# Properties

 Label 9075.2.a.v.1.2 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -1.82843 q^{12} +5.65685 q^{13} -2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} +0.414214 q^{18} +1.17157 q^{19} -4.82843 q^{21} +4.00000 q^{23} -1.58579 q^{24} +2.34315 q^{26} +1.00000 q^{27} +8.82843 q^{28} -0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} -1.82843 q^{36} -0.343146 q^{37} +0.485281 q^{38} +5.65685 q^{39} +0.828427 q^{41} -2.00000 q^{42} -3.17157 q^{43} +1.65685 q^{46} +4.00000 q^{47} +3.00000 q^{48} +16.3137 q^{49} -6.82843 q^{51} -10.3431 q^{52} +13.3137 q^{53} +0.414214 q^{54} +7.65685 q^{56} +1.17157 q^{57} -0.343146 q^{58} -4.00000 q^{59} +0.343146 q^{61} -4.82843 q^{63} -4.17157 q^{64} -5.65685 q^{67} +12.4853 q^{68} +4.00000 q^{69} +13.6569 q^{71} -1.58579 q^{72} -11.3137 q^{73} -0.142136 q^{74} -2.14214 q^{76} +2.34315 q^{78} +8.48528 q^{79} +1.00000 q^{81} +0.343146 q^{82} -10.0000 q^{83} +8.82843 q^{84} -1.31371 q^{86} -0.828427 q^{87} -7.65685 q^{89} -27.3137 q^{91} -7.31371 q^{92} +1.65685 q^{94} +4.41421 q^{96} -0.343146 q^{97} +6.75736 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + 2q^{12} - 4q^{14} + 6q^{16} - 8q^{17} - 2q^{18} + 8q^{19} - 4q^{21} + 8q^{23} - 6q^{24} + 16q^{26} + 2q^{27} + 12q^{28} + 4q^{29} + 6q^{32} + 2q^{36} - 12q^{37} - 16q^{38} - 4q^{41} - 4q^{42} - 12q^{43} - 8q^{46} + 8q^{47} + 6q^{48} + 10q^{49} - 8q^{51} - 32q^{52} + 4q^{53} - 2q^{54} + 4q^{56} + 8q^{57} - 12q^{58} - 8q^{59} + 12q^{61} - 4q^{63} - 14q^{64} + 8q^{68} + 8q^{69} + 16q^{71} - 6q^{72} + 28q^{74} + 24q^{76} + 16q^{78} + 2q^{81} + 12q^{82} - 20q^{83} + 12q^{84} + 20q^{86} + 4q^{87} - 4q^{89} - 32q^{91} + 8q^{92} - 8q^{94} + 6q^{96} - 12q^{97} + 22q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.414214 0.292893 0.146447 0.989219i $$-0.453216\pi$$
0.146447 + 0.989219i $$0.453216\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.82843 −0.914214
$$5$$ 0 0
$$6$$ 0.414214 0.169102
$$7$$ −4.82843 −1.82497 −0.912487 0.409106i $$-0.865841\pi$$
−0.912487 + 0.409106i $$0.865841\pi$$
$$8$$ −1.58579 −0.560660
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ −1.82843 −0.527821
$$13$$ 5.65685 1.56893 0.784465 0.620174i $$-0.212938\pi$$
0.784465 + 0.620174i $$0.212938\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ −6.82843 −1.65614 −0.828068 0.560627i $$-0.810560\pi$$
−0.828068 + 0.560627i $$0.810560\pi$$
$$18$$ 0.414214 0.0976311
$$19$$ 1.17157 0.268777 0.134389 0.990929i $$-0.457093\pi$$
0.134389 + 0.990929i $$0.457093\pi$$
$$20$$ 0 0
$$21$$ −4.82843 −1.05365
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.58579 −0.323697
$$25$$ 0 0
$$26$$ 2.34315 0.459529
$$27$$ 1.00000 0.192450
$$28$$ 8.82843 1.66842
$$29$$ −0.828427 −0.153835 −0.0769175 0.997037i $$-0.524508\pi$$
−0.0769175 + 0.997037i $$0.524508\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 4.41421 0.780330
$$33$$ 0 0
$$34$$ −2.82843 −0.485071
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ −0.343146 −0.0564128 −0.0282064 0.999602i $$-0.508980\pi$$
−0.0282064 + 0.999602i $$0.508980\pi$$
$$38$$ 0.485281 0.0787230
$$39$$ 5.65685 0.905822
$$40$$ 0 0
$$41$$ 0.828427 0.129379 0.0646893 0.997905i $$-0.479394\pi$$
0.0646893 + 0.997905i $$0.479394\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −3.17157 −0.483660 −0.241830 0.970319i $$-0.577748\pi$$
−0.241830 + 0.970319i $$0.577748\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.65685 0.244290
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 3.00000 0.433013
$$49$$ 16.3137 2.33053
$$50$$ 0 0
$$51$$ −6.82843 −0.956171
$$52$$ −10.3431 −1.43434
$$53$$ 13.3137 1.82878 0.914389 0.404836i $$-0.132671\pi$$
0.914389 + 0.404836i $$0.132671\pi$$
$$54$$ 0.414214 0.0563673
$$55$$ 0 0
$$56$$ 7.65685 1.02319
$$57$$ 1.17157 0.155179
$$58$$ −0.343146 −0.0450572
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 0.343146 0.0439353 0.0219677 0.999759i $$-0.493007\pi$$
0.0219677 + 0.999759i $$0.493007\pi$$
$$62$$ 0 0
$$63$$ −4.82843 −0.608325
$$64$$ −4.17157 −0.521447
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −5.65685 −0.691095 −0.345547 0.938401i $$-0.612307\pi$$
−0.345547 + 0.938401i $$0.612307\pi$$
$$68$$ 12.4853 1.51406
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 13.6569 1.62077 0.810385 0.585897i $$-0.199258\pi$$
0.810385 + 0.585897i $$0.199258\pi$$
$$72$$ −1.58579 −0.186887
$$73$$ −11.3137 −1.32417 −0.662085 0.749429i $$-0.730328\pi$$
−0.662085 + 0.749429i $$0.730328\pi$$
$$74$$ −0.142136 −0.0165229
$$75$$ 0 0
$$76$$ −2.14214 −0.245720
$$77$$ 0 0
$$78$$ 2.34315 0.265309
