# Properties

 Label 8470.2.a.cl.1.3 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.892.1 Defining polynomial: $$x^{3} - x^{2} - 8 x + 10$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.59774$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} -1.00000 q^{10} +3.34596 q^{12} -6.69193 q^{13} +1.00000 q^{14} -3.34596 q^{15} +1.00000 q^{16} +7.19547 q^{17} +8.19547 q^{18} +1.84951 q^{19} -1.00000 q^{20} +3.34596 q^{21} +1.84951 q^{23} +3.34596 q^{24} +1.00000 q^{25} -6.69193 q^{26} +17.3839 q^{27} +1.00000 q^{28} -6.84242 q^{29} -3.34596 q^{30} +6.00000 q^{31} +1.00000 q^{32} +7.19547 q^{34} -1.00000 q^{35} +8.19547 q^{36} -6.54143 q^{37} +1.84951 q^{38} -22.3909 q^{39} -1.00000 q^{40} +9.34596 q^{41} +3.34596 q^{42} -0.503544 q^{43} -8.19547 q^{45} +1.84951 q^{46} -1.49646 q^{47} +3.34596 q^{48} +1.00000 q^{49} +1.00000 q^{50} +24.0758 q^{51} -6.69193 q^{52} +6.84242 q^{53} +17.3839 q^{54} +1.00000 q^{56} +6.18838 q^{57} -6.84242 q^{58} -7.88740 q^{59} -3.34596 q^{60} -4.50354 q^{61} +6.00000 q^{62} +8.19547 q^{63} +1.00000 q^{64} +6.69193 q^{65} +8.00000 q^{67} +7.19547 q^{68} +6.18838 q^{69} -1.00000 q^{70} +0.300986 q^{71} +8.19547 q^{72} +10.1884 q^{73} -6.54143 q^{74} +3.34596 q^{75} +1.84951 q^{76} -22.3909 q^{78} +12.2404 q^{79} -1.00000 q^{80} +33.5793 q^{81} +9.34596 q^{82} +1.30807 q^{83} +3.34596 q^{84} -7.19547 q^{85} -0.503544 q^{86} -22.8945 q^{87} -8.69193 q^{89} -8.19547 q^{90} -6.69193 q^{91} +1.84951 q^{92} +20.0758 q^{93} -1.49646 q^{94} -1.84951 q^{95} +3.34596 q^{96} +3.84951 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10})$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 12 q^{27} + 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 11 q^{36} + 4 q^{37} + 2 q^{38} - 40 q^{39} - 3 q^{40} + 18 q^{41} - 8 q^{43} - 11 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{49} + 3 q^{50} + 12 q^{51} + 4 q^{53} + 12 q^{54} + 3 q^{56} - 8 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 8 q^{69} - 3 q^{70} + 8 q^{71} + 11 q^{72} + 4 q^{73} + 4 q^{74} + 2 q^{76} - 40 q^{78} + 6 q^{79} - 3 q^{80} + 47 q^{81} + 18 q^{82} + 24 q^{83} - 8 q^{85} - 8 q^{86} - 48 q^{87} - 6 q^{89} - 11 q^{90} + 2 q^{92} + 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98} + O(q^{100})$$

## 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
$$3$$ 3.34596 1.93179 0.965896 0.258929i $$-0.0833695\pi$$
0.965896 + 0.258929i $$0.0833695\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 3.34596 1.36598
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 8.19547 2.73182
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 3.34596 0.965896
$$13$$ −6.69193 −1.85601 −0.928003 0.372572i $$-0.878476\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −3.34596 −0.863924
$$16$$ 1.00000 0.250000
$$17$$ 7.19547 1.74516 0.872579 0.488473i $$-0.162446\pi$$
0.872579 + 0.488473i $$0.162446\pi$$
$$18$$ 8.19547 1.93169
$$19$$ 1.84951 0.424306 0.212153 0.977236i $$-0.431952\pi$$
0.212153 + 0.977236i $$0.431952\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 3.34596 0.730149
$$22$$ 0 0
$$23$$ 1.84951 0.385649 0.192824 0.981233i $$-0.438235\pi$$
0.192824 + 0.981233i $$0.438235\pi$$
$$24$$ 3.34596 0.682992
$$25$$ 1.00000 0.200000
$$26$$ −6.69193 −1.31239
$$27$$ 17.3839 3.34552
$$28$$ 1.00000 0.188982
$$29$$ −6.84242 −1.27061 −0.635303 0.772263i $$-0.719125\pi$$
−0.635303 + 0.772263i $$0.719125\pi$$
$$30$$ −3.34596 −0.610887
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 7.19547 1.23401
$$35$$ −1.00000 −0.169031
$$36$$ 8.19547 1.36591
$$37$$ −6.54143 −1.07541 −0.537703 0.843135i $$-0.680708\pi$$
−0.537703 + 0.843135i $$0.680708\pi$$
$$38$$ 1.84951 0.300030
$$39$$ −22.3909 −3.58542
$$40$$ −1.00000 −0.158114
$$41$$ 9.34596 1.45959 0.729797 0.683664i $$-0.239615\pi$$
0.729797 + 0.683664i $$0.239615\pi$$
$$42$$ 3.34596 0.516293
$$43$$ −0.503544 −0.0767897 −0.0383949 0.999263i $$-0.512224\pi$$
−0.0383949 + 0.999263i $$0.512224\pi$$
$$44$$ 0 0
$$45$$ −8.19547 −1.22171
$$46$$ 1.84951 0.272695
$$47$$ −1.49646 −0.218281 −0.109140 0.994026i $$-0.534810\pi$$
−0.109140 + 0.994026i $$0.534810\pi$$
$$48$$ 3.34596 0.482948
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 24.0758 3.37128
$$52$$ −6.69193 −0.928003
$$53$$ 6.84242 0.939879 0.469939 0.882699i $$-0.344276\pi$$
0.469939 + 0.882699i $$0.344276\pi$$
$$54$$ 17.3839 2.36564
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 6.18838 0.819671
$$58$$ −6.84242 −0.898454
$$59$$ −7.88740 −1.02685 −0.513426 0.858134i $$-0.671624\pi$$
−0.513426 + 0.858134i $$0.671624\pi$$
$$60$$ −3.34596 −0.431962
$$61$$ −4.50354 −0.576620 −0.288310 0.957537i $$-0.593093\pi$$
−0.288310 + 0.957537i $$0.593093\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 8.19547 1.03253
$$64$$ 1.00000 0.125000
$$65$$ 6.69193 0.830031
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 7.19547 0.872579
$$69$$ 6.18838 0.744994
$$70$$ −1.00000 −0.119523
