# Properties

 Label 8281.2.a.co.1.1 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1241179138$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.58860$$ of defining polynomial Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} +1.61205 q^{5} +1.34210 q^{6} -6.99143 q^{8} -2.73119 q^{9} +O(q^{10})$$ $$q-2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} +1.61205 q^{5} +1.34210 q^{6} -6.99143 q^{8} -2.73119 q^{9} -4.17296 q^{10} -2.70496 q^{11} -2.43723 q^{12} -0.835795 q^{15} +8.69632 q^{16} -3.12661 q^{17} +7.06997 q^{18} +3.68150 q^{19} +7.57803 q^{20} +7.00205 q^{22} +1.98604 q^{23} +3.62482 q^{24} -2.40128 q^{25} +2.97143 q^{27} -5.37271 q^{29} +2.16354 q^{30} +10.4780 q^{31} -8.52843 q^{32} +1.40243 q^{33} +8.09354 q^{34} -12.8389 q^{36} -5.95346 q^{37} -9.52994 q^{38} -11.2706 q^{40} -7.70150 q^{41} -3.35600 q^{43} -12.7156 q^{44} -4.40283 q^{45} -5.14106 q^{46} +1.05508 q^{47} -4.50874 q^{48} +6.21596 q^{50} +1.62104 q^{51} +7.26568 q^{53} -7.69184 q^{54} -4.36054 q^{55} -1.90873 q^{57} +13.9078 q^{58} +11.4241 q^{59} -3.92895 q^{60} +2.92507 q^{61} -27.1235 q^{62} +4.68406 q^{64} -3.63033 q^{66} +13.5818 q^{67} -14.6977 q^{68} -1.02969 q^{69} +1.35111 q^{71} +19.0949 q^{72} +9.10335 q^{73} +15.4111 q^{74} +1.24498 q^{75} +17.3062 q^{76} -6.20578 q^{79} +14.0189 q^{80} +6.65300 q^{81} +19.9361 q^{82} -2.69672 q^{83} -5.04026 q^{85} +8.68734 q^{86} +2.78557 q^{87} +18.9115 q^{88} +1.75988 q^{89} +11.3972 q^{90} +9.33607 q^{92} -5.43251 q^{93} -2.73119 q^{94} +5.93478 q^{95} +4.42170 q^{96} +15.4820 q^{97} +7.38776 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} + O(q^{10})$$ $$12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} - 24 q^{10} + 2 q^{12} + 16 q^{16} - 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} - 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} - 38 q^{38} - 2 q^{40} + 22 q^{43} + 38 q^{48} + 8 q^{51} + 16 q^{53} - 30 q^{55} + 10 q^{61} - 82 q^{62} - 2 q^{64} - 68 q^{66} - 22 q^{68} - 14 q^{69} + 66 q^{74} - 2 q^{75} + 70 q^{79} - 28 q^{81} - 10 q^{82} + 20 q^{87} - 28 q^{88} - 66 q^{92} + 2 q^{94} + 4 q^{95} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.58860 −1.83042 −0.915209 0.402981i $$-0.867974\pi$$
−0.915209 + 0.402981i $$0.867974\pi$$
$$3$$ −0.518466 −0.299336 −0.149668 0.988736i $$-0.547821\pi$$
−0.149668 + 0.988736i $$0.547821\pi$$
$$4$$ 4.70085 2.35043
$$5$$ 1.61205 0.720932 0.360466 0.932772i $$-0.382618\pi$$
0.360466 + 0.932772i $$0.382618\pi$$
$$6$$ 1.34210 0.547910
$$7$$ 0 0
$$8$$ −6.99143 −2.47184
$$9$$ −2.73119 −0.910398
$$10$$ −4.17296 −1.31961
$$11$$ −2.70496 −0.815575 −0.407788 0.913077i $$-0.633700\pi$$
−0.407788 + 0.913077i $$0.633700\pi$$
$$12$$ −2.43723 −0.703568
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −0.835795 −0.215801
$$16$$ 8.69632 2.17408
$$17$$ −3.12661 −0.758314 −0.379157 0.925332i $$-0.623786\pi$$
−0.379157 + 0.925332i $$0.623786\pi$$
$$18$$ 7.06997 1.66641
$$19$$ 3.68150 0.844595 0.422297 0.906457i $$-0.361224\pi$$
0.422297 + 0.906457i $$0.361224\pi$$
$$20$$ 7.57803 1.69450
$$21$$ 0 0
$$22$$ 7.00205 1.49284
$$23$$ 1.98604 0.414117 0.207059 0.978329i $$-0.433611\pi$$
0.207059 + 0.978329i $$0.433611\pi$$
$$24$$ 3.62482 0.739913
$$25$$ −2.40128 −0.480257
$$26$$ 0 0
$$27$$ 2.97143 0.571852
$$28$$ 0 0
$$29$$ −5.37271 −0.997687 −0.498844 0.866692i $$-0.666242\pi$$
−0.498844 + 0.866692i $$0.666242\pi$$
$$30$$ 2.16354 0.395006
$$31$$ 10.4780 1.88191 0.940956 0.338529i $$-0.109929\pi$$
0.940956 + 0.338529i $$0.109929\pi$$
$$32$$ −8.52843 −1.50763
$$33$$ 1.40243 0.244131
$$34$$ 8.09354 1.38803
$$35$$ 0 0
$$36$$ −12.8389 −2.13982
$$37$$ −5.95346 −0.978743 −0.489371 0.872075i $$-0.662774\pi$$
−0.489371 + 0.872075i $$0.662774\pi$$
$$38$$ −9.52994 −1.54596
$$39$$ 0 0
$$40$$ −11.2706 −1.78203
$$41$$ −7.70150 −1.20277 −0.601386 0.798958i $$-0.705385\pi$$
−0.601386 + 0.798958i $$0.705385\pi$$
$$42$$ 0 0
$$43$$ −3.35600 −0.511785 −0.255892 0.966705i $$-0.582369\pi$$
−0.255892 + 0.966705i $$0.582369\pi$$
$$44$$ −12.7156 −1.91695
$$45$$ −4.40283 −0.656335
$$46$$ −5.14106 −0.758008
$$47$$ 1.05508 0.153900 0.0769500 0.997035i $$-0.475482\pi$$
0.0769500 + 0.997035i $$0.475482\pi$$
$$48$$ −4.50874 −0.650781
$$49$$ 0 0
$$50$$ 6.21596 0.879070
$$51$$ 1.62104 0.226991
$$52$$ 0 0
$$53$$ 7.26568 0.998017 0.499009 0.866597i $$-0.333698\pi$$
0.499009 + 0.866597i $$0.333698\pi$$
$$54$$ −7.69184 −1.04673
$$55$$ −4.36054 −0.587975
$$56$$ 0 0
$$57$$ −1.90873 −0.252818
$$58$$ 13.9078 1.82618
$$59$$ 11.4241 1.48729 0.743643 0.668577i $$-0.233096\pi$$
0.743643 + 0.668577i $$0.233096\pi$$
$$60$$ −3.92895 −0.507225
$$61$$ 2.92507 0.374517 0.187259 0.982311i $$-0.440040\pi$$
0.187259 + 0.982311i $$0.440040\pi$$
$$62$$ −27.1235 −3.44468
$$63$$ 0 0
$$64$$ 4.68406 0.585507
$$65$$ 0 0
$$66$$ −3.63033 −0.446862
$$67$$ 13.5818 1.65928 0.829642 0.558296i $$-0.188545\pi$$
0.829642 + 0.558296i $$0.188545\pi$$
$$68$$ −14.6977 −1.78236
$$69$$ −1.02969 −0.123960