$$79$$ 8.48528 0.954669 0.477334 0.878722i $$-0.341603\pi$$
0.477334 + 0.878722i $$0.341603\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0.343146 0.0378941
$$83$$ −10.0000 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$84$$ 8.82843 0.963260
$$85$$ 0 0
$$86$$ −1.31371 −0.141661
$$87$$ −0.828427 −0.0888167
$$88$$ 0 0
$$89$$ −7.65685 −0.811625 −0.405812 0.913956i $$-0.633011\pi$$
−0.405812 + 0.913956i $$0.633011\pi$$
$$90$$ 0 0
$$91$$ −27.3137 −2.86325
$$92$$ −7.31371 −0.762507
$$93$$ 0 0
$$94$$ 1.65685 0.170891
$$95$$ 0 0
$$96$$ 4.41421 0.450524
$$97$$ −0.343146 −0.0348412 −0.0174206 0.999848i $$-0.505545\pi$$
−0.0174206 + 0.999848i $$0.505545\pi$$
$$98$$ 6.75736 0.682596
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.82843 −0.480446 −0.240223 0.970718i $$-0.577221\pi$$
−0.240223 + 0.970718i $$0.577221\pi$$
$$102$$ −2.82843 −0.280056
$$103$$ −19.3137 −1.90304 −0.951518 0.307593i $$-0.900477\pi$$
−0.951518 + 0.307593i $$0.900477\pi$$
$$104$$ −8.97056 −0.879636
$$105$$ 0 0
$$106$$ 5.51472 0.535637
$$107$$ −5.31371 −0.513696 −0.256848 0.966452i $$-0.582684\pi$$
−0.256848 + 0.966452i $$0.582684\pi$$
$$108$$ −1.82843 −0.175940
$$109$$ 5.31371 0.508961 0.254480 0.967078i $$-0.418096\pi$$
0.254480 + 0.967078i $$0.418096\pi$$
$$110$$ 0 0
$$111$$ −0.343146 −0.0325700
$$112$$ −14.4853 −1.36873
$$113$$ −14.9706 −1.40831 −0.704156 0.710045i $$-0.748674\pi$$
−0.704156 + 0.710045i $$0.748674\pi$$
$$114$$ 0.485281 0.0454508
$$115$$ 0 0
$$116$$ 1.51472 0.140638
$$117$$ 5.65685 0.522976
$$118$$ −1.65685 −0.152526
$$119$$ 32.9706 3.02241
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0.142136 0.0128684
$$123$$ 0.828427 0.0746968
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 2.48528 0.220533 0.110267 0.993902i $$-0.464830\pi$$
0.110267 + 0.993902i $$0.464830\pi$$
$$128$$ −10.5563 −0.933058
$$129$$ −3.17157 −0.279241
$$130$$ 0 0
$$131$$ 19.3137 1.68745 0.843723 0.536778i $$-0.180359\pi$$
0.843723 + 0.536778i $$0.180359\pi$$
$$132$$ 0 0
$$133$$ −5.65685 −0.490511
$$134$$ −2.34315 −0.202417
$$135$$ 0 0
$$136$$ 10.8284 0.928530
$$137$$ −9.31371 −0.795724 −0.397862 0.917445i $$-0.630248\pi$$
−0.397862 + 0.917445i $$0.630248\pi$$
$$138$$ 1.65685 0.141041
$$139$$ 16.4853 1.39826 0.699132 0.714993i $$-0.253570\pi$$
0.699132 + 0.714993i $$0.253570\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 5.65685 0.474713
$$143$$ 0 0
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ −4.68629 −0.387840
$$147$$ 16.3137 1.34553
$$148$$ 0.627417 0.0515734
$$149$$ −18.4853 −1.51437 −0.757187 0.653199i $$-0.773427\pi$$
−0.757187 + 0.653199i $$0.773427\pi$$
$$150$$ 0 0
$$151$$ 0.485281 0.0394916 0.0197458 0.999805i $$-0.493714\pi$$
0.0197458 + 0.999805i $$0.493714\pi$$
$$152$$ −1.85786 −0.150693
$$153$$ −6.82843 −0.552046
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −10.3431 −0.828114
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 3.51472 0.279616
$$159$$ 13.3137 1.05585
$$160$$ 0 0
$$161$$ −19.3137 −1.52213
$$162$$ 0.414214 0.0325437
$$163$$ −15.3137 −1.19946 −0.599731 0.800202i $$-0.704726\pi$$
−0.599731 + 0.800202i $$0.704726\pi$$
$$164$$ −1.51472 −0.118280
$$165$$ 0 0
$$166$$ −4.14214 −0.321492
$$167$$ 9.31371 0.720716 0.360358 0.932814i $$-0.382654\pi$$
0.360358 + 0.932814i $$0.382654\pi$$
$$168$$ 7.65685 0.590739
$$169$$ 19.0000 1.46154
$$170$$ 0 0
$$171$$ 1.17157 0.0895924
$$172$$ 5.79899 0.442169
$$173$$ −2.82843 −0.215041 −0.107521 0.994203i $$-0.534291\pi$$
−0.107521 + 0.994203i $$0.534291\pi$$
$$174$$ −0.343146 −0.0260138
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ −3.17157 −0.237719
$$179$$ −6.34315 −0.474109 −0.237054 0.971496i $$-0.576182\pi$$
−0.237054 + 0.971496i $$0.576182\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ −11.3137 −0.838628
$$183$$ 0.343146 0.0253661
$$184$$ −6.34315 −0.467623
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −7.31371 −0.533407
$$189$$ −4.82843 −0.351216
$$190$$ 0 0
$$191$$ −5.65685 −0.409316 −0.204658 0.978834i $$-0.565608\pi$$
−0.204658 + 0.978834i $$0.565608\pi$$
$$192$$ −4.17157 −0.301057
$$193$$ 2.34315 0.168663 0.0843317 0.996438i $$-0.473124\pi$$
0.0843317 + 0.996438i $$0.473124\pi$$
$$194$$ −0.142136 −0.0102047
$$195$$ 0 0
$$196$$ −29.8284 −2.13060
$$197$$ 8.48528 0.604551 0.302276 0.953221i $$-0.402254\pi$$
0.302276 + 0.953221i $$0.402254\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ −5.65685 −0.399004
$$202$$ −2.00000 −0.140720
$$203$$ 4.00000 0.280745
$$204$$ 12.4853 0.874145
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 4.00000 0.278019
$$208$$ 16.9706 1.17670
