# Properties

 Label 3525.2.a.bd.1.4 Level $3525$ Weight $2$ Character 3525.1 Self dual yes Analytic conductor $28.147$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$0.936719$$ of defining polynomial Character $$\chi$$ $$=$$ 3525.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.936719 q^{2} +1.00000 q^{3} -1.12256 q^{4} -0.936719 q^{6} +3.65526 q^{7} +2.92496 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.936719 q^{2} +1.00000 q^{3} -1.12256 q^{4} -0.936719 q^{6} +3.65526 q^{7} +2.92496 q^{8} +1.00000 q^{9} -5.57376 q^{11} -1.12256 q^{12} +2.92869 q^{13} -3.42395 q^{14} -0.494746 q^{16} +2.26461 q^{17} -0.936719 q^{18} -2.64416 q^{19} +3.65526 q^{21} +5.22104 q^{22} -7.72850 q^{23} +2.92496 q^{24} -2.74336 q^{26} +1.00000 q^{27} -4.10324 q^{28} -7.79408 q^{29} -6.03466 q^{31} -5.38648 q^{32} -5.57376 q^{33} -2.12130 q^{34} -1.12256 q^{36} +6.60338 q^{37} +2.47683 q^{38} +2.92869 q^{39} -8.81927 q^{41} -3.42395 q^{42} -9.69632 q^{43} +6.25687 q^{44} +7.23943 q^{46} -1.00000 q^{47} -0.494746 q^{48} +6.36091 q^{49} +2.26461 q^{51} -3.28763 q^{52} +2.97538 q^{53} -0.936719 q^{54} +10.6915 q^{56} -2.64416 q^{57} +7.30086 q^{58} +0.571458 q^{59} +8.30187 q^{61} +5.65278 q^{62} +3.65526 q^{63} +6.03511 q^{64} +5.22104 q^{66} +8.03594 q^{67} -2.54216 q^{68} -7.72850 q^{69} +11.7268 q^{71} +2.92496 q^{72} -5.94373 q^{73} -6.18551 q^{74} +2.96822 q^{76} -20.3735 q^{77} -2.74336 q^{78} -16.6200 q^{79} +1.00000 q^{81} +8.26117 q^{82} -13.8354 q^{83} -4.10324 q^{84} +9.08272 q^{86} -7.79408 q^{87} -16.3030 q^{88} +4.31121 q^{89} +10.7051 q^{91} +8.67569 q^{92} -6.03466 q^{93} +0.936719 q^{94} -5.38648 q^{96} -5.24644 q^{97} -5.95838 q^{98} -5.57376 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.936719 −0.662360 −0.331180 0.943568i $$-0.607447\pi$$
−0.331180 + 0.943568i $$0.607447\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.12256 −0.561279
$$5$$ 0 0
$$6$$ −0.936719 −0.382414
$$7$$ 3.65526 1.38156 0.690779 0.723066i $$-0.257268\pi$$
0.690779 + 0.723066i $$0.257268\pi$$
$$8$$ 2.92496 1.03413
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.57376 −1.68055 −0.840275 0.542160i $$-0.817607\pi$$
−0.840275 + 0.542160i $$0.817607\pi$$
$$12$$ −1.12256 −0.324055
$$13$$ 2.92869 0.812274 0.406137 0.913812i $$-0.366876\pi$$
0.406137 + 0.913812i $$0.366876\pi$$
$$14$$ −3.42395 −0.915089
$$15$$ 0 0
$$16$$ −0.494746 −0.123686
$$17$$ 2.26461 0.549249 0.274624 0.961552i $$-0.411446\pi$$
0.274624 + 0.961552i $$0.411446\pi$$
$$18$$ −0.936719 −0.220787
$$19$$ −2.64416 −0.606612 −0.303306 0.952893i $$-0.598090\pi$$
−0.303306 + 0.952893i $$0.598090\pi$$
$$20$$ 0 0
$$21$$ 3.65526 0.797643
$$22$$ 5.22104 1.11313
$$23$$ −7.72850 −1.61150 −0.805751 0.592254i $$-0.798238\pi$$
−0.805751 + 0.592254i $$0.798238\pi$$
$$24$$ 2.92496 0.597055
$$25$$ 0 0
$$26$$ −2.74336 −0.538018
$$27$$ 1.00000 0.192450
$$28$$ −4.10324 −0.775440
$$29$$ −7.79408 −1.44732 −0.723662 0.690154i $$-0.757543\pi$$
−0.723662 + 0.690154i $$0.757543\pi$$
$$30$$ 0 0
$$31$$ −6.03466 −1.08386 −0.541928 0.840425i $$-0.682306\pi$$
−0.541928 + 0.840425i $$0.682306\pi$$
$$32$$ −5.38648 −0.952204
$$33$$ −5.57376 −0.970266
$$34$$ −2.12130 −0.363801
$$35$$ 0 0
$$36$$ −1.12256 −0.187093
$$37$$ 6.60338 1.08559 0.542794 0.839866i $$-0.317366\pi$$
0.542794 + 0.839866i $$0.317366\pi$$
$$38$$ 2.47683 0.401796
$$39$$ 2.92869 0.468966
$$40$$ 0 0
$$41$$ −8.81927 −1.37734 −0.688669 0.725076i $$-0.741805\pi$$
−0.688669 + 0.725076i $$0.741805\pi$$
$$42$$ −3.42395 −0.528327
$$43$$ −9.69632 −1.47867 −0.739337 0.673335i $$-0.764861\pi$$
−0.739337 + 0.673335i $$0.764861\pi$$
$$44$$ 6.25687 0.943258
$$45$$ 0 0
$$46$$ 7.23943 1.06740
$$47$$ −1.00000 −0.145865
$$48$$ −0.494746 −0.0714104
$$49$$ 6.36091 0.908702
$$50$$ 0 0
$$51$$ 2.26461 0.317109
$$52$$ −3.28763 −0.455912
$$53$$ 2.97538 0.408700 0.204350 0.978898i $$-0.434492\pi$$
0.204350 + 0.978898i $$0.434492\pi$$
$$54$$ −0.936719 −0.127471
$$55$$ 0 0
$$56$$ 10.6915 1.42871
$$57$$ −2.64416 −0.350228
$$58$$ 7.30086 0.958650
$$59$$ 0.571458 0.0743975 0.0371987 0.999308i $$-0.488157\pi$$
0.0371987 + 0.999308i $$0.488157\pi$$
$$60$$ 0 0
$$61$$ 8.30187 1.06295 0.531473 0.847075i $$-0.321639\pi$$
0.531473 + 0.847075i $$0.321639\pi$$
$$62$$ 5.65278 0.717903
$$63$$ 3.65526 0.460519
$$64$$ 6.03511 0.754388
$$65$$ 0 0
$$66$$ 5.22104 0.642666
$$67$$ 8.03594 0.981746 0.490873 0.871231i $$-0.336678\pi$$
0.490873 + 0.871231i $$0.336678\pi$$
$$68$$ −2.54216 −0.308282
$$69$$ −7.72850 −0.930402
$$70$$ 0 0
$$71$$ 11.7268 1.39171 0.695855 0.718183i $$-0.255026\pi$$
0.695855 + 0.718183i $$0.255026\pi$$
$$72$$ 2.92496 0.344710
$$73$$ −5.94373 −0.695661 −0.347830 0.937558i $$-0.613081\pi$$
