# Properties

 Label 4704.2.a.bz.1.2 Level $4704$ Weight $2$ Character 4704.1 Self dual yes Analytic conductor $37.562$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4704 = 2^{5} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4704.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.5616291108$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ x^4 - 2*x^3 - 3*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.22833$$ of defining polynomial Character $$\chi$$ $$=$$ 4704.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.04244 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.04244 q^{5} +1.00000 q^{9} +5.93089 q^{11} -0.888450 q^{13} -1.04244 q^{15} +4.51668 q^{17} +3.47424 q^{19} -1.93089 q^{23} -3.91331 q^{25} +1.00000 q^{27} -8.38755 q^{29} +5.13109 q^{31} +5.93089 q^{33} +5.65685 q^{37} -0.888450 q^{39} +8.39663 q^{41} -0.743541 q^{43} -1.04244 q^{45} -4.21778 q^{47} +4.51668 q^{51} -6.91331 q^{53} -6.18262 q^{55} +3.47424 q^{57} +5.43908 q^{59} +10.9733 q^{61} +0.926159 q^{65} -6.20493 q^{67} -1.93089 q^{69} +3.72596 q^{71} -3.49910 q^{73} -3.91331 q^{75} +12.6053 q^{79} +1.00000 q^{81} -2.94847 q^{83} -4.70838 q^{85} -8.38755 q^{87} -5.00196 q^{89} +5.13109 q^{93} -3.62169 q^{95} +10.1578 q^{97} +5.93089 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{5} + 4 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{39} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 16 q^{47} + 4 q^{51} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{73} + 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} + 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 4 * q^17 + 8 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 16 * q^47 + 4 * q^51 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 12 * q^69 + 12 * q^71 + 8 * q^73 + 12 * q^75 + 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 + 24 * q^95 + 40 * q^97 + 4 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.04244 −0.466195 −0.233097 0.972453i $$-0.574886\pi$$
−0.233097 + 0.972453i $$0.574886\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.93089 1.78823 0.894116 0.447836i $$-0.147805\pi$$
0.894116 + 0.447836i $$0.147805\pi$$
$$12$$ 0 0
$$13$$ −0.888450 −0.246412 −0.123206 0.992381i $$-0.539318\pi$$
−0.123206 + 0.992381i $$0.539318\pi$$
$$14$$ 0 0
$$15$$ −1.04244 −0.269158
$$16$$ 0 0
$$17$$ 4.51668 1.09546 0.547728 0.836657i $$-0.315493\pi$$
0.547728 + 0.836657i $$0.315493\pi$$
$$18$$ 0 0
$$19$$ 3.47424 0.797045 0.398522 0.917159i $$-0.369523\pi$$
0.398522 + 0.917159i $$0.369523\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.93089 −0.402619 −0.201310 0.979528i $$-0.564520\pi$$
−0.201310 + 0.979528i $$0.564520\pi$$
$$24$$ 0 0
$$25$$ −3.91331 −0.782663
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −8.38755 −1.55753 −0.778764 0.627316i $$-0.784153\pi$$
−0.778764 + 0.627316i $$0.784153\pi$$
$$30$$ 0 0
$$31$$ 5.13109 0.921571 0.460786 0.887511i $$-0.347568\pi$$
0.460786 + 0.887511i $$0.347568\pi$$
$$32$$ 0 0
$$33$$ 5.93089 1.03244
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.65685 0.929981 0.464991 0.885316i $$-0.346058\pi$$
0.464991 + 0.885316i $$0.346058\pi$$
$$38$$ 0 0
$$39$$ −0.888450 −0.142266
$$40$$ 0 0
$$41$$ 8.39663 1.31133 0.655667 0.755050i $$-0.272388\pi$$
0.655667 + 0.755050i $$0.272388\pi$$
$$42$$ 0 0
$$43$$ −0.743541 −0.113389 −0.0566945 0.998392i $$-0.518056\pi$$
−0.0566945 + 0.998392i $$0.518056\pi$$
$$44$$ 0 0
$$45$$ −1.04244 −0.155398
$$46$$ 0 0
$$47$$ −4.21778 −0.615226 −0.307613 0.951512i $$-0.599530\pi$$
−0.307613 + 0.951512i $$0.599530\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 4.51668 0.632462
$$52$$ 0 0
$$53$$ −6.91331 −0.949617 −0.474808 0.880089i $$-0.657483\pi$$
−0.474808 + 0.880089i $$0.657483\pi$$
$$54$$ 0 0
$$55$$ −6.18262 −0.833664
$$56$$ 0 0
$$57$$ 3.47424 0.460174
$$58$$ 0 0
$$59$$ 5.43908 0.708107 0.354054 0.935225i $$-0.384803\pi$$
0.354054 + 0.935225i $$0.384803\pi$$
$$60$$ 0 0
$$61$$ 10.9733 1.40499 0.702496 0.711688i $$-0.252069\pi$$
0.702496 + 0.711688i $$0.252069\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.926159 0.114876
$$66$$ 0 0