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −6.82843 −0.470088 −0.235044 0.971985i $$-0.575523\pi$$
−0.235044 + 0.971985i $$0.575523\pi$$
$$212$$ −24.3431 −1.67189
$$213$$ 13.6569 0.935752
$$214$$ −2.20101 −0.150458
$$215$$ 0 0
$$216$$ −1.58579 −0.107899
$$217$$ 0 0
$$218$$ 2.20101 0.149071
$$219$$ −11.3137 −0.764510
$$220$$ 0 0
$$221$$ −38.6274 −2.59836
$$222$$ −0.142136 −0.00953952
$$223$$ 17.6569 1.18239 0.591195 0.806529i $$-0.298656\pi$$
0.591195 + 0.806529i $$0.298656\pi$$
$$224$$ −21.3137 −1.42408
$$225$$ 0 0
$$226$$ −6.20101 −0.412485
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ −2.14214 −0.141866
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.31371 0.0862492
$$233$$ −13.1716 −0.862898 −0.431449 0.902137i $$-0.641998\pi$$
−0.431449 + 0.902137i $$0.641998\pi$$
$$234$$ 2.34315 0.153176
$$235$$ 0 0
$$236$$ 7.31371 0.476082
$$237$$ 8.48528 0.551178
$$238$$ 13.6569 0.885242
$$239$$ −6.34315 −0.410304 −0.205152 0.978730i $$-0.565769\pi$$
−0.205152 + 0.978730i $$0.565769\pi$$
$$240$$ 0 0
$$241$$ 23.6569 1.52387 0.761936 0.647652i $$-0.224249\pi$$
0.761936 + 0.647652i $$0.224249\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ −0.627417 −0.0401663
$$245$$ 0 0
$$246$$ 0.343146 0.0218782
$$247$$ 6.62742 0.421692
$$248$$ 0 0
$$249$$ −10.0000 −0.633724
$$250$$ 0 0
$$251$$ −12.9706 −0.818695 −0.409347 0.912379i $$-0.634244\pi$$
−0.409347 + 0.912379i $$0.634244\pi$$
$$252$$ 8.82843 0.556139
$$253$$ 0 0
$$254$$ 1.02944 0.0645926
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ 27.6569 1.72519 0.862594 0.505898i $$-0.168839\pi$$
0.862594 + 0.505898i $$0.168839\pi$$
$$258$$ −1.31371 −0.0817879
$$259$$ 1.65685 0.102952
$$260$$ 0 0
$$261$$ −0.828427 −0.0512784
$$262$$ 8.00000 0.494242
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.34315 −0.143667
$$267$$ −7.65685 −0.468592
$$268$$ 10.3431 0.631808
$$269$$ −24.6274 −1.50156 −0.750780 0.660552i $$-0.770322\pi$$
−0.750780 + 0.660552i $$0.770322\pi$$
$$270$$ 0 0
$$271$$ −27.7990 −1.68867 −0.844334 0.535817i $$-0.820004\pi$$
−0.844334 + 0.535817i $$0.820004\pi$$
$$272$$ −20.4853 −1.24210
$$273$$ −27.3137 −1.65310
$$274$$ −3.85786 −0.233062
$$275$$ 0 0
$$276$$ −7.31371 −0.440234
$$277$$ −13.6569 −0.820561 −0.410280 0.911959i $$-0.634569\pi$$
−0.410280 + 0.911959i $$0.634569\pi$$
$$278$$ 6.82843 0.409542
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.8284 1.00390 0.501950 0.864897i $$-0.332616\pi$$
0.501950 + 0.864897i $$0.332616\pi$$
$$282$$ 1.65685 0.0986642
$$283$$ −3.17157 −0.188530 −0.0942652 0.995547i $$-0.530050\pi$$
−0.0942652 + 0.995547i $$0.530050\pi$$
$$284$$ −24.9706 −1.48173
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ 4.41421 0.260110
$$289$$ 29.6274 1.74279
$$290$$ 0 0
$$291$$ −0.343146 −0.0201156
$$292$$ 20.6863 1.21057
$$293$$ −1.17157 −0.0684440 −0.0342220 0.999414i $$-0.510895\pi$$
−0.0342220 + 0.999414i $$0.510895\pi$$
$$294$$ 6.75736 0.394097
$$295$$ 0 0
$$296$$ 0.544156 0.0316284
$$297$$ 0 0
$$298$$ −7.65685 −0.443550
$$299$$ 22.6274 1.30858
$$300$$ 0 0
$$301$$ 15.3137 0.882667
$$302$$ 0.201010 0.0115668
$$303$$ −4.82843 −0.277386
$$304$$ 3.51472 0.201583
$$305$$ 0 0
$$306$$ −2.82843 −0.161690
$$307$$ −8.82843 −0.503865 −0.251932 0.967745i $$-0.581066\pi$$
−0.251932 + 0.967745i $$0.581066\pi$$
$$308$$ 0 0
$$309$$ −19.3137 −1.09872
$$310$$ 0 0
$$311$$ 19.3137 1.09518 0.547590 0.836747i $$-0.315545\pi$$
0.547590 + 0.836747i $$0.315545\pi$$
$$312$$ −8.97056 −0.507858
$$313$$ −4.34315 −0.245489 −0.122745 0.992438i $$-0.539170\pi$$
−0.122745 + 0.992438i $$0.539170\pi$$
$$314$$ −7.45584 −0.420758
$$315$$ 0 0
$$316$$ −15.5147 −0.872771
$$317$$ −30.2843 −1.70093 −0.850467 0.526028i $$-0.823681\pi$$
−0.850467 + 0.526028i $$0.823681\pi$$
$$318$$ 5.51472 0.309250
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −5.31371 −0.296582
$$322$$ −8.00000 −0.445823
$$323$$ −8.00000 −0.445132
$$324$$ −1.82843 −0.101579
$$325$$ 0 0
$$326$$ −6.34315 −0.351314
$$327$$ 5.31371 0.293849
$$328$$ −1.31371 −0.0725374
$$329$$ −19.3137 −1.06480
$$330$$ 0 0
$$331$$ 17.6569 0.970508 0.485254 0.874373i $$-0.338727\pi$$
0.485254 + 0.874373i $$0.338727\pi$$
$$332$$ 18.2843 1.00348
$$333$$ −0.343146 −0.0188043
$$334$$ 3.85786 0.211093
$$335$$ 0 0
$$336$$ −14.4853 −0.790237
$$337$$ −19.3137 −1.05208 −0.526042 0.850458i $$-0.676325\pi$$
−0.526042 + 0.850458i $$0.676325\pi$$
$$338$$ 7.87006 0.428075
$$339$$ −14.9706 −0.813089
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0.485281 0.0262410
$$343$$ −44.9706 −2.42818
$$344$$ 5.02944 0.271169
$$345$$ 0 0