$$71$$ 0.300986 0.0357204 0.0178602 0.999840i $$-0.494315\pi$$
0.0178602 + 0.999840i $$0.494315\pi$$
$$72$$ 8.19547 0.965845
$$73$$ 10.1884 1.19246 0.596230 0.802814i $$-0.296665\pi$$
0.596230 + 0.802814i $$0.296665\pi$$
$$74$$ −6.54143 −0.760426
$$75$$ 3.34596 0.386359
$$76$$ 1.84951 0.212153
$$77$$ 0 0
$$78$$ −22.3909 −2.53527
$$79$$ 12.2404 1.37716 0.688579 0.725161i $$-0.258235\pi$$
0.688579 + 0.725161i $$0.258235\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 33.5793 3.73104
$$82$$ 9.34596 1.03209
$$83$$ 1.30807 0.143580 0.0717899 0.997420i $$-0.477129\pi$$
0.0717899 + 0.997420i $$0.477129\pi$$
$$84$$ 3.34596 0.365075
$$85$$ −7.19547 −0.780458
$$86$$ −0.503544 −0.0542985
$$87$$ −22.8945 −2.45455
$$88$$ 0 0
$$89$$ −8.69193 −0.921342 −0.460671 0.887571i $$-0.652391\pi$$
−0.460671 + 0.887571i $$0.652391\pi$$
$$90$$ −8.19547 −0.863878
$$91$$ −6.69193 −0.701505
$$92$$ 1.84951 0.192824
$$93$$ 20.0758 2.08176
$$94$$ −1.49646 −0.154348
$$95$$ −1.84951 −0.189755
$$96$$ 3.34596 0.341496
$$97$$ 3.84951 0.390858 0.195429 0.980718i $$-0.437390\pi$$
0.195429 + 0.980718i $$0.437390\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −6.89448 −0.686027 −0.343013 0.939331i $$-0.611448\pi$$
−0.343013 + 0.939331i $$0.611448\pi$$
$$102$$ 24.0758 2.38386
$$103$$ −10.5793 −1.04241 −0.521206 0.853431i $$-0.674518\pi$$
−0.521206 + 0.853431i $$0.674518\pi$$
$$104$$ −6.69193 −0.656197
$$105$$ −3.34596 −0.326533
$$106$$ 6.84242 0.664595
$$107$$ 7.49646 0.724710 0.362355 0.932040i $$-0.381973\pi$$
0.362355 + 0.932040i $$0.381973\pi$$
$$108$$ 17.3839 1.67276
$$109$$ −1.45857 −0.139705 −0.0698527 0.997557i $$-0.522253\pi$$
−0.0698527 + 0.997557i $$0.522253\pi$$
$$110$$ 0 0
$$111$$ −21.8874 −2.07746
$$112$$ 1.00000 0.0944911
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 6.18838 0.579595
$$115$$ −1.84951 −0.172467
$$116$$ −6.84242 −0.635303
$$117$$ −54.8435 −5.07028
$$118$$ −7.88740 −0.726094
$$119$$ 7.19547 0.659608
$$120$$ −3.34596 −0.305443
$$121$$ 0 0
$$122$$ −4.50354 −0.407732
$$123$$ 31.2713 2.81963
$$124$$ 6.00000 0.538816
$$125$$ −1.00000 −0.0894427
$$126$$ 8.19547 0.730111
$$127$$ 19.7748 1.75473 0.877365 0.479824i $$-0.159300\pi$$
0.877365 + 0.479824i $$0.159300\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −1.68484 −0.148342
$$130$$ 6.69193 0.586921
$$131$$ −2.15049 −0.187889 −0.0939447 0.995577i $$-0.529948\pi$$
−0.0939447 + 0.995577i $$0.529948\pi$$
$$132$$ 0 0
$$133$$ 1.84951 0.160373
$$134$$ 8.00000 0.691095
$$135$$ −17.3839 −1.49616
$$136$$ 7.19547 0.617006
$$137$$ −8.39094 −0.716886 −0.358443 0.933552i $$-0.616692\pi$$
−0.358443 + 0.933552i $$0.616692\pi$$
$$138$$ 6.18838 0.526790
$$139$$ −17.9253 −1.52040 −0.760201 0.649687i $$-0.774900\pi$$
−0.760201 + 0.649687i $$0.774900\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −5.00709 −0.421673
$$142$$ 0.300986 0.0252582
$$143$$ 0 0
$$144$$ 8.19547 0.682956
$$145$$ 6.84242 0.568232
$$146$$ 10.1884 0.843197
$$147$$ 3.34596 0.275970
$$148$$ −6.54143 −0.537703
$$149$$ 0.451479 0.0369866 0.0184933 0.999829i $$-0.494113\pi$$
0.0184933 + 0.999829i $$0.494113\pi$$
$$150$$ 3.34596 0.273197
$$151$$ −19.2334 −1.56519 −0.782594 0.622532i $$-0.786104\pi$$
−0.782594 + 0.622532i $$0.786104\pi$$
$$152$$ 1.84951 0.150015
$$153$$ 58.9703 4.76746
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ −22.3909 −1.79271
$$157$$ −9.69901 −0.774066 −0.387033 0.922066i $$-0.626500\pi$$
−0.387033 + 0.922066i $$0.626500\pi$$
$$158$$ 12.2404 0.973798
$$159$$ 22.8945 1.81565
$$160$$ −1.00000 −0.0790569
$$161$$ 1.84951 0.145762
$$162$$ 33.5793 2.63824
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ 9.34596 0.729797
$$165$$ 0 0
$$166$$ 1.30807 0.101526
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 3.34596 0.258147
$$169$$ 31.7819 2.44476
$$170$$ −7.19547 −0.551867
$$171$$ 15.1576 1.15913
$$172$$ −0.503544 −0.0383949
$$173$$ 15.4965 1.17817 0.589087 0.808070i $$-0.299488\pi$$
0.589087 + 0.808070i $$0.299488\pi$$
$$174$$ −22.8945 −1.73563
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −26.3909 −1.98366
$$178$$ −8.69193 −0.651487
$$179$$ −13.8874 −1.03799 −0.518996 0.854776i $$-0.673694\pi$$
−0.518996 + 0.854776i $$0.673694\pi$$
$$180$$ −8.19547 −0.610854
$$181$$ 18.7819 1.39605 0.698023 0.716075i $$-0.254063\pi$$
0.698023 + 0.716075i $$0.254063\pi$$
$$182$$ −6.69193 −0.496039
$$183$$ −15.0687 −1.11391
$$184$$ 1.84951 0.136347
$$185$$ 6.54143 0.480936
$$186$$ 20.0758 1.47203
$$187$$ 0 0
$$188$$ −1.49646 −0.109140
$$189$$ 17.3839 1.26449
$$190$$ −1.84951 −0.134177
$$191$$ −6.39094 −0.462432 −0.231216 0.972902i $$-0.574270\pi$$
−0.231216 + 0.972902i $$0.574270\pi$$
$$192$$ 3.34596 0.241474
$$193$$ 13.1955 0.949831 0.474915 0.880031i $$-0.342479\pi$$
0.474915 + 0.880031i $$0.342479\pi$$
$$194$$ 3.84951 0.276379
$$195$$ 22.3909 1.60345
$$196$$ 1.00000 0.0714286
$$197$$ 4.69193 0.334286 0.167143 0.985933i $$-0.446546\pi$$
0.167143 + 0.985933i $$0.446546\pi$$
$$198$$ 0 0