$$67$$ −6.20493 −0.758053 −0.379027 0.925386i $$-0.623741\pi$$
−0.379027 + 0.925386i $$0.623741\pi$$
$$68$$ 0 0
$$69$$ −1.93089 −0.232452
$$70$$ 0 0
$$71$$ 3.72596 0.442190 0.221095 0.975252i $$-0.429037\pi$$
0.221095 + 0.975252i $$0.429037\pi$$
$$72$$ 0 0
$$73$$ −3.49910 −0.409539 −0.204769 0.978810i $$-0.565644\pi$$
−0.204769 + 0.978810i $$0.565644\pi$$
$$74$$ 0 0
$$75$$ −3.91331 −0.451870
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.6053 1.41821 0.709105 0.705103i $$-0.249099\pi$$
0.709105 + 0.705103i $$0.249099\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.94847 −0.323637 −0.161819 0.986821i $$-0.551736\pi$$
−0.161819 + 0.986821i $$0.551736\pi$$
$$84$$ 0 0
$$85$$ −4.70838 −0.510696
$$86$$ 0 0
$$87$$ −8.38755 −0.899240
$$88$$ 0 0
$$89$$ −5.00196 −0.530207 −0.265103 0.964220i $$-0.585406\pi$$
−0.265103 + 0.964220i $$0.585406\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 5.13109 0.532069
$$94$$ 0 0
$$95$$ −3.62169 −0.371578
$$96$$ 0 0
$$97$$ 10.1578 1.03136 0.515682 0.856780i $$-0.327539\pi$$
0.515682 + 0.856780i $$0.327539\pi$$
$$98$$ 0 0
$$99$$ 5.93089 0.596077
$$100$$ 0 0
$$101$$ 1.78598 0.177712 0.0888560 0.996044i $$-0.471679\pi$$
0.0888560 + 0.996044i $$0.471679\pi$$
$$102$$ 0 0
$$103$$ −18.5925 −1.83197 −0.915986 0.401211i $$-0.868590\pi$$
−0.915986 + 0.401211i $$0.868590\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.23888 0.603135 0.301568 0.953445i $$-0.402490\pi$$
0.301568 + 0.953445i $$0.402490\pi$$
$$108$$ 0 0
$$109$$ −4.74354 −0.454349 −0.227174 0.973854i $$-0.572949\pi$$
−0.227174 + 0.973854i $$0.572949\pi$$
$$110$$ 0 0
$$111$$ 5.65685 0.536925
$$112$$ 0 0
$$113$$ −13.3137 −1.25245 −0.626224 0.779643i $$-0.715401\pi$$
−0.626224 + 0.779643i $$0.715401\pi$$
$$114$$ 0 0
$$115$$ 2.01285 0.187699
$$116$$ 0 0
$$117$$ −0.888450 −0.0821373
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 24.1755 2.19777
$$122$$ 0 0
$$123$$ 8.39663 0.757099
$$124$$ 0 0
$$125$$ 9.29162 0.831068
$$126$$ 0 0
$$127$$ −0.307985 −0.0273292 −0.0136646 0.999907i $$-0.504350\pi$$
−0.0136646 + 0.999907i $$0.504350\pi$$
$$128$$ 0 0
$$129$$ −0.743541 −0.0654652
$$130$$ 0 0
$$131$$ 9.46139 0.826646 0.413323 0.910585i $$-0.364368\pi$$
0.413323 + 0.910585i $$0.364368\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −1.04244 −0.0897192
$$136$$ 0 0
$$137$$ −1.87463 −0.160161 −0.0800803 0.996788i $$-0.525518\pi$$
−0.0800803 + 0.996788i $$0.525518\pi$$
$$138$$ 0 0
$$139$$ −7.31371 −0.620341 −0.310170 0.950681i $$-0.600386\pi$$
−0.310170 + 0.950681i $$0.600386\pi$$
$$140$$ 0 0
$$141$$ −4.21778 −0.355201
$$142$$ 0 0
$$143$$ −5.26930 −0.440641
$$144$$ 0 0
$$145$$ 8.74354 0.726112
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 22.2270 1.82091 0.910454 0.413610i $$-0.135732\pi$$
0.910454 + 0.413610i $$0.135732\pi$$
$$150$$ 0 0
$$151$$ −14.4004 −1.17189 −0.585944 0.810352i $$-0.699276\pi$$
−0.585944 + 0.810352i $$0.699276\pi$$
$$152$$ 0 0
$$153$$ 4.51668 0.365152
$$154$$ 0 0
$$155$$ −5.34887 −0.429632
$$156$$ 0 0
$$157$$ 17.9715 1.43428 0.717142 0.696927i $$-0.245450\pi$$
0.717142 + 0.696927i $$0.245450\pi$$
$$158$$ 0 0
$$159$$ −6.91331 −0.548261
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −23.1755 −1.81524 −0.907622 0.419787i $$-0.862105\pi$$
−0.907622 + 0.419787i $$0.862105\pi$$
$$164$$ 0 0
$$165$$ −6.18262 −0.481316
$$166$$ 0 0
$$167$$ 12.9929 1.00542 0.502710 0.864455i $$-0.332337\pi$$
0.502710 + 0.864455i $$0.332337\pi$$
$$168$$ 0 0
$$169$$ −12.2107 −0.939281
$$170$$ 0 0
$$171$$ 3.47424 0.265682
$$172$$ 0 0
$$173$$ 23.1846 1.76269 0.881345 0.472472i $$-0.156638\pi$$
0.881345 + 0.472472i $$0.156638\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.43908 0.408826
$$178$$ 0 0
$$179$$ 24.9366 1.86385 0.931925 0.362651i $$-0.118128\pi$$
0.931925 + 0.362651i $$0.118128\pi$$
$$180$$ 0 0
$$181$$ 22.3498 1.66125 0.830625 0.556832i $$-0.187983\pi$$
0.830625 + 0.556832i $$0.187983\pi$$
$$182$$ 0 0
$$183$$ 10.9733 0.811172
$$184$$ 0 0
$$185$$ −5.89695 −0.433552
$$186$$ 0 0
$$187$$ 26.7879 1.95893
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.03967 0.509373 0.254686 0.967024i $$-0.418028\pi$$
0.254686 + 0.967024i $$0.418028\pi$$