$$346$$ −1.17157 −0.0629841
$$347$$ −6.68629 −0.358939 −0.179469 0.983764i $$-0.557438\pi$$
−0.179469 + 0.983764i $$0.557438\pi$$
$$348$$ 1.51472 0.0811974
$$349$$ 22.9706 1.22959 0.614793 0.788688i $$-0.289240\pi$$
0.614793 + 0.788688i $$0.289240\pi$$
$$350$$ 0 0
$$351$$ 5.65685 0.301941
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ −1.65685 −0.0880608
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ 32.9706 1.74499
$$358$$ −2.62742 −0.138863
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ −17.6274 −0.927759
$$362$$ −5.79899 −0.304788
$$363$$ 0 0
$$364$$ 49.9411 2.61763
$$365$$ 0 0
$$366$$ 0.142136 0.00742955
$$367$$ 1.65685 0.0864871 0.0432435 0.999065i $$-0.486231\pi$$
0.0432435 + 0.999065i $$0.486231\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 0.828427 0.0431262
$$370$$ 0 0
$$371$$ −64.2843 −3.33747
$$372$$ 0 0
$$373$$ −34.6274 −1.79294 −0.896470 0.443105i $$-0.853877\pi$$
−0.896470 + 0.443105i $$0.853877\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.34315 −0.327123
$$377$$ −4.68629 −0.241356
$$378$$ −2.00000 −0.102869
$$379$$ −0.686292 −0.0352524 −0.0176262 0.999845i $$-0.505611\pi$$
−0.0176262 + 0.999845i $$0.505611\pi$$
$$380$$ 0 0
$$381$$ 2.48528 0.127325
$$382$$ −2.34315 −0.119886
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ −10.5563 −0.538701
$$385$$ 0 0
$$386$$ 0.970563 0.0494003
$$387$$ −3.17157 −0.161220
$$388$$ 0.627417 0.0318523
$$389$$ −12.3431 −0.625822 −0.312911 0.949782i $$-0.601304\pi$$
−0.312911 + 0.949782i $$0.601304\pi$$
$$390$$ 0 0
$$391$$ −27.3137 −1.38131
$$392$$ −25.8701 −1.30664
$$393$$ 19.3137 0.974248
$$394$$ 3.51472 0.177069
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −18.9706 −0.952105 −0.476053 0.879417i $$-0.657933\pi$$
−0.476053 + 0.879417i $$0.657933\pi$$
$$398$$ −4.28427 −0.214751
$$399$$ −5.65685 −0.283197
$$400$$ 0 0
$$401$$ −29.3137 −1.46386 −0.731928 0.681382i $$-0.761379\pi$$
−0.731928 + 0.681382i $$0.761379\pi$$
$$402$$ −2.34315 −0.116865
$$403$$ 0 0
$$404$$ 8.82843 0.439231
$$405$$ 0 0
$$406$$ 1.65685 0.0822283
$$407$$ 0 0
$$408$$ 10.8284 0.536087
$$409$$ 8.34315 0.412542 0.206271 0.978495i $$-0.433867\pi$$
0.206271 + 0.978495i $$0.433867\pi$$
$$410$$ 0 0
$$411$$ −9.31371 −0.459411
$$412$$ 35.3137 1.73978
$$413$$ 19.3137 0.950365
$$414$$ 1.65685 0.0814299
$$415$$ 0 0
$$416$$ 24.9706 1.22428
$$417$$ 16.4853 0.807288
$$418$$ 0 0
$$419$$ −3.02944 −0.147998 −0.0739988 0.997258i $$-0.523576\pi$$
−0.0739988 + 0.997258i $$0.523576\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ −2.82843 −0.137686
$$423$$ 4.00000 0.194487
$$424$$ −21.1127 −1.02532
$$425$$ 0 0
$$426$$ 5.65685 0.274075
$$427$$ −1.65685 −0.0801808
$$428$$ 9.71573 0.469627
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −10.3431 −0.498212 −0.249106 0.968476i $$-0.580137\pi$$
−0.249106 + 0.968476i $$0.580137\pi$$
$$432$$ 3.00000 0.144338
$$433$$ 4.34315 0.208718 0.104359 0.994540i $$-0.466721\pi$$
0.104359 + 0.994540i $$0.466721\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −9.71573 −0.465299
$$437$$ 4.68629 0.224176
$$438$$ −4.68629 −0.223920
$$439$$ 3.51472 0.167748 0.0838742 0.996476i $$-0.473271\pi$$
0.0838742 + 0.996476i $$0.473271\pi$$
$$440$$ 0 0
$$441$$ 16.3137 0.776843
$$442$$ −16.0000 −0.761042
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0.627417 0.0297759
$$445$$ 0 0
$$446$$ 7.31371 0.346314
$$447$$ −18.4853 −0.874324
$$448$$ 20.1421 0.951626
$$449$$ −2.97056 −0.140190 −0.0700948 0.997540i $$-0.522330\pi$$
−0.0700948 + 0.997540i $$0.522330\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 27.3726 1.28750
$$453$$ 0.485281 0.0228005
$$454$$ −5.79899 −0.272160
$$455$$ 0 0
$$456$$ −1.85786 −0.0870025
$$457$$ −0.686292 −0.0321034 −0.0160517 0.999871i $$-0.505110\pi$$
−0.0160517 + 0.999871i $$0.505110\pi$$
$$458$$ −0.828427 −0.0387099
$$459$$ −6.82843 −0.318724
$$460$$ 0 0
$$461$$ 28.1421 1.31071 0.655355 0.755321i $$-0.272519\pi$$
0.655355 + 0.755321i $$0.272519\pi$$
$$462$$ 0 0
$$463$$ 28.9706 1.34638 0.673188 0.739471i $$-0.264924\pi$$
0.673188 + 0.739471i $$0.264924\pi$$
$$464$$ −2.48528 −0.115376
$$465$$ 0 0
$$466$$ −5.45584 −0.252737
$$467$$ −22.6274 −1.04707 −0.523536 0.852004i $$-0.675387\pi$$
−0.523536 + 0.852004i $$0.675387\pi$$
$$468$$ −10.3431 −0.478112
$$469$$ 27.3137 1.26123
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 6.34315 0.291967
$$473$$ 0 0
$$474$$ 3.51472 0.161436
$$475$$ 0 0
$$476$$ −60.2843 −2.76313
$$477$$ 13.3137 0.609593
$$478$$ −2.62742 −0.120175
$$479$$ −3.02944 −0.138419 −0.0692093 0.997602i $$-0.522048\pi$$
−0.0692093 + 0.997602i $$0.522048\pi$$
