# Properties

 Label 8281.2.a.co.1.12 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 1.12 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 $$-0.132026\pi$$
0.915209 + 0.402981i $$0.132026\pi$$
$$3$$ −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 $$-0.617382\pi$$
−0.360466 + 0.932772i $$0.617382\pi$$
$$6$$ −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 $$-0.366300\pi$$
0.407788 + 0.913077i $$0.366300\pi$$
$$12$$ −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 $$-0.638776\pi$$
−0.422297 + 0.906457i $$0.638776\pi$$
$$20$$ −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 $$-0.890071\pi$$
−0.940956 + 0.338529i $$0.890071\pi$$
$$32$$ 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 $$-0.337226\pi$$
0.489371 + 0.872075i $$0.337226\pi$$
$$38$$ −9.52994 −1.54596
$$39$$ 0 0
$$40$$ −11.2706 −1.78203
$$41$$ 7.70150 1.20277 0.601386 0.798958i $$-0.294615\pi$$
0.601386 + 0.798958i $$0.294615\pi$$
$$42$$ 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 $$-0.524518\pi$$
−0.0769500 + 0.997035i $$0.524518\pi$$
$$48$$ −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 $$-0.766904\pi$$
−0.743643 + 0.668577i $$0.766904\pi$$
$$60$$ 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 $$-0.811455\pi$$
−0.829642 + 0.558296i $$0.811455\pi$$
$$68$$ −14.6977 −1.78236
$$69$$ −1.02969 −0.123960
$$70$$ 0 0
$$71$$ −1.35111 −0.160347 −0.0801736 0.996781i $$-0.525547\pi$$
−0.0801736 + 0.996781i $$0.525547\pi$$
$$72$$ −19.0949 −2.25036
$$73$$ −9.10335 −1.06547 −0.532733 0.846283i $$-0.678835\pi$$
−0.532733 + 0.846283i $$0.678835\pi$$
$$74$$ 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 $$-0.452716\pi$$
0.148002 + 0.988987i $$0.452716\pi$$
$$84$$ 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 $$-0.529733\pi$$
−0.0932732 + 0.995641i $$0.529733\pi$$
$$90$$ 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 $$-0.787842\pi$$
−0.785981 + 0.618250i $$0.787842\pi$$
$$98$$ 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 $$-0.473630\pi$$
0.0827481 + 0.996570i $$0.473630\pi$$
$$110$$ −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 $$-0.640381\pi$$
−0.426863 + 0.904316i $$0.640381\pi$$
$$138$$ −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 $$-0.198946\pi$$
0.810959 + 0.585103i $$0.198946\pi$$
$$150$$ 3.22276 0.263138
$$151$$ 7.53493 0.613184 0.306592 0.951841i $$-0.400811\pi$$
0.306592 + 0.951841i $$0.400811\pi$$
$$152$$ −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 $$-0.590599\pi$$
−0.280797 + 0.959767i $$0.590599\pi$$
$$164$$ 36.2036 2.82703
$$165$$ 2.26079 0.176002
$$166$$ 6.98072 0.541809
$$167$$ 17.9805 1.39138 0.695688 0.718344i $$-0.255099\pi$$
0.695688 + 0.718344i $$0.255099\pi$$
$$168$$ 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 $$-0.465344\pi$$
0.108660 + 0.994079i $$0.465344\pi$$
$$194$$ −40.0768 −2.87735
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4.64991 0.331292 0.165646 0.986185i $$-0.447029\pi$$
0.165646 + 0.986185i $$0.447029\pi$$
$$198$$ −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 $$-0.227143\pi$$
0.756016 + 0.654553i $$0.227143\pi$$
$$224$$ 0 0
$$225$$ 6.55837 0.437225
$$226$$ −22.2390 −1.47932
$$227$$ −13.6717 −0.907424 −0.453712 0.891148i $$-0.649901\pi$$
−0.453712 + 0.891148i $$0.649901\pi$$
$$228$$ 8.97268 0.594230
$$229$$ −7.93086 −0.524086 −0.262043 0.965056i $$-0.584396\pi$$
−0.262043 + 0.965056i $$0.584396\pi$$
$$230$$ −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 $$-0.598251\pi$$
−0.303786 + 0.952740i $$0.598251\pi$$
$$240$$ 7.26833 0.469169
$$241$$ 10.0858 0.649686 0.324843 0.945768i $$-0.394689\pi$$
0.324843 + 0.945768i $$0.394689\pi$$
$$242$$ −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 $$-0.418609\pi$$
0.252921 + 0.967487i $$0.418609\pi$$
$$272$$ −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 $$-0.199413\pi$$
0.810100 + 0.586292i $$0.199413\pi$$
$$282$$ 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 $$-0.359292\pi$$
0.427791 + 0.903878i $$0.359292\pi$$
$$294$$ 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 $$-0.417525\pi$$
0.256213 + 0.966620i $$0.417525\pi$$
$$308$$ 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 $$-0.655536\pi$$