$$70$$ 0 0
$$71$$ 1.35111 0.160347 0.0801736 0.996781i $$-0.474453\pi$$
0.0801736 + 0.996781i $$0.474453\pi$$
$$72$$ 19.0949 2.25036
$$73$$ 9.10335 1.06547 0.532733 0.846283i $$-0.321165\pi$$
0.532733 + 0.846283i $$0.321165\pi$$
$$74$$ 15.4111 1.79151
$$75$$ 1.24498 0.143758
$$76$$ 17.3062 1.98516
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −6.20578 −0.698205 −0.349102 0.937085i $$-0.613513\pi$$
−0.349102 + 0.937085i $$0.613513\pi$$
$$80$$ 14.0189 1.56736
$$81$$ 6.65300 0.739222
$$82$$ 19.9361 2.20158
$$83$$ −2.69672 −0.296003 −0.148002 0.988987i $$-0.547284\pi$$
−0.148002 + 0.988987i $$0.547284\pi$$
$$84$$ 0 0
$$85$$ −5.04026 −0.546693
$$86$$ 8.68734 0.936780
$$87$$ 2.78557 0.298644
$$88$$ 18.9115 2.01597
$$89$$ 1.75988 0.186546 0.0932732 0.995641i $$-0.470267\pi$$
0.0932732 + 0.995641i $$0.470267\pi$$
$$90$$ 11.3972 1.20137
$$91$$ 0 0
$$92$$ 9.33607 0.973353
$$93$$ −5.43251 −0.563325
$$94$$ −2.73119 −0.281701
$$95$$ 5.93478 0.608896
$$96$$ 4.42170 0.451288
$$97$$ 15.4820 1.57196 0.785981 0.618250i $$-0.212158\pi$$
0.785981 + 0.618250i $$0.212158\pi$$
$$98$$ 0 0
$$99$$ 7.38776 0.742498
$$100$$ −11.2881 −1.12881
$$101$$ 1.27930 0.127295 0.0636477 0.997972i $$-0.479727\pi$$
0.0636477 + 0.997972i $$0.479727\pi$$
$$102$$ −4.19622 −0.415488
$$103$$ 11.4673 1.12991 0.564956 0.825121i $$-0.308893\pi$$
0.564956 + 0.825121i $$0.308893\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −18.8079 −1.82679
$$107$$ −5.13525 −0.496444 −0.248222 0.968703i $$-0.579846\pi$$
−0.248222 + 0.968703i $$0.579846\pi$$
$$108$$ 13.9682 1.34410
$$109$$ −1.72783 −0.165496 −0.0827481 0.996570i $$-0.526370\pi$$
−0.0827481 + 0.996570i $$0.526370\pi$$
$$110$$ 11.2877 1.07624
$$111$$ 3.08667 0.292973
$$112$$ 0 0
$$113$$ −8.59113 −0.808185 −0.404093 0.914718i $$-0.632413\pi$$
−0.404093 + 0.914718i $$0.632413\pi$$
$$114$$ 4.94095 0.462762
$$115$$ 3.20160 0.298551
$$116$$ −25.2563 −2.34499
$$117$$ 0 0
$$118$$ −29.5723 −2.72235
$$119$$ 0 0
$$120$$ 5.84340 0.533427
$$121$$ −3.68321 −0.334837
$$122$$ −7.57184 −0.685522
$$123$$ 3.99297 0.360034
$$124$$ 49.2557 4.42330
$$125$$ −11.9313 −1.06716
$$126$$ 0 0
$$127$$ −3.12412 −0.277221 −0.138610 0.990347i $$-0.544264\pi$$
−0.138610 + 0.990347i $$0.544264\pi$$
$$128$$ 4.93170 0.435904
$$129$$ 1.73997 0.153196
$$130$$ 0 0
$$131$$ −10.2092 −0.891982 −0.445991 0.895038i $$-0.647149\pi$$
−0.445991 + 0.895038i $$0.647149\pi$$
$$132$$ 6.59261 0.573813
$$133$$ 0 0
$$134$$ −35.1579 −3.03718
$$135$$ 4.79010 0.412266
$$136$$ 21.8595 1.87443
$$137$$ 9.99261 0.853726 0.426863 0.904316i $$-0.359619\pi$$
0.426863 + 0.904316i $$0.359619\pi$$
$$138$$ 2.66546 0.226899
$$139$$ 1.66420 0.141156 0.0705778 0.997506i $$-0.477516\pi$$
0.0705778 + 0.997506i $$0.477516\pi$$
$$140$$ 0 0
$$141$$ −0.547025 −0.0460679
$$142$$ −3.49748 −0.293502
$$143$$ 0 0
$$144$$ −23.7513 −1.97928
$$145$$ −8.66110 −0.719265
$$146$$ −23.5649 −1.95025
$$147$$ 0 0
$$148$$ −27.9863 −2.30046
$$149$$ −19.7980 −1.62192 −0.810959 0.585103i $$-0.801054\pi$$
−0.810959 + 0.585103i $$0.801054\pi$$
$$150$$ −3.22276 −0.263138
$$151$$ −7.53493 −0.613184 −0.306592 0.951841i $$-0.599189\pi$$
−0.306592 + 0.951841i $$0.599189\pi$$
$$152$$ −25.7390 −2.08771
$$153$$ 8.53937 0.690367
$$154$$ 0 0
$$155$$ 16.8912 1.35673
$$156$$ 0 0
$$157$$ −14.0045 −1.11768 −0.558839 0.829276i $$-0.688753\pi$$
−0.558839 + 0.829276i $$0.688753\pi$$
$$158$$ 16.0643 1.27801
$$159$$ −3.76700 −0.298743
$$160$$ −13.7483 −1.08690
$$161$$ 0 0
$$162$$ −17.2219 −1.35308
$$163$$ 7.16995 0.561594 0.280797 0.959767i $$-0.409401\pi$$
0.280797 + 0.959767i $$0.409401\pi$$
$$164$$ −36.2036 −2.82703
$$165$$ 2.26079 0.176002
$$166$$ 6.98072 0.541809
$$167$$ −17.9805 −1.39138 −0.695688 0.718344i $$-0.744901\pi$$
−0.695688 + 0.718344i $$0.744901\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 13.0472 1.00068
$$171$$ −10.0549 −0.768917
$$172$$ −15.7761 −1.20291
$$173$$ 12.8116 0.974047 0.487023 0.873389i $$-0.338083\pi$$
0.487023 + 0.873389i $$0.338083\pi$$
$$174$$ −7.21072 −0.546643
$$175$$ 0 0
$$176$$ −23.5232 −1.77312
$$177$$ −5.92298 −0.445199
$$178$$ −4.55561 −0.341458
$$179$$ 1.84022 0.137545 0.0687723 0.997632i $$-0.478092\pi$$
0.0687723 + 0.997632i $$0.478092\pi$$
$$180$$ −20.6971 −1.54267
$$181$$ −3.29928 −0.245234 −0.122617 0.992454i $$-0.539129\pi$$
−0.122617 + 0.992454i $$0.539129\pi$$
$$182$$ 0 0
$$183$$ −1.51655 −0.112107
$$184$$ −13.8852 −1.02363
$$185$$ −9.59730 −0.705607
$$186$$ 14.0626 1.03112
$$187$$ 8.45734 0.618462
$$188$$ 4.95980 0.361731
$$189$$ 0 0
$$190$$ −15.3628 −1.11453
$$191$$ 4.89614 0.354272 0.177136 0.984186i $$-0.443317\pi$$
0.177136 + 0.984186i $$0.443317\pi$$
$$192$$ −2.42852 −0.175264
$$193$$ −3.01910 −0.217320 −0.108660 0.994079i $$-0.534656\pi$$
−0.108660 + 0.994079i $$0.534656\pi$$
$$194$$ −40.0768 −2.87735
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.64991 −0.331292 −0.165646 0.986185i $$-0.552971\pi$$
−0.165646 + 0.986185i $$0.552971\pi$$
$$198$$ −19.1240 −1.35908