$$192$$ 0 0
$$193$$ 12.3747 0.890751 0.445375 0.895344i $$-0.353070\pi$$
0.445375 + 0.895344i $$0.353070\pi$$
$$194$$ 0 0
$$195$$ 0.926159 0.0663236
$$196$$ 0 0
$$197$$ 13.4262 0.956579 0.478290 0.878202i $$-0.341257\pi$$
0.478290 + 0.878202i $$0.341257\pi$$
$$198$$ 0 0
$$199$$ 9.82663 0.696591 0.348296 0.937385i $$-0.386761\pi$$
0.348296 + 0.937385i $$0.386761\pi$$
$$200$$ 0 0
$$201$$ −6.20493 −0.437662
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −8.75301 −0.611337
$$206$$ 0 0
$$207$$ −1.93089 −0.134206
$$208$$ 0 0
$$209$$ 20.6053 1.42530
$$210$$ 0 0
$$211$$ −20.1698 −1.38854 −0.694272 0.719713i $$-0.744274\pi$$
−0.694272 + 0.719713i $$0.744274\pi$$
$$212$$ 0 0
$$213$$ 3.72596 0.255299
$$214$$ 0 0
$$215$$ 0.775099 0.0528613
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −3.49910 −0.236447
$$220$$ 0 0
$$221$$ −4.01285 −0.269933
$$222$$ 0 0
$$223$$ −27.0373 −1.81055 −0.905275 0.424826i $$-0.860335\pi$$
−0.905275 + 0.424826i $$0.860335\pi$$
$$224$$ 0 0
$$225$$ −3.91331 −0.260888
$$226$$ 0 0
$$227$$ 8.02231 0.532460 0.266230 0.963910i $$-0.414222\pi$$
0.266230 + 0.963910i $$0.414222\pi$$
$$228$$ 0 0
$$229$$ −12.4920 −0.825493 −0.412747 0.910846i $$-0.635430\pi$$
−0.412747 + 0.910846i $$0.635430\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 19.7013 1.29067 0.645336 0.763899i $$-0.276717\pi$$
0.645336 + 0.763899i $$0.276717\pi$$
$$234$$ 0 0
$$235$$ 4.39679 0.286815
$$236$$ 0 0
$$237$$ 12.6053 0.818804
$$238$$ 0 0
$$239$$ −1.93089 −0.124899 −0.0624496 0.998048i $$-0.519891\pi$$
−0.0624496 + 0.998048i $$0.519891\pi$$
$$240$$ 0 0
$$241$$ −15.1026 −0.972846 −0.486423 0.873724i $$-0.661699\pi$$
−0.486423 + 0.873724i $$0.661699\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.08669 −0.196401
$$248$$ 0 0
$$249$$ −2.94847 −0.186852
$$250$$ 0 0
$$251$$ 29.5982 1.86822 0.934111 0.356982i $$-0.116194\pi$$
0.934111 + 0.356982i $$0.116194\pi$$
$$252$$ 0 0
$$253$$ −11.4519 −0.719976
$$254$$ 0 0
$$255$$ −4.70838 −0.294850
$$256$$ 0 0
$$257$$ 13.8580 0.864440 0.432220 0.901768i $$-0.357730\pi$$
0.432220 + 0.901768i $$0.357730\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.38755 −0.519176
$$262$$ 0 0
$$263$$ 2.78696 0.171851 0.0859255 0.996302i $$-0.472615\pi$$
0.0859255 + 0.996302i $$0.472615\pi$$
$$264$$ 0 0
$$265$$ 7.20673 0.442706
$$266$$ 0 0
$$267$$ −5.00196 −0.306115
$$268$$ 0 0
$$269$$ −6.44104 −0.392717 −0.196358 0.980532i $$-0.562912\pi$$
−0.196358 + 0.980532i $$0.562912\pi$$
$$270$$ 0 0
$$271$$ 20.8547 1.26683 0.633415 0.773812i $$-0.281652\pi$$
0.633415 + 0.773812i $$0.281652\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −23.2094 −1.39958
$$276$$ 0 0
$$277$$ −27.0724 −1.62663 −0.813313 0.581827i $$-0.802338\pi$$
−0.813313 + 0.581827i $$0.802338\pi$$
$$278$$ 0 0
$$279$$ 5.13109 0.307190
$$280$$ 0 0
$$281$$ 6.28450 0.374902 0.187451 0.982274i $$-0.439977\pi$$
0.187451 + 0.982274i $$0.439977\pi$$
$$282$$ 0 0
$$283$$ 5.57729 0.331535 0.165768 0.986165i $$-0.446990\pi$$
0.165768 + 0.986165i $$0.446990\pi$$
$$284$$ 0 0
$$285$$ −3.62169 −0.214531
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 3.40040 0.200023
$$290$$ 0 0
$$291$$ 10.1578 0.595458
$$292$$ 0 0
$$293$$ −12.6460 −0.738785 −0.369393 0.929273i $$-0.620434\pi$$
−0.369393 + 0.929273i $$0.620434\pi$$
$$294$$ 0 0
$$295$$ −5.66993 −0.330116
$$296$$ 0 0
$$297$$ 5.93089 0.344145
$$298$$ 0 0
$$299$$ 1.71550 0.0992101
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1.78598 0.102602
$$304$$ 0 0
$$305$$ −11.4391 −0.655000
$$306$$ 0 0
$$307$$ −16.2750 −0.928865 −0.464432 0.885609i $$-0.653742\pi$$
−0.464432 + 0.885609i $$0.653742\pi$$
$$308$$ 0 0
$$309$$ −18.5925 −1.05769
$$310$$ 0 0
$$311$$ −28.9929 −1.64404 −0.822018 0.569462i $$-0.807152\pi$$
−0.822018 + 0.569462i $$0.807152\pi$$
$$312$$ 0 0
$$313$$ 13.5840 0.767812 0.383906 0.923372i $$-0.374579\pi$$
0.383906 + 0.923372i $$0.374579\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.1240 1.35494 0.677469 0.735552i $$-0.263077\pi$$
0.677469 + 0.735552i $$0.263077\pi$$
$$318$$ 0 0
$$319$$ −49.7457 −2.78522
$$320$$ 0 0
$$321$$ 6.23888 0.348220
$$322$$ 0 0
$$323$$ 15.6920 0.873127
$$324$$ 0 0