−0.347830 + 0.937558i $$0.613081\pi$$
$$74$$ −6.18551 −0.719050
$$75$$ 0 0
$$76$$ 2.96822 0.340479
$$77$$ −20.3735 −2.32178
$$78$$ −2.74336 −0.310625
$$79$$ −16.6200 −1.86990 −0.934950 0.354780i $$-0.884556\pi$$
−0.934950 + 0.354780i $$0.884556\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.26117 0.912294
$$83$$ −13.8354 −1.51863 −0.759316 0.650722i $$-0.774466\pi$$
−0.759316 + 0.650722i $$0.774466\pi$$
$$84$$ −4.10324 −0.447700
$$85$$ 0 0
$$86$$ 9.08272 0.979415
$$87$$ −7.79408 −0.835613
$$88$$ −16.3030 −1.73791
$$89$$ 4.31121 0.456988 0.228494 0.973545i $$-0.426620\pi$$
0.228494 + 0.973545i $$0.426620\pi$$
$$90$$ 0 0
$$91$$ 10.7051 1.12220
$$92$$ 8.67569 0.904503
$$93$$ −6.03466 −0.625765
$$94$$ 0.936719 0.0966151
$$95$$ 0 0
$$96$$ −5.38648 −0.549755
$$97$$ −5.24644 −0.532696 −0.266348 0.963877i $$-0.585817\pi$$
−0.266348 + 0.963877i $$0.585817\pi$$
$$98$$ −5.95838 −0.601888
$$99$$ −5.57376 −0.560184
$$100$$ 0 0
$$101$$ −17.9567 −1.78675 −0.893377 0.449308i $$-0.851671\pi$$
−0.893377 + 0.449308i $$0.851671\pi$$
$$102$$ −2.12130 −0.210040
$$103$$ −13.8334 −1.36305 −0.681525 0.731795i $$-0.738683\pi$$
−0.681525 + 0.731795i $$0.738683\pi$$
$$104$$ 8.56631 0.839996
$$105$$ 0 0
$$106$$ −2.78710 −0.270707
$$107$$ −6.17861 −0.597309 −0.298655 0.954361i $$-0.596538\pi$$
−0.298655 + 0.954361i $$0.596538\pi$$
$$108$$ −1.12256 −0.108018
$$109$$ 14.0831 1.34892 0.674458 0.738314i $$-0.264378\pi$$
0.674458 + 0.738314i $$0.264378\pi$$
$$110$$ 0 0
$$111$$ 6.60338 0.626765
$$112$$ −1.80842 −0.170880
$$113$$ 16.6357 1.56496 0.782480 0.622676i $$-0.213955\pi$$
0.782480 + 0.622676i $$0.213955\pi$$
$$114$$ 2.47683 0.231977
$$115$$ 0 0
$$116$$ 8.74931 0.812353
$$117$$ 2.92869 0.270758
$$118$$ −0.535295 −0.0492779
$$119$$ 8.27774 0.758819
$$120$$ 0 0
$$121$$ 20.0668 1.82425
$$122$$ −7.77652 −0.704053
$$123$$ −8.81927 −0.795207
$$124$$ 6.77426 0.608346
$$125$$ 0 0
$$126$$ −3.42395 −0.305030
$$127$$ 12.1084 1.07445 0.537225 0.843439i $$-0.319473\pi$$
0.537225 + 0.843439i $$0.319473\pi$$
$$128$$ 5.11976 0.452527
$$129$$ −9.69632 −0.853713
$$130$$ 0 0
$$131$$ 1.58271 0.138282 0.0691409 0.997607i $$-0.477974\pi$$
0.0691409 + 0.997607i $$0.477974\pi$$
$$132$$ 6.25687 0.544590
$$133$$ −9.66509 −0.838069
$$134$$ −7.52741 −0.650269
$$135$$ 0 0
$$136$$ 6.62389 0.567994
$$137$$ 20.5052 1.75188 0.875938 0.482424i $$-0.160243\pi$$
0.875938 + 0.482424i $$0.160243\pi$$
$$138$$ 7.23943 0.616261
$$139$$ −16.7180 −1.41800 −0.709002 0.705207i $$-0.750854\pi$$
−0.709002 + 0.705207i $$0.750854\pi$$
$$140$$ 0 0
$$141$$ −1.00000 −0.0842152
$$142$$ −10.9847 −0.921813
$$143$$ −16.3238 −1.36507
$$144$$ −0.494746 −0.0412288
$$145$$ 0 0
$$146$$ 5.56760 0.460778
$$147$$ 6.36091 0.524639
$$148$$ −7.41268 −0.609318
$$149$$ 10.2172 0.837025 0.418513 0.908211i $$-0.362552\pi$$
0.418513 + 0.908211i $$0.362552\pi$$
$$150$$ 0 0
$$151$$ −20.9048 −1.70121 −0.850603 0.525808i $$-0.823763\pi$$
−0.850603 + 0.525808i $$0.823763\pi$$
$$152$$ −7.73406 −0.627315
$$153$$ 2.26461 0.183083
$$154$$ 19.0843 1.53785
$$155$$ 0 0
$$156$$ −3.28763 −0.263221
$$157$$ 22.8938 1.82712 0.913561 0.406702i $$-0.133321\pi$$
0.913561 + 0.406702i $$0.133321\pi$$
$$158$$ 15.5683 1.23855
$$159$$ 2.97538 0.235963
$$160$$ 0 0
$$161$$ −28.2496 −2.22638
$$162$$ −0.936719 −0.0735956
$$163$$ −11.2612 −0.882042 −0.441021 0.897497i $$-0.645384\pi$$
−0.441021 + 0.897497i $$0.645384\pi$$
$$164$$ 9.90014 0.773071
$$165$$ 0 0
$$166$$ 12.9599 1.00588
$$167$$ −16.8461 −1.30359 −0.651794 0.758396i $$-0.725983\pi$$
−0.651794 + 0.758396i $$0.725983\pi$$
$$168$$ 10.6915 0.824865
$$169$$ −4.42275 −0.340212
$$170$$ 0 0
$$171$$ −2.64416 −0.202204
$$172$$ 10.8847 0.829949
$$173$$ −13.5553 −1.03059 −0.515294 0.857014i $$-0.672317\pi$$
−0.515294 + 0.857014i $$0.672317\pi$$
$$174$$ 7.30086 0.553477
$$175$$ 0 0
$$176$$ 2.75759 0.207861
$$177$$ 0.571458 0.0429534
$$178$$ −4.03839 −0.302690
$$179$$ −5.29177 −0.395525 −0.197763 0.980250i $$-0.563368\pi$$
−0.197763 + 0.980250i $$0.563368\pi$$
$$180$$ 0 0
$$181$$ −6.93711 −0.515632 −0.257816 0.966194i $$-0.583003\pi$$
−0.257816 + 0.966194i $$0.583003\pi$$
$$182$$ −10.0277 −0.743302
$$183$$ 8.30187 0.613692
$$184$$ −22.6055 −1.66650
$$185$$ 0 0
$$186$$ 5.65278 0.414482
$$187$$ −12.6224 −0.923041
$$188$$ 1.12256 0.0818710
$$189$$ 3.65526 0.265881
$$190$$ 0 0
$$191$$ 3.27524 0.236988 0.118494 0.992955i $$-0.462193\pi$$
0.118494 + 0.992955i $$0.462193\pi$$
$$192$$ 6.03511 0.435546
$$193$$ 6.91822 0.497984 0.248992 0.968506i $$-0.419901\pi$$
0.248992 + 0.968506i $$0.419901\pi$$
$$194$$ 4.91444 0.352836
$$195$$ 0 0
$$196$$ −7.14050 −0.510035
$$197$$ −13.2026 −0.940650 −0.470325 0.882493i $$-0.655863\pi$$
−0.470325 + 0.882493i $$0.655863\pi$$