$$199$$ −9.49646 −0.673186 −0.336593 0.941650i $$-0.609275\pi$$
−0.336593 + 0.941650i $$0.609275\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 26.7677 1.88805
$$202$$ −6.89448 −0.485094
$$203$$ −6.84242 −0.480244
$$204$$ 24.0758 1.68564
$$205$$ −9.34596 −0.652750
$$206$$ −10.5793 −0.737096
$$207$$ 15.1576 1.05352
$$208$$ −6.69193 −0.464002
$$209$$ 0 0
$$210$$ −3.34596 −0.230893
$$211$$ 2.39094 0.164599 0.0822996 0.996608i $$-0.473774\pi$$
0.0822996 + 0.996608i $$0.473774\pi$$
$$212$$ 6.84242 0.469939
$$213$$ 1.00709 0.0690045
$$214$$ 7.49646 0.512447
$$215$$ 0.503544 0.0343414
$$216$$ 17.3839 1.18282
$$217$$ 6.00000 0.407307
$$218$$ −1.45857 −0.0987866
$$219$$ 34.0900 2.30359
$$220$$ 0 0
$$221$$ −48.1516 −3.23902
$$222$$ −21.8874 −1.46899
$$223$$ 0.188383 0.0126150 0.00630752 0.999980i $$-0.497992\pi$$
0.00630752 + 0.999980i $$0.497992\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 8.19547 0.546365
$$226$$ −14.0000 −0.931266
$$227$$ 6.99291 0.464136 0.232068 0.972700i $$-0.425451\pi$$
0.232068 + 0.972700i $$0.425451\pi$$
$$228$$ 6.18838 0.409836
$$229$$ −9.19547 −0.607654 −0.303827 0.952727i $$-0.598264\pi$$
−0.303827 + 0.952727i $$0.598264\pi$$
$$230$$ −1.84951 −0.121953
$$231$$ 0 0
$$232$$ −6.84242 −0.449227
$$233$$ 0.992912 0.0650479 0.0325239 0.999471i $$-0.489645\pi$$
0.0325239 + 0.999471i $$0.489645\pi$$
$$234$$ −54.8435 −3.58523
$$235$$ 1.49646 0.0976180
$$236$$ −7.88740 −0.513426
$$237$$ 40.9561 2.66038
$$238$$ 7.19547 0.466413
$$239$$ 1.14341 0.0739607 0.0369804 0.999316i $$-0.488226\pi$$
0.0369804 + 0.999316i $$0.488226\pi$$
$$240$$ −3.34596 −0.215981
$$241$$ −8.33888 −0.537154 −0.268577 0.963258i $$-0.586553\pi$$
−0.268577 + 0.963258i $$0.586553\pi$$
$$242$$ 0 0
$$243$$ 60.2036 3.86206
$$244$$ −4.50354 −0.288310
$$245$$ −1.00000 −0.0638877
$$246$$ 31.2713 1.99378
$$247$$ −12.3768 −0.787515
$$248$$ 6.00000 0.381000
$$249$$ 4.37677 0.277366
$$250$$ −1.00000 −0.0632456
$$251$$ 23.8874 1.50776 0.753880 0.657013i $$-0.228180\pi$$
0.753880 + 0.657013i $$0.228180\pi$$
$$252$$ 8.19547 0.516266
$$253$$ 0 0
$$254$$ 19.7748 1.24078
$$255$$ −24.0758 −1.50768
$$256$$ 1.00000 0.0625000
$$257$$ −31.6243 −1.97267 −0.986335 0.164753i $$-0.947317\pi$$
−0.986335 + 0.164753i $$0.947317\pi$$
$$258$$ −1.68484 −0.104893
$$259$$ −6.54143 −0.406465
$$260$$ 6.69193 0.415016
$$261$$ −56.0768 −3.47107
$$262$$ −2.15049 −0.132858
$$263$$ −13.3839 −0.825284 −0.412642 0.910893i $$-0.635394\pi$$
−0.412642 + 0.910893i $$0.635394\pi$$
$$264$$ 0 0
$$265$$ −6.84242 −0.420326
$$266$$ 1.84951 0.113401
$$267$$ −29.0829 −1.77984
$$268$$ 8.00000 0.488678
$$269$$ 5.79744 0.353476 0.176738 0.984258i $$-0.443445\pi$$
0.176738 + 0.984258i $$0.443445\pi$$
$$270$$ −17.3839 −1.05795
$$271$$ 20.0758 1.21952 0.609758 0.792587i $$-0.291266\pi$$
0.609758 + 0.792587i $$0.291266\pi$$
$$272$$ 7.19547 0.436289
$$273$$ −22.3909 −1.35516
$$274$$ −8.39094 −0.506915
$$275$$ 0 0
$$276$$ 6.18838 0.372497
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ −17.9253 −1.07509
$$279$$ 49.1728 2.94390
$$280$$ −1.00000 −0.0597614
$$281$$ 6.30099 0.375885 0.187943 0.982180i $$-0.439818\pi$$
0.187943 + 0.982180i $$0.439818\pi$$
$$282$$ −5.00709 −0.298168
$$283$$ −22.3909 −1.33100 −0.665502 0.746396i $$-0.731782\pi$$
−0.665502 + 0.746396i $$0.731782\pi$$
$$284$$ 0.300986 0.0178602
$$285$$ −6.18838 −0.366568
$$286$$ 0 0
$$287$$ 9.34596 0.551675
$$288$$ 8.19547 0.482923
$$289$$ 34.7748 2.04558
$$290$$ 6.84242 0.401801
$$291$$ 12.8803 0.755057
$$292$$ 10.1884 0.596230
$$293$$ 10.6919 0.624629 0.312315 0.949979i $$-0.398896\pi$$
0.312315 + 0.949979i $$0.398896\pi$$
$$294$$ 3.34596 0.195141
$$295$$ 7.88740 0.459222
$$296$$ −6.54143 −0.380213
$$297$$ 0 0
$$298$$ 0.451479 0.0261535
$$299$$ −12.3768 −0.715767
$$300$$ 3.34596 0.193179
$$301$$ −0.503544 −0.0290238
$$302$$ −19.2334 −1.10676
$$303$$ −23.0687 −1.32526
$$304$$ 1.84951 0.106077
$$305$$ 4.50354 0.257872
$$306$$ 58.9703 3.37111
$$307$$ 9.08287 0.518387 0.259193 0.965825i $$-0.416543\pi$$
0.259193 + 0.965825i $$0.416543\pi$$
$$308$$ 0 0
$$309$$ −35.3980 −2.01372
$$310$$ −6.00000 −0.340777
$$311$$ −11.3839 −0.645519 −0.322760 0.946481i $$-0.604611\pi$$
−0.322760 + 0.946481i $$0.604611\pi$$
$$312$$ −22.3909 −1.26764
$$313$$ −23.6243 −1.33532 −0.667662 0.744464i $$-0.732705\pi$$
−0.667662 + 0.744464i $$0.732705\pi$$
$$314$$ −9.69901 −0.547347
$$315$$ −8.19547 −0.461762
$$316$$ 12.2404 0.688579
$$317$$ −9.53435 −0.535502 −0.267751 0.963488i $$-0.586280\pi$$
−0.267751 + 0.963488i $$0.586280\pi$$
$$318$$ 22.8945 1.28386
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 25.0829 1.39999
$$322$$ 1.84951 0.103069
$$323$$ 13.3081 0.740481
$$324$$ 33.5793 1.86552
$$325$$ −6.69193 −0.371201
$$326$$ −8.00000 −0.443079
$$327$$ −4.88031 −0.269882
$$328$$ 9.34596 0.516044
$$329$$ −1.49646 −0.0825023
$$330$$ 0 0
$$331$$ 0.503544 0.0276773 0.0138386 0.999904i $$-0.495595\pi$$
0.0138386 + 0.999904i $$0.495595\pi$$
$$332$$ 1.30807 0.0717899