$$325$$ 3.47678 0.192857
$$326$$ 0 0
$$327$$ −4.74354 −0.262318
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 7.72357 0.424526 0.212263 0.977213i $$-0.431917\pi$$
0.212263 + 0.977213i $$0.431917\pi$$
$$332$$ 0 0
$$333$$ 5.65685 0.309994
$$334$$ 0 0
$$335$$ 6.46829 0.353400
$$336$$ 0 0
$$337$$ 1.05153 0.0572803 0.0286401 0.999590i $$-0.490882\pi$$
0.0286401 + 0.999590i $$0.490882\pi$$
$$338$$ 0 0
$$339$$ −13.3137 −0.723101
$$340$$ 0 0
$$341$$ 30.4320 1.64798
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2.01285 0.108368
$$346$$ 0 0
$$347$$ 1.52103 0.0816531 0.0408265 0.999166i $$-0.487001\pi$$
0.0408265 + 0.999166i $$0.487001\pi$$
$$348$$ 0 0
$$349$$ 18.1670 0.972457 0.486229 0.873832i $$-0.338372\pi$$
0.486229 + 0.873832i $$0.338372\pi$$
$$350$$ 0 0
$$351$$ −0.888450 −0.0474220
$$352$$ 0 0
$$353$$ 11.2069 0.596483 0.298241 0.954490i $$-0.403600\pi$$
0.298241 + 0.954490i $$0.403600\pi$$
$$354$$ 0 0
$$355$$ −3.88410 −0.206147
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8.44381 0.445647 0.222824 0.974859i $$-0.428473\pi$$
0.222824 + 0.974859i $$0.428473\pi$$
$$360$$ 0 0
$$361$$ −6.92968 −0.364720
$$362$$ 0 0
$$363$$ 24.1755 1.26888
$$364$$ 0 0
$$365$$ 3.64761 0.190925
$$366$$ 0 0
$$367$$ −23.2880 −1.21562 −0.607812 0.794081i $$-0.707953\pi$$
−0.607812 + 0.794081i $$0.707953\pi$$
$$368$$ 0 0
$$369$$ 8.39663 0.437111
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 35.5502 1.84072 0.920360 0.391073i $$-0.127896\pi$$
0.920360 + 0.391073i $$0.127896\pi$$
$$374$$ 0 0
$$375$$ 9.29162 0.479817
$$376$$ 0 0
$$377$$ 7.45192 0.383794
$$378$$ 0 0
$$379$$ −13.4168 −0.689173 −0.344586 0.938755i $$-0.611981\pi$$
−0.344586 + 0.938755i $$0.611981\pi$$
$$380$$ 0 0
$$381$$ −0.307985 −0.0157785
$$382$$ 0 0
$$383$$ −35.6533 −1.82180 −0.910898 0.412632i $$-0.864610\pi$$
−0.910898 + 0.412632i $$0.864610\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.743541 −0.0377963
$$388$$ 0 0
$$389$$ 23.3360 1.18318 0.591592 0.806238i $$-0.298500\pi$$
0.591592 + 0.806238i $$0.298500\pi$$
$$390$$ 0 0
$$391$$ −8.72123 −0.441051
$$392$$ 0 0
$$393$$ 9.46139 0.477264
$$394$$ 0 0
$$395$$ −13.1403 −0.661162
$$396$$ 0 0
$$397$$ −9.63199 −0.483416 −0.241708 0.970349i $$-0.577708\pi$$
−0.241708 + 0.970349i $$0.577708\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7.61245 −0.380148 −0.190074 0.981770i $$-0.560873\pi$$
−0.190074 + 0.981770i $$0.560873\pi$$
$$402$$ 0 0
$$403$$ −4.55872 −0.227086
$$404$$ 0 0
$$405$$ −1.04244 −0.0517994
$$406$$ 0 0
$$407$$ 33.5502 1.66302
$$408$$ 0 0
$$409$$ 34.6564 1.71365 0.856825 0.515607i $$-0.172434\pi$$
0.856825 + 0.515607i $$0.172434\pi$$
$$410$$ 0 0
$$411$$ −1.87463 −0.0924688
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3.07362 0.150878
$$416$$ 0 0
$$417$$ −7.31371 −0.358154
$$418$$ 0 0
$$419$$ −6.53523 −0.319267 −0.159633 0.987176i $$-0.551031\pi$$
−0.159633 + 0.987176i $$0.551031\pi$$
$$420$$ 0 0
$$421$$ −10.4778 −0.510655 −0.255327 0.966855i $$-0.582183\pi$$
−0.255327 + 0.966855i $$0.582183\pi$$
$$422$$ 0 0
$$423$$ −4.21778 −0.205075
$$424$$ 0 0
$$425$$ −17.6752 −0.857372
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −5.26930 −0.254404
$$430$$ 0 0
$$431$$ −28.9682 −1.39535 −0.697674 0.716415i $$-0.745782\pi$$
−0.697674 + 0.716415i $$0.745782\pi$$
$$432$$ 0 0
$$433$$ −7.31116 −0.351352 −0.175676 0.984448i $$-0.556211\pi$$
−0.175676 + 0.984448i $$0.556211\pi$$
$$434$$ 0 0
$$435$$ 8.74354 0.419221
$$436$$ 0 0
$$437$$ −6.70838 −0.320905
$$438$$ 0 0
$$439$$ −12.6863 −0.605484 −0.302742 0.953073i $$-0.597902\pi$$
−0.302742 + 0.953073i $$0.597902\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16.6136 −0.789335 −0.394668 0.918824i $$-0.629140\pi$$
−0.394668 + 0.918824i $$0.629140\pi$$
$$444$$ 0 0
$$445$$ 5.21426 0.247180
$$446$$ 0 0
$$447$$ 22.2270 1.05130
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 49.7995 2.34497
$$452$$ 0 0
$$453$$ −14.4004 −0.676590
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9.72357 −0.454850 −0.227425 0.973796i $$-0.573031\pi$$
−0.227425 + 0.973796i $$0.573031\pi$$
$$458$$ 0 0
$$459$$ 4.51668 0.210821
$$460$$ 0 0
$$461$$ −17.9451 −0.835787 −0.417894 0.908496i $$-0.637232\pi$$