$$199$$ −0.410721 −0.0291152 −0.0145576 0.999894i $$-0.504634\pi$$
−0.0145576 + 0.999894i $$0.504634\pi$$
$$200$$ 16.7884 1.18712
$$201$$ −7.04171 −0.496684
$$202$$ −3.31160 −0.233004
$$203$$ 0 0
$$204$$ 7.62027 0.533525
$$205$$ −12.4152 −0.867118
$$206$$ −29.6844 −2.06821
$$207$$ −5.42425 −0.377012
$$208$$ 0 0
$$209$$ −9.95831 −0.688831
$$210$$ 0 0
$$211$$ −7.51600 −0.517423 −0.258711 0.965955i $$-0.583298\pi$$
−0.258711 + 0.965955i $$0.583298\pi$$
$$212$$ 34.1549 2.34577
$$213$$ −0.700504 −0.0479977
$$214$$ 13.2931 0.908699
$$215$$ −5.41005 −0.368962
$$216$$ −20.7745 −1.41353
$$217$$ 0 0
$$218$$ 4.47267 0.302927
$$219$$ −4.71978 −0.318933
$$220$$ −20.4982 −1.38199
$$221$$ 0 0
$$222$$ −7.99014 −0.536263
$$223$$ −22.5794 −1.51203 −0.756016 0.654553i $$-0.772857\pi$$
−0.756016 + 0.654553i $$0.772857\pi$$
$$224$$ 0 0
$$225$$ 6.55837 0.437225
$$226$$ 22.2390 1.47932
$$227$$ 13.6717 0.907424 0.453712 0.891148i $$-0.350099\pi$$
0.453712 + 0.891148i $$0.350099\pi$$
$$228$$ −8.97268 −0.594230
$$229$$ 7.93086 0.524086 0.262043 0.965056i $$-0.415604\pi$$
0.262043 + 0.965056i $$0.415604\pi$$
$$230$$ −8.28766 −0.546472
$$231$$ 0 0
$$232$$ 37.5629 2.46613
$$233$$ −6.57171 −0.430527 −0.215263 0.976556i $$-0.569061\pi$$
−0.215263 + 0.976556i $$0.569061\pi$$
$$234$$ 0 0
$$235$$ 1.70085 0.110951
$$236$$ 53.7028 3.49576
$$237$$ 3.21749 0.208998
$$238$$ 0 0
$$239$$ 9.39284 0.607572 0.303786 0.952740i $$-0.401749\pi$$
0.303786 + 0.952740i $$0.401749\pi$$
$$240$$ −7.26833 −0.469169
$$241$$ −10.0858 −0.649686 −0.324843 0.945768i $$-0.605311\pi$$
−0.324843 + 0.945768i $$0.605311\pi$$
$$242$$ 9.53435 0.612891
$$243$$ −12.3636 −0.793128
$$244$$ 13.7503 0.880275
$$245$$ 0 0
$$246$$ −10.3362 −0.659012
$$247$$ 0 0
$$248$$ −73.2565 −4.65179
$$249$$ 1.39816 0.0886045
$$250$$ 30.8853 1.95336
$$251$$ 10.3485 0.653194 0.326597 0.945164i $$-0.394098\pi$$
0.326597 + 0.945164i $$0.394098\pi$$
$$252$$ 0 0
$$253$$ −5.37215 −0.337744
$$254$$ 8.08709 0.507429
$$255$$ 2.61320 0.163645
$$256$$ −22.1343 −1.38339
$$257$$ −7.98658 −0.498189 −0.249095 0.968479i $$-0.580133\pi$$
−0.249095 + 0.968479i $$0.580133\pi$$
$$258$$ −4.50409 −0.280412
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 14.6739 0.908292
$$262$$ 26.4275 1.63270
$$263$$ 5.05934 0.311972 0.155986 0.987759i $$-0.450144\pi$$
0.155986 + 0.987759i $$0.450144\pi$$
$$264$$ −9.80498 −0.603455
$$265$$ 11.7127 0.719503
$$266$$ 0 0
$$267$$ −0.912435 −0.0558401
$$268$$ 63.8461 3.90002
$$269$$ −13.8902 −0.846902 −0.423451 0.905919i $$-0.639181\pi$$
−0.423451 + 0.905919i $$0.639181\pi$$
$$270$$ −12.3997 −0.754619
$$271$$ −8.32721 −0.505842 −0.252921 0.967487i $$-0.581391\pi$$
−0.252921 + 0.967487i $$0.581391\pi$$
$$272$$ −27.1900 −1.64863
$$273$$ 0 0
$$274$$ −25.8669 −1.56267
$$275$$ 6.49537 0.391685
$$276$$ −4.84043 −0.291360
$$277$$ −23.2116 −1.39465 −0.697325 0.716755i $$-0.745626\pi$$
−0.697325 + 0.716755i $$0.745626\pi$$
$$278$$ −4.30795 −0.258374
$$279$$ −28.6176 −1.71329
$$280$$ 0 0
$$281$$ −27.1595 −1.62020 −0.810100 0.586292i $$-0.800587\pi$$
−0.810100 + 0.586292i $$0.800587\pi$$
$$282$$ 1.41603 0.0843234
$$283$$ 16.1513 0.960092 0.480046 0.877243i $$-0.340620\pi$$
0.480046 + 0.877243i $$0.340620\pi$$
$$284$$ 6.35136 0.376884
$$285$$ −3.07698 −0.182265
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 23.2928 1.37254
$$289$$ −7.22433 −0.424960
$$290$$ 22.4201 1.31656
$$291$$ −8.02691 −0.470546
$$292$$ 42.7935 2.50430
$$293$$ −14.6452 −0.855582 −0.427791 0.903878i $$-0.640708\pi$$
−0.427791 + 0.903878i $$0.640708\pi$$
$$294$$ 0 0
$$295$$ 18.4162 1.07223
$$296$$ 41.6232 2.41930
$$297$$ −8.03758 −0.466388
$$298$$ 51.2492 2.96879
$$299$$ 0 0
$$300$$ 5.85248 0.337893
$$301$$ 0 0
$$302$$ 19.5049 1.12238
$$303$$ −0.663274 −0.0381041
$$304$$ 32.0155 1.83622
$$305$$ 4.71537 0.270001
$$306$$ −22.1050 −1.26366
$$307$$ −8.97844 −0.512427 −0.256213 0.966620i $$-0.582475\pi$$
−0.256213 + 0.966620i $$0.582475\pi$$
$$308$$ 0 0
$$309$$ −5.94543 −0.338224
$$310$$ −43.7245 −2.48338
$$311$$ −12.1816 −0.690755 −0.345378 0.938464i $$-0.612249\pi$$
−0.345378 + 0.938464i $$0.612249\pi$$
$$312$$ 0 0
$$313$$ −13.1240 −0.741810 −0.370905 0.928671i $$-0.620953\pi$$
−0.370905 + 0.928671i $$0.620953\pi$$
$$314$$ 36.2520 2.04582
$$315$$ 0 0
$$316$$ −29.1725 −1.64108
$$317$$ 16.7155 0.938836 0.469418 0.882976i $$-0.344464\pi$$
0.469418 + 0.882976i $$0.344464\pi$$
$$318$$ 9.75127 0.546824
$$319$$ 14.5330 0.813689
$$320$$ 7.55095 0.422111
$$321$$ 2.66245 0.148604
$$322$$ 0 0
$$323$$ −11.5106 −0.640468
$$324$$ 31.2748 1.73749
$$325$$ 0 0
$$326$$ −18.5601 −1.02795
$$327$$ 0.895821 0.0495390
$$328$$ 53.8445 2.97307
$$329$$ 0 0
$$330$$ −5.85228 −0.322157
$$331$$ 3.96665 0.218027 0.109013 0.994040i $$-0.465231\pi$$
0.109013 + 0.994040i $$0.465231\pi$$
$$332$$ −12.6769 −0.695733
$$333$$ 16.2600 0.891045
$$334$$ 46.5445 2.54680
$$335$$ 21.8946 1.19623
$$336$$ 0 0
$$337$$ 13.7032 0.746461 0.373230 0.927739i $$-0.378250\pi$$