$$480$$ 0 0
$$481$$ −1.94113 −0.0885077
$$482$$ 9.79899 0.446332
$$483$$ −19.3137 −0.878804
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0.414214 0.0187891
$$487$$ −20.9706 −0.950267 −0.475133 0.879914i $$-0.657600\pi$$
−0.475133 + 0.879914i $$0.657600\pi$$
$$488$$ −0.544156 −0.0246328
$$489$$ −15.3137 −0.692510
$$490$$ 0 0
$$491$$ −25.6569 −1.15788 −0.578939 0.815371i $$-0.696533\pi$$
−0.578939 + 0.815371i $$0.696533\pi$$
$$492$$ −1.51472 −0.0682888
$$493$$ 5.65685 0.254772
$$494$$ 2.74517 0.123511
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −65.9411 −2.95786
$$498$$ −4.14214 −0.185614
$$499$$ −33.6569 −1.50669 −0.753344 0.657627i $$-0.771560\pi$$
−0.753344 + 0.657627i $$0.771560\pi$$
$$500$$ 0 0
$$501$$ 9.31371 0.416106
$$502$$ −5.37258 −0.239790
$$503$$ −5.31371 −0.236927 −0.118463 0.992958i $$-0.537797\pi$$
−0.118463 + 0.992958i $$0.537797\pi$$
$$504$$ 7.65685 0.341063
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 19.0000 0.843820
$$508$$ −4.54416 −0.201614
$$509$$ 41.3137 1.83120 0.915599 0.402093i $$-0.131717\pi$$
0.915599 + 0.402093i $$0.131717\pi$$
$$510$$ 0 0
$$511$$ 54.6274 2.41657
$$512$$ 22.7574 1.00574
$$513$$ 1.17157 0.0517262
$$514$$ 11.4558 0.505296
$$515$$ 0 0
$$516$$ 5.79899 0.255286
$$517$$ 0 0
$$518$$ 0.686292 0.0301539
$$519$$ −2.82843 −0.124154
$$520$$ 0 0
$$521$$ 12.6274 0.553217 0.276609 0.960983i $$-0.410789\pi$$
0.276609 + 0.960983i $$0.410789\pi$$
$$522$$ −0.343146 −0.0150191
$$523$$ −26.4853 −1.15812 −0.579060 0.815285i $$-0.696580\pi$$
−0.579060 + 0.815285i $$0.696580\pi$$
$$524$$ −35.3137 −1.54269
$$525$$ 0 0
$$526$$ 7.45584 0.325090
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 10.3431 0.448432
$$533$$ 4.68629 0.202986
$$534$$ −3.17157 −0.137247
$$535$$ 0 0
$$536$$ 8.97056 0.387469
$$537$$ −6.34315 −0.273727
$$538$$ −10.2010 −0.439797
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.31371 0.228454 0.114227 0.993455i $$-0.463561\pi$$
0.114227 + 0.993455i $$0.463561\pi$$
$$542$$ −11.5147 −0.494600
$$543$$ −14.0000 −0.600798
$$544$$ −30.1421 −1.29233
$$545$$ 0 0
$$546$$ −11.3137 −0.484182
$$547$$ 20.1421 0.861216 0.430608 0.902539i $$-0.358299\pi$$
0.430608 + 0.902539i $$0.358299\pi$$
$$548$$ 17.0294 0.727462
$$549$$ 0.343146 0.0146451
$$550$$ 0 0
$$551$$ −0.970563 −0.0413474
$$552$$ −6.34315 −0.269982
$$553$$ −40.9706 −1.74225
$$554$$ −5.65685 −0.240337
$$555$$ 0 0
$$556$$ −30.1421 −1.27831
$$557$$ 10.8284 0.458815 0.229408 0.973330i $$-0.426321\pi$$
0.229408 + 0.973330i $$0.426321\pi$$
$$558$$ 0 0
$$559$$ −17.9411 −0.758829
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.97056 0.294035
$$563$$ 20.3431 0.857361 0.428681 0.903456i $$-0.358979\pi$$
0.428681 + 0.903456i $$0.358979\pi$$
$$564$$ −7.31371 −0.307963
$$565$$ 0 0
$$566$$ −1.31371 −0.0552193
$$567$$ −4.82843 −0.202775
$$568$$ −21.6569 −0.908701
$$569$$ −15.4558 −0.647943 −0.323971 0.946067i $$-0.605018\pi$$
−0.323971 + 0.946067i $$0.605018\pi$$
$$570$$ 0 0
$$571$$ −0.485281 −0.0203084 −0.0101542 0.999948i $$-0.503232\pi$$
−0.0101542 + 0.999948i $$0.503232\pi$$
$$572$$ 0 0
$$573$$ −5.65685 −0.236318
$$574$$ −1.65685 −0.0691558
$$575$$ 0 0
$$576$$ −4.17157 −0.173816
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 12.2721 0.510451
$$579$$ 2.34315 0.0973778
$$580$$ 0 0
$$581$$ 48.2843 2.00317
$$582$$ −0.142136 −0.00589171
$$583$$ 0 0
$$584$$ 17.9411 0.742409
$$585$$ 0 0
$$586$$ −0.485281 −0.0200468
$$587$$ 30.6274 1.26413 0.632064 0.774916i $$-0.282208\pi$$
0.632064 + 0.774916i $$0.282208\pi$$
$$588$$ −29.8284 −1.23010
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 8.48528 0.349038
$$592$$ −1.02944 −0.0423096
$$593$$ −17.1716 −0.705152 −0.352576 0.935783i $$-0.614694\pi$$
−0.352576 + 0.935783i $$0.614694\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 33.7990 1.38446
$$597$$ −10.3431 −0.423317
$$598$$ 9.37258 0.383273
$$599$$ 4.68629 0.191477 0.0957383 0.995407i $$-0.469479\pi$$
0.0957383 + 0.995407i $$0.469479\pi$$
$$600$$ 0 0
$$601$$ −17.3137 −0.706241 −0.353120 0.935578i $$-0.614879\pi$$
−0.353120 + 0.935578i $$0.614879\pi$$
$$602$$ 6.34315 0.258527
$$603$$ −5.65685 −0.230365
$$604$$ −0.887302 −0.0361038
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 18.4853 0.750294 0.375147 0.926965i $$-0.377592\pi$$
0.375147 + 0.926965i $$0.377592\pi$$
$$608$$ 5.17157 0.209735
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ 22.6274 0.915407
$$612$$ 12.4853 0.504688
$$613$$ 21.9411 0.886194 0.443097 0.896474i $$-0.353880\pi$$
0.443097 + 0.896474i $$0.353880\pi$$
$$614$$ −3.65685 −0.147579