−0.417894 + 0.908496i $$0.637232\pi$$
$$462$$ 0 0
$$463$$ 2.37470 0.110362 0.0551809 0.998476i $$-0.482426\pi$$
0.0551809 + 0.998476i $$0.482426\pi$$
$$464$$ 0 0
$$465$$ −5.34887 −0.248048
$$466$$ 0 0
$$467$$ 15.3617 0.710855 0.355428 0.934704i $$-0.384335\pi$$
0.355428 + 0.934704i $$0.384335\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 17.9715 0.828085
$$472$$ 0 0
$$473$$ −4.40986 −0.202766
$$474$$ 0 0
$$475$$ −13.5958 −0.623817
$$476$$ 0 0
$$477$$ −6.91331 −0.316539
$$478$$ 0 0
$$479$$ −14.0444 −0.641705 −0.320853 0.947129i $$-0.603969\pi$$
−0.320853 + 0.947129i $$0.603969\pi$$
$$480$$ 0 0
$$481$$ −5.02583 −0.229158
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.5889 −0.480816
$$486$$ 0 0
$$487$$ −30.4004 −1.37757 −0.688787 0.724964i $$-0.741856\pi$$
−0.688787 + 0.724964i $$0.741856\pi$$
$$488$$ 0 0
$$489$$ −23.1755 −1.04803
$$490$$ 0 0
$$491$$ −25.5526 −1.15317 −0.576586 0.817036i $$-0.695615\pi$$
−0.576586 + 0.817036i $$0.695615\pi$$
$$492$$ 0 0
$$493$$ −37.8839 −1.70620
$$494$$ 0 0
$$495$$ −6.18262 −0.277888
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −16.2270 −0.726421 −0.363211 0.931707i $$-0.618319\pi$$
−0.363211 + 0.931707i $$0.618319\pi$$
$$500$$ 0 0
$$501$$ 12.9929 0.580479
$$502$$ 0 0
$$503$$ −39.2880 −1.75177 −0.875883 0.482523i $$-0.839720\pi$$
−0.875883 + 0.482523i $$0.839720\pi$$
$$504$$ 0 0
$$505$$ −1.86179 −0.0828484
$$506$$ 0 0
$$507$$ −12.2107 −0.542294
$$508$$ 0 0
$$509$$ −2.93187 −0.129953 −0.0649763 0.997887i $$-0.520697\pi$$
−0.0649763 + 0.997887i $$0.520697\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 3.47424 0.153391
$$514$$ 0 0
$$515$$ 19.3816 0.854055
$$516$$ 0 0
$$517$$ −25.0152 −1.10017
$$518$$ 0 0
$$519$$ 23.1846 1.01769
$$520$$ 0 0
$$521$$ 26.3508 1.15445 0.577225 0.816585i $$-0.304135\pi$$
0.577225 + 0.816585i $$0.304135\pi$$
$$522$$ 0 0
$$523$$ 40.7048 1.77990 0.889948 0.456062i $$-0.150741\pi$$
0.889948 + 0.456062i $$0.150741\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 23.1755 1.00954
$$528$$ 0 0
$$529$$ −19.2717 −0.837898
$$530$$ 0 0
$$531$$ 5.43908 0.236036
$$532$$ 0 0
$$533$$ −7.45999 −0.323128
$$534$$ 0 0
$$535$$ −6.50367 −0.281178
$$536$$ 0 0
$$537$$ 24.9366 1.07609
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −15.3489 −0.659899 −0.329950 0.943999i $$-0.607032\pi$$
−0.329950 + 0.943999i $$0.607032\pi$$
$$542$$ 0 0
$$543$$ 22.3498 0.959123
$$544$$ 0 0
$$545$$ 4.94487 0.211815
$$546$$ 0 0
$$547$$ −34.2258 −1.46339 −0.731696 0.681631i $$-0.761271\pi$$
−0.731696 + 0.681631i $$0.761271\pi$$
$$548$$ 0 0
$$549$$ 10.9733 0.468331
$$550$$ 0 0
$$551$$ −29.1403 −1.24142
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −5.89695 −0.250311
$$556$$ 0 0
$$557$$ −18.6369 −0.789670 −0.394835 0.918752i $$-0.629198\pi$$
−0.394835 + 0.918752i $$0.629198\pi$$
$$558$$ 0 0
$$559$$ 0.660600 0.0279404
$$560$$ 0 0
$$561$$ 26.7879 1.13099
$$562$$ 0 0
$$563$$ −2.90047 −0.122240 −0.0611201 0.998130i $$-0.519467\pi$$
−0.0611201 + 0.998130i $$0.519467\pi$$
$$564$$ 0 0
$$565$$ 13.8788 0.583885
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −27.0853 −1.13547 −0.567737 0.823210i $$-0.692181\pi$$
−0.567737 + 0.823210i $$0.692181\pi$$
$$570$$ 0 0
$$571$$ −3.39467 −0.142063 −0.0710313 0.997474i $$-0.522629\pi$$
−0.0710313 + 0.997474i $$0.522629\pi$$
$$572$$ 0 0
$$573$$ 7.03967 0.294086
$$574$$ 0 0
$$575$$ 7.55619 0.315115
$$576$$ 0 0
$$577$$ −22.5545 −0.938958 −0.469479 0.882944i $$-0.655558\pi$$
−0.469479 + 0.882944i $$0.655558\pi$$
$$578$$ 0 0
$$579$$ 12.3747 0.514275
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −41.0021 −1.69813
$$584$$ 0 0
$$585$$ 0.926159 0.0382920
$$586$$ 0 0
$$587$$ 29.4391 1.21508 0.607540 0.794289i $$-0.292156\pi$$
0.607540 + 0.794289i $$0.292156\pi$$
$$588$$ 0 0
$$589$$ 17.8266 0.734533
$$590$$ 0 0
$$591$$ 13.4262 0.552281
$$592$$ 0 0
$$593$$ −33.9153 −1.39273 −0.696367 0.717686i $$-0.745201\pi$$
−0.696367 + 0.717686i $$0.745201\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 9.82663 0.402177
$$598$$ 0 0
$$599$$ 28.0655 1.14673 0.573363 0.819302i $$-0.305639\pi$$
0.573363 + 0.819302i $$0.305639\pi$$
$$600$$ 0 0