$$333$$ −53.6101 −2.93782
$$334$$ −8.00000 −0.437741
$$335$$ −8.00000 −0.437087
$$336$$ 3.34596 0.182537
$$337$$ −12.3909 −0.674978 −0.337489 0.941330i $$-0.609578\pi$$
−0.337489 + 0.941330i $$0.609578\pi$$
$$338$$ 31.7819 1.72871
$$339$$ −46.8435 −2.54419
$$340$$ −7.19547 −0.390229
$$341$$ 0 0
$$342$$ 15.1576 0.819628
$$343$$ 1.00000 0.0539949
$$344$$ −0.503544 −0.0271493
$$345$$ −6.18838 −0.333171
$$346$$ 15.4965 0.833095
$$347$$ −20.8803 −1.12091 −0.560457 0.828184i $$-0.689374\pi$$
−0.560457 + 0.828184i $$0.689374\pi$$
$$348$$ −22.8945 −1.22727
$$349$$ 22.1884 1.18772 0.593858 0.804570i $$-0.297604\pi$$
0.593858 + 0.804570i $$0.297604\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −116.331 −6.20931
$$352$$ 0 0
$$353$$ −31.6243 −1.68319 −0.841596 0.540108i $$-0.818383\pi$$
−0.841596 + 0.540108i $$0.818383\pi$$
$$354$$ −26.3909 −1.40266
$$355$$ −0.300986 −0.0159747
$$356$$ −8.69193 −0.460671
$$357$$ 24.0758 1.27423
$$358$$ −13.8874 −0.733972
$$359$$ −30.5273 −1.61117 −0.805584 0.592482i $$-0.798148\pi$$
−0.805584 + 0.592482i $$0.798148\pi$$
$$360$$ −8.19547 −0.431939
$$361$$ −15.5793 −0.819964
$$362$$ 18.7819 0.987154
$$363$$ 0 0
$$364$$ −6.69193 −0.350752
$$365$$ −10.1884 −0.533284
$$366$$ −15.0687 −0.787653
$$367$$ −10.5793 −0.552236 −0.276118 0.961124i $$-0.589048\pi$$
−0.276118 + 0.961124i $$0.589048\pi$$
$$368$$ 1.84951 0.0964122
$$369$$ 76.5946 3.98735
$$370$$ 6.54143 0.340073
$$371$$ 6.84242 0.355241
$$372$$ 20.0758 1.04088
$$373$$ −36.4667 −1.88818 −0.944088 0.329695i $$-0.893054\pi$$
−0.944088 + 0.329695i $$0.893054\pi$$
$$374$$ 0 0
$$375$$ −3.34596 −0.172785
$$376$$ −1.49646 −0.0771738
$$377$$ 45.7890 2.35825
$$378$$ 17.3839 0.894129
$$379$$ 14.9929 0.770134 0.385067 0.922889i $$-0.374178\pi$$
0.385067 + 0.922889i $$0.374178\pi$$
$$380$$ −1.84951 −0.0948777
$$381$$ 66.1657 3.38977
$$382$$ −6.39094 −0.326989
$$383$$ 35.1813 1.79768 0.898840 0.438277i $$-0.144411\pi$$
0.898840 + 0.438277i $$0.144411\pi$$
$$384$$ 3.34596 0.170748
$$385$$ 0 0
$$386$$ 13.1955 0.671632
$$387$$ −4.12678 −0.209776
$$388$$ 3.84951 0.195429
$$389$$ 2.30099 0.116665 0.0583323 0.998297i $$-0.481422\pi$$
0.0583323 + 0.998297i $$0.481422\pi$$
$$390$$ 22.3909 1.13381
$$391$$ 13.3081 0.673018
$$392$$ 1.00000 0.0505076
$$393$$ −7.19547 −0.362963
$$394$$ 4.69193 0.236376
$$395$$ −12.2404 −0.615884
$$396$$ 0 0
$$397$$ 18.0758 0.907197 0.453599 0.891206i $$-0.350140\pi$$
0.453599 + 0.891206i $$0.350140\pi$$
$$398$$ −9.49646 −0.476014
$$399$$ 6.18838 0.309807
$$400$$ 1.00000 0.0500000
$$401$$ 22.5793 1.12756 0.563779 0.825926i $$-0.309347\pi$$
0.563779 + 0.825926i $$0.309347\pi$$
$$402$$ 26.7677 1.33505
$$403$$ −40.1516 −2.00009
$$404$$ −6.89448 −0.343013
$$405$$ −33.5793 −1.66857
$$406$$ −6.84242 −0.339584
$$407$$ 0 0
$$408$$ 24.0758 1.19193
$$409$$ −1.04498 −0.0516708 −0.0258354 0.999666i $$-0.508225\pi$$
−0.0258354 + 0.999666i $$0.508225\pi$$
$$410$$ −9.34596 −0.461564
$$411$$ −28.0758 −1.38488
$$412$$ −10.5793 −0.521206
$$413$$ −7.88740 −0.388113
$$414$$ 15.1576 0.744954
$$415$$ −1.30807 −0.0642108
$$416$$ −6.69193 −0.328099
$$417$$ −59.9774 −2.93710
$$418$$ 0 0
$$419$$ 5.49646 0.268519 0.134260 0.990946i $$-0.457134\pi$$
0.134260 + 0.990946i $$0.457134\pi$$
$$420$$ −3.34596 −0.163266
$$421$$ 2.60197 0.126812 0.0634062 0.997988i $$-0.479804\pi$$
0.0634062 + 0.997988i $$0.479804\pi$$
$$422$$ 2.39094 0.116389
$$423$$ −12.2642 −0.596304
$$424$$ 6.84242 0.332297
$$425$$ 7.19547 0.349032
$$426$$ 1.00709 0.0487936
$$427$$ −4.50354 −0.217942
$$428$$ 7.49646 0.362355
$$429$$ 0 0
$$430$$ 0.503544 0.0242830
$$431$$ −19.2334 −0.926438 −0.463219 0.886244i $$-0.653306\pi$$
−0.463219 + 0.886244i $$0.653306\pi$$
$$432$$ 17.3839 0.836381
$$433$$ −8.93237 −0.429263 −0.214631 0.976695i $$-0.568855\pi$$
−0.214631 + 0.976695i $$0.568855\pi$$
$$434$$ 6.00000 0.288009
$$435$$ 22.8945 1.09771
$$436$$ −1.45857 −0.0698527
$$437$$ 3.42068 0.163633
$$438$$ 34.0900 1.62888
$$439$$ 0.300986 0.0143653 0.00718264 0.999974i $$-0.497714\pi$$
0.00718264 + 0.999974i $$0.497714\pi$$
$$440$$ 0 0
$$441$$ 8.19547 0.390260
$$442$$ −48.1516 −2.29034
$$443$$ 30.7677 1.46182 0.730909 0.682475i $$-0.239096\pi$$
0.730909 + 0.682475i $$0.239096\pi$$
$$444$$ −21.8874 −1.03873
$$445$$ 8.69193 0.412037
$$446$$ 0.188383 0.00892018
$$447$$ 1.51063 0.0714504
$$448$$ 1.00000 0.0472456
$$449$$ −1.19547 −0.0564177 −0.0282089 0.999602i $$-0.508980\pi$$
−0.0282089 + 0.999602i $$0.508980\pi$$
$$450$$ 8.19547 0.386338
$$451$$ 0 0
$$452$$ −14.0000 −0.658505
$$453$$ −64.3541 −3.02362
$$454$$ 6.99291 0.328194
$$455$$ 6.69193 0.313722
$$456$$ 6.18838 0.289798
$$457$$ 25.7748 1.20569 0.602847 0.797857i $$-0.294033\pi$$
0.602847 + 0.797857i $$0.294033\pi$$
$$458$$ −9.19547 −0.429676
$$459$$ 125.085 5.83847
$$460$$ −1.84951 −0.0862337
$$461$$ −11.5722 −0.538973 −0.269486 0.963004i $$-0.586854\pi$$
−0.269486 + 0.963004i $$0.586854\pi$$
$$462$$ 0 0
$$463$$ −32.9182 −1.52984 −0.764919 0.644126i $$-0.777221\pi$$