$$198$$ 5.22104 0.371043
$$199$$ −10.3855 −0.736210 −0.368105 0.929784i $$-0.619993\pi$$
−0.368105 + 0.929784i $$0.619993\pi$$
$$200$$ 0 0
$$201$$ 8.03594 0.566811
$$202$$ 16.8203 1.18347
$$203$$ −28.4894 −1.99956
$$204$$ −2.54216 −0.177987
$$205$$ 0 0
$$206$$ 12.9580 0.902830
$$207$$ −7.72850 −0.537168
$$208$$ −1.44896 −0.100467
$$209$$ 14.7379 1.01944
$$210$$ 0 0
$$211$$ 19.5002 1.34245 0.671226 0.741253i $$-0.265768\pi$$
0.671226 + 0.741253i $$0.265768\pi$$
$$212$$ −3.34004 −0.229395
$$213$$ 11.7268 0.803504
$$214$$ 5.78762 0.395634
$$215$$ 0 0
$$216$$ 2.92496 0.199018
$$217$$ −22.0582 −1.49741
$$218$$ −13.1919 −0.893467
$$219$$ −5.94373 −0.401640
$$220$$ 0 0
$$221$$ 6.63235 0.446140
$$222$$ −6.18551 −0.415144
$$223$$ 5.43059 0.363659 0.181830 0.983330i $$-0.441798\pi$$
0.181830 + 0.983330i $$0.441798\pi$$
$$224$$ −19.6890 −1.31552
$$225$$ 0 0
$$226$$ −15.5830 −1.03657
$$227$$ −2.68717 −0.178354 −0.0891770 0.996016i $$-0.528424\pi$$
−0.0891770 + 0.996016i $$0.528424\pi$$
$$228$$ 2.96822 0.196575
$$229$$ 17.6212 1.16444 0.582221 0.813030i $$-0.302184\pi$$
0.582221 + 0.813030i $$0.302184\pi$$
$$230$$ 0 0
$$231$$ −20.3735 −1.34048
$$232$$ −22.7974 −1.49672
$$233$$ −14.3094 −0.937437 −0.468719 0.883348i $$-0.655284\pi$$
−0.468719 + 0.883348i $$0.655284\pi$$
$$234$$ −2.74336 −0.179339
$$235$$ 0 0
$$236$$ −0.641495 −0.0417578
$$237$$ −16.6200 −1.07959
$$238$$ −7.75391 −0.502611
$$239$$ −0.608101 −0.0393348 −0.0196674 0.999807i $$-0.506261\pi$$
−0.0196674 + 0.999807i $$0.506261\pi$$
$$240$$ 0 0
$$241$$ −0.888912 −0.0572598 −0.0286299 0.999590i $$-0.509114\pi$$
−0.0286299 + 0.999590i $$0.509114\pi$$
$$242$$ −18.7969 −1.20831
$$243$$ 1.00000 0.0641500
$$244$$ −9.31934 −0.596609
$$245$$ 0 0
$$246$$ 8.26117 0.526713
$$247$$ −7.74394 −0.492735
$$248$$ −17.6511 −1.12085
$$249$$ −13.8354 −0.876782
$$250$$ 0 0
$$251$$ −0.0483724 −0.00305324 −0.00152662 0.999999i $$-0.500486\pi$$
−0.00152662 + 0.999999i $$0.500486\pi$$
$$252$$ −4.10324 −0.258480
$$253$$ 43.0768 2.70821
$$254$$ −11.3422 −0.711672
$$255$$ 0 0
$$256$$ −16.8660 −1.05412
$$257$$ 3.58017 0.223325 0.111662 0.993746i $$-0.464382\pi$$
0.111662 + 0.993746i $$0.464382\pi$$
$$258$$ 9.08272 0.565465
$$259$$ 24.1370 1.49980
$$260$$ 0 0
$$261$$ −7.79408 −0.482441
$$262$$ −1.48255 −0.0915924
$$263$$ 17.6132 1.08608 0.543040 0.839707i $$-0.317273\pi$$
0.543040 + 0.839707i $$0.317273\pi$$
$$264$$ −16.3030 −1.00338
$$265$$ 0 0
$$266$$ 9.05347 0.555104
$$267$$ 4.31121 0.263842
$$268$$ −9.02081 −0.551034
$$269$$ −14.7262 −0.897870 −0.448935 0.893564i $$-0.648196\pi$$
−0.448935 + 0.893564i $$0.648196\pi$$
$$270$$ 0 0
$$271$$ 7.26580 0.441366 0.220683 0.975346i $$-0.429171\pi$$
0.220683 + 0.975346i $$0.429171\pi$$
$$272$$ −1.12041 −0.0679347
$$273$$ 10.7051 0.647904
$$274$$ −19.2076 −1.16037
$$275$$ 0 0
$$276$$ 8.67569 0.522215
$$277$$ −16.0913 −0.966833 −0.483416 0.875391i $$-0.660604\pi$$
−0.483416 + 0.875391i $$0.660604\pi$$
$$278$$ 15.6601 0.939229
$$279$$ −6.03466 −0.361286
$$280$$ 0 0
$$281$$ −2.13975 −0.127647 −0.0638234 0.997961i $$-0.520329\pi$$
−0.0638234 + 0.997961i $$0.520329\pi$$
$$282$$ 0.936719 0.0557808
$$283$$ −23.8300 −1.41655 −0.708274 0.705937i $$-0.750526\pi$$
−0.708274 + 0.705937i $$0.750526\pi$$
$$284$$ −13.1640 −0.781138
$$285$$ 0 0
$$286$$ 15.2908 0.904166
$$287$$ −32.2367 −1.90287
$$288$$ −5.38648 −0.317401
$$289$$ −11.8715 −0.698326
$$290$$ 0 0
$$291$$ −5.24644 −0.307552
$$292$$ 6.67218 0.390460
$$293$$ −17.5931 −1.02780 −0.513901 0.857850i $$-0.671800\pi$$
−0.513901 + 0.857850i $$0.671800\pi$$
$$294$$ −5.95838 −0.347500
$$295$$ 0 0
$$296$$ 19.3146 1.12264
$$297$$ −5.57376 −0.323422
$$298$$ −9.57064 −0.554412
$$299$$ −22.6344 −1.30898
$$300$$ 0 0
$$301$$ −35.4425 −2.04287
$$302$$ 19.5819 1.12681
$$303$$ −17.9567 −1.03158
$$304$$ 1.30819 0.0750297
$$305$$ 0 0
$$306$$ −2.12130 −0.121267
$$307$$ −20.7229 −1.18272 −0.591360 0.806407i $$-0.701409\pi$$
−0.591360 + 0.806407i $$0.701409\pi$$
$$308$$ 22.8705 1.30317
$$309$$ −13.8334 −0.786957
$$310$$ 0 0
$$311$$ −24.3065 −1.37829 −0.689146 0.724622i $$-0.742014\pi$$
−0.689146 + 0.724622i $$0.742014\pi$$
$$312$$ 8.56631 0.484972
$$313$$ 17.3665 0.981615 0.490808 0.871268i $$-0.336702\pi$$
0.490808 + 0.871268i $$0.336702\pi$$
$$314$$ −21.4450 −1.21021
$$315$$ 0 0
$$316$$ 18.6570 1.04954
$$317$$ 2.00730 0.112741 0.0563706 0.998410i $$-0.482047\pi$$
0.0563706 + 0.998410i $$0.482047\pi$$
$$318$$ −2.78710 −0.156293
$$319$$ 43.4423 2.43230
$$320$$ 0 0
$$321$$ −6.17861 −0.344857
$$322$$ 26.4620 1.47467
$$323$$ −5.98799 −0.333181
$$324$$ −1.12256 −0.0623644
$$325$$ 0 0
$$326$$ 10.5485 0.584230
$$327$$ 14.0831 0.778797
$$328$$ −25.7960 −1.42435
$$329$$ −3.65526 −0.201521