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11.6569 −0.469287 −0.234644 0.972081i $$-0.575392\pi$$
−0.234644 + 0.972081i $$0.575392\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −25.6569 −1.03124 −0.515618 0.856819i $$-0.672438\pi$$
−0.515618 + 0.856819i $$0.672438\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 8.00000 0.320771
$$623$$ 36.9706 1.48119
$$624$$ 16.9706 0.679366
$$625$$ 0 0
$$626$$ −1.79899 −0.0719021
$$627$$ 0 0
$$628$$ 32.9117 1.31332
$$629$$ 2.34315 0.0934273
$$630$$ 0 0
$$631$$ 34.3431 1.36718 0.683590 0.729867i $$-0.260418\pi$$
0.683590 + 0.729867i $$0.260418\pi$$
$$632$$ −13.4558 −0.535245
$$633$$ −6.82843 −0.271406
$$634$$ −12.5442 −0.498192
$$635$$ 0 0
$$636$$ −24.3431 −0.965269
$$637$$ 92.2843 3.65644
$$638$$ 0 0
$$639$$ 13.6569 0.540257
$$640$$ 0 0
$$641$$ −26.9706 −1.06527 −0.532637 0.846344i $$-0.678799\pi$$
−0.532637 + 0.846344i $$0.678799\pi$$
$$642$$ −2.20101 −0.0868669
$$643$$ 29.9411 1.18076 0.590381 0.807124i $$-0.298977\pi$$
0.590381 + 0.807124i $$0.298977\pi$$
$$644$$ 35.3137 1.39156
$$645$$ 0 0
$$646$$ −3.31371 −0.130376
$$647$$ −27.3137 −1.07381 −0.536906 0.843642i $$-0.680407\pi$$
−0.536906 + 0.843642i $$0.680407\pi$$
$$648$$ −1.58579 −0.0622956
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 28.0000 1.09656
$$653$$ 26.9706 1.05544 0.527720 0.849418i $$-0.323047\pi$$
0.527720 + 0.849418i $$0.323047\pi$$
$$654$$ 2.20101 0.0860663
$$655$$ 0 0
$$656$$ 2.48528 0.0970339
$$657$$ −11.3137 −0.441390
$$658$$ −8.00000 −0.311872
$$659$$ −7.31371 −0.284902 −0.142451 0.989802i $$-0.545498\pi$$
−0.142451 + 0.989802i $$0.545498\pi$$
$$660$$ 0 0
$$661$$ −13.3137 −0.517843 −0.258922 0.965898i $$-0.583367\pi$$
−0.258922 + 0.965898i $$0.583367\pi$$
$$662$$ 7.31371 0.284255
$$663$$ −38.6274 −1.50016
$$664$$ 15.8579 0.615404
$$665$$ 0 0
$$666$$ −0.142136 −0.00550764
$$667$$ −3.31371 −0.128307
$$668$$ −17.0294 −0.658889
$$669$$ 17.6569 0.682653
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −21.3137 −0.822194
$$673$$ 29.6569 1.14319 0.571594 0.820537i $$-0.306325\pi$$
0.571594 + 0.820537i $$0.306325\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ −34.7401 −1.33616
$$677$$ −21.4558 −0.824615 −0.412308 0.911045i $$-0.635277\pi$$
−0.412308 + 0.911045i $$0.635277\pi$$
$$678$$ −6.20101 −0.238148
$$679$$ 1.65685 0.0635842
$$680$$ 0 0
$$681$$ −14.0000 −0.536481
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ −2.14214 −0.0819066
$$685$$ 0 0
$$686$$ −18.6274 −0.711198
$$687$$ −2.00000 −0.0763048
$$688$$ −9.51472 −0.362745
$$689$$ 75.3137 2.86922
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 5.17157 0.196594
$$693$$ 0 0
$$694$$ −2.76955 −0.105131
$$695$$ 0 0
$$696$$ 1.31371 0.0497960
$$697$$ −5.65685 −0.214269
$$698$$ 9.51472 0.360137
$$699$$ −13.1716 −0.498195
$$700$$ 0 0
$$701$$ −7.85786 −0.296787 −0.148394 0.988928i $$-0.547410\pi$$
−0.148394 + 0.988928i $$0.547410\pi$$
$$702$$ 2.34315 0.0884363
$$703$$ −0.402020 −0.0151625
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −10.7696 −0.405317
$$707$$ 23.3137 0.876802
$$708$$ 7.31371 0.274866
$$709$$ 29.3137 1.10090 0.550450 0.834868i $$-0.314456\pi$$
0.550450 + 0.834868i $$0.314456\pi$$
$$710$$ 0 0
$$711$$ 8.48528 0.318223
$$712$$ 12.1421 0.455046
$$713$$ 0 0
$$714$$ 13.6569 0.511095
$$715$$ 0 0
$$716$$ 11.5980 0.433437
$$717$$ −6.34315 −0.236889
$$718$$ −4.97056 −0.185500
$$719$$ 31.5980 1.17841 0.589203 0.807985i $$-0.299442\pi$$
0.589203 + 0.807985i $$0.299442\pi$$
$$720$$ 0 0
$$721$$ 93.2548 3.47299
$$722$$ −7.30152 −0.271734
$$723$$ 23.6569 0.879808
$$724$$ 25.5980 0.951341
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 33.9411 1.25881 0.629403 0.777079i $$-0.283299\pi$$
0.629403 + 0.777079i $$0.283299\pi$$
$$728$$ 43.3137 1.60531
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 21.6569 0.801008
$$732$$ −0.627417 −0.0231900
$$733$$ −17.6569 −0.652171 −0.326085 0.945340i $$-0.605730\pi$$
−0.326085 + 0.945340i $$0.605730\pi$$
$$734$$ 0.686292 0.0253315
$$735$$ 0 0
$$736$$ 17.6569 0.650840
$$737$$ 0 0
$$738$$ 0.343146 0.0126314
$$739$$ −47.1127 −1.73307 −0.866534 0.499118i $$-0.833658\pi$$
−0.866534 + 0.499118i $$0.833658\pi$$
$$740$$ 0 0
$$741$$ 6.62742 0.243464
$$742$$ −26.6274 −0.977523
$$743$$ 47.6569 1.74836 0.874180 0.485602i $$-0.161399\pi$$
0.874180 + 0.485602i $$0.161399\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −14.3431 −0.525140
$$747$$ −10.0000 −0.365881
$$748$$ 0 0
$$749$$ 25.6569 0.937481
$$750$$ 0 0
$$751$$ −36.2843 −1.32403 −0.662016 0.749490i $$-0.730299\pi$$