−0.764919 + 0.644126i $$0.777221\pi$$
$$464$$ −6.84242 −0.317651
$$465$$ −20.0758 −0.930992
$$466$$ 0.992912 0.0459958
$$467$$ −14.6399 −0.677452 −0.338726 0.940885i $$-0.609996\pi$$
−0.338726 + 0.940885i $$0.609996\pi$$
$$468$$ −54.8435 −2.53514
$$469$$ 8.00000 0.369406
$$470$$ 1.49646 0.0690264
$$471$$ −32.4525 −1.49533
$$472$$ −7.88740 −0.363047
$$473$$ 0 0
$$474$$ 40.9561 1.88118
$$475$$ 1.84951 0.0848612
$$476$$ 7.19547 0.329804
$$477$$ 56.0768 2.56758
$$478$$ 1.14341 0.0522981
$$479$$ −10.3909 −0.474774 −0.237387 0.971415i $$-0.576291\pi$$
−0.237387 + 0.971415i $$0.576291\pi$$
$$480$$ −3.34596 −0.152722
$$481$$ 43.7748 1.99596
$$482$$ −8.33888 −0.379825
$$483$$ 6.18838 0.281581
$$484$$ 0 0
$$485$$ −3.84951 −0.174797
$$486$$ 60.2036 2.73089
$$487$$ 39.3091 1.78127 0.890634 0.454722i $$-0.150261\pi$$
0.890634 + 0.454722i $$0.150261\pi$$
$$488$$ −4.50354 −0.203866
$$489$$ −26.7677 −1.21048
$$490$$ −1.00000 −0.0451754
$$491$$ −23.7748 −1.07294 −0.536471 0.843919i $$-0.680243\pi$$
−0.536471 + 0.843919i $$0.680243\pi$$
$$492$$ 31.2713 1.40982
$$493$$ −49.2344 −2.21741
$$494$$ −12.3768 −0.556857
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ 0.300986 0.0135011
$$498$$ 4.37677 0.196128
$$499$$ −6.11260 −0.273638 −0.136819 0.990596i $$-0.543688\pi$$
−0.136819 + 0.990596i $$0.543688\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −26.7677 −1.19589
$$502$$ 23.8874 1.06615
$$503$$ 13.1586 0.586715 0.293358 0.956003i $$-0.405227\pi$$
0.293358 + 0.956003i $$0.405227\pi$$
$$504$$ 8.19547 0.365055
$$505$$ 6.89448 0.306801
$$506$$ 0 0
$$507$$ 106.341 4.72277
$$508$$ 19.7748 0.877365
$$509$$ 34.1799 1.51500 0.757499 0.652836i $$-0.226421\pi$$
0.757499 + 0.652836i $$0.226421\pi$$
$$510$$ −24.0758 −1.06609
$$511$$ 10.1884 0.450708
$$512$$ 1.00000 0.0441942
$$513$$ 32.1516 1.41953
$$514$$ −31.6243 −1.39489
$$515$$ 10.5793 0.466181
$$516$$ −1.68484 −0.0741709
$$517$$ 0 0
$$518$$ −6.54143 −0.287414
$$519$$ 51.8506 2.27599
$$520$$ 6.69193 0.293460
$$521$$ −26.6778 −1.16877 −0.584387 0.811475i $$-0.698665\pi$$
−0.584387 + 0.811475i $$0.698665\pi$$
$$522$$ −56.0768 −2.45442
$$523$$ −25.4880 −1.11451 −0.557256 0.830341i $$-0.688146\pi$$
−0.557256 + 0.830341i $$0.688146\pi$$
$$524$$ −2.15049 −0.0939447
$$525$$ 3.34596 0.146030
$$526$$ −13.3839 −0.583564
$$527$$ 43.1728 1.88064
$$528$$ 0 0
$$529$$ −19.5793 −0.851275
$$530$$ −6.84242 −0.297216
$$531$$ −64.6409 −2.80518
$$532$$ 1.84951 0.0801863
$$533$$ −62.5425 −2.70902
$$534$$ −29.0829 −1.25854
$$535$$ −7.49646 −0.324100
$$536$$ 8.00000 0.345547
$$537$$ −46.4667 −2.00519
$$538$$ 5.79744 0.249945
$$539$$ 0 0
$$540$$ −17.3839 −0.748082
$$541$$ −4.45148 −0.191384 −0.0956920 0.995411i $$-0.530506\pi$$
−0.0956920 + 0.995411i $$0.530506\pi$$
$$542$$ 20.0758 0.862329
$$543$$ 62.8435 2.69687
$$544$$ 7.19547 0.308503
$$545$$ 1.45857 0.0624781
$$546$$ −22.3909 −0.958244
$$547$$ 1.38385 0.0591693 0.0295846 0.999562i $$-0.490582\pi$$
0.0295846 + 0.999562i $$0.490582\pi$$
$$548$$ −8.39094 −0.358443
$$549$$ −36.9087 −1.57522
$$550$$ 0 0
$$551$$ −12.6551 −0.539126
$$552$$ 6.18838 0.263395
$$553$$ 12.2404 0.520517
$$554$$ −22.0000 −0.934690
$$555$$ 21.8874 0.929068
$$556$$ −17.9253 −0.760201
$$557$$ 6.70610 0.284147 0.142073 0.989856i $$-0.454623\pi$$
0.142073 + 0.989856i $$0.454623\pi$$
$$558$$ 49.1728 2.08165
$$559$$ 3.36968 0.142522
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ 6.30099 0.265791
$$563$$ −3.59488 −0.151506 −0.0757532 0.997127i $$-0.524136\pi$$
−0.0757532 + 0.997127i $$0.524136\pi$$
$$564$$ −5.00709 −0.210836
$$565$$ 14.0000 0.588984
$$566$$ −22.3909 −0.941161
$$567$$ 33.5793 1.41020
$$568$$ 0.300986 0.0126291
$$569$$ −31.7606 −1.33147 −0.665737 0.746186i $$-0.731883\pi$$
−0.665737 + 0.746186i $$0.731883\pi$$
$$570$$ −6.18838 −0.259203
$$571$$ 5.68484 0.237903 0.118952 0.992900i $$-0.462047\pi$$
0.118952 + 0.992900i $$0.462047\pi$$
$$572$$ 0 0
$$573$$ −21.3839 −0.893323
$$574$$ 9.34596 0.390093
$$575$$ 1.84951 0.0771298
$$576$$ 8.19547 0.341478
$$577$$ 13.5343 0.563442 0.281721 0.959496i $$-0.409095\pi$$
0.281721 + 0.959496i $$0.409095\pi$$
$$578$$ 34.7748 1.44644
$$579$$ 44.1516 1.83488
$$580$$ 6.84242 0.284116
$$581$$ 1.30807 0.0542680
$$582$$ 12.8803 0.533906
$$583$$ 0 0
$$584$$ 10.1884 0.421598
$$585$$ 54.8435 2.26750
$$586$$ 10.6919 0.441679
$$587$$ −0.0520650 −0.00214895 −0.00107448 0.999999i $$-0.500342\pi$$
−0.00107448 + 0.999999i $$0.500342\pi$$
$$588$$ 3.34596 0.137985
$$589$$ 11.0970 0.457246
$$590$$ 7.88740 0.324719
$$591$$ 15.6990 0.645771
$$592$$ −6.54143 −0.268851
$$593$$ −31.9774 −1.31315 −0.656576 0.754260i $$-0.727996\pi$$
−0.656576 + 0.754260i $$0.727996\pi$$
$$594$$ 0 0
$$595$$ −7.19547 −0.294986
$$596$$ 0.451479 0.0184933
$$597$$ −31.7748 −1.30046
$$598$$ −12.3768 −0.506124
$$599$$ 21.0829 0.861423 0.430711 0.902490i $$-0.358263\pi$$
0.430711 + 0.902490i $$0.358263\pi$$
$$600$$ 3.34596 0.136598
$$601$$ 21.6469 0.882997 0.441499 0.897262i $$-0.354447\pi$$