$$330$$ 0 0
$$331$$ 23.0557 1.26725 0.633627 0.773638i $$-0.281565\pi$$
0.633627 + 0.773638i $$0.281565\pi$$
$$332$$ 15.5310 0.852376
$$333$$ 6.60338 0.361863
$$334$$ 15.7800 0.863445
$$335$$ 0 0
$$336$$ −1.80842 −0.0986576
$$337$$ 0.977276 0.0532356 0.0266178 0.999646i $$-0.491526\pi$$
0.0266178 + 0.999646i $$0.491526\pi$$
$$338$$ 4.14287 0.225343
$$339$$ 16.6357 0.903530
$$340$$ 0 0
$$341$$ 33.6357 1.82148
$$342$$ 2.47683 0.133932
$$343$$ −2.33603 −0.126134
$$344$$ −28.3613 −1.52914
$$345$$ 0 0
$$346$$ 12.6975 0.682620
$$347$$ 9.11111 0.489110 0.244555 0.969635i $$-0.421358\pi$$
0.244555 + 0.969635i $$0.421358\pi$$
$$348$$ 8.74931 0.469012
$$349$$ 2.64665 0.141672 0.0708359 0.997488i $$-0.477433\pi$$
0.0708359 + 0.997488i $$0.477433\pi$$
$$350$$ 0 0
$$351$$ 2.92869 0.156322
$$352$$ 30.0229 1.60023
$$353$$ 15.8458 0.843386 0.421693 0.906739i $$-0.361436\pi$$
0.421693 + 0.906739i $$0.361436\pi$$
$$354$$ −0.535295 −0.0284506
$$355$$ 0 0
$$356$$ −4.83959 −0.256498
$$357$$ 8.27774 0.438104
$$358$$ 4.95690 0.261980
$$359$$ 11.8670 0.626318 0.313159 0.949701i $$-0.398613\pi$$
0.313159 + 0.949701i $$0.398613\pi$$
$$360$$ 0 0
$$361$$ −12.0084 −0.632022
$$362$$ 6.49812 0.341534
$$363$$ 20.0668 1.05323
$$364$$ −12.0171 −0.629869
$$365$$ 0 0
$$366$$ −7.77652 −0.406485
$$367$$ −5.06080 −0.264172 −0.132086 0.991238i $$-0.542167\pi$$
−0.132086 + 0.991238i $$0.542167\pi$$
$$368$$ 3.82364 0.199321
$$369$$ −8.81927 −0.459113
$$370$$ 0 0
$$371$$ 10.8758 0.564643
$$372$$ 6.77426 0.351229
$$373$$ 1.19434 0.0618405 0.0309203 0.999522i $$-0.490156\pi$$
0.0309203 + 0.999522i $$0.490156\pi$$
$$374$$ 11.8236 0.611385
$$375$$ 0 0
$$376$$ −2.92496 −0.150843
$$377$$ −22.8265 −1.17562
$$378$$ −3.42395 −0.176109
$$379$$ 16.0321 0.823511 0.411756 0.911294i $$-0.364916\pi$$
0.411756 + 0.911294i $$0.364916\pi$$
$$380$$ 0 0
$$381$$ 12.1084 0.620334
$$382$$ −3.06798 −0.156972
$$383$$ 23.4599 1.19875 0.599374 0.800469i $$-0.295416\pi$$
0.599374 + 0.800469i $$0.295416\pi$$
$$384$$ 5.11976 0.261267
$$385$$ 0 0
$$386$$ −6.48042 −0.329845
$$387$$ −9.69632 −0.492891
$$388$$ 5.88944 0.298991
$$389$$ −27.8951 −1.41434 −0.707169 0.707045i $$-0.750028\pi$$
−0.707169 + 0.707045i $$0.750028\pi$$
$$390$$ 0 0
$$391$$ −17.5020 −0.885116
$$392$$ 18.6054 0.939715
$$393$$ 1.58271 0.0798371
$$394$$ 12.3672 0.623049
$$395$$ 0 0
$$396$$ 6.25687 0.314419
$$397$$ −11.3412 −0.569199 −0.284600 0.958646i $$-0.591861\pi$$
−0.284600 + 0.958646i $$0.591861\pi$$
$$398$$ 9.72830 0.487636
$$399$$ −9.66509 −0.483860
$$400$$ 0 0
$$401$$ −12.6214 −0.630283 −0.315142 0.949045i $$-0.602052\pi$$
−0.315142 + 0.949045i $$0.602052\pi$$
$$402$$ −7.52741 −0.375433
$$403$$ −17.6737 −0.880388
$$404$$ 20.1574 1.00287
$$405$$ 0 0
$$406$$ 26.6865 1.32443
$$407$$ −36.8056 −1.82439
$$408$$ 6.62389 0.327932
$$409$$ −2.98302 −0.147501 −0.0737505 0.997277i $$-0.523497\pi$$
−0.0737505 + 0.997277i $$0.523497\pi$$
$$410$$ 0 0
$$411$$ 20.5052 1.01145
$$412$$ 15.5288 0.765051
$$413$$ 2.08883 0.102784
$$414$$ 7.23943 0.355798
$$415$$ 0 0
$$416$$ −15.7754 −0.773450
$$417$$ −16.7180 −0.818685
$$418$$ −13.8053 −0.675238
$$419$$ 15.5076 0.757595 0.378798 0.925480i $$-0.376338\pi$$
0.378798 + 0.925480i $$0.376338\pi$$
$$420$$ 0 0
$$421$$ −22.2089 −1.08240 −0.541198 0.840895i $$-0.682029\pi$$
−0.541198 + 0.840895i $$0.682029\pi$$
$$422$$ −18.2662 −0.889186
$$423$$ −1.00000 −0.0486217
$$424$$ 8.70287 0.422649
$$425$$ 0 0
$$426$$ −10.9847 −0.532209
$$427$$ 30.3455 1.46852
$$428$$ 6.93585 0.335257
$$429$$ −16.3238 −0.788122
$$430$$ 0 0
$$431$$ −20.1349 −0.969865 −0.484932 0.874552i $$-0.661156\pi$$
−0.484932 + 0.874552i $$0.661156\pi$$
$$432$$ −0.494746 −0.0238035
$$433$$ 17.8238 0.856557 0.428278 0.903647i $$-0.359120\pi$$
0.428278 + 0.903647i $$0.359120\pi$$
$$434$$ 20.6624 0.991825
$$435$$ 0 0
$$436$$ −15.8091 −0.757118
$$437$$ 20.4354 0.977557
$$438$$ 5.56760 0.266030
$$439$$ 11.9909 0.572293 0.286146 0.958186i $$-0.407626\pi$$
0.286146 + 0.958186i $$0.407626\pi$$
$$440$$ 0 0
$$441$$ 6.36091 0.302901
$$442$$ −6.21265 −0.295506
$$443$$ −25.2442 −1.19939 −0.599694 0.800230i $$-0.704711\pi$$
−0.599694 + 0.800230i $$0.704711\pi$$
$$444$$ −7.41268 −0.351790
$$445$$ 0 0
$$446$$ −5.08693 −0.240873
$$447$$ 10.2172 0.483257
$$448$$ 22.0599 1.04223
$$449$$ −37.0349 −1.74778 −0.873891 0.486121i $$-0.838411\pi$$
−0.873891 + 0.486121i $$0.838411\pi$$
$$450$$ 0 0
$$451$$ 49.1564 2.31469
$$452$$ −18.6746 −0.878379
$$453$$ −20.9048 −0.982192
$$454$$ 2.51712 0.118134
$$455$$ 0 0
$$456$$ −7.73406 −0.362180
$$457$$ 13.9630 0.653162 0.326581 0.945169i $$-0.394103\pi$$
0.326581 + 0.945169i $$0.394103\pi$$
$$458$$ −16.5061 −0.771280
$$459$$ 2.26461 0.105703
$$460$$ 0 0