0.373230 + 0.927739i $$0.378250\pi$$
$$338$$ 0 0
$$339$$ 4.45421 0.241919
$$340$$ −23.6935 −1.28496
$$341$$ −28.3426 −1.53484
$$342$$ 26.0281 1.40744
$$343$$ 0 0
$$344$$ 23.4632 1.26505
$$345$$ −1.65992 −0.0893671
$$346$$ −33.1641 −1.78291
$$347$$ −26.3979 −1.41711 −0.708556 0.705655i $$-0.750653\pi$$
−0.708556 + 0.705655i $$0.750653\pi$$
$$348$$ 13.0945 0.701941
$$349$$ 4.89024 0.261769 0.130884 0.991398i $$-0.458218\pi$$
0.130884 + 0.991398i $$0.458218\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 23.0690 1.22958
$$353$$ 13.5577 0.721605 0.360802 0.932642i $$-0.382503\pi$$
0.360802 + 0.932642i $$0.382503\pi$$
$$354$$ 15.3322 0.814899
$$355$$ 2.17806 0.115599
$$356$$ 8.27291 0.438464
$$357$$ 0 0
$$358$$ −4.76360 −0.251764
$$359$$ −8.58568 −0.453135 −0.226567 0.973996i $$-0.572750\pi$$
−0.226567 + 0.973996i $$0.572750\pi$$
$$360$$ 30.7821 1.62236
$$361$$ −5.44653 −0.286659
$$362$$ 8.54053 0.448880
$$363$$ 1.90962 0.100229
$$364$$ 0 0
$$365$$ 14.6751 0.768130
$$366$$ 3.92574 0.205202
$$367$$ 1.66322 0.0868196 0.0434098 0.999057i $$-0.486178\pi$$
0.0434098 + 0.999057i $$0.486178\pi$$
$$368$$ 17.2712 0.900324
$$369$$ 21.0343 1.09500
$$370$$ 24.8436 1.29156
$$371$$ 0 0
$$372$$ −25.5374 −1.32405
$$373$$ 13.9635 0.723002 0.361501 0.932372i $$-0.382264\pi$$
0.361501 + 0.932372i $$0.382264\pi$$
$$374$$ −21.8927 −1.13204
$$375$$ 6.18595 0.319441
$$376$$ −7.37655 −0.380417
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 31.5758 1.62194 0.810969 0.585089i $$-0.198941\pi$$
0.810969 + 0.585089i $$0.198941\pi$$
$$380$$ 27.8985 1.43116
$$381$$ 1.61975 0.0829822
$$382$$ −12.6741 −0.648466
$$383$$ −31.9082 −1.63043 −0.815217 0.579156i $$-0.803382\pi$$
−0.815217 + 0.579156i $$0.803382\pi$$
$$384$$ −2.55692 −0.130482
$$385$$ 0 0
$$386$$ 7.81525 0.397786
$$387$$ 9.16588 0.465928
$$388$$ 72.7788 3.69478
$$389$$ −25.4150 −1.28859 −0.644296 0.764776i $$-0.722850\pi$$
−0.644296 + 0.764776i $$0.722850\pi$$
$$390$$ 0 0
$$391$$ −6.20956 −0.314031
$$392$$ 0 0
$$393$$ 5.29312 0.267003
$$394$$ 12.0368 0.606403
$$395$$ −10.0041 −0.503358
$$396$$ 34.7288 1.74519
$$397$$ 4.15897 0.208733 0.104366 0.994539i $$-0.466719\pi$$
0.104366 + 0.994539i $$0.466719\pi$$
$$398$$ 1.06319 0.0532930
$$399$$ 0 0
$$400$$ −20.8823 −1.04412
$$401$$ 19.6013 0.978844 0.489422 0.872047i $$-0.337208\pi$$
0.489422 + 0.872047i $$0.337208\pi$$
$$402$$ 18.2282 0.909139
$$403$$ 0 0
$$404$$ 6.01381 0.299198
$$405$$ 10.7250 0.532929
$$406$$ 0 0
$$407$$ 16.1039 0.798238
$$408$$ −11.3334 −0.561086
$$409$$ −17.6337 −0.871930 −0.435965 0.899964i $$-0.643593\pi$$
−0.435965 + 0.899964i $$0.643593\pi$$
$$410$$ 32.1381 1.58719
$$411$$ −5.18083 −0.255551
$$412$$ 53.9063 2.65577
$$413$$ 0 0
$$414$$ 14.0412 0.690088
$$415$$ −4.34725 −0.213398
$$416$$ 0 0
$$417$$ −0.862831 −0.0422530
$$418$$ 25.7781 1.26085
$$419$$ −29.8911 −1.46027 −0.730137 0.683301i $$-0.760544\pi$$
−0.730137 + 0.683301i $$0.760544\pi$$
$$420$$ 0 0
$$421$$ 12.8528 0.626407 0.313203 0.949686i $$-0.398598\pi$$
0.313203 + 0.949686i $$0.398598\pi$$
$$422$$ 19.4559 0.947100
$$423$$ −2.88164 −0.140110
$$424$$ −50.7975 −2.46694
$$425$$ 7.50787 0.364185
$$426$$ 1.81332 0.0878559
$$427$$ 0 0
$$428$$ −24.1401 −1.16685
$$429$$ 0 0
$$430$$ 14.0045 0.675355
$$431$$ 8.97060 0.432098 0.216049 0.976382i $$-0.430683\pi$$
0.216049 + 0.976382i $$0.430683\pi$$
$$432$$ 25.8405 1.24325
$$433$$ 3.45062 0.165826 0.0829132 0.996557i $$-0.473578\pi$$
0.0829132 + 0.996557i $$0.473578\pi$$
$$434$$ 0 0
$$435$$ 4.49048 0.215302
$$436$$ −8.12228 −0.388987
$$437$$ 7.31161 0.349762
$$438$$ 12.2176 0.583780
$$439$$ 38.5144 1.83819 0.919096 0.394034i $$-0.128921\pi$$
0.919096 + 0.394034i $$0.128921\pi$$
$$440$$ 30.4864 1.45338
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −15.0399 −0.714569 −0.357284 0.933996i $$-0.616297\pi$$
−0.357284 + 0.933996i $$0.616297\pi$$
$$444$$ 14.5100 0.688612
$$445$$ 2.83701 0.134487
$$446$$ 58.4492 2.76765
$$447$$ 10.2646 0.485499
$$448$$ 0 0
$$449$$ −38.9235 −1.83691 −0.918456 0.395522i $$-0.870564\pi$$
−0.918456 + 0.395522i $$0.870564\pi$$
$$450$$ −16.9770 −0.800303
$$451$$ 20.8322 0.980952
$$452$$ −40.3856 −1.89958
$$453$$ 3.90660 0.183548
$$454$$ −35.3906 −1.66097
$$455$$ 0 0
$$456$$ 13.3448 0.624927
$$457$$ −13.9396 −0.652069 −0.326034 0.945358i $$-0.605713\pi$$
−0.326034 + 0.945358i $$0.605713\pi$$
$$458$$ −20.5298 −0.959295
$$459$$ −9.29049 −0.433643
$$460$$ 15.0502 0.701721
$$461$$ −37.4635 −1.74485 −0.872424 0.488749i $$-0.837453\pi$$
−0.872424 + 0.488749i $$0.837453\pi$$
$$462$$ 0 0
$$463$$ 6.75275 0.313827 0.156913 0.987612i $$-0.449846\pi$$
0.156913 + 0.987612i $$0.449846\pi$$
$$464$$ −46.7228 −2.16905
$$465$$ −8.75749 −0.406119
$$466$$ 17.0115 0.788044
$$467$$ −5.05032 −0.233701 −0.116851 0.993150i $$-0.537280\pi$$
−0.116851 + 0.993150i $$0.537280\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −4.40283 −0.203087
$$471$$ 7.26084 0.334562
$$472$$ −79.8705 −3.67634
$$473$$ 9.07783 0.417399
$$474$$ −8.32878 −0.382554