0.441499 + 0.897262i $$0.354447\pi$$
$$602$$ −0.503544 −0.0205229
$$603$$ 65.5638 2.66996
$$604$$ −19.2334 −0.782594
$$605$$ 0 0
$$606$$ −23.0687 −0.937102
$$607$$ −21.6622 −0.879241 −0.439621 0.898184i $$-0.644887\pi$$
−0.439621 + 0.898184i $$0.644887\pi$$
$$608$$ 1.84951 0.0750074
$$609$$ −22.8945 −0.927731
$$610$$ 4.50354 0.182343
$$611$$ 10.0142 0.405130
$$612$$ 58.9703 2.38373
$$613$$ −8.31516 −0.335846 −0.167923 0.985800i $$-0.553706\pi$$
−0.167923 + 0.985800i $$0.553706\pi$$
$$614$$ 9.08287 0.366555
$$615$$ −31.2713 −1.26098
$$616$$ 0 0
$$617$$ 12.4667 0.501891 0.250946 0.968001i $$-0.419258\pi$$
0.250946 + 0.968001i $$0.419258\pi$$
$$618$$ −35.3980 −1.42392
$$619$$ 5.27125 0.211869 0.105935 0.994373i $$-0.466217\pi$$
0.105935 + 0.994373i $$0.466217\pi$$
$$620$$ −6.00000 −0.240966
$$621$$ 32.1516 1.29020
$$622$$ −11.3839 −0.456451
$$623$$ −8.69193 −0.348235
$$624$$ −22.3909 −0.896355
$$625$$ 1.00000 0.0400000
$$626$$ −23.6243 −0.944217
$$627$$ 0 0
$$628$$ −9.69901 −0.387033
$$629$$ −47.0687 −1.87675
$$630$$ −8.19547 −0.326515
$$631$$ −17.1586 −0.683075 −0.341537 0.939868i $$-0.610948\pi$$
−0.341537 + 0.939868i $$0.610948\pi$$
$$632$$ 12.2404 0.486899
$$633$$ 8.00000 0.317971
$$634$$ −9.53435 −0.378657
$$635$$ −19.7748 −0.784739
$$636$$ 22.8945 0.907825
$$637$$ −6.69193 −0.265144
$$638$$ 0 0
$$639$$ 2.46672 0.0975820
$$640$$ −1.00000 −0.0395285
$$641$$ 24.7677 0.978266 0.489133 0.872209i $$-0.337313\pi$$
0.489133 + 0.872209i $$0.337313\pi$$
$$642$$ 25.0829 0.989942
$$643$$ 1.03080 0.0406509 0.0203254 0.999793i $$-0.493530\pi$$
0.0203254 + 0.999793i $$0.493530\pi$$
$$644$$ 1.84951 0.0728808
$$645$$ 1.68484 0.0663405
$$646$$ 13.3081 0.523599
$$647$$ 12.7904 0.502841 0.251420 0.967878i $$-0.419102\pi$$
0.251420 + 0.967878i $$0.419102\pi$$
$$648$$ 33.5793 1.31912
$$649$$ 0 0
$$650$$ −6.69193 −0.262479
$$651$$ 20.0758 0.786832
$$652$$ −8.00000 −0.313304
$$653$$ −22.3162 −0.873301 −0.436651 0.899631i $$-0.643835\pi$$
−0.436651 + 0.899631i $$0.643835\pi$$
$$654$$ −4.88031 −0.190835
$$655$$ 2.15049 0.0840267
$$656$$ 9.34596 0.364899
$$657$$ 83.4986 3.25759
$$658$$ −1.49646 −0.0583379
$$659$$ −22.6919 −0.883952 −0.441976 0.897027i $$-0.645722\pi$$
−0.441976 + 0.897027i $$0.645722\pi$$
$$660$$ 0 0
$$661$$ 43.5638 1.69443 0.847217 0.531247i $$-0.178276\pi$$
0.847217 + 0.531247i $$0.178276\pi$$
$$662$$ 0.503544 0.0195708
$$663$$ −161.113 −6.25712
$$664$$ 1.30807 0.0507631
$$665$$ −1.84951 −0.0717208
$$666$$ −53.6101 −2.07735
$$667$$ −12.6551 −0.490008
$$668$$ −8.00000 −0.309529
$$669$$ 0.630322 0.0243697
$$670$$ −8.00000 −0.309067
$$671$$ 0 0
$$672$$ 3.34596 0.129073
$$673$$ −38.0000 −1.46479 −0.732396 0.680879i $$-0.761598\pi$$
−0.732396 + 0.680879i $$0.761598\pi$$
$$674$$ −12.3909 −0.477281
$$675$$ 17.3839 0.669105
$$676$$ 31.7819 1.22238
$$677$$ 16.0984 0.618713 0.309356 0.950946i $$-0.399886\pi$$
0.309356 + 0.950946i $$0.399886\pi$$
$$678$$ −46.8435 −1.79901
$$679$$ 3.84951 0.147731
$$680$$ −7.19547 −0.275934
$$681$$ 23.3980 0.896614
$$682$$ 0 0
$$683$$ 29.0829 1.11282 0.556412 0.830906i $$-0.312177\pi$$
0.556412 + 0.830906i $$0.312177\pi$$
$$684$$ 15.1576 0.579565
$$685$$ 8.39094 0.320601
$$686$$ 1.00000 0.0381802
$$687$$ −30.7677 −1.17386
$$688$$ −0.503544 −0.0191974
$$689$$ −45.7890 −1.74442
$$690$$ −6.18838 −0.235588
$$691$$ 7.58641 0.288601 0.144300 0.989534i $$-0.453907\pi$$
0.144300 + 0.989534i $$0.453907\pi$$
$$692$$ 15.4965 0.589087
$$693$$ 0 0
$$694$$ −20.8803 −0.792606
$$695$$ 17.9253 0.679945
$$696$$ −22.8945 −0.867813
$$697$$ 67.2486 2.54722
$$698$$ 22.1884 0.839843
$$699$$ 3.32225 0.125659
$$700$$ 1.00000 0.0377964
$$701$$ −4.07471 −0.153900 −0.0769499 0.997035i $$-0.524518\pi$$
−0.0769499 + 0.997035i $$0.524518\pi$$
$$702$$ −116.331 −4.39065
$$703$$ −12.0984 −0.456301
$$704$$ 0 0
$$705$$ 5.00709 0.188578
$$706$$ −31.6243 −1.19020
$$707$$ −6.89448 −0.259294
$$708$$ −26.3909 −0.991832
$$709$$ −46.4525 −1.74456 −0.872281 0.489005i $$-0.837360\pi$$
−0.872281 + 0.489005i $$0.837360\pi$$
$$710$$ −0.300986 −0.0112958
$$711$$ 100.316 3.76215
$$712$$ −8.69193 −0.325744
$$713$$ 11.0970 0.415588
$$714$$ 24.0758 0.901013
$$715$$ 0 0
$$716$$ −13.8874 −0.518996
$$717$$ 3.82579 0.142877
$$718$$ −30.5273 −1.13927
$$719$$ −17.8732 −0.666559 −0.333279 0.942828i $$-0.608155\pi$$
−0.333279 + 0.942828i $$0.608155\pi$$
$$720$$ −8.19547 −0.305427
$$721$$ −10.5793 −0.393995
$$722$$ −15.5793 −0.579802
$$723$$ −27.9016 −1.03767
$$724$$ 18.7819 0.698023
$$725$$ −6.84242 −0.254121
$$726$$ 0 0
$$727$$ −11.8874 −0.440879 −0.220440 0.975401i $$-0.570749\pi$$
−0.220440 + 0.975401i $$0.570749\pi$$
$$728$$ −6.69193 −0.248019
$$729$$ 100.701 3.72967
$$730$$ −10.1884 −0.377089
$$731$$ −3.62323 −0.134010
$$732$$ −15.0687 −0.556955
$$733$$ 31.6764 1.16999 0.584997 0.811036i $$-0.301096\pi$$
0.584997 + 0.811036i $$0.301096\pi$$
$$734$$ −10.5793 −0.390490
$$735$$ −3.34596 −0.123418
$$736$$ 1.84951 0.0681737
$$737$$ 0 0