$$461$$ −41.9963 −1.95596 −0.977982 0.208689i $$-0.933080\pi$$
−0.977982 + 0.208689i $$0.933080\pi$$
$$462$$ 19.0843 0.887880
$$463$$ 23.1267 1.07479 0.537394 0.843331i $$-0.319409\pi$$
0.537394 + 0.843331i $$0.319409\pi$$
$$464$$ 3.85609 0.179014
$$465$$ 0 0
$$466$$ 13.4038 0.620921
$$467$$ 13.4945 0.624451 0.312225 0.950008i $$-0.398926\pi$$
0.312225 + 0.950008i $$0.398926\pi$$
$$468$$ −3.28763 −0.151971
$$469$$ 29.3734 1.35634
$$470$$ 0 0
$$471$$ 22.8938 1.05489
$$472$$ 1.67149 0.0769366
$$473$$ 54.0449 2.48499
$$474$$ 15.5683 0.715075
$$475$$ 0 0
$$476$$ −9.29224 −0.425909
$$477$$ 2.97538 0.136233
$$478$$ 0.569620 0.0260538
$$479$$ 6.69141 0.305739 0.152869 0.988246i $$-0.451149\pi$$
0.152869 + 0.988246i $$0.451149\pi$$
$$480$$ 0 0
$$481$$ 19.3393 0.881795
$$482$$ 0.832660 0.0379266
$$483$$ −28.2496 −1.28540
$$484$$ −22.5261 −1.02391
$$485$$ 0 0
$$486$$ −0.936719 −0.0424904
$$487$$ −28.6972 −1.30039 −0.650197 0.759765i $$-0.725314\pi$$
−0.650197 + 0.759765i $$0.725314\pi$$
$$488$$ 24.2826 1.09922
$$489$$ −11.2612 −0.509247
$$490$$ 0 0
$$491$$ −6.00762 −0.271120 −0.135560 0.990769i $$-0.543283\pi$$
−0.135560 + 0.990769i $$0.543283\pi$$
$$492$$ 9.90014 0.446333
$$493$$ −17.6506 −0.794941
$$494$$ 7.25389 0.326368
$$495$$ 0 0
$$496$$ 2.98562 0.134058
$$497$$ 42.8643 1.92273
$$498$$ 12.9599 0.580746
$$499$$ 40.6329 1.81898 0.909489 0.415727i $$-0.136473\pi$$
0.909489 + 0.415727i $$0.136473\pi$$
$$500$$ 0 0
$$501$$ −16.8461 −0.752627
$$502$$ 0.0453114 0.00202235
$$503$$ 15.2536 0.680124 0.340062 0.940403i $$-0.389552\pi$$
0.340062 + 0.940403i $$0.389552\pi$$
$$504$$ 10.6915 0.476236
$$505$$ 0 0
$$506$$ −40.3508 −1.79381
$$507$$ −4.42275 −0.196421
$$508$$ −13.5924 −0.603066
$$509$$ −12.9027 −0.571904 −0.285952 0.958244i $$-0.592310\pi$$
−0.285952 + 0.958244i $$0.592310\pi$$
$$510$$ 0 0
$$511$$ −21.7259 −0.961095
$$512$$ 5.55916 0.245683
$$513$$ −2.64416 −0.116743
$$514$$ −3.35361 −0.147921
$$515$$ 0 0
$$516$$ 10.8847 0.479171
$$517$$ 5.57376 0.245134
$$518$$ −22.6096 −0.993410
$$519$$ −13.5553 −0.595010
$$520$$ 0 0
$$521$$ 8.30560 0.363875 0.181937 0.983310i $$-0.441763\pi$$
0.181937 + 0.983310i $$0.441763\pi$$
$$522$$ 7.30086 0.319550
$$523$$ 36.5557 1.59847 0.799235 0.601019i $$-0.205238\pi$$
0.799235 + 0.601019i $$0.205238\pi$$
$$524$$ −1.77668 −0.0776147
$$525$$ 0 0
$$526$$ −16.4987 −0.719375
$$527$$ −13.6662 −0.595307
$$528$$ 2.75759 0.120009
$$529$$ 36.7297 1.59694
$$530$$ 0 0
$$531$$ 0.571458 0.0247992
$$532$$ 10.8496 0.470391
$$533$$ −25.8289 −1.11878
$$534$$ −4.03839 −0.174758
$$535$$ 0 0
$$536$$ 23.5048 1.01525
$$537$$ −5.29177 −0.228357
$$538$$ 13.7943 0.594713
$$539$$ −35.4542 −1.52712
$$540$$ 0 0
$$541$$ 17.6937 0.760713 0.380356 0.924840i $$-0.375801\pi$$
0.380356 + 0.924840i $$0.375801\pi$$
$$542$$ −6.80601 −0.292343
$$543$$ −6.93711 −0.297700
$$544$$ −12.1983 −0.522997
$$545$$ 0 0
$$546$$ −10.0277 −0.429146
$$547$$ −15.8840 −0.679150 −0.339575 0.940579i $$-0.610283\pi$$
−0.339575 + 0.940579i $$0.610283\pi$$
$$548$$ −23.0183 −0.983291
$$549$$ 8.30187 0.354315
$$550$$ 0 0
$$551$$ 20.6088 0.877964
$$552$$ −22.6055 −0.962155
$$553$$ −60.7505 −2.58337
$$554$$ 15.0730 0.640391
$$555$$ 0 0
$$556$$ 18.7669 0.795896
$$557$$ −8.53325 −0.361566 −0.180783 0.983523i $$-0.557863\pi$$
−0.180783 + 0.983523i $$0.557863\pi$$
$$558$$ 5.65278 0.239301
$$559$$ −28.3975 −1.20109
$$560$$ 0 0
$$561$$ −12.6224 −0.532918
$$562$$ 2.00434 0.0845482
$$563$$ 20.6854 0.871785 0.435893 0.899999i $$-0.356433\pi$$
0.435893 + 0.899999i $$0.356433\pi$$
$$564$$ 1.12256 0.0472682
$$565$$ 0 0
$$566$$ 22.3220 0.938265
$$567$$ 3.65526 0.153506
$$568$$ 34.3003 1.43921
$$569$$ 10.7355 0.450057 0.225029 0.974352i $$-0.427752\pi$$
0.225029 + 0.974352i $$0.427752\pi$$
$$570$$ 0 0
$$571$$ −11.4293 −0.478303 −0.239152 0.970982i $$-0.576869\pi$$
−0.239152 + 0.970982i $$0.576869\pi$$
$$572$$ 18.3245 0.766184
$$573$$ 3.27524 0.136825
$$574$$ 30.1967 1.26039
$$575$$ 0 0
$$576$$ 6.03511 0.251463
$$577$$ −18.3590 −0.764296 −0.382148 0.924101i $$-0.624816\pi$$
−0.382148 + 0.924101i $$0.624816\pi$$
$$578$$ 11.1203 0.462543
$$579$$ 6.91822 0.287511
$$580$$ 0 0
$$581$$ −50.5719 −2.09808
$$582$$ 4.91444 0.203710
$$583$$ −16.5841 −0.686841
$$584$$ −17.3852 −0.719403
$$585$$ 0 0
$$586$$ 16.4798 0.680774
$$587$$ 15.6571 0.646237 0.323119 0.946358i $$-0.395269\pi$$
0.323119 + 0.946358i $$0.395269\pi$$
$$588$$ −7.14050 −0.294469
$$589$$ 15.9566 0.657480
$$590$$ 0 0
$$591$$ −13.2026 −0.543084
$$592$$ −3.26699 −0.134273
$$593$$ 15.5623 0.639067 0.319533 0.947575i $$-0.396474\pi$$
0.319533 + 0.947575i $$0.396474\pi$$
$$594$$ 5.22104 0.214222
$$595$$ 0 0
$$596$$ −11.4694 −0.469805
$$597$$ −10.3855 −0.425051
$$598$$ 21.2021 0.867017