−0.469418 + 0.882976i $$0.655536\pi$$
$$318$$ −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 $$-0.534769\pi$$
−0.109013 + 0.994040i $$0.534769\pi$$
$$332$$ 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 $$-0.541782\pi$$
−0.130884 + 0.991398i $$0.541782\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 23.0690 1.22958
$$353$$ −13.5577 −0.721605 −0.360802 0.932642i $$-0.617497\pi$$
−0.360802 + 0.932642i $$0.617497\pi$$
$$354$$ 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 $$-0.427250\pi$$
0.226567 + 0.973996i $$0.427250\pi$$
$$360$$ 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 $$-0.801059\pi$$
−0.810969 + 0.585089i $$0.801059\pi$$
$$380$$ 27.8985 1.43116
$$381$$ 1.61975 0.0829822
$$382$$ 12.6741 0.648466
$$383$$ 31.9082 1.63043 0.815217 0.579156i $$-0.196618\pi$$
0.815217 + 0.579156i $$0.196618\pi$$
$$384$$ 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 $$-0.533281\pi$$
−0.104366 + 0.994539i $$0.533281\pi$$
$$398$$ −1.06319 −0.0532930
$$399$$ 0 0
$$400$$ −20.8823 −1.04412
$$401$$ −19.6013 −0.978844 −0.489422 0.872047i $$-0.662792\pi$$
−0.489422 + 0.872047i $$0.662792\pi$$
$$402$$ 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 $$-0.356407\pi$$
0.435965 + 0.899964i $$0.356407\pi$$
$$410$$ −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 $$-0.601402\pi$$
−0.313203 + 0.949686i $$0.601402\pi$$
$$422$$ −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 $$-0.569317\pi$$
−0.216049 + 0.976382i $$0.569317\pi$$
$$432$$ 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 $$-0.129436\pi$$
0.918456 + 0.395522i $$0.129436\pi$$
$$450$$ 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 $$-0.394287\pi$$
0.326034 + 0.945358i $$0.394287\pi$$
$$458$$ −20.5298 −0.959295
$$459$$ −9.29049 −0.433643
$$460$$ −15.0502 −0.701721
$$461$$ 37.4635 1.74485 0.872424 0.488749i $$-0.162547\pi$$
0.872424 + 0.488749i $$0.162547\pi$$
$$462$$ 0 0
$$463$$ −6.75275 −0.313827 −0.156913 0.987612i $$-0.550154\pi$$
−0.156913 + 0.987612i $$0.550154\pi$$
$$464$$ −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 $$-0.569289\pi$$
−0.215964 + 0.976401i $$0.569289\pi$$
$$480$$ 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 $$-0.138912\pi$$
0.906277 + 0.422684i $$0.138912\pi$$
$$488$$ 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 $$-0.418182\pi$$
0.254217 + 0.967147i $$0.418182\pi$$
$$500$$ 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 $$-0.355283\pi$$
0.439141 + 0.898418i $$0.355283\pi$$
$$510$$ −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 $$-0.485634\pi$$
0.0451155 + 0.998982i $$0.485634\pi$$
$$542$$ 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 $$-0.886841\pi$$
−0.937473 + 0.348058i $$0.886841\pi$$
$$558$$ 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 $$-0.676780\pi$$
−0.527259 + 0.849705i $$0.676780\pi$$
$$578$$ −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 $$-0.523426\pi$$
−0.0735278 + 0.997293i $$0.523426\pi$$
$$588$$ 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 $$-0.674282\pi$$
−0.520573 + 0.853817i $$0.674282\pi$$
$$594$$ 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 $$-0.476342\pi$$
0.0742540 + 0.997239i $$0.476342\pi$$
$$614$$ 23.2416 0.937955
$$615$$ 6.43688 0.259560
$$616$$ 0 0
$$617$$ −18.7468 −0.754718 −0.377359 0.926067i $$-0.623168\pi$$
−0.377359 + 0.926067i $$0.623168\pi$$
$$618$$ −15.3903 −0.619090
$$619$$ 15.8945 0.638854 0.319427 0.947611i $$-0.396510\pi$$
0.319427 + 0.947611i $$0.396510\pi$$
$$620$$ 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 $$-0.628570\pi$$
−0.393020 + 0.919530i $$0.628570\pi$$
$$632$$ −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 $$-0.426720\pi$$
0.228187 + 0.973617i $$0.426720\pi$$
$$644$$ 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 $$-0.359998\pi$$
0.425785 + 0.904825i $$0.359998\pi$$
$$662$$ −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 $$-0.260337\pi$$
0.683775 + 0.729693i $$0.260337\pi$$
$$684$$ 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 $$-0.335132\pi$$
0.495099 + 0.868837i $$0.335132\pi$$
$$692$$ 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 $$-0.463758\pi$$