−0.662016 + 0.749490i $$0.730299\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −12.9706 −0.472674
$$754$$ −1.94113 −0.0706916
$$755$$ 0 0
$$756$$ 8.82843 0.321087
$$757$$ −8.62742 −0.313569 −0.156784 0.987633i $$-0.550113\pi$$
−0.156784 + 0.987633i $$0.550113\pi$$
$$758$$ −0.284271 −0.0103252
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 23.1716 0.839969 0.419984 0.907531i $$-0.362036\pi$$
0.419984 + 0.907531i $$0.362036\pi$$
$$762$$ 1.02944 0.0372926
$$763$$ −25.6569 −0.928840
$$764$$ 10.3431 0.374202
$$765$$ 0 0
$$766$$ 3.31371 0.119729
$$767$$ −22.6274 −0.817029
$$768$$ 3.97056 0.143275
$$769$$ −33.3137 −1.20132 −0.600662 0.799503i $$-0.705096\pi$$
−0.600662 + 0.799503i $$0.705096\pi$$
$$770$$ 0 0
$$771$$ 27.6569 0.996037
$$772$$ −4.28427 −0.154194
$$773$$ 7.65685 0.275398 0.137699 0.990474i $$-0.456029\pi$$
0.137699 + 0.990474i $$0.456029\pi$$
$$774$$ −1.31371 −0.0472203
$$775$$ 0 0
$$776$$ 0.544156 0.0195341
$$777$$ 1.65685 0.0594393
$$778$$ −5.11270 −0.183299
$$779$$ 0.970563 0.0347740
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −11.3137 −0.404577
$$783$$ −0.828427 −0.0296056
$$784$$ 48.9411 1.74790
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ 8.14214 0.290236 0.145118 0.989414i $$-0.453644\pi$$
0.145118 + 0.989414i $$0.453644\pi$$
$$788$$ −15.5147 −0.552689
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ 72.2843 2.57013
$$792$$ 0 0
$$793$$ 1.94113 0.0689314
$$794$$ −7.85786 −0.278865
$$795$$ 0 0
$$796$$ 18.9117 0.670307
$$797$$ −1.02944 −0.0364645 −0.0182323 0.999834i $$-0.505804\pi$$
−0.0182323 + 0.999834i $$0.505804\pi$$
$$798$$ −2.34315 −0.0829465
$$799$$ −27.3137 −0.966290
$$800$$ 0 0
$$801$$ −7.65685 −0.270542
$$802$$ −12.1421 −0.428754
$$803$$ 0 0
$$804$$ 10.3431 0.364775
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −24.6274 −0.866926
$$808$$ 7.65685 0.269367
$$809$$ 56.4264 1.98385 0.991923 0.126838i $$-0.0404829\pi$$
0.991923 + 0.126838i $$0.0404829\pi$$
$$810$$ 0 0
$$811$$ 16.4853 0.578877 0.289438 0.957197i $$-0.406532\pi$$
0.289438 + 0.957197i $$0.406532\pi$$
$$812$$ −7.31371 −0.256661
$$813$$ −27.7990 −0.974953
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −20.4853 −0.717128
$$817$$ −3.71573 −0.129997
$$818$$ 3.45584 0.120831
$$819$$ −27.3137 −0.954418
$$820$$ 0 0
$$821$$ 7.17157 0.250290 0.125145 0.992138i $$-0.460060\pi$$
0.125145 + 0.992138i $$0.460060\pi$$
$$822$$ −3.85786 −0.134558
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 30.6274 1.06696
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 18.6863 0.649786 0.324893 0.945751i $$-0.394672\pi$$
0.324893 + 0.945751i $$0.394672\pi$$
$$828$$ −7.31371 −0.254169
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ −13.6569 −0.473751
$$832$$ −23.5980 −0.818113
$$833$$ −111.397 −3.85968
$$834$$ 6.82843 0.236449
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −1.25483 −0.0433475
$$839$$ 22.6274 0.781185 0.390593 0.920564i $$-0.372270\pi$$
0.390593 + 0.920564i $$0.372270\pi$$
$$840$$ 0 0
$$841$$ −28.3137 −0.976335
$$842$$ −2.48528 −0.0856485
$$843$$ 16.8284 0.579602
$$844$$ 12.4853 0.429761
$$845$$ 0 0
$$846$$ 1.65685 0.0569638
$$847$$ 0 0
$$848$$ 39.9411 1.37158
$$849$$ −3.17157 −0.108848
$$850$$ 0 0
$$851$$ −1.37258 −0.0470515
$$852$$ −24.9706 −0.855477
$$853$$ −31.3137 −1.07216 −0.536080 0.844167i $$-0.680096\pi$$
−0.536080 + 0.844167i $$0.680096\pi$$
$$854$$ −0.686292 −0.0234844
$$855$$ 0 0
$$856$$ 8.42641 0.288009
$$857$$ −11.5147 −0.393335 −0.196668 0.980470i $$-0.563012\pi$$
−0.196668 + 0.980470i $$0.563012\pi$$
$$858$$ 0 0
$$859$$ −19.0294 −0.649276 −0.324638 0.945838i $$-0.605242\pi$$
−0.324638 + 0.945838i $$0.605242\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ −4.28427 −0.145923
$$863$$ 43.3137 1.47442 0.737208 0.675666i $$-0.236144\pi$$
0.737208 + 0.675666i $$0.236144\pi$$
$$864$$ 4.41421 0.150175
$$865$$ 0 0
$$866$$ 1.79899 0.0611322
$$867$$ 29.6274 1.00620
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ −8.42641 −0.285354
$$873$$ −0.343146 −0.0116137
$$874$$ 1.94113 0.0656595
$$875$$ 0 0
$$876$$ 20.6863 0.698925
$$877$$ −42.6274 −1.43943 −0.719713 0.694272i $$-0.755727\pi$$
−0.719713 + 0.694272i $$0.755727\pi$$
$$878$$ 1.45584 0.0491324
$$879$$ −1.17157 −0.0395162
$$880$$ 0 0
$$881$$ −13.0294 −0.438973 −0.219486 0.975616i $$-0.570438\pi$$
−0.219486 + 0.975616i $$0.570438\pi$$
$$882$$ 6.75736 0.227532
$$883$$ 50.6274 1.70375 0.851874 0.523747i $$-0.175466\pi$$
0.851874 + 0.523747i $$0.175466\pi$$
$$884$$ 70.6274 2.37546
$$885$$ 0 0
$$886$$ −4.97056 −0.166989
$$887$$ −4.34315 −0.145829 −0.0729143 0.997338i $$-0.523230\pi$$