$$599$$ −30.3904 −1.24172 −0.620859 0.783922i $$-0.713216\pi$$
−0.620859 + 0.783922i $$0.713216\pi$$
$$600$$ 0 0
$$601$$ −30.1251 −1.22883 −0.614414 0.788984i $$-0.710608\pi$$
−0.614414 + 0.788984i $$0.710608\pi$$
$$602$$ 33.1997 1.35312
$$603$$ 8.03594 0.327249
$$604$$ 23.4668 0.954852
$$605$$ 0 0
$$606$$ 16.8203 0.683279
$$607$$ 44.1714 1.79286 0.896430 0.443184i $$-0.146151\pi$$
0.896430 + 0.443184i $$0.146151\pi$$
$$608$$ 14.2427 0.577618
$$609$$ −28.4894 −1.15445
$$610$$ 0 0
$$611$$ −2.92869 −0.118482
$$612$$ −2.54216 −0.102761
$$613$$ 24.8671 1.00437 0.502186 0.864759i $$-0.332529\pi$$
0.502186 + 0.864759i $$0.332529\pi$$
$$614$$ 19.4116 0.783387
$$615$$ 0 0
$$616$$ −59.5917 −2.40102
$$617$$ 24.8433 1.00015 0.500077 0.865981i $$-0.333305\pi$$
0.500077 + 0.865981i $$0.333305\pi$$
$$618$$ 12.9580 0.521249
$$619$$ 18.9581 0.761988 0.380994 0.924577i $$-0.375582\pi$$
0.380994 + 0.924577i $$0.375582\pi$$
$$620$$ 0 0
$$621$$ −7.72850 −0.310134
$$622$$ 22.7683 0.912926
$$623$$ 15.7586 0.631355
$$624$$ −1.44896 −0.0580048
$$625$$ 0 0
$$626$$ −16.2676 −0.650183
$$627$$ 14.7379 0.588575
$$628$$ −25.6996 −1.02553
$$629$$ 14.9541 0.596258
$$630$$ 0 0
$$631$$ 39.5889 1.57601 0.788004 0.615670i $$-0.211115\pi$$
0.788004 + 0.615670i $$0.211115\pi$$
$$632$$ −48.6129 −1.93372
$$633$$ 19.5002 0.775065
$$634$$ −1.88027 −0.0746752
$$635$$ 0 0
$$636$$ −3.34004 −0.132441
$$637$$ 18.6292 0.738114
$$638$$ −40.6932 −1.61106
$$639$$ 11.7268 0.463903
$$640$$ 0 0
$$641$$ −7.69114 −0.303782 −0.151891 0.988397i $$-0.548536\pi$$
−0.151891 + 0.988397i $$0.548536\pi$$
$$642$$ 5.78762 0.228419
$$643$$ −32.5033 −1.28180 −0.640902 0.767623i $$-0.721439\pi$$
−0.640902 + 0.767623i $$0.721439\pi$$
$$644$$ 31.7119 1.24962
$$645$$ 0 0
$$646$$ 5.60907 0.220686
$$647$$ −33.1357 −1.30270 −0.651349 0.758778i $$-0.725797\pi$$
−0.651349 + 0.758778i $$0.725797\pi$$
$$648$$ 2.92496 0.114903
$$649$$ −3.18517 −0.125029
$$650$$ 0 0
$$651$$ −22.0582 −0.864530
$$652$$ 12.6413 0.495072
$$653$$ 1.78903 0.0700102 0.0350051 0.999387i $$-0.488855\pi$$
0.0350051 + 0.999387i $$0.488855\pi$$
$$654$$ −13.1919 −0.515844
$$655$$ 0 0
$$656$$ 4.36330 0.170358
$$657$$ −5.94373 −0.231887
$$658$$ 3.42395 0.133479
$$659$$ −13.1926 −0.513912 −0.256956 0.966423i $$-0.582720\pi$$
−0.256956 + 0.966423i $$0.582720\pi$$
$$660$$ 0 0
$$661$$ 48.8401 1.89966 0.949830 0.312766i $$-0.101256\pi$$
0.949830 + 0.312766i $$0.101256\pi$$
$$662$$ −21.5967 −0.839379
$$663$$ 6.63235 0.257579
$$664$$ −40.4679 −1.57046
$$665$$ 0 0
$$666$$ −6.18551 −0.239683
$$667$$ 60.2365 2.33237
$$668$$ 18.9107 0.731677
$$669$$ 5.43059 0.209959
$$670$$ 0 0
$$671$$ −46.2726 −1.78633
$$672$$ −19.6890 −0.759519
$$673$$ −7.24521 −0.279282 −0.139641 0.990202i $$-0.544595\pi$$
−0.139641 + 0.990202i $$0.544595\pi$$
$$674$$ −0.915433 −0.0352611
$$675$$ 0 0
$$676$$ 4.96479 0.190954
$$677$$ 17.0194 0.654109 0.327054 0.945006i $$-0.393944\pi$$
0.327054 + 0.945006i $$0.393944\pi$$
$$678$$ −15.5830 −0.598462
$$679$$ −19.1771 −0.735950
$$680$$ 0 0
$$681$$ −2.68717 −0.102973
$$682$$ −31.5072 −1.20647
$$683$$ 1.41734 0.0542331 0.0271166 0.999632i $$-0.491367\pi$$
0.0271166 + 0.999632i $$0.491367\pi$$
$$684$$ 2.96822 0.113493
$$685$$ 0 0
$$686$$ 2.18820 0.0835460
$$687$$ 17.6212 0.672291
$$688$$ 4.79721 0.182892
$$689$$ 8.71398 0.331976
$$690$$ 0 0
$$691$$ 8.82922 0.335879 0.167940 0.985797i $$-0.446289\pi$$
0.167940 + 0.985797i $$0.446289\pi$$
$$692$$ 15.2166 0.578447
$$693$$ −20.3735 −0.773926
$$694$$ −8.53455 −0.323967
$$695$$ 0 0
$$696$$ −22.7974 −0.864132
$$697$$ −19.9722 −0.756501
$$698$$ −2.47916 −0.0938377
$$699$$ −14.3094 −0.541230
$$700$$ 0 0
$$701$$ −29.1496 −1.10096 −0.550482 0.834847i $$-0.685556\pi$$
−0.550482 + 0.834847i $$0.685556\pi$$
$$702$$ −2.74336 −0.103542
$$703$$ −17.4604 −0.658531
$$704$$ −33.6382 −1.26779
$$705$$ 0 0
$$706$$ −14.8430 −0.558625
$$707$$ −65.6362 −2.46850
$$708$$ −0.641495 −0.0241088
$$709$$ −19.7491 −0.741694 −0.370847 0.928694i $$-0.620933\pi$$
−0.370847 + 0.928694i $$0.620933\pi$$
$$710$$ 0 0
$$711$$ −16.6200 −0.623300
$$712$$ 12.6101 0.472584
$$713$$ 46.6388 1.74664
$$714$$ −7.75391 −0.290183
$$715$$ 0 0
$$716$$ 5.94032 0.222000
$$717$$ −0.608101 −0.0227100
$$718$$ −11.1161 −0.414848
$$719$$ 16.8239 0.627426 0.313713 0.949518i $$-0.398427\pi$$
0.313713 + 0.949518i $$0.398427\pi$$
$$720$$ 0 0
$$721$$ −50.5648 −1.88313
$$722$$ 11.2485 0.418626
$$723$$ −0.888912 −0.0330590
$$724$$ 7.78732 0.289413
$$725$$ 0 0
$$726$$ −18.7969 −0.697619
$$727$$ 21.9005 0.812244 0.406122 0.913819i $$-0.366881\pi$$
0.406122 + 0.913819i $$0.366881\pi$$
$$728$$ 31.3121 1.16050
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −21.9584 −0.812160