0.113612 + 0.993525i $$0.463758\pi$$
$$710$$ 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 $$-0.815121\pi$$
−0.836016 + 0.548705i $$0.815121\pi$$
$$734$$ 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 $$-0.384730\pi$$
0.354270 + 0.935143i $$0.384730\pi$$
$$740$$ −45.1155 −1.65848
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 34.8853 1.27982 0.639908 0.768452i $$-0.278972\pi$$
0.639908 + 0.768452i $$0.278972\pi$$
$$744$$ 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 $$-0.298505\pi$$
0.591578 + 0.806248i $$0.298505\pi$$
$$762$$ 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 $$-0.110129\pi$$
0.940744 + 0.339119i $$0.110129\pi$$
$$770$$ 0 0
$$771$$ 4.14077 0.149126
$$772$$ 14.1924 0.510794
$$773$$ 35.7057 1.28425 0.642123 0.766602i $$-0.278054\pi$$
0.642123 + 0.766602i $$0.278054\pi$$
$$774$$ 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 $$-0.465289\pi$$
0.108831 + 0.994060i $$0.465289\pi$$
$$788$$ 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 $$-0.461306\pi$$
0.121262 + 0.992620i $$0.461306\pi$$
$$812$$ 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 $$-0.489386\pi$$
0.0333382 + 0.999444i $$0.489386\pi$$
$$822$$ 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 $$-0.308816\pi$$
0.565155 + 0.824985i $$0.308816\pi$$
$$828$$ −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 $$-0.529465\pi$$
−0.0924354 + 0.995719i $$0.529465\pi$$
$$840$$ 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 $$-0.823155\pi$$
−0.849598 + 0.527431i $$0.823155\pi$$
$$854$$ 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 $$-0.677730\pi$$
−0.529792 + 0.848128i $$0.677730\pi$$
$$864$$ 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 $$-0.331491\pi$$
0.505004 + 0.863117i $$0.331491\pi$$
$$878$$ 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 $$-0.289585\pi$$
0.613936 + 0.789355i $$0.289585\pi$$
$$930$$ −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 $$-0.352022\pi$$
0.448323 + 0.893872i $$0.352022\pi$$
$$942$$ 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 $$-0.473680\pi$$
0.0825925 + 0.996583i $$0.473680\pi$$
$$948$$ 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 $$-0.513804\pi$$
−0.0433518 + 0.999060i $$0.513804\pi$$
$$968$$ −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 $$-0.649613\pi$$
−0.452906 + 0.891558i $$0.649613\pi$$
$$978$$ 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 $$-0.706284\pi$$
−0.603641 + 0.797256i $$0.706284\pi$$
$$984$$ −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.1.12 12
7.3 odd 6 1183.2.e.j.170.1 24
7.5 odd 6 1183.2.e.j.508.1 24
7.6 odd 2 8281.2.a.cp.1.12 12
13.2 odd 12 637.2.q.i.589.6 12
13.7 odd 12 637.2.q.i.491.6 12
13.12 even 2 inner 8281.2.a.co.1.1 12
91.2 odd 12 637.2.k.i.459.1 12
91.12 odd 6 1183.2.e.j.508.12 24
91.20 even 12 637.2.q.g.491.6 12
91.33 even 12 91.2.u.b.88.1 yes 12
91.38 odd 6 1183.2.e.j.170.12 24
91.41 even 12 637.2.q.g.589.6 12
91.46 odd 12 637.2.k.i.569.6 12
91.54 even 12 91.2.k.b.4.1 12
91.59 even 12 91.2.k.b.23.6 yes 12
91.67 odd 12 637.2.u.g.30.1 12
91.72 odd 12 637.2.u.g.361.1 12
91.80 even 12 91.2.u.b.30.1 yes 12
91.90 odd 2 8281.2.a.cp.1.1 12
273.59 odd 12 819.2.bm.f.478.1 12
273.80 odd 12 819.2.do.e.667.6 12
273.215 odd 12 819.2.do.e.361.6 12
273.236 odd 12 819.2.bm.f.550.6 12

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.1 12 91.54 even 12
91.2.k.b.23.6 yes 12 91.59 even 12
91.2.u.b.30.1 yes 12 91.80 even 12
91.2.u.b.88.1 yes 12 91.33 even 12
637.2.k.i.459.1 12 91.2 odd 12
637.2.k.i.569.6 12 91.46 odd 12
637.2.q.g.491.6 12 91.20 even 12
637.2.q.g.589.6 12 91.41 even 12
637.2.q.i.491.6 12 13.7 odd 12
637.2.q.i.589.6 12 13.2 odd 12
637.2.u.g.30.1 12 91.67 odd 12
637.2.u.g.361.1 12 91.72 odd 12
819.2.bm.f.478.1 12 273.59 odd 12
819.2.bm.f.550.6 12 273.236 odd 12
819.2.do.e.361.6 12 273.215 odd 12
819.2.do.e.667.6 12 273.80 odd 12
1183.2.e.j.170.1 24 7.3 odd 6
1183.2.e.j.170.12 24 91.38 odd 6
1183.2.e.j.508.1 24 7.5 odd 6
1183.2.e.j.508.12 24 91.12 odd 6
8281.2.a.co.1.1 12 13.12 even 2 inner
8281.2.a.co.1.12 12 1.1 even 1 trivial
8281.2.a.cp.1.1 12 91.90 odd 2
8281.2.a.cp.1.12 12 7.6 odd 2