$$475$$ −8.84033 −0.405622
$$476$$ 0 0
$$477$$ −19.8440 −0.908593
$$478$$ −24.3143 −1.11211
$$479$$ 9.45319 0.431927 0.215964 0.976401i $$-0.430711\pi$$
0.215964 + 0.976401i $$0.430711\pi$$
$$480$$ 7.12801 0.325348
$$481$$ 0 0
$$482$$ 26.1082 1.18920
$$483$$ 0 0
$$484$$ −17.3142 −0.787010
$$485$$ 24.9579 1.13328
$$486$$ 32.0045 1.45175
$$487$$ −39.9996 −1.81255 −0.906277 0.422684i $$-0.861088\pi$$
−0.906277 + 0.422684i $$0.861088\pi$$
$$488$$ −20.4504 −0.925748
$$489$$ −3.71737 −0.168105
$$490$$ 0 0
$$491$$ −6.76097 −0.305118 −0.152559 0.988294i $$-0.548751\pi$$
−0.152559 + 0.988294i $$0.548751\pi$$
$$492$$ 18.7704 0.846233
$$493$$ 16.7984 0.756560
$$494$$ 0 0
$$495$$ 11.9095 0.535291
$$496$$ 91.1203 4.09142
$$497$$ 0 0
$$498$$ −3.61927 −0.162183
$$499$$ −11.3575 −0.508433 −0.254217 0.967147i $$-0.581818\pi$$
−0.254217 + 0.967147i $$0.581818\pi$$
$$500$$ −56.0871 −2.50829
$$501$$ 9.32230 0.416490
$$502$$ −26.7882 −1.19562
$$503$$ 13.9285 0.621040 0.310520 0.950567i $$-0.399497\pi$$
0.310520 + 0.950567i $$0.399497\pi$$
$$504$$ 0 0
$$505$$ 2.06230 0.0917713
$$506$$ 13.9063 0.618212
$$507$$ 0 0
$$508$$ −14.6860 −0.651587
$$509$$ −19.8149 −0.878281 −0.439141 0.898418i $$-0.644717\pi$$
−0.439141 + 0.898418i $$0.644717\pi$$
$$510$$ −6.76454 −0.299539
$$511$$ 0 0
$$512$$ 47.4335 2.09628
$$513$$ 10.9393 0.482983
$$514$$ 20.6741 0.911894
$$515$$ 18.4860 0.814590
$$516$$ 8.17934 0.360076
$$517$$ −2.85396 −0.125517
$$518$$ 0 0
$$519$$ −6.64237 −0.291568
$$520$$ 0 0
$$521$$ 31.0951 1.36230 0.681151 0.732143i $$-0.261480\pi$$
0.681151 + 0.732143i $$0.261480\pi$$
$$522$$ −37.9849 −1.66255
$$523$$ −22.7202 −0.993485 −0.496742 0.867898i $$-0.665471\pi$$
−0.496742 + 0.867898i $$0.665471\pi$$
$$524$$ −47.9919 −2.09654
$$525$$ 0 0
$$526$$ −13.0966 −0.571040
$$527$$ −32.7607 −1.42708
$$528$$ 12.1960 0.530761
$$529$$ −19.0557 −0.828507
$$530$$ −30.3194 −1.31699
$$531$$ −31.2013 −1.35402
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 2.36193 0.102211
$$535$$ −8.27830 −0.357902
$$536$$ −94.9564 −4.10149
$$537$$ −0.954091 −0.0411721
$$538$$ 35.9563 1.55018
$$539$$ 0 0
$$540$$ 22.5176 0.969002
$$541$$ −2.09872 −0.0902310 −0.0451155 0.998982i $$-0.514366\pi$$
−0.0451155 + 0.998982i $$0.514366\pi$$
$$542$$ 21.5558 0.925902
$$543$$ 1.71057 0.0734074
$$544$$ 26.6650 1.14325
$$545$$ −2.78536 −0.119312
$$546$$ 0 0
$$547$$ 25.3770 1.08504 0.542521 0.840042i $$-0.317470\pi$$
0.542521 + 0.840042i $$0.317470\pi$$
$$548$$ 46.9738 2.00662
$$549$$ −7.98894 −0.340959
$$550$$ −16.8139 −0.716948
$$551$$ −19.7797 −0.842642
$$552$$ 7.19902 0.306411
$$553$$ 0 0
$$554$$ 60.0855 2.55279
$$555$$ 4.97587 0.211214
$$556$$ 7.82316 0.331776
$$557$$ 44.2503 1.87495 0.937473 0.348058i $$-0.113159\pi$$
0.937473 + 0.348058i $$0.113159\pi$$
$$558$$ 74.0794 3.13603
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.38484 −0.185128
$$562$$ 70.3051 2.96564
$$563$$ 38.8907 1.63905 0.819523 0.573046i $$-0.194238\pi$$
0.819523 + 0.573046i $$0.194238\pi$$
$$564$$ −2.57149 −0.108279
$$565$$ −13.8494 −0.582647
$$566$$ −41.8092 −1.75737
$$567$$ 0 0
$$568$$ −9.44618 −0.396353
$$569$$ 46.1579 1.93504 0.967520 0.252796i $$-0.0813500\pi$$
0.967520 + 0.252796i $$0.0813500\pi$$
$$570$$ 7.96508 0.333620
$$571$$ 21.1368 0.884548 0.442274 0.896880i $$-0.354172\pi$$
0.442274 + 0.896880i $$0.354172\pi$$
$$572$$ 0 0
$$573$$ −2.53848 −0.106047
$$574$$ 0 0
$$575$$ −4.76904 −0.198883
$$576$$ −12.7931 −0.533045
$$577$$ 25.3304 1.05452 0.527259 0.849705i $$-0.323220\pi$$
0.527259 + 0.849705i $$0.323220\pi$$
$$578$$ 18.7009 0.777855
$$579$$ 1.56530 0.0650517
$$580$$ −40.7146 −1.69058
$$581$$ 0 0
$$582$$ 20.7785 0.861295
$$583$$ −19.6533 −0.813958
$$584$$ −63.6455 −2.63367
$$585$$ 0 0
$$586$$ 37.9106 1.56607
$$587$$ 3.56287 0.147056 0.0735278 0.997293i $$-0.476574\pi$$
0.0735278 + 0.997293i $$0.476574\pi$$
$$588$$ 0 0
$$589$$ 38.5749 1.58945
$$590$$ −47.6722 −1.96263
$$591$$ 2.41082 0.0991679
$$592$$ −51.7732 −2.12786
$$593$$ 25.3536 1.04115 0.520573 0.853817i $$-0.325718\pi$$
0.520573 + 0.853817i $$0.325718\pi$$
$$594$$ 20.8061 0.853685
$$595$$ 0 0
$$596$$ −93.0677 −3.81220
$$597$$ 0.212945 0.00871524
$$598$$ 0 0
$$599$$ 10.9216 0.446243 0.223122 0.974791i $$-0.428375\pi$$
0.223122 + 0.974791i $$0.428375\pi$$
$$600$$ −8.70421 −0.355348
$$601$$ −24.2564 −0.989439 −0.494720 0.869053i $$-0.664729\pi$$
−0.494720 + 0.869053i $$0.664729\pi$$
$$602$$ 0 0
$$603$$ −37.0946 −1.51061
$$604$$ −35.4206 −1.44124
$$605$$ −5.93753 −0.241395
$$606$$ 1.71695 0.0697464
$$607$$ 9.85447 0.399981 0.199990 0.979798i $$-0.435909\pi$$
0.199990 + 0.979798i $$0.435909\pi$$
$$608$$ −31.3974 −1.27333
$$609$$ 0 0
$$610$$ −12.2062 −0.494215
$$611$$ 0 0
$$612$$ 40.1423 1.62266
$$613$$ −3.67688 −0.148508 −0.0742540 0.997239i $$-0.523658\pi$$
−0.0742540 + 0.997239i $$0.523658\pi$$
$$614$$ 23.2416 0.937955
$$615$$ 6.43688 0.259560
$$616$$ 0 0
$$617$$ 18.7468 0.754718 0.377359 0.926067i $$-0.376832\pi$$
0.377359 + 0.926067i $$0.376832\pi$$