$$732$$ −9.31934 −0.344453
$$733$$ −36.6289 −1.35292 −0.676460 0.736479i $$-0.736487\pi$$
−0.676460 + 0.736479i $$0.736487\pi$$
$$734$$ 4.74055 0.174977
$$735$$ 0 0
$$736$$ 41.6294 1.53448
$$737$$ −44.7904 −1.64987
$$738$$ 8.26117 0.304098
$$739$$ −9.51491 −0.350011 −0.175006 0.984567i $$-0.555994\pi$$
−0.175006 + 0.984567i $$0.555994\pi$$
$$740$$ 0 0
$$741$$ −7.74394 −0.284481
$$742$$ −10.1876 −0.373997
$$743$$ −4.47470 −0.164161 −0.0820805 0.996626i $$-0.526156\pi$$
−0.0820805 + 0.996626i $$0.526156\pi$$
$$744$$ −17.6511 −0.647122
$$745$$ 0 0
$$746$$ −1.11876 −0.0409607
$$747$$ −13.8354 −0.506211
$$748$$ 14.1694 0.518084
$$749$$ −22.5844 −0.825217
$$750$$ 0 0
$$751$$ 33.1075 1.20811 0.604055 0.796943i $$-0.293551\pi$$
0.604055 + 0.796943i $$0.293551\pi$$
$$752$$ 0.494746 0.0180415
$$753$$ −0.0483724 −0.00176279
$$754$$ 21.3820 0.778686
$$755$$ 0 0
$$756$$ −4.10324 −0.149233
$$757$$ 12.0208 0.436902 0.218451 0.975848i $$-0.429900\pi$$
0.218451 + 0.975848i $$0.429900\pi$$
$$758$$ −15.0175 −0.545461
$$759$$ 43.0768 1.56359
$$760$$ 0 0
$$761$$ 26.4998 0.960616 0.480308 0.877100i $$-0.340525\pi$$
0.480308 + 0.877100i $$0.340525\pi$$
$$762$$ −11.3422 −0.410884
$$763$$ 51.4773 1.86360
$$764$$ −3.67665 −0.133017
$$765$$ 0 0
$$766$$ −21.9754 −0.794002
$$767$$ 1.67363 0.0604311
$$768$$ −16.8660 −0.608599
$$769$$ −7.70999 −0.278029 −0.139015 0.990290i $$-0.544394\pi$$
−0.139015 + 0.990290i $$0.544394\pi$$
$$770$$ 0 0
$$771$$ 3.58017 0.128937
$$772$$ −7.76610 −0.279508
$$773$$ −20.8018 −0.748190 −0.374095 0.927390i $$-0.622047\pi$$
−0.374095 + 0.927390i $$0.622047\pi$$
$$774$$ 9.08272 0.326472
$$775$$ 0 0
$$776$$ −15.3456 −0.550876
$$777$$ 24.1370 0.865912
$$778$$ 26.1299 0.936801
$$779$$ 23.3196 0.835510
$$780$$ 0 0
$$781$$ −65.3621 −2.33884
$$782$$ 16.3945 0.586266
$$783$$ −7.79408 −0.278538
$$784$$ −3.14704 −0.112394
$$785$$ 0 0
$$786$$ −1.48255 −0.0528809
$$787$$ −46.7576 −1.66673 −0.833364 0.552725i $$-0.813588\pi$$
−0.833364 + 0.552725i $$0.813588\pi$$
$$788$$ 14.8207 0.527967
$$789$$ 17.6132 0.627048
$$790$$ 0 0
$$791$$ 60.8079 2.16208
$$792$$ −16.3030 −0.579302
$$793$$ 24.3136 0.863403
$$794$$ 10.6235 0.377015
$$795$$ 0 0
$$796$$ 11.6583 0.413219
$$797$$ 24.6265 0.872314 0.436157 0.899870i $$-0.356339\pi$$
0.436157 + 0.899870i $$0.356339\pi$$
$$798$$ 9.05347 0.320489
$$799$$ −2.26461 −0.0801162
$$800$$ 0 0
$$801$$ 4.31121 0.152329
$$802$$ 11.8227 0.417474
$$803$$ 33.1289 1.16909
$$804$$ −9.02081 −0.318139
$$805$$ 0 0
$$806$$ 16.5553 0.583134
$$807$$ −14.7262 −0.518385
$$808$$ −52.5225 −1.84773
$$809$$ 32.1274 1.12954 0.564770 0.825248i $$-0.308965\pi$$
0.564770 + 0.825248i $$0.308965\pi$$
$$810$$ 0 0
$$811$$ −0.000411906 0 −1.44640e−5 0 −7.23199e−6 1.00000i $$-0.500002\pi$$
−7.23199e−6 1.00000i $$0.500002\pi$$
$$812$$ 31.9810 1.12231
$$813$$ 7.26580 0.254823
$$814$$ 34.4765 1.20840
$$815$$ 0 0
$$816$$ −1.12041 −0.0392221
$$817$$ 25.6386 0.896981
$$818$$ 2.79425 0.0976988
$$819$$ 10.7051 0.374068
$$820$$ 0 0
$$821$$ 53.7005 1.87416 0.937081 0.349113i $$-0.113517\pi$$
0.937081 + 0.349113i $$0.113517\pi$$
$$822$$ −19.2076 −0.669941
$$823$$ −30.1058 −1.04942 −0.524711 0.851280i $$-0.675827\pi$$
−0.524711 + 0.851280i $$0.675827\pi$$
$$824$$ −40.4622 −1.40957
$$825$$ 0 0
$$826$$ −1.95664 −0.0680803
$$827$$ 4.74104 0.164862 0.0824310 0.996597i $$-0.473732\pi$$
0.0824310 + 0.996597i $$0.473732\pi$$
$$828$$ 8.67569 0.301501
$$829$$ 24.2241 0.841337 0.420669 0.907214i $$-0.361796\pi$$
0.420669 + 0.907214i $$0.361796\pi$$
$$830$$ 0 0
$$831$$ −16.0913 −0.558201
$$832$$ 17.6750 0.612770
$$833$$ 14.4050 0.499103
$$834$$ 15.6601 0.542264
$$835$$ 0 0
$$836$$ −16.5442 −0.572192
$$837$$ −6.03466 −0.208588
$$838$$ −14.5262 −0.501801
$$839$$ 13.1024 0.452345 0.226172 0.974087i $$-0.427379\pi$$
0.226172 + 0.974087i $$0.427379\pi$$
$$840$$ 0 0
$$841$$ 31.7477 1.09475
$$842$$ 20.8035 0.716936
$$843$$ −2.13975 −0.0736969
$$844$$ −21.8902 −0.753490
$$845$$ 0 0
$$846$$ 0.936719 0.0322050
$$847$$ 73.3492 2.52031
$$848$$ −1.47206 −0.0505507
$$849$$ −23.8300 −0.817845
$$850$$ 0 0
$$851$$ −51.0342 −1.74943
$$852$$ −13.1640 −0.450990
$$853$$ −50.2328 −1.71994 −0.859969 0.510347i $$-0.829517\pi$$
−0.859969 + 0.510347i $$0.829517\pi$$
$$854$$ −28.4252 −0.972690
$$855$$ 0 0
$$856$$ −18.0722 −0.617695
$$857$$ 11.5763 0.395438 0.197719 0.980259i $$-0.436647\pi$$
0.197719 + 0.980259i $$0.436647\pi$$
$$858$$ 15.2908 0.522020
$$859$$ 5.35315 0.182647 0.0913236 0.995821i $$-0.470890\pi$$
0.0913236 + 0.995821i $$0.470890\pi$$
$$860$$ 0 0
$$861$$ −32.2367 −1.09862
$$862$$ 18.8608 0.642400
$$863$$ 27.8072 0.946569 0.473285 0.880910i $$-0.343068\pi$$
0.473285 + 0.880910i $$0.343068\pi$$
$$864$$ −5.38648 −0.183252
$$865$$ 0 0
