.729220i $$-0.739883\pi$$
−0.684279 + 0.729220i $$0.739883\pi$$
$$798$$ 7.32664 0.259360
$$799$$ 0.814804 0.0288257
$$800$$ −1.00000 −0.0353553
$$801$$ 54.1743 1.91416
$$802$$ 14.4578 0.510524
$$803$$ 0 0
$$804$$ 5.83897 0.205925
$$805$$ −1.83120 −0.0645413
$$806$$ 12.6262 0.444738
$$807$$ −39.9406 −1.40598
$$808$$ −5.53961 −0.194883
$$809$$ −30.9594 −1.08847 −0.544237 0.838932i $$-0.683181\pi$$
−0.544237 + 0.838932i $$0.683181\pi$$
$$810$$ 5.03990 0.177084
$$811$$ 49.4329 1.73582 0.867912 0.496717i $$-0.165461\pi$$
0.867912 + 0.496717i $$0.165461\pi$$
$$812$$ 3.30937 0.116136
$$813$$ −1.32092 −0.0463268
$$814$$ 0 0
$$815$$ −3.70882 −0.129914
$$816$$ −0.180757 −0.00632776
$$817$$ 7.87053 0.275355
$$818$$ −3.54419 −0.123920
$$819$$ 12.2170 0.426896
$$820$$ 5.13257 0.179237
$$821$$ −23.1278 −0.807167 −0.403584 0.914943i $$-0.632236\pi$$
−0.403584 + 0.914943i $$0.632236\pi$$
$$822$$ −50.3882 −1.75749
$$823$$ −36.0554 −1.25681 −0.628406 0.777885i $$-0.716292\pi$$
−0.628406 + 0.777885i $$0.716292\pi$$
$$824$$ 3.61697 0.126003
$$825$$ 0 0
$$826$$ −11.1929 −0.389452
$$827$$ 10.3899 0.361293 0.180647 0.983548i $$-0.442181\pi$$
0.180647 + 0.983548i $$0.442181\pi$$
$$828$$ 7.31016 0.254046
$$829$$ 37.4379 1.30027 0.650136 0.759818i $$-0.274712\pi$$
0.650136 + 0.759818i $$0.274712\pi$$
$$830$$ −3.05366 −0.105994
$$831$$ −29.3672 −1.01874
$$832$$ −3.06037 −0.106099
$$833$$ 0.0683587 0.00236849
$$834$$ −20.2369 −0.700745
$$835$$ 8.75429 0.302955
$$836$$ 0 0
$$837$$ −10.8222 −0.374069
$$838$$ 35.5085 1.22662
$$839$$ −32.6640 −1.12769 −0.563843 0.825882i $$-0.690678\pi$$
−0.563843 + 0.825882i $$0.690678\pi$$
$$840$$ −2.64424 −0.0912350
$$841$$ −18.0481 −0.622348
$$842$$ 26.6190 0.917351
$$843$$ 40.8567 1.40718
$$844$$ 16.4483 0.566173
$$845$$ −3.63416 −0.125019
$$846$$ −47.5829 −1.63593
$$847$$ 0 0
$$848$$ 0.123629 0.00424544
$$849$$ 37.8495 1.29899
$$850$$ −0.0683587 −0.00234468
$$851$$ −10.3179 −0.353692
$$852$$ −36.2915 −1.24333
$$853$$ 21.4274 0.733659 0.366830 0.930288i $$-0.380443\pi$$
0.366830 + 0.930288i $$0.380443\pi$$
$$854$$ −2.53653 −0.0867983
$$855$$ −11.0610 −0.378279
$$856$$ 6.58182 0.224962
$$857$$ 46.6805 1.59457 0.797287 0.603600i $$-0.206268\pi$$
0.797287 + 0.603600i $$0.206268\pi$$
$$858$$ 0 0
$$859$$ 34.0149 1.16057 0.580287 0.814412i $$-0.302940\pi$$
0.580287 + 0.814412i $$0.302940\pi$$
$$860$$ −2.84053 −0.0968614
$$861$$ 13.5718 0.462524
$$862$$ 26.0439 0.887057
$$863$$ −5.55865 −0.189219 −0.0946094 0.995514i $$-0.530160\pi$$
−0.0946094 + 0.995514i $$0.530160\pi$$
$$864$$ 2.62310 0.0892398