−0.0729143 + 0.997338i $$0.523230\pi$$
$$888$$ 0.544156 0.0182607
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −32.2843 −1.08096
$$893$$ 4.68629 0.156821
$$894$$ −7.65685 −0.256084
$$895$$ 0 0
$$896$$ 50.9706 1.70281
$$897$$ 22.6274 0.755507
$$898$$ −1.23045 −0.0410606
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −90.9117 −3.02871
$$902$$ 0 0
$$903$$ 15.3137 0.509608
$$904$$ 23.7401 0.789584
$$905$$ 0 0
$$906$$ 0.201010 0.00667811
$$907$$ −7.02944 −0.233409 −0.116704 0.993167i $$-0.537233\pi$$
−0.116704 + 0.993167i $$0.537233\pi$$
$$908$$ 25.5980 0.849499
$$909$$ −4.82843 −0.160149
$$910$$ 0 0
$$911$$ −15.0294 −0.497947 −0.248974 0.968510i $$-0.580093\pi$$
−0.248974 + 0.968510i $$0.580093\pi$$
$$912$$ 3.51472 0.116384
$$913$$ 0 0
$$914$$ −0.284271 −0.00940286
$$915$$ 0 0
$$916$$ 3.65685 0.120826
$$917$$ −93.2548 −3.07955
$$918$$ −2.82843 −0.0933520
$$919$$ −28.4853 −0.939643 −0.469821 0.882762i $$-0.655682\pi$$
−0.469821 + 0.882762i $$0.655682\pi$$
$$920$$ 0 0
$$921$$ −8.82843 −0.290907
$$922$$ 11.6569 0.383898
$$923$$ 77.2548 2.54287
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 12.0000 0.394344
$$927$$ −19.3137 −0.634345
$$928$$ −3.65685 −0.120042
$$929$$ 33.5980 1.10231 0.551157 0.834402i $$-0.314187\pi$$
0.551157 + 0.834402i $$0.314187\pi$$
$$930$$ 0 0
$$931$$ 19.1127 0.626393
$$932$$ 24.0833 0.788873
$$933$$ 19.3137 0.632302
$$934$$ −9.37258 −0.306680
$$935$$ 0 0
$$936$$ −8.97056 −0.293212
$$937$$ 44.9706 1.46912 0.734562 0.678541i $$-0.237388\pi$$
0.734562 + 0.678541i $$0.237388\pi$$
$$938$$ 11.3137 0.369406
$$939$$ −4.34315 −0.141733
$$940$$ 0 0
$$941$$ −38.7696 −1.26385 −0.631926 0.775029i $$-0.717735\pi$$
−0.631926 + 0.775029i $$0.717735\pi$$
$$942$$ −7.45584 −0.242925
$$943$$ 3.31371 0.107909
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 38.6274 1.25522 0.627611 0.778527i $$-0.284033\pi$$
0.627611 + 0.778527i $$0.284033\pi$$
$$948$$ −15.5147 −0.503895
$$949$$ −64.0000 −2.07753
$$950$$ 0 0
$$951$$ −30.2843 −0.982035
$$952$$ −52.2843 −1.69454
$$953$$ 27.7990 0.900498 0.450249 0.892903i $$-0.351335\pi$$
0.450249 + 0.892903i $$0.351335\pi$$
$$954$$ 5.51472 0.178546
$$955$$ 0 0
$$956$$ 11.5980 0.375105
$$957$$ 0 0
$$958$$ −1.25483 −0.0405418
$$959$$ 44.9706 1.45218
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −0.804041 −0.0259233
$$963$$ −5.31371 −0.171232
$$964$$ −43.2548 −1.39314
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ −39.4558 −1.26881 −0.634407 0.772999i $$-0.718756\pi$$
−0.634407 + 0.772999i $$0.718756\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ 10.6274 0.341050 0.170525 0.985353i $$-0.445454\pi$$
0.170525 + 0.985353i $$0.445454\pi$$
$$972$$ −1.82843 −0.0586468
$$973$$ −79.5980 −2.55179
$$974$$ −8.68629 −0.278327
$$975$$ 0 0
$$976$$ 1.02944 0.0329515
$$977$$ −25.3137 −0.809857 −0.404929 0.914348i $$-0.632704\pi$$
−0.404929 + 0.914348i $$0.632704\pi$$
$$978$$ −6.34315 −0.202831
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 5.31371 0.169654
$$982$$ −10.6274 −0.339135
$$983$$ −14.6274 −0.466542 −0.233271 0.972412i $$-0.574943\pi$$
−0.233271 + 0.972412i $$0.574943\pi$$
$$984$$ −1.31371 −0.0418795
$$985$$ 0 0
$$986$$ 2.34315 0.0746210
$$987$$ −19.3137 −0.614762
$$988$$ −12.1177 −0.385517
$$989$$ −12.6863 −0.403401
$$990$$ 0 0
$$991$$ 14.6274 0.464655 0.232328 0.972638i $$-0.425366\pi$$
0.232328 + 0.972638i $$0.425366\pi$$
$$992$$ 0 0
$$993$$ 17.6569 0.560323
$$994$$ −27.3137 −0.866338
$$995$$ 0 0
$$996$$ 18.2843 0.579359
$$997$$ −16.6863 −0.528460 −0.264230 0.964460i $$-0.585118\pi$$
−0.264230 + 0.964460i $$0.585118\pi$$
$$998$$ −13.9411 −0.441299
$$999$$ −0.343146 −0.0108567
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 9075.2.a.v.1.2 2
5.4 even 2 1815.2.a.k.1.1 2
11.10 odd 2 825.2.a.g.1.1 2
15.14 odd 2 5445.2.a.m.1.2 2
33.32 even 2 2475.2.a.m.1.2 2
55.32 even 4 825.2.c.e.199.2 4
55.43 even 4 825.2.c.e.199.3 4
55.54 odd 2 165.2.a.a.1.2 2
165.32 odd 4 2475.2.c.m.199.3 4
165.98 odd 4 2475.2.c.m.199.2 4
165.164 even 2 495.2.a.d.1.1 2
220.219 even 2 2640.2.a.bb.1.2 2
385.384 even 2 8085.2.a.ba.1.2 2
660.659 odd 2 7920.2.a.cg.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 55.54 odd 2
495.2.a.d.1.1 2 165.164 even 2
825.2.a.g.1.1 2 11.10 odd 2
825.2.c.e.199.2 4 55.32 even 4
825.2.c.e.199.3 4 55.43 even 4
1815.2.a.k.1.1 2 5.4 even 2
2475.2.a.m.1.2 2 33.32 even 2
2475.2.c.m.199.2 4 165.98 odd 4
2475.2.c.m.199.3 4 165.32 odd 4
2640.2.a.bb.1.2 2 220.219 even 2
5445.2.a.m.1.2 2 15.14 odd 2
7920.2.a.cg.1.2 2 660.659 odd 2
8085.2.a.ba.1.2 2 385.384 even 2
9075.2.a.v.1.2 2 1.1 even 1 trivial