$$866$$ −16.6959 −0.567349
$$867$$ −11.8715 −0.403178
$$868$$ 24.7617 0.840465
$$869$$ 92.6360 3.14246
$$870$$ 0 0
$$871$$ 23.5348 0.797447
$$872$$ 41.1924 1.39495
$$873$$ −5.24644 −0.177565
$$874$$ −19.1422 −0.647495
$$875$$ 0 0
$$876$$ 6.67218 0.225432
$$877$$ −24.8181 −0.838048 −0.419024 0.907975i $$-0.637628\pi$$
−0.419024 + 0.907975i $$0.637628\pi$$
$$878$$ −11.2321 −0.379064
$$879$$ −17.5931 −0.593401
$$880$$ 0 0
$$881$$ 35.1845 1.18540 0.592698 0.805424i $$-0.298063\pi$$
0.592698 + 0.805424i $$0.298063\pi$$
$$882$$ −5.95838 −0.200629
$$883$$ 18.2761 0.615039 0.307520 0.951542i $$-0.400501\pi$$
0.307520 + 0.951542i $$0.400501\pi$$
$$884$$ −7.44520 −0.250409
$$885$$ 0 0
$$886$$ 23.6467 0.794426
$$887$$ 21.5403 0.723254 0.361627 0.932323i $$-0.382221\pi$$
0.361627 + 0.932323i $$0.382221\pi$$
$$888$$ 19.3146 0.648156
$$889$$ 44.2594 1.48441
$$890$$ 0 0
$$891$$ −5.57376 −0.186728
$$892$$ −6.09615 −0.204114
$$893$$ 2.64416 0.0884834
$$894$$ −9.57064 −0.320090
$$895$$ 0 0
$$896$$ 18.7140 0.625193
$$897$$ −22.6344 −0.755741
$$898$$ 34.6912 1.15766
$$899$$ 47.0346 1.56869
$$900$$ 0 0
$$901$$ 6.73808 0.224478
$$902$$ −46.0458 −1.53316
$$903$$ −35.4425 −1.17945
$$904$$ 48.6589 1.61837
$$905$$ 0 0
$$906$$ 19.5819 0.650565
$$907$$ −17.7267 −0.588605 −0.294302 0.955712i $$-0.595087\pi$$
−0.294302 + 0.955712i $$0.595087\pi$$
$$908$$ 3.01651 0.100106
$$909$$ −17.9567 −0.595585
$$910$$ 0 0
$$911$$ −5.58606 −0.185074 −0.0925372 0.995709i $$-0.529498\pi$$
−0.0925372 + 0.995709i $$0.529498\pi$$
$$912$$ 1.30819 0.0433184
$$913$$ 77.1151 2.55214
$$914$$ −13.0794 −0.432628
$$915$$ 0 0
$$916$$ −19.7808 −0.653577
$$917$$ 5.78521 0.191044
$$918$$ −2.12130 −0.0700134
$$919$$ 9.14763 0.301752 0.150876 0.988553i $$-0.451791\pi$$
0.150876 + 0.988553i $$0.451791\pi$$
$$920$$ 0 0
$$921$$ −20.7229 −0.682844
$$922$$ 39.3387 1.29555
$$923$$ 34.3441 1.13045
$$924$$ 22.8705 0.752383
$$925$$ 0 0
$$926$$ −21.6632 −0.711896
$$927$$ −13.8334 −0.454350
$$928$$ 41.9826 1.37815
$$929$$ 19.9482 0.654480 0.327240 0.944941i $$-0.393881\pi$$
0.327240 + 0.944941i $$0.393881\pi$$
$$930$$ 0 0
$$931$$ −16.8193 −0.551229
$$932$$ 16.0631 0.526164
$$933$$ −24.3065 −0.795758
$$934$$ −12.6405 −0.413611
$$935$$ 0 0
$$936$$ 8.56631 0.279999
$$937$$ −28.6210 −0.935007 −0.467504 0.883991i $$-0.654847\pi$$
−0.467504 + 0.883991i $$0.654847\pi$$
$$938$$ −27.5146 −0.898385
$$939$$ 17.3665 0.566736
$$940$$ 0 0
$$941$$ 17.6221 0.574464 0.287232 0.957861i $$-0.407265\pi$$
0.287232 + 0.957861i $$0.407265\pi$$
$$942$$ −21.4450 −0.698716
$$943$$ 68.1597 2.21958
$$944$$ −0.282726 −0.00920196
$$945$$ 0 0
$$946$$ −50.6249 −1.64596
$$947$$ 41.9598 1.36351 0.681754 0.731581i $$-0.261217\pi$$
0.681754 + 0.731581i $$0.261217\pi$$
$$948$$ 18.6570 0.605950
$$949$$ −17.4074 −0.565067
$$950$$ 0 0
$$951$$ 2.00730 0.0650911
$$952$$ 24.2120 0.784717
$$953$$ −26.1334 −0.846545 −0.423273 0.906002i $$-0.639119\pi$$
−0.423273 + 0.906002i $$0.639119\pi$$
$$954$$ −2.78710 −0.0902355
$$955$$ 0 0
$$956$$ 0.682629 0.0220778
$$957$$ 43.4423 1.40429
$$958$$ −6.26797 −0.202509
$$959$$ 74.9517 2.42032
$$960$$ 0 0
$$961$$ 5.41711 0.174745
$$962$$ −18.1155 −0.584066
$$963$$ −6.17861 −0.199103
$$964$$ 0.997855 0.0321388
$$965$$ 0 0
$$966$$ 26.4620 0.851400
$$967$$ 9.86542 0.317251 0.158625 0.987339i $$-0.449294\pi$$
0.158625 + 0.987339i $$0.449294\pi$$
$$968$$ 58.6944 1.88651
$$969$$ −5.98799 −0.192362
$$970$$ 0 0
$$971$$ 3.91113 0.125514 0.0627571 0.998029i $$-0.480011\pi$$
0.0627571 + 0.998029i $$0.480011\pi$$
$$972$$ −1.12256 −0.0360061
$$973$$ −61.1087 −1.95905
$$974$$ 26.8812 0.861329
$$975$$ 0 0
$$976$$ −4.10732 −0.131472
$$977$$ 8.80390 0.281662 0.140831 0.990034i $$-0.455023\pi$$
0.140831 + 0.990034i $$0.455023\pi$$
$$978$$ 10.5485 0.337305
$$979$$ −24.0297 −0.767991
$$980$$ 0 0
$$981$$ 14.0831 0.449638
$$982$$ 5.62745 0.179579
$$983$$ −2.42901 −0.0774735 −0.0387368 0.999249i $$-0.512333\pi$$
−0.0387368 + 0.999249i $$0.512333\pi$$
$$984$$ −25.7960 −0.822346
$$985$$ 0 0
$$986$$ 16.5336 0.526537
$$987$$ −3.65526 −0.116348
$$988$$ 8.69302 0.276562
$$989$$ 74.9379 2.38289
$$990$$ 0 0
$$991$$ −12.4176 −0.394459 −0.197230 0.980357i $$-0.563194\pi$$
−0.197230 + 0.980357i $$0.563194\pi$$
$$992$$ 32.5056 1.03205
$$993$$ 23.0557 0.731650
$$994$$ −40.1518 −1.27354
$$995$$ 0 0
$$996$$ 15.5310 0.492120
$$997$$ −14.3515 −0.454518 −0.227259 0.973834i $$-0.572976\pi$$
−0.227259 + 0.973834i $$0.572976\pi$$
$$998$$ −38.0616 −1.20482
$$999$$ 6.60338 0.208922
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 3525.2.a.bd.1.4 8
5.4 even 2 3525.2.a.be.1.5 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.4 8 1.1 even 1 trivial
3525.2.a.be.1.5 yes 8 5.4 even 2