$$738$$ 76.5946 2.81948
$$739$$ −28.5567 −1.05047 −0.525237 0.850956i $$-0.676023\pi$$
−0.525237 + 0.850956i $$0.676023\pi$$
$$740$$ 6.54143 0.240468
$$741$$ −41.4122 −1.52132
$$742$$ 6.84242 0.251193
$$743$$ −13.9858 −0.513090 −0.256545 0.966532i $$-0.582584\pi$$
−0.256545 + 0.966532i $$0.582584\pi$$
$$744$$ 20.0758 0.736014
$$745$$ −0.451479 −0.0165409
$$746$$ −36.4667 −1.33514
$$747$$ 10.7203 0.392234
$$748$$ 0 0
$$749$$ 7.49646 0.273915
$$750$$ −3.34596 −0.122177
$$751$$ 18.9171 0.690296 0.345148 0.938548i $$-0.387829\pi$$
0.345148 + 0.938548i $$0.387829\pi$$
$$752$$ −1.49646 −0.0545701
$$753$$ 79.9264 2.91268
$$754$$ 45.7890 1.66754
$$755$$ 19.2334 0.699974
$$756$$ 17.3839 0.632245
$$757$$ 10.9182 0.396829 0.198414 0.980118i $$-0.436421\pi$$
0.198414 + 0.980118i $$0.436421\pi$$
$$758$$ 14.9929 0.544567
$$759$$ 0 0
$$760$$ −1.84951 −0.0670887
$$761$$ 30.3531 1.10030 0.550149 0.835067i $$-0.314571\pi$$
0.550149 + 0.835067i $$0.314571\pi$$
$$762$$ 66.1657 2.39693
$$763$$ −1.45857 −0.0528036
$$764$$ −6.39094 −0.231216
$$765$$ −58.9703 −2.13207
$$766$$ 35.1813 1.27115
$$767$$ 52.7819 1.90584
$$768$$ 3.34596 0.120737
$$769$$ 19.7369 0.711731 0.355865 0.934537i $$-0.384186\pi$$
0.355865 + 0.934537i $$0.384186\pi$$
$$770$$ 0 0
$$771$$ −105.814 −3.81079
$$772$$ 13.1955 0.474915
$$773$$ −24.3909 −0.877281 −0.438641 0.898663i $$-0.644540\pi$$
−0.438641 + 0.898663i $$0.644540\pi$$
$$774$$ −4.12678 −0.148334
$$775$$ 6.00000 0.215526
$$776$$ 3.84951 0.138189
$$777$$ −21.8874 −0.785206
$$778$$ 2.30099 0.0824943
$$779$$ 17.2854 0.619315
$$780$$ 22.3909 0.801724
$$781$$ 0 0
$$782$$ 13.3081 0.475896
$$783$$ −118.948 −4.25084
$$784$$ 1.00000 0.0357143
$$785$$ 9.69901 0.346173
$$786$$ −7.19547 −0.256654
$$787$$ −33.7890 −1.20445 −0.602223 0.798328i $$-0.705718\pi$$
−0.602223 + 0.798328i $$0.705718\pi$$
$$788$$ 4.69193 0.167143
$$789$$ −44.7819 −1.59428
$$790$$ −12.2404 −0.435496
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 30.1374 1.07021
$$794$$ 18.0758 0.641485
$$795$$ −22.8945 −0.811984
$$796$$ −9.49646 −0.336593
$$797$$ 36.8435 1.30506 0.652532 0.757761i $$-0.273707\pi$$
0.652532 + 0.757761i $$0.273707\pi$$
$$798$$ 6.18838 0.219066
$$799$$ −10.7677 −0.380934
$$800$$ 1.00000 0.0353553
$$801$$ −71.2344 −2.51694
$$802$$ 22.5793 0.797304
$$803$$ 0 0
$$804$$ 26.7677 0.944024
$$805$$ −1.84951 −0.0651866
$$806$$ −40.1516 −1.41428
$$807$$ 19.3980 0.682843
$$808$$ −6.89448 −0.242547
$$809$$ −37.6990 −1.32543 −0.662713 0.748873i $$-0.730595\pi$$
−0.662713 + 0.748873i $$0.730595\pi$$
$$810$$ −33.5793 −1.17986
$$811$$ −38.3304 −1.34596 −0.672981 0.739660i $$-0.734987\pi$$
−0.672981 + 0.739660i $$0.734987\pi$$
$$812$$ −6.84242 −0.240122
$$813$$ 67.1728 2.35585
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 24.0758 0.842821
$$817$$ −0.931308 −0.0325823
$$818$$ −1.04498 −0.0365368
$$819$$ −54.8435 −1.91639
$$820$$ −9.34596 −0.326375
$$821$$ 18.1363 0.632962 0.316481 0.948599i $$-0.397499\pi$$
0.316481 + 0.948599i $$0.397499\pi$$
$$822$$ −28.0758 −0.979255
$$823$$ −7.86368 −0.274111 −0.137055 0.990563i $$-0.543764\pi$$
−0.137055 + 0.990563i $$0.543764\pi$$
$$824$$ −10.5793 −0.368548
$$825$$ 0 0
$$826$$ −7.88740 −0.274438
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 15.1576 0.526762
$$829$$ 2.80453 0.0974053 0.0487027 0.998813i $$-0.484491\pi$$
0.0487027 + 0.998813i $$0.484491\pi$$
$$830$$ −1.30807 −0.0454039
$$831$$ −73.6112 −2.55354
$$832$$ −6.69193 −0.232001
$$833$$ 7.19547 0.249308
$$834$$ −59.9774 −2.07685
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 104.303 3.60524
$$838$$ 5.49646 0.189872
$$839$$ 36.9929 1.27714 0.638569 0.769565i $$-0.279527\pi$$
0.638569 + 0.769565i $$0.279527\pi$$
$$840$$ −3.34596 −0.115447
$$841$$ 17.8187 0.614438
$$842$$ 2.60197 0.0896699
$$843$$ 21.0829 0.726133
$$844$$ 2.39094 0.0822996
$$845$$ −31.7819 −1.09333
$$846$$ −12.2642 −0.421651
$$847$$ 0 0
$$848$$ 6.84242 0.234970
$$849$$ −74.9193 −2.57122
$$850$$ 7.19547 0.246803
$$851$$ −12.0984 −0.414729
$$852$$ 1.00709 0.0345023
$$853$$ 49.2571 1.68653 0.843265 0.537498i $$-0.180630\pi$$
0.843265 + 0.537498i $$0.180630\pi$$
$$854$$ −4.50354 −0.154108
$$855$$ −15.1576 −0.518378
$$856$$ 7.49646 0.256224
$$857$$ 20.3541 0.695283 0.347642 0.937627i $$-0.386983\pi$$
0.347642 + 0.937627i $$0.386983\pi$$
$$858$$ 0 0
$$859$$ −34.2783 −1.16956 −0.584781 0.811191i $$-0.698820\pi$$
−0.584781 + 0.811191i $$0.698820\pi$$
$$860$$ 0.503544 0.0171707
$$861$$ 31.2713 1.06572
$$862$$ −19.2334 −0.655091
$$863$$ −12.4373 −0.423371 −0.211685 0.977338i $$-0.567895\pi$$
−0.211685 + 0.977338i $$0.567895\pi$$
$$864$$ 17.3839 0.591411
$$865$$ −15.4965 −0.526895
$$866$$ −8.93237 −0.303534
$$867$$ 116.355 3.95163
$$868$$ 6.00000 0.203653
$$869$$ 0 0
$$870$$ 22.8945 0.776196
$$871$$ −53.5354 −1.81398
$$872$$ −1.45857 −0.0493933
$$873$$ 31.5485 1.06776
$$874$$ 3.42068 0.115706
$$875$$ −1.00000 −0.0338062
$$876$$ 34.0900 1.15179
$$877$$ −34.4809 −1.16434 −0.582169 0.813068i $$-0.697796\pi$$