$$618$$ 15.3903 0.619090
$$619$$ −15.8945 −0.638854 −0.319427 0.947611i $$-0.603490\pi$$
−0.319427 + 0.947611i $$0.603490\pi$$
$$620$$ 79.4029 3.18890
$$621$$ 5.90137 0.236814
$$622$$ 31.5333 1.26437
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −7.22743 −0.289097
$$626$$ 33.9727 1.35782
$$627$$ 5.16304 0.206192
$$628$$ −65.8330 −2.62702
$$629$$ 18.6141 0.742194
$$630$$ 0 0
$$631$$ 19.7451 0.786040 0.393020 0.919530i $$-0.371430\pi$$
0.393020 + 0.919530i $$0.371430\pi$$
$$632$$ 43.3873 1.72585
$$633$$ 3.89679 0.154883
$$634$$ −43.2698 −1.71846
$$635$$ −5.03624 −0.199857
$$636$$ −17.7081 −0.702173
$$637$$ 0 0
$$638$$ −37.6200 −1.48939
$$639$$ −3.69014 −0.145980
$$640$$ 7.95016 0.314258
$$641$$ −29.7786 −1.17618 −0.588092 0.808794i $$-0.700121\pi$$
−0.588092 + 0.808794i $$0.700121\pi$$
$$642$$ −6.89203 −0.272007
$$643$$ −11.5725 −0.456373 −0.228187 0.973617i $$-0.573280\pi$$
−0.228187 + 0.973617i $$0.573280\pi$$
$$644$$ 0 0
$$645$$ 2.80493 0.110444
$$646$$ 29.7964 1.17232
$$647$$ −25.5065 −1.00276 −0.501382 0.865226i $$-0.667175\pi$$
−0.501382 + 0.865226i $$0.667175\pi$$
$$648$$ −46.5140 −1.82724
$$649$$ −30.9016 −1.21299
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 33.7049 1.31999
$$653$$ −44.8293 −1.75430 −0.877152 0.480212i $$-0.840560\pi$$
−0.877152 + 0.480212i $$0.840560\pi$$
$$654$$ −2.31892 −0.0906771
$$655$$ −16.4578 −0.643058
$$656$$ −66.9747 −2.61492
$$657$$ −24.8630 −0.969999
$$658$$ 0 0
$$659$$ 41.1734 1.60389 0.801944 0.597399i $$-0.203799\pi$$
0.801944 + 0.597399i $$0.203799\pi$$
$$660$$ 10.6276 0.413680
$$661$$ −21.8938 −0.851569 −0.425785 0.904825i $$-0.640002\pi$$
−0.425785 + 0.904825i $$0.640002\pi$$
$$662$$ −10.2681 −0.399080
$$663$$ 0 0
$$664$$ 18.8539 0.731673
$$665$$ 0 0
$$666$$ −42.0908 −1.63098
$$667$$ −10.6704 −0.413160
$$668$$ −84.5239 −3.27033
$$669$$ 11.7067 0.452606
$$670$$ −56.6764 −2.18960
$$671$$ −7.91219 −0.305447
$$672$$ 0 0
$$673$$ 35.6688 1.37493 0.687466 0.726217i $$-0.258723\pi$$
0.687466 + 0.726217i $$0.258723\pi$$
$$674$$ −35.4721 −1.36633
$$675$$ −7.13524 −0.274635
$$676$$ 0 0
$$677$$ −2.55532 −0.0982089 −0.0491044 0.998794i $$-0.515637\pi$$
−0.0491044 + 0.998794i $$0.515637\pi$$
$$678$$ −11.5302 −0.442813
$$679$$ 0 0
$$680$$ 35.2386 1.35134
$$681$$ −7.08832 −0.271625
$$682$$ 73.3678 2.80940
$$683$$ −35.7399 −1.36755 −0.683775 0.729693i $$-0.739663\pi$$
−0.683775 + 0.729693i $$0.739663\pi$$
$$684$$ −47.2666 −1.80728
$$685$$ 16.1086 0.615479
$$686$$ 0 0
$$687$$ −4.11188 −0.156878
$$688$$ −29.1848 −1.11266
$$689$$ 0 0
$$690$$ 4.29687 0.163579
$$691$$ −26.0292 −0.990197 −0.495099 0.868837i $$-0.664868\pi$$
−0.495099 + 0.868837i $$0.664868\pi$$
$$692$$ 60.2254 2.28943
$$693$$ 0 0
$$694$$ 68.3335 2.59391
$$695$$ 2.68278 0.101764
$$696$$ −19.4751 −0.738202
$$697$$ 24.0796 0.912079
$$698$$ −12.6589 −0.479146
$$699$$ 3.40721 0.128872
$$700$$ 0 0
$$701$$ 1.12731 0.0425779 0.0212890 0.999773i $$-0.493223\pi$$
0.0212890 + 0.999773i $$0.493223\pi$$
$$702$$ 0 0
$$703$$ −21.9177 −0.826641
$$704$$ −12.6702 −0.477525
$$705$$ −0.881834 −0.0332118
$$706$$ −35.0955 −1.32084
$$707$$ 0 0
$$708$$ −27.8431 −1.04641
$$709$$ −6.05031 −0.227224 −0.113612 0.993525i $$-0.536242\pi$$
−0.113612 + 0.993525i $$0.536242\pi$$
$$710$$ −5.63813 −0.211595
$$711$$ 16.9492 0.635644
$$712$$ −12.3040 −0.461114
$$713$$ 20.8098 0.779332
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 8.65061 0.323288
$$717$$ −4.86986 −0.181868
$$718$$ 22.2249 0.829425
$$719$$ −47.1177 −1.75719 −0.878597 0.477563i $$-0.841520\pi$$
−0.878597 + 0.477563i $$0.841520\pi$$
$$720$$ −38.2884 −1.42692
$$721$$ 0 0
$$722$$ 14.0989 0.524706
$$723$$ 5.22916 0.194475
$$724$$ −15.5095 −0.576404
$$725$$ 12.9014 0.479146
$$726$$ −4.94324 −0.183461
$$727$$ 17.9215 0.664671 0.332335 0.943161i $$-0.392163\pi$$
0.332335 + 0.943161i $$0.392163\pi$$
$$728$$ 0 0
$$729$$ −13.5489 −0.501810
$$730$$ −37.9880 −1.40600
$$731$$ 10.4929 0.388093
$$732$$ −7.12908 −0.263498
$$733$$ 45.2685 1.67203 0.836016 0.548705i $$-0.184879\pi$$
0.836016 + 0.548705i $$0.184879\pi$$
$$734$$ −4.30542 −0.158916
$$735$$ 0 0
$$736$$ −16.9378 −0.624335
$$737$$ −36.7382 −1.35327
$$738$$ −54.4494 −2.00431
$$739$$ −19.2613 −0.708539 −0.354270 0.935143i $$-0.615270\pi$$
−0.354270 + 0.935143i $$0.615270\pi$$
$$740$$ −45.1155 −1.65848
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −34.8853 −1.27982 −0.639908 0.768452i $$-0.721028\pi$$
−0.639908 + 0.768452i $$0.721028\pi$$
$$744$$ 37.9810 1.39245
$$745$$ −31.9155 −1.16929
$$746$$ −36.1459 −1.32339
$$747$$ 7.36525 0.269480
$$748$$ 39.7567 1.45365
$$749$$ 0 0
$$750$$ −16.0130 −0.584711
$$751$$ −24.9668 −0.911051 −0.455526 0.890223i $$-0.650549\pi$$
−0.455526 + 0.890223i $$0.650549\pi$$
$$752$$ 9.17535 0.334591
$$753$$ −5.36536 −0.195525
$$754$$ 0 0
$$755$$ −12.1467 −0.442064
$$756$$ 0 0
$$757$$ −10.6049 −0.385440 −0.192720 0.981254i $$-0.561731\pi$$
−0.192720 + 0.981254i $$0.561731\pi$$
$$758$$ −81.7370 −2.96882