$$601$$ −17.0434 −0.695216 −0.347608 0.937640i $$-0.613006\pi$$
−0.347608 + 0.937640i $$0.613006\pi$$
$$602$$ 0 0
$$603$$ −6.20493 −0.252684
$$604$$ 0 0
$$605$$ −25.2016 −1.02459
$$606$$ 0 0
$$607$$ 36.8639 1.49626 0.748130 0.663552i $$-0.230952\pi$$
0.748130 + 0.663552i $$0.230952\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.74729 0.151599
$$612$$ 0 0
$$613$$ 25.4483 1.02785 0.513924 0.857836i $$-0.328191\pi$$
0.513924 + 0.857836i $$0.328191\pi$$
$$614$$ 0 0
$$615$$ −8.75301 −0.352955
$$616$$ 0 0
$$617$$ −35.3103 −1.42154 −0.710770 0.703424i $$-0.751653\pi$$
−0.710770 + 0.703424i $$0.751653\pi$$
$$618$$ 0 0
$$619$$ 40.9599 1.64632 0.823159 0.567811i $$-0.192209\pi$$
0.823159 + 0.567811i $$0.192209\pi$$
$$620$$ 0 0
$$621$$ −1.93089 −0.0774841
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 9.88058 0.395223
$$626$$ 0 0
$$627$$ 20.6053 0.822898
$$628$$ 0 0
$$629$$ 25.5502 1.01875
$$630$$ 0 0
$$631$$ −21.9648 −0.874406 −0.437203 0.899363i $$-0.644031\pi$$
−0.437203 + 0.899363i $$0.644031\pi$$
$$632$$ 0 0
$$633$$ −20.1698 −0.801676
$$634$$ 0 0
$$635$$ 0.321057 0.0127407
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 3.72596 0.147397
$$640$$ 0 0
$$641$$ −31.9520 −1.26203 −0.631014 0.775772i $$-0.717361\pi$$
−0.631014 + 0.775772i $$0.717361\pi$$
$$642$$ 0 0
$$643$$ −41.0501 −1.61886 −0.809430 0.587217i $$-0.800223\pi$$
−0.809430 + 0.587217i $$0.800223\pi$$
$$644$$ 0 0
$$645$$ 0.775099 0.0305195
$$646$$ 0 0
$$647$$ 9.67917 0.380527 0.190264 0.981733i $$-0.439066\pi$$
0.190264 + 0.981733i $$0.439066\pi$$
$$648$$ 0 0
$$649$$ 32.2586 1.26626
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −17.5982 −0.688671 −0.344336 0.938847i $$-0.611896\pi$$
−0.344336 + 0.938847i $$0.611896\pi$$
$$654$$ 0 0
$$655$$ −9.86296 −0.385378
$$656$$ 0 0
$$657$$ −3.49910 −0.136513
$$658$$ 0 0
$$659$$ 4.10427 0.159880 0.0799398 0.996800i $$-0.474527\pi$$
0.0799398 + 0.996800i $$0.474527\pi$$
$$660$$ 0 0
$$661$$ 47.0124 1.82857 0.914286 0.405070i $$-0.132753\pi$$
0.914286 + 0.405070i $$0.132753\pi$$
$$662$$ 0 0
$$663$$ −4.01285 −0.155846
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.1955 0.627091
$$668$$ 0 0
$$669$$ −27.0373 −1.04532
$$670$$ 0 0
$$671$$ 65.0817 2.51245
$$672$$ 0 0
$$673$$ 21.7587 0.838738 0.419369 0.907816i $$-0.362251\pi$$
0.419369 + 0.907816i $$0.362251\pi$$
$$674$$ 0 0
$$675$$ −3.91331 −0.150623
$$676$$ 0 0
$$677$$ −23.6829 −0.910209 −0.455105 0.890438i $$-0.650398\pi$$
−0.455105 + 0.890438i $$0.650398\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 8.02231 0.307416
$$682$$ 0 0
$$683$$ 32.0655 1.22695 0.613476 0.789713i $$-0.289771\pi$$
0.613476 + 0.789713i $$0.289771\pi$$
$$684$$ 0 0
$$685$$ 1.95420 0.0746660
$$686$$ 0 0
$$687$$ −12.4920 −0.476599
$$688$$ 0 0
$$689$$ 6.14214 0.233997
$$690$$ 0 0
$$691$$ −14.8782 −0.565992 −0.282996 0.959121i $$-0.591328\pi$$
−0.282996 + 0.959121i $$0.591328\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 7.62412 0.289199
$$696$$ 0 0
$$697$$ 37.9249 1.43651
$$698$$ 0 0
$$699$$ 19.7013 0.745170
$$700$$ 0 0
$$701$$ 31.0150 1.17142 0.585710 0.810521i $$-0.300816\pi$$
0.585710 + 0.810521i $$0.300816\pi$$
$$702$$ 0 0
$$703$$ 19.6533 0.741236
$$704$$ 0 0
$$705$$ 4.39679 0.165593
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −51.7105 −1.94203 −0.971014 0.239021i $$-0.923173\pi$$
−0.971014 + 0.239021i $$0.923173\pi$$
$$710$$ 0 0
$$711$$ 12.6053 0.472737
$$712$$ 0 0
$$713$$ −9.90759 −0.371042
$$714$$ 0 0
$$715$$ 5.49295 0.205425
$$716$$ 0 0
$$717$$ −1.93089 −0.0721105
$$718$$ 0 0
$$719$$ 19.7493 0.736523 0.368262 0.929722i $$-0.379953\pi$$
0.368262 + 0.929722i $$0.379953\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −15.1026 −0.561673
$$724$$ 0 0
$$725$$ 32.8231 1.21902
$$726$$ 0 0
$$727$$ −51.3230 −1.90346 −0.951731 0.306932i $$-0.900698\pi$$
−0.951731 + 0.306932i $$0.900698\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −3.35834 −0.124213
$$732$$ 0 0
$$733$$ −29.1230 −1.07568 −0.537841 0.843046i $$-0.680760\pi$$
−0.537841 + 0.843046i $$0.680760\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −36.8008 −1.35557
$$738$$ 0 0
$$739$$ 9.27855 0.341317 0.170658 0.985330i $$-0.445411\pi$$