$$865$$ 3.03876 0.103321
$$866$$ −1.69218 −0.0575025
$$867$$ 44.9397 1.52623
$$868$$ −4.12571 −0.140036
$$869$$ 0 0
$$870$$ −8.75076 −0.296679
$$871$$ 6.75785 0.228981
$$872$$ 1.14986 0.0389392
$$873$$ −10.9571 −0.370843
$$874$$ 5.07387 0.171626
$$875$$ −1.00000 −0.0338062
$$876$$ 18.4807 0.624406
$$877$$ 18.3183 0.618564 0.309282 0.950970i $$-0.399911\pi$$
0.309282 + 0.950970i $$0.399911\pi$$
$$878$$ 12.8065 0.432200
$$879$$ −46.4721 −1.56747
$$880$$ 0 0
$$881$$ −0.0269589 −0.000908270 0 −0.000454135 1.00000i $$-0.500145\pi$$
−0.000454135 1.00000i $$0.500145\pi$$
$$882$$ −3.99201 −0.134418
$$883$$ −5.28336 −0.177799 −0.0888997 0.996041i $$-0.528335\pi$$
−0.0888997 + 0.996041i $$0.528335\pi$$
$$884$$ −0.209203 −0.00703625
$$885$$ 29.5968 0.994886
$$886$$ −0.704398 −0.0236647
$$887$$ −39.7555 −1.33486 −0.667430 0.744672i $$-0.732606\pi$$
−0.667430 + 0.744672i $$0.732606\pi$$
$$888$$ −14.8989 −0.499975
$$889$$ 13.7815 0.462218
$$890$$ −13.5707 −0.454891
$$891$$ 0 0
$$892$$ 20.4675 0.685302
$$893$$ −33.0266 −1.10519
$$894$$ 38.8105 1.29802
$$895$$ −7.29987 −0.244008
$$896$$ 1.00000 0.0334077
$$897$$ 14.8187 0.494782
$$898$$ 18.4066 0.614235
$$899$$ −13.6535 −0.455369
$$900$$ 3.99201 0.133067
$$901$$ 0.00845113 0.000281548 0
$$902$$ 0 0
$$903$$ −7.51106 −0.249952
$$904$$ 12.1536 0.404223
$$905$$ 10.9351 0.363495
$$906$$ 29.3338 0.974551
$$907$$ −5.61591 −0.186473 −0.0932367 0.995644i $$-0.529721\pi$$
−0.0932367 + 0.995644i $$0.529721\pi$$
$$908$$ −23.7041 −0.786647
$$909$$ 22.1142 0.733481
$$910$$ −3.06037 −0.101450
$$911$$ −28.1487 −0.932609 −0.466304 0.884624i $$-0.654415\pi$$
−0.466304 + 0.884624i $$0.654415\pi$$
$$912$$ 7.32664 0.242609
$$913$$ 0 0
$$914$$ 16.9955 0.562161
$$915$$ 6.70720 0.221733
$$916$$ −26.6368 −0.880106
$$917$$ 4.89161 0.161535
$$918$$ 0.179312 0.00591818
$$919$$ −47.6030 −1.57028 −0.785140 0.619319i $$-0.787409\pi$$
−0.785140 + 0.619319i $$0.787409\pi$$
$$920$$ −1.83120 −0.0603728
$$921$$ 14.2269 0.468791
$$922$$ 13.3276 0.438921
$$923$$ −42.0027 −1.38254
$$924$$ 0 0
$$925$$ −5.63448 −0.185261
$$926$$ 34.9814 1.14956
$$927$$ −14.4390 −0.474238
$$928$$ 3.30937 0.108635
$$929$$ −38.1848 −1.25280 −0.626401 0.779501i $$-0.715473\pi$$
−0.626401 + 0.779501i $$0.715473\pi$$
$$930$$ 10.9094 0.357732
$$931$$ −2.77079 −0.0908091
$$932$$ −30.3368 −0.993714
$$933$$ −44.6951 −1.46325
$$934$$ 3.57861 0.117096
$$935$$ 0 0
$$936$$ 12.2170 0.399325
$$937$$ 0.397595 0.0129889 0.00649443 0.999979i $$-0.497933\pi$$
0.00649443 + 0.999979i $$0.497933\pi$$
$$938$$ −2.20818 −0.0720998