−0.582169 + 0.813068i $$0.697796\pi$$
$$878$$ 0.300986 0.0101578
$$879$$ 35.7748 1.20665
$$880$$ 0 0
$$881$$ 27.3839 0.922585 0.461293 0.887248i $$-0.347386\pi$$
0.461293 + 0.887248i $$0.347386\pi$$
$$882$$ 8.19547 0.275956
$$883$$ 22.0900 0.743386 0.371693 0.928356i $$-0.378777\pi$$
0.371693 + 0.928356i $$0.378777\pi$$
$$884$$ −48.1516 −1.61951
$$885$$ 26.3909 0.887122
$$886$$ 30.7677 1.03366
$$887$$ −25.7890 −0.865909 −0.432954 0.901416i $$-0.642529\pi$$
−0.432954 + 0.901416i $$0.642529\pi$$
$$888$$ −21.8874 −0.734493
$$889$$ 19.7748 0.663225
$$890$$ 8.69193 0.291354
$$891$$ 0 0
$$892$$ 0.188383 0.00630752
$$893$$ −2.76771 −0.0926178
$$894$$ 1.51063 0.0505231
$$895$$ 13.8874 0.464204
$$896$$ 1.00000 0.0334077
$$897$$ −41.4122 −1.38271
$$898$$ −1.19547 −0.0398934
$$899$$ −41.0545 −1.36924
$$900$$ 8.19547 0.273182
$$901$$ 49.2344 1.64024
$$902$$ 0 0
$$903$$ −1.68484 −0.0560679
$$904$$ −14.0000 −0.465633
$$905$$ −18.7819 −0.624331
$$906$$ −64.3541 −2.13802
$$907$$ −45.4596 −1.50946 −0.754731 0.656034i $$-0.772233\pi$$
−0.754731 + 0.656034i $$0.772233\pi$$
$$908$$ 6.99291 0.232068
$$909$$ −56.5035 −1.87410
$$910$$ 6.69193 0.221835
$$911$$ −15.8506 −0.525153 −0.262576 0.964911i $$-0.584572\pi$$
−0.262576 + 0.964911i $$0.584572\pi$$
$$912$$ 6.18838 0.204918
$$913$$ 0 0
$$914$$ 25.7748 0.852554
$$915$$ 15.0687 0.498156
$$916$$ −9.19547 −0.303827
$$917$$ −2.15049 −0.0710155
$$918$$ 125.085 4.12842
$$919$$ −18.6314 −0.614593 −0.307296 0.951614i $$-0.599424\pi$$
−0.307296 + 0.951614i $$0.599424\pi$$
$$920$$ −1.84951 −0.0609764
$$921$$ 30.3909 1.00142
$$922$$ −11.5722 −0.381111
$$923$$ −2.01418 −0.0662974
$$924$$ 0 0
$$925$$ −6.54143 −0.215081
$$926$$ −32.9182 −1.08176
$$927$$ −86.7025 −2.84768
$$928$$ −6.84242 −0.224613
$$929$$ −29.2486 −0.959616 −0.479808 0.877374i $$-0.659294\pi$$
−0.479808 + 0.877374i $$0.659294\pi$$
$$930$$ −20.0758 −0.658311
$$931$$ 1.84951 0.0606151
$$932$$ 0.992912 0.0325239
$$933$$ −38.0900 −1.24701
$$934$$ −14.6399 −0.479031
$$935$$ 0 0
$$936$$ −54.8435 −1.79262
$$937$$ 46.4441 1.51726 0.758631 0.651521i $$-0.225869\pi$$
0.758631 + 0.651521i $$0.225869\pi$$
$$938$$ 8.00000 0.261209
$$939$$ −79.0460 −2.57957
$$940$$ 1.49646 0.0488090
$$941$$ 39.2713 1.28021 0.640103 0.768289i $$-0.278892\pi$$
0.640103 + 0.768289i $$0.278892\pi$$
$$942$$ −32.4525 −1.05736
$$943$$ 17.2854 0.562891
$$944$$ −7.88740 −0.256713
$$945$$ −17.3839 −0.565497
$$946$$ 0 0
$$947$$ 54.2415 1.76261 0.881306 0.472546i $$-0.156665\pi$$
0.881306 + 0.472546i $$0.156665\pi$$
$$948$$ 40.9561 1.33019
$$949$$ −68.1799 −2.21321
$$950$$ 1.84951 0.0600059
$$951$$ −31.9016 −1.03448
$$952$$ 7.19547 0.233207
$$953$$ −59.3612 −1.92290 −0.961449 0.274983i $$-0.911328\pi$$
−0.961449 + 0.274983i $$0.911328\pi$$
$$954$$ 56.0768 1.81555
$$955$$ 6.39094 0.206806
$$956$$ 1.14341 0.0369804
$$957$$ 0 0
$$958$$ −10.3909 −0.335716
$$959$$ −8.39094 −0.270958
$$960$$ −3.34596 −0.107991
$$961$$ 5.00000 0.161290
$$962$$ 43.7748 1.41136
$$963$$ 61.4370 1.97978
$$964$$ −8.33888 −0.268577
$$965$$ −13.1955 −0.424777
$$966$$ 6.18838 0.199108
$$967$$ 18.7677 0.603529 0.301764 0.953383i $$-0.402424\pi$$
0.301764 + 0.953383i $$0.402424\pi$$
$$968$$ 0 0
$$969$$ 44.5283 1.43046
$$970$$ −3.84951 −0.123600
$$971$$ −59.4370 −1.90742 −0.953712 0.300722i $$-0.902772\pi$$
−0.953712 + 0.300722i $$0.902772\pi$$
$$972$$ 60.2036 1.93103
$$973$$ −17.9253 −0.574658
$$974$$ 39.3091 1.25955
$$975$$ −22.3909 −0.717084
$$976$$ −4.50354 −0.144155
$$977$$ −32.7677 −1.04833 −0.524166 0.851616i $$-0.675623\pi$$
−0.524166 + 0.851616i $$0.675623\pi$$
$$978$$ −26.7677 −0.855937
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −11.9536 −0.381650
$$982$$ −23.7748 −0.758684
$$983$$ 13.4207 0.428053 0.214027 0.976828i $$-0.431342\pi$$
0.214027 + 0.976828i $$0.431342\pi$$
$$984$$ 31.2713 0.996891
$$985$$ −4.69193 −0.149497
$$986$$ −49.2344 −1.56794
$$987$$ −5.00709 −0.159377
$$988$$ −12.3768 −0.393757
$$989$$ −0.931308 −0.0296139
$$990$$ 0 0
$$991$$ −38.1657 −1.21237 −0.606187 0.795322i $$-0.707302\pi$$
−0.606187 + 0.795322i $$0.707302\pi$$
$$992$$ 6.00000 0.190500
$$993$$ 1.68484 0.0534668
$$994$$ 0.300986 0.00954669
$$995$$ 9.49646 0.301058
$$996$$ 4.37677 0.138683
$$997$$ 33.6622 1.06609 0.533046 0.846086i $$-0.321047\pi$$
0.533046 + 0.846086i $$0.321047\pi$$
$$998$$ −6.11260 −0.193491
$$999$$ −113.715 −3.59779
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.cl.1.3 3
11.10 odd 2 770.2.a.l.1.3 3
33.32 even 2 6930.2.a.cl.1.3 3
44.43 even 2 6160.2.a.bi.1.1 3
55.32 even 4 3850.2.c.z.1849.1 6
55.43 even 4 3850.2.c.z.1849.6 6
55.54 odd 2 3850.2.a.bu.1.1 3
77.76 even 2 5390.2.a.bz.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.l.1.3 3 11.10 odd 2
3850.2.a.bu.1.1 3 55.54 odd 2
3850.2.c.z.1849.1 6 55.32 even 4
3850.2.c.z.1849.6 6 55.43 even 4
5390.2.a.bz.1.1 3 77.76 even 2
6160.2.a.bi.1.1 3 44.43 even 2
6930.2.a.cl.1.3 3 33.32 even 2
8470.2.a.cl.1.3 3 1.1 even 1 trivial