0.170658 + 0.985330i $$0.445411\pi$$
$$740$$ 0 0
$$741$$ −3.08669 −0.113392
$$742$$ 0 0
$$743$$ 26.5935 0.975620 0.487810 0.872950i $$-0.337796\pi$$
0.487810 + 0.872950i $$0.337796\pi$$
$$744$$ 0 0
$$745$$ −23.1704 −0.848898
$$746$$ 0 0
$$747$$ −2.94847 −0.107879
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.2843 1.03211 0.516054 0.856556i $$-0.327400\pi$$
0.516054 + 0.856556i $$0.327400\pi$$
$$752$$ 0 0
$$753$$ 29.5982 1.07862
$$754$$ 0 0
$$755$$ 15.0116 0.546328
$$756$$ 0 0
$$757$$ 17.8839 0.650001 0.325000 0.945714i $$-0.394636\pi$$
0.325000 + 0.945714i $$0.394636\pi$$
$$758$$ 0 0
$$759$$ −11.4519 −0.415678
$$760$$ 0 0
$$761$$ −49.5263 −1.79533 −0.897664 0.440681i $$-0.854737\pi$$
−0.897664 + 0.440681i $$0.854737\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −4.70838 −0.170232
$$766$$ 0 0
$$767$$ −4.83235 −0.174486
$$768$$ 0 0
$$769$$ −45.0675 −1.62517 −0.812587 0.582840i $$-0.801942\pi$$
−0.812587 + 0.582840i $$0.801942\pi$$
$$770$$ 0 0
$$771$$ 13.8580 0.499085
$$772$$ 0 0
$$773$$ 4.45051 0.160074 0.0800368 0.996792i $$-0.474496\pi$$
0.0800368 + 0.996792i $$0.474496\pi$$
$$774$$ 0 0
$$775$$ −20.0796 −0.721279
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 29.1719 1.04519
$$780$$ 0 0
$$781$$ 22.0983 0.790739
$$782$$ 0 0
$$783$$ −8.38755 −0.299747
$$784$$ 0 0
$$785$$ −18.7343 −0.668656
$$786$$ 0 0
$$787$$ 18.1919 0.648470 0.324235 0.945977i $$-0.394893\pi$$
0.324235 + 0.945977i $$0.394893\pi$$
$$788$$ 0 0
$$789$$ 2.78696 0.0992183
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −9.74926 −0.346207
$$794$$ 0 0
$$795$$ 7.20673 0.255597
$$796$$ 0 0
$$797$$ −23.8005 −0.843059 −0.421529 0.906815i $$-0.638507\pi$$
−0.421529 + 0.906815i $$0.638507\pi$$
$$798$$ 0 0
$$799$$ −19.0504 −0.673953
$$800$$ 0 0
$$801$$ −5.00196 −0.176736
$$802$$ 0 0
$$803$$ −20.7528 −0.732350
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.44104 −0.226735
$$808$$ 0 0
$$809$$ 51.8897 1.82435 0.912173 0.409805i $$-0.134403\pi$$
0.912173 + 0.409805i $$0.134403\pi$$
$$810$$ 0 0
$$811$$ −13.6462 −0.479183 −0.239592 0.970874i $$-0.577014\pi$$
−0.239592 + 0.970874i $$0.577014\pi$$
$$812$$ 0 0
$$813$$ 20.8547 0.731405
$$814$$ 0 0
$$815$$ 24.1591 0.846257
$$816$$ 0 0
$$817$$ −2.58324 −0.0903761
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 4.69576 0.163883 0.0819416 0.996637i $$-0.473888\pi$$
0.0819416 + 0.996637i $$0.473888\pi$$
$$822$$ 0 0
$$823$$ −15.8045 −0.550912 −0.275456 0.961314i $$-0.588829\pi$$
−0.275456 + 0.961314i $$0.588829\pi$$
$$824$$ 0 0
$$825$$ −23.2094 −0.808049
$$826$$ 0 0
$$827$$ 32.8221 1.14134 0.570668 0.821181i $$-0.306684\pi$$
0.570668 + 0.821181i $$0.306684\pi$$
$$828$$ 0 0
$$829$$ 3.71688 0.129092 0.0645462 0.997915i $$-0.479440\pi$$
0.0645462 + 0.997915i $$0.479440\pi$$
$$830$$ 0 0
$$831$$ −27.0724 −0.939133
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −13.5443 −0.468721
$$836$$ 0 0
$$837$$ 5.13109 0.177356
$$838$$ 0 0
$$839$$ 3.16625 0.109311 0.0546556 0.998505i $$-0.482594\pi$$
0.0546556 + 0.998505i $$0.482594\pi$$
$$840$$ 0 0
$$841$$ 41.3510 1.42590
$$842$$ 0 0
$$843$$ 6.28450 0.216450
$$844$$ 0 0
$$845$$ 12.7289 0.437888
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 5.57729 0.191412
$$850$$ 0 0
$$851$$ −10.9228 −0.374428
$$852$$ 0 0
$$853$$ 8.30404 0.284325 0.142162 0.989843i $$-0.454594\pi$$
0.142162 + 0.989843i $$0.454594\pi$$
$$854$$ 0 0
$$855$$ −3.62169 −0.123859
$$856$$ 0 0
$$857$$ 14.3117 0.488880 0.244440 0.969664i $$-0.421396\pi$$
0.244440 + 0.969664i $$0.421396\pi$$
$$858$$ 0 0
$$859$$ 34.1463 1.16506 0.582528 0.812811i $$-0.302064\pi$$
0.582528 + 0.812811i $$0.302064\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 39.0397 1.32893 0.664463 0.747321i $$-0.268660\pi$$
0.664463 + 0.747321i $$0.268660\pi$$
$$864$$ 0 0
$$865$$ −24.1686 −0.821757
$$866$$ 0 0
$$867$$ 3.40040 0.115483
$$868$$ 0 0
$$869$$ 74.7609 2.53609
$$870$$ 0 0
$$871$$ 5.51277 0.186793
$$872$$ 0 0
$$873$$ 10.1578 0.343788
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12.4225 −0.419478 −0.209739 0.977757i $$-0.567261\pi$$
−0.209739 + 0.977757i $$0.567261\pi$$
$$878$$ 0 0