$$759$$ 2.78527 0.101099
$$760$$ −41.4926 −1.50510
$$761$$ −32.6388 −1.18316 −0.591578 0.806248i $$-0.701495\pi$$
−0.591578 + 0.806248i $$0.701495\pi$$
$$762$$ −4.19288 −0.151892
$$763$$ 0 0
$$764$$ 23.0160 0.832691
$$765$$ 13.7659 0.497708
$$766$$ 82.5976 2.98437
$$767$$ 0 0
$$768$$ 11.4759 0.414100
$$769$$ −52.1752 −1.88149 −0.940744 0.339119i $$-0.889871\pi$$
−0.940744 + 0.339119i $$0.889871\pi$$
$$770$$ 0 0
$$771$$ 4.14077 0.149126
$$772$$ −14.1924 −0.510794
$$773$$ −35.7057 −1.28425 −0.642123 0.766602i $$-0.721946\pi$$
−0.642123 + 0.766602i $$0.721946\pi$$
$$774$$ −23.7268 −0.852842
$$775$$ −25.1607 −0.903800
$$776$$ −108.242 −3.88565
$$777$$ 0 0
$$778$$ 65.7893 2.35866
$$779$$ −28.3531 −1.01586
$$780$$ 0 0
$$781$$ −3.65469 −0.130775
$$782$$ 16.0741 0.574808
$$783$$ −15.9646 −0.570529
$$784$$ 0 0
$$785$$ −22.5760 −0.805770
$$786$$ −13.7018 −0.488726
$$787$$ −6.10621 −0.217663 −0.108831 0.994060i $$-0.534711\pi$$
−0.108831 + 0.994060i $$0.534711\pi$$
$$788$$ −21.8586 −0.778679
$$789$$ −2.62310 −0.0933847
$$790$$ 25.8965 0.921356
$$791$$ 0 0
$$792$$ −51.6510 −1.83534
$$793$$ 0 0
$$794$$ −10.7659 −0.382068
$$795$$ −6.07261 −0.215373
$$796$$ −1.93074 −0.0684332
$$797$$ 46.2299 1.63755 0.818773 0.574117i $$-0.194654\pi$$
0.818773 + 0.574117i $$0.194654\pi$$
$$798$$ 0 0
$$799$$ −3.29884 −0.116704
$$800$$ 20.4792 0.724048
$$801$$ −4.80656 −0.169831
$$802$$ −50.7400 −1.79169
$$803$$ −24.6242 −0.868968
$$804$$ −33.1020 −1.16742
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 7.20161 0.253509
$$808$$ −8.94415 −0.314654
$$809$$ 39.2879 1.38129 0.690644 0.723195i $$-0.257327\pi$$
0.690644 + 0.723195i $$0.257327\pi$$
$$810$$ −27.7627 −0.975482
$$811$$ −6.90664 −0.242525 −0.121262 0.992620i $$-0.538694\pi$$
−0.121262 + 0.992620i $$0.538694\pi$$
$$812$$ 0 0
$$813$$ 4.31738 0.151417
$$814$$ −41.6864 −1.46111
$$815$$ 11.5583 0.404871
$$816$$ 14.0971 0.493496
$$817$$ −12.3551 −0.432251
$$818$$ 45.6466 1.59600
$$819$$ 0 0
$$820$$ −58.3622 −2.03810
$$821$$ −1.91049 −0.0666765 −0.0333382 0.999444i $$-0.510614\pi$$
−0.0333382 + 0.999444i $$0.510614\pi$$
$$822$$ 13.4111 0.467765
$$823$$ 1.57969 0.0550645 0.0275322 0.999621i $$-0.491235\pi$$
0.0275322 + 0.999621i $$0.491235\pi$$
$$824$$ −80.1732 −2.79296
$$825$$ −3.36763 −0.117246
$$826$$ 0 0
$$827$$ −32.5050 −1.13031 −0.565155 0.824985i $$-0.691184\pi$$
−0.565155 + 0.824985i $$0.691184\pi$$
$$828$$ −25.4986 −0.886138
$$829$$ 35.0538 1.21747 0.608735 0.793373i $$-0.291677\pi$$
0.608735 + 0.793373i $$0.291677\pi$$
$$830$$ 11.2533 0.390608
$$831$$ 12.0344 0.417469
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 2.23353 0.0773407
$$835$$ −28.9856 −1.00309
$$836$$ −46.8126 −1.61905
$$837$$ 31.1347 1.07617
$$838$$ 77.3760 2.67291
$$839$$ 5.35487 0.184871 0.0924354 0.995719i $$-0.470535\pi$$
0.0924354 + 0.995719i $$0.470535\pi$$
$$840$$ 0 0
$$841$$ −0.133978 −0.00461993
$$842$$ −33.2708 −1.14659
$$843$$ 14.0813 0.484985
$$844$$ −35.3316 −1.21616
$$845$$ 0 0
$$846$$ 7.45942 0.256460
$$847$$ 0 0
$$848$$ 63.1846 2.16977
$$849$$ −8.37387 −0.287391
$$850$$ −19.4349 −0.666611
$$851$$ −11.8238 −0.405314
$$852$$ −3.29297 −0.112815
$$853$$ 49.6270 1.69920 0.849598 0.527431i $$-0.176845\pi$$
0.849598 + 0.527431i $$0.176845\pi$$
$$854$$ 0 0
$$855$$ −16.2090 −0.554337
$$856$$ 35.9028 1.22713
$$857$$ −5.88392 −0.200991 −0.100496 0.994938i $$-0.532043\pi$$
−0.100496 + 0.994938i $$0.532043\pi$$
$$858$$ 0 0
$$859$$ −43.3862 −1.48032 −0.740159 0.672432i $$-0.765250\pi$$
−0.740159 + 0.672432i $$0.765250\pi$$
$$860$$ −25.4318 −0.867219
$$861$$ 0 0
$$862$$ −23.2213 −0.790920
$$863$$ 31.1272 1.05958 0.529792 0.848128i $$-0.322270\pi$$
0.529792 + 0.848128i $$0.322270\pi$$
$$864$$ −25.3416 −0.862139
$$865$$ 20.6530 0.702222
$$866$$ −8.93228 −0.303531
$$867$$ 3.74557 0.127206
$$868$$ 0 0
$$869$$ 16.7864 0.569439
$$870$$ −11.6241 −0.394093
$$871$$ 0 0
$$872$$ 12.0800 0.409081
$$873$$ −42.2844 −1.43111
$$874$$ −18.9268 −0.640209
$$875$$ 0 0
$$876$$ −22.1870 −0.749629
$$877$$ −29.9106 −1.01001 −0.505004 0.863117i $$-0.668509\pi$$
−0.505004 + 0.863117i $$0.668509\pi$$
$$878$$ −99.6984 −3.36466
$$879$$ 7.59304 0.256107
$$880$$ −37.9206 −1.27830
$$881$$ −14.5695 −0.490860 −0.245430 0.969414i $$-0.578929\pi$$
−0.245430 + 0.969414i $$0.578929\pi$$
$$882$$ 0 0
$$883$$ −48.9296 −1.64661 −0.823307 0.567597i $$-0.807873\pi$$
−0.823307 + 0.567597i $$0.807873\pi$$
$$884$$ 0 0
$$885$$ −9.54817 −0.320958
$$886$$ 38.9324 1.30796
$$887$$ −54.5902 −1.83296 −0.916480 0.400080i $$-0.868982\pi$$
−0.916480 + 0.400080i $$0.868982\pi$$
$$888$$ −21.5802 −0.724184
$$889$$ 0 0
$$890$$ −7.34389 −0.246168
$$891$$ −17.9961 −0.602891
$$892$$ −106.143 −3.55392
$$893$$ 3.88430 0.129983
$$894$$ −26.5710 −0.888666
$$895$$ 2.96653 0.0991603
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 100.757 3.36232
$$899$$ −56.2955 −1.87756
$$900$$ 30.8299 1.02766
$$901$$ −22.7169 −0.756810
$$902$$ −53.9263 −1.79555
$$903$$ 0 0
$$904$$ 60.0643 1.99771