$$939$$ 3.58355 0.116945
$$940$$ 11.9195 0.388772
$$941$$ 6.77789 0.220953 0.110477 0.993879i $$-0.464762\pi$$
0.110477 + 0.993879i $$0.464762\pi$$
$$942$$ 34.8884 1.13673
$$943$$ 9.39876 0.306066
$$944$$ −11.1929 −0.364299
$$945$$ 2.62310 0.0853296
$$946$$ 0 0
$$947$$ −10.5378 −0.342432 −0.171216 0.985234i $$-0.554770\pi$$
−0.171216 + 0.985234i $$0.554770\pi$$
$$948$$ −32.8262 −1.06614
$$949$$ 21.3891 0.694318
$$950$$ 2.77079 0.0898964
$$951$$ −77.5335 −2.51420
$$952$$ 0.0683587 0.00221552
$$953$$ 9.44810 0.306054 0.153027 0.988222i $$-0.451098\pi$$
0.153027 + 0.988222i $$0.451098\pi$$
$$954$$ −0.493528 −0.0159786
$$955$$ 22.6520 0.733000
$$956$$ 9.95180 0.321864
$$957$$ 0 0
$$958$$ 18.3300 0.592215
$$959$$ 19.0559 0.615346
$$960$$ −2.64424 −0.0853425
$$961$$ −13.9785 −0.450920
$$962$$ −17.2436 −0.555955
$$963$$ −26.2747 −0.846689
$$964$$ 30.1849 0.972191
$$965$$ 16.9764 0.546490
$$966$$ −4.84213 −0.155793
$$967$$ −47.0752 −1.51384 −0.756918 0.653509i $$-0.773296\pi$$
−0.756918 + 0.653509i $$0.773296\pi$$
$$968$$ 0 0
$$969$$ 0.500840 0.0160893
$$970$$ 2.74477 0.0881293
$$971$$ 17.2012 0.552012 0.276006 0.961156i $$-0.410989\pi$$
0.276006 + 0.961156i $$0.410989\pi$$
$$972$$ 21.1960 0.679863
$$973$$ 7.65318 0.245350
$$974$$ −0.487041 −0.0156058
$$975$$ 8.09234 0.259162
$$976$$ −2.53653 −0.0811924
$$977$$ −6.75798 −0.216207 −0.108103 0.994140i $$-0.534478\pi$$
−0.108103 + 0.994140i $$0.534478\pi$$
$$978$$ −9.80700 −0.313593
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ −4.59025 −0.146555
$$982$$ 6.09167 0.194393
$$983$$ −45.4819 −1.45065 −0.725323 0.688408i $$-0.758310\pi$$
−0.725323 + 0.688408i $$0.758310\pi$$
$$984$$ 13.5718 0.432652
$$985$$ −26.3241 −0.838755
$$986$$ 0.226224 0.00720444
$$987$$ 31.5181 1.00323
$$988$$ 8.47964 0.269773
$$989$$ −5.20158 −0.165401
$$990$$ 0 0
$$991$$ −11.7391 −0.372906 −0.186453 0.982464i $$-0.559699\pi$$
−0.186453 + 0.982464i $$0.559699\pi$$
$$992$$ −4.12571 −0.130991
$$993$$ −18.7885 −0.596236
$$994$$ 13.7247 0.435322
$$995$$ 16.9127 0.536170
$$996$$ −8.07461 −0.255854
$$997$$ 17.1623 0.543536 0.271768 0.962363i $$-0.412392\pi$$
0.271768 + 0.962363i $$0.412392\pi$$
$$998$$ −42.3141 −1.33943
$$999$$ 14.7798 0.467613
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.cy.1.2 6
11.7 odd 10 770.2.n.i.71.3 12
11.8 odd 10 770.2.n.i.141.3 yes 12
11.10 odd 2 8470.2.a.de.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.71.3 12 11.7 odd 10
770.2.n.i.141.3 yes 12 11.8 odd 10
8470.2.a.cy.1.2 6 1.1 even 1 trivial
8470.2.a.de.1.2 6 11.10 odd 2