$$879$$ −12.6460 −0.426538
$$880$$ 0 0
$$881$$ −32.4282 −1.09253 −0.546267 0.837611i $$-0.683952\pi$$
−0.546267 + 0.837611i $$0.683952\pi$$
$$882$$ 0 0
$$883$$ 21.8475 0.735228 0.367614 0.929978i $$-0.380175\pi$$
0.367614 + 0.929978i $$0.380175\pi$$
$$884$$ 0 0
$$885$$ −5.66993 −0.190592
$$886$$ 0 0
$$887$$ −54.2035 −1.81998 −0.909988 0.414634i $$-0.863910\pi$$
−0.909988 + 0.414634i $$0.863910\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 5.93089 0.198692
$$892$$ 0 0
$$893$$ −14.6536 −0.490363
$$894$$ 0 0
$$895$$ −25.9950 −0.868917
$$896$$ 0 0
$$897$$ 1.71550 0.0572790
$$898$$ 0 0
$$899$$ −43.0373 −1.43537
$$900$$ 0 0
$$901$$ −31.2252 −1.04026
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −23.2984 −0.774466
$$906$$ 0 0
$$907$$ −19.8172 −0.658018 −0.329009 0.944327i $$-0.606715\pi$$
−0.329009 + 0.944327i $$0.606715\pi$$
$$908$$ 0 0
$$909$$ 1.78598 0.0592374
$$910$$ 0 0
$$911$$ −32.5583 −1.07870 −0.539352 0.842080i $$-0.681331\pi$$
−0.539352 + 0.842080i $$0.681331\pi$$
$$912$$ 0 0
$$913$$ −17.4871 −0.578738
$$914$$ 0 0
$$915$$ −11.4391 −0.378164
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1.95420 −0.0644630 −0.0322315 0.999480i $$-0.510261\pi$$
−0.0322315 + 0.999480i $$0.510261\pi$$
$$920$$ 0 0
$$921$$ −16.2750 −0.536280
$$922$$ 0 0
$$923$$ −3.31033 −0.108961
$$924$$ 0 0
$$925$$ −22.1370 −0.727861
$$926$$ 0 0
$$927$$ −18.5925 −0.610657
$$928$$ 0 0
$$929$$ −23.2641 −0.763272 −0.381636 0.924313i $$-0.624639\pi$$
−0.381636 + 0.924313i $$0.624639\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −28.9929 −0.949184
$$934$$ 0 0
$$935$$ −27.9249 −0.913242
$$936$$ 0 0
$$937$$ −25.1244 −0.820778 −0.410389 0.911911i $$-0.634607\pi$$
−0.410389 + 0.911911i $$0.634607\pi$$
$$938$$ 0 0
$$939$$ 13.5840 0.443297
$$940$$ 0 0
$$941$$ 4.67721 0.152473 0.0762363 0.997090i $$-0.475710\pi$$
0.0762363 + 0.997090i $$0.475710\pi$$
$$942$$ 0 0
$$943$$ −16.2130 −0.527968
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.24460 −0.0404441 −0.0202221 0.999796i $$-0.506437\pi$$
−0.0202221 + 0.999796i $$0.506437\pi$$
$$948$$ 0 0
$$949$$ 3.10878 0.100915
$$950$$ 0 0
$$951$$ 24.1240 0.782273
$$952$$ 0 0
$$953$$ −33.0373 −1.07018 −0.535091 0.844794i $$-0.679723\pi$$
−0.535091 + 0.844794i $$0.679723\pi$$
$$954$$ 0 0
$$955$$ −7.33845 −0.237467
$$956$$ 0 0
$$957$$ −49.7457 −1.60805
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −4.67190 −0.150707
$$962$$ 0 0
$$963$$ 6.23888 0.201045
$$964$$ 0 0
$$965$$ −12.8999 −0.415263
$$966$$ 0 0
$$967$$ 13.9799 0.449563 0.224781 0.974409i $$-0.427833\pi$$
0.224781 + 0.974409i $$0.427833\pi$$
$$968$$ 0 0
$$969$$ 15.6920 0.504100
$$970$$ 0 0
$$971$$ 5.48708 0.176089 0.0880444 0.996117i $$-0.471938\pi$$
0.0880444 + 0.996117i $$0.471938\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 3.47678 0.111346
$$976$$ 0 0
$$977$$ −0.324434 −0.0103796 −0.00518978 0.999987i $$-0.501652\pi$$
−0.00518978 + 0.999987i $$0.501652\pi$$
$$978$$ 0 0
$$979$$ −29.6661 −0.948133
$$980$$ 0 0
$$981$$ −4.74354 −0.151450
$$982$$ 0 0
$$983$$ −22.5129 −0.718051 −0.359025 0.933328i $$-0.616891\pi$$
−0.359025 + 0.933328i $$0.616891\pi$$
$$984$$ 0 0
$$985$$ −13.9961 −0.445952
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.43570 0.0456526
$$990$$ 0 0
$$991$$ −43.1649 −1.37118 −0.685588 0.727989i $$-0.740455\pi$$
−0.685588 + 0.727989i $$0.740455\pi$$
$$992$$ 0 0
$$993$$ 7.72357 0.245100
$$994$$ 0 0
$$995$$ −10.2437 −0.324747
$$996$$ 0 0
$$997$$ −24.7743 −0.784609 −0.392305 0.919835i $$-0.628322\pi$$
−0.392305 + 0.919835i $$0.628322\pi$$
$$998$$ 0 0
$$999$$ 5.65685 0.178975
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 4704.2.a.bz.1.2 yes 4
4.3 odd 2 4704.2.a.bx.1.2 yes 4
7.6 odd 2 4704.2.a.bw.1.3 4
8.3 odd 2 9408.2.a.em.1.3 4
8.5 even 2 9408.2.a.ek.1.3 4
28.27 even 2 4704.2.a.by.1.3 yes 4
56.13 odd 2 9408.2.a.en.1.2 4
56.27 even 2 9408.2.a.el.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.3 4 7.6 odd 2
4704.2.a.bx.1.2 yes 4 4.3 odd 2
4704.2.a.by.1.3 yes 4 28.27 even 2
4704.2.a.bz.1.2 yes 4 1.1 even 1 trivial
9408.2.a.ek.1.3 4 8.5 even 2
9408.2.a.el.1.2 4 56.27 even 2
9408.2.a.em.1.3 4 8.3 odd 2
9408.2.a.en.1.2 4 56.13 odd 2