$$905$$ −5.31862 −0.176797
$$906$$ −10.1126 −0.335970
$$907$$ 22.7255 0.754589 0.377295 0.926093i $$-0.376854\pi$$
0.377295 + 0.926093i $$0.376854\pi$$
$$908$$ 64.2688 2.13283
$$909$$ −3.49402 −0.115889
$$910$$ 0 0
$$911$$ −42.2359 −1.39934 −0.699669 0.714467i $$-0.746669\pi$$
−0.699669 + 0.714467i $$0.746669\pi$$
$$912$$ −16.5990 −0.549646
$$913$$ 7.29450 0.241413
$$914$$ 36.0842 1.19356
$$915$$ −2.44476 −0.0808213
$$916$$ 37.2818 1.23182
$$917$$ 0 0
$$918$$ 24.0494 0.793747
$$919$$ 30.6940 1.01250 0.506251 0.862386i $$-0.331031\pi$$
0.506251 + 0.862386i $$0.331031\pi$$
$$920$$ −22.3838 −0.737971
$$921$$ 4.65502 0.153388
$$922$$ 96.9780 3.19380
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 14.2959 0.470048
$$926$$ −17.4802 −0.574434
$$927$$ −31.3195 −1.02867
$$928$$ 45.8208 1.50414
$$929$$ −37.4250 −1.22787 −0.613936 0.789355i $$-0.710415\pi$$
−0.613936 + 0.789355i $$0.710415\pi$$
$$930$$ 22.6696 0.743367
$$931$$ 0 0
$$932$$ −30.8926 −1.01192
$$933$$ 6.31574 0.206768
$$934$$ 13.0733 0.427770
$$935$$ 13.6337 0.445869
$$936$$ 0 0
$$937$$ −44.3386 −1.44848 −0.724239 0.689549i $$-0.757809\pi$$
−0.724239 + 0.689549i $$0.757809\pi$$
$$938$$ 0 0
$$939$$ 6.80433 0.222051
$$940$$ 7.99546 0.260783
$$941$$ −27.5052 −0.896646 −0.448323 0.893872i $$-0.647978\pi$$
−0.448323 + 0.893872i $$0.647978\pi$$
$$942$$ −18.7954 −0.612388
$$943$$ −15.2955 −0.498089
$$944$$ 99.3472 3.23348
$$945$$ 0 0
$$946$$ −23.4989 −0.764014
$$947$$ −5.08330 −0.165185 −0.0825925 0.996583i $$-0.526320\pi$$
−0.0825925 + 0.996583i $$0.526320\pi$$
$$948$$ 15.1249 0.491235
$$949$$ 0 0
$$950$$ 22.8841 0.742458
$$951$$ −8.66642 −0.281028
$$952$$ 0 0
$$953$$ 9.81437 0.317919 0.158959 0.987285i $$-0.449186\pi$$
0.158959 + 0.987285i $$0.449186\pi$$
$$954$$ 51.3681 1.66310
$$955$$ 7.89284 0.255406
$$956$$ 44.1543 1.42805
$$957$$ −7.53484 −0.243567
$$958$$ −24.4705 −0.790607
$$959$$ 0 0
$$960$$ −3.91491 −0.126353
$$961$$ 78.7893 2.54159
$$962$$ 0 0
$$963$$ 14.0254 0.451961
$$964$$ −47.4121 −1.52704
$$965$$ −4.86695 −0.156673
$$966$$ 0 0
$$967$$ 2.69619 0.0867036 0.0433518 0.999060i $$-0.486196\pi$$
0.0433518 + 0.999060i $$0.486196\pi$$
$$968$$ 25.7509 0.827665
$$969$$ 5.96786 0.191715
$$970$$ −64.6060 −2.07437
$$971$$ −24.9240 −0.799850 −0.399925 0.916548i $$-0.630964\pi$$
−0.399925 + 0.916548i $$0.630964\pi$$
$$972$$ −58.1196 −1.86419
$$973$$ 0 0
$$974$$ 103.543 3.31773
$$975$$ 0 0
$$976$$ 25.4373 0.814230
$$977$$ 28.3129 0.905811 0.452906 0.891558i $$-0.350387\pi$$
0.452906 + 0.891558i $$0.350387\pi$$
$$978$$ 9.62280 0.307703
$$979$$ −4.76039 −0.152143
$$980$$ 0 0
$$981$$ 4.71904 0.150667
$$982$$ 17.5015 0.558494
$$983$$ 37.8517 1.20728 0.603641 0.797256i $$-0.293716\pi$$
0.603641 + 0.797256i $$0.293716\pi$$
$$984$$ −27.9166 −0.889947
$$985$$ −7.49591 −0.238839
$$986$$ −43.4842 −1.38482
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.66514 −0.211939
$$990$$ −30.8289 −0.979805
$$991$$ 58.4158 1.85564 0.927820 0.373028i $$-0.121681\pi$$
0.927820 + 0.373028i $$0.121681\pi$$
$$992$$ −89.3612 −2.83722
$$993$$ −2.05657 −0.0652633
$$994$$ 0 0
$$995$$ −0.662104 −0.0209901
$$996$$ 6.57252 0.208258
$$997$$ −28.0588 −0.888632 −0.444316 0.895870i $$-0.646553\pi$$
−0.444316 + 0.895870i $$0.646553\pi$$
$$998$$ 29.4001 0.930645
$$999$$ −17.6903 −0.559696
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 8281.2.a.co.1.1 12
7.3 odd 6 1183.2.e.j.170.12 24
7.5 odd 6 1183.2.e.j.508.12 24
7.6 odd 2 8281.2.a.cp.1.1 12
13.6 odd 12 637.2.q.i.491.6 12
13.11 odd 12 637.2.q.i.589.6 12
13.12 even 2 inner 8281.2.a.co.1.12 12
91.6 even 12 637.2.q.g.491.6 12
91.11 odd 12 637.2.u.g.30.1 12
91.12 odd 6 1183.2.e.j.508.1 24
91.19 even 12 91.2.u.b.88.1 yes 12
91.24 even 12 91.2.u.b.30.1 yes 12
91.32 odd 12 637.2.k.i.569.6 12
91.37 odd 12 637.2.k.i.459.1 12
91.38 odd 6 1183.2.e.j.170.1 24
91.45 even 12 91.2.k.b.23.6 yes 12
91.58 odd 12 637.2.u.g.361.1 12
91.76 even 12 637.2.q.g.589.6 12
91.89 even 12 91.2.k.b.4.1 12
91.90 odd 2 8281.2.a.cp.1.12 12
273.89 odd 12 819.2.bm.f.550.6 12
273.110 odd 12 819.2.do.e.361.6 12
273.206 odd 12 819.2.do.e.667.6 12
273.227 odd 12 819.2.bm.f.478.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.1 12 91.89 even 12
91.2.k.b.23.6 yes 12 91.45 even 12
91.2.u.b.30.1 yes 12 91.24 even 12
91.2.u.b.88.1 yes 12 91.19 even 12
637.2.k.i.459.1 12 91.37 odd 12
637.2.k.i.569.6 12 91.32 odd 12
637.2.q.g.491.6 12 91.6 even 12
637.2.q.g.589.6 12 91.76 even 12
637.2.q.i.491.6 12 13.6 odd 12
637.2.q.i.589.6 12 13.11 odd 12
637.2.u.g.30.1 12 91.11 odd 12
637.2.u.g.361.1 12 91.58 odd 12
819.2.bm.f.478.1 12 273.227 odd 12
819.2.bm.f.550.6 12 273.89 odd 12
819.2.do.e.361.6 12 273.110 odd 12
819.2.do.e.667.6 12 273.206 odd 12
1183.2.e.j.170.1 24 91.38 odd 6
1183.2.e.j.170.12 24 7.3 odd 6
1183.2.e.j.508.1 24 91.12 odd 6
1183.2.e.j.508.12 24 7.5 odd 6
8281.2.a.co.1.1 12 1.1 even 1 trivial
8281.2.a.co.1.12 12 13.12 even 2 inner
8281.2.a.cp.1.1 12 7.6 odd 2
8281.2.a.cp.1.12 12 91.90 odd 2