# Properties

 Label 1850.2.a.bd.1.4 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.1791440.1 Defining polynomial: $$x^{5} - 9x^{3} + 13x - 4$$ x^5 - 9*x^3 + 13*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$1.09441$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +3.82327 q^{11} +1.09441 q^{12} -0.147332 q^{13} -3.20984 q^{14} +1.00000 q^{16} -0.978989 q^{17} +1.80226 q^{18} +2.67594 q^{19} +3.51289 q^{21} -3.82327 q^{22} +2.33616 q^{23} -1.09441 q^{24} +0.147332 q^{26} -5.25565 q^{27} +3.20984 q^{28} +6.30425 q^{29} -3.62372 q^{31} -1.00000 q^{32} +4.18424 q^{33} +0.978989 q^{34} -1.80226 q^{36} -1.00000 q^{37} -2.67594 q^{38} -0.161242 q^{39} +11.8265 q^{41} -3.51289 q^{42} +4.53390 q^{43} +3.82327 q^{44} -2.33616 q^{46} -6.23085 q^{47} +1.09441 q^{48} +3.30305 q^{49} -1.07142 q^{51} -0.147332 q^{52} -11.2978 q^{53} +5.25565 q^{54} -3.20984 q^{56} +2.92858 q^{57} -6.30425 q^{58} +6.92858 q^{59} +10.4885 q^{61} +3.62372 q^{62} -5.78496 q^{63} +1.00000 q^{64} -4.18424 q^{66} -2.80936 q^{67} -0.978989 q^{68} +2.55672 q^{69} -12.3189 q^{71} +1.80226 q^{72} +13.9966 q^{73} +1.00000 q^{74} +2.67594 q^{76} +12.2721 q^{77} +0.161242 q^{78} +15.6057 q^{79} -0.345071 q^{81} -11.8265 q^{82} -13.5371 q^{83} +3.51289 q^{84} -4.53390 q^{86} +6.89945 q^{87} -3.82327 q^{88} -6.46929 q^{89} -0.472913 q^{91} +2.33616 q^{92} -3.96585 q^{93} +6.23085 q^{94} -1.09441 q^{96} +3.07063 q^{97} -3.30305 q^{98} -6.89054 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 $$5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100})$$ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.09441 0.631859 0.315930 0.948783i $$-0.397684\pi$$
0.315930 + 0.948783i $$0.397684\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.09441 −0.446792
$$7$$ 3.20984 1.21320 0.606602 0.795006i $$-0.292532\pi$$
0.606602 + 0.795006i $$0.292532\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −1.80226 −0.600754
$$10$$ 0 0
$$11$$ 3.82327 1.15276 0.576380 0.817182i $$-0.304465\pi$$
0.576380 + 0.817182i $$0.304465\pi$$
$$12$$ 1.09441 0.315930
$$13$$ −0.147332 −0.0408626 −0.0204313 0.999791i $$-0.506504\pi$$
−0.0204313 + 0.999791i $$0.506504\pi$$
$$14$$ −3.20984 −0.857865
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.978989 −0.237440 −0.118720 0.992928i $$-0.537879\pi$$
−0.118720 + 0.992928i $$0.537879\pi$$
$$18$$ 1.80226 0.424797
$$19$$ 2.67594 0.613903 0.306951 0.951725i $$-0.400691\pi$$
0.306951 + 0.951725i $$0.400691\pi$$
$$20$$ 0 0
$$21$$ 3.51289 0.766574
$$22$$ −3.82327 −0.815124
$$23$$ 2.33616 0.487122 0.243561 0.969886i $$-0.421684\pi$$
0.243561 + 0.969886i $$0.421684\pi$$
$$24$$ −1.09441 −0.223396
$$25$$ 0 0
$$26$$ 0.147332 0.0288942
$$27$$ −5.25565 −1.01145
$$28$$ 3.20984 0.606602
$$29$$ 6.30425 1.17067 0.585335 0.810792i $$-0.300963\pi$$
0.585335 + 0.810792i $$0.300963\pi$$
$$30$$ 0 0
$$31$$ −3.62372 −0.650839 −0.325420 0.945570i $$-0.605506\pi$$
−0.325420 + 0.945570i $$0.605506\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.18424 0.728382
$$34$$ 0.978989 0.167895
$$35$$ 0 0
$$36$$ −1.80226 −0.300377
$$37$$ −1.00000 −0.164399
$$38$$ −2.67594 −0.434095
$$39$$ −0.161242 −0.0258194
$$40$$ 0 0
$$41$$ 11.8265 1.84698 0.923491 0.383620i $$-0.125323\pi$$
0.923491 + 0.383620i $$0.125323\pi$$
$$42$$ −3.51289 −0.542050
$$43$$ 4.53390 0.691413 0.345706 0.938343i $$-0.387639\pi$$
0.345706 + 0.938343i $$0.387639\pi$$
$$44$$ 3.82327 0.576380
$$45$$ 0 0
$$46$$ −2.33616 −0.344448
$$47$$ −6.23085 −0.908863 −0.454431 0.890782i $$-0.650157\pi$$
−0.454431 + 0.890782i $$0.650157\pi$$
$$48$$ 1.09441 0.157965
$$49$$ 3.30305 0.471864
$$50$$ 0 0
$$51$$ −1.07142 −0.150028
$$52$$ −0.147332 −0.0204313
$$53$$ −11.2978 −1.55188 −0.775939 0.630807i $$-0.782724\pi$$
−0.775939 + 0.630807i $$0.782724\pi$$
$$54$$ 5.25565 0.715204
$$55$$ 0 0
$$56$$ −3.20984 −0.428932
$$57$$ 2.92858 0.387900
$$58$$ −6.30425 −0.827788
$$59$$ 6.92858 0.902025 0.451012 0.892518i $$-0.351063\pi$$
0.451012 + 0.892518i $$0.351063\pi$$
$$60$$ 0 0
$$61$$ 10.4885 1.34291 0.671457 0.741044i $$-0.265669\pi$$
0.671457 + 0.741044i $$0.265669\pi$$
$$62$$ 3.62372 0.460213
$$63$$ −5.78496 −0.728837
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.18424 −0.515044
$$67$$ −2.80936 −0.343218 −0.171609 0.985165i $$-0.554897\pi$$
−0.171609 + 0.985165i $$0.554897\pi$$
$$68$$ −0.978989 −0.118720
$$69$$ 2.55672 0.307793
$$70$$ 0 0
$$71$$ −12.3189 −1.46198 −0.730990 0.682388i $$-0.760941\pi$$
−0.730990 + 0.682388i $$0.760941\pi$$
$$72$$ 1.80226 0.212399
$$73$$ 13.9966 1.63818 0.819090 0.573665i $$-0.194479\pi$$
0.819090 + 0.573665i $$0.194479\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 2.67594 0.306951
$$77$$ 12.2721 1.39853
$$78$$ 0.161242 0.0182571
$$79$$ 15.6057 1.75578 0.877890 0.478861i $$-0.158950\pi$$
0.877890 + 0.478861i $$0.158950\pi$$
$$80$$ 0 0
$$81$$ −0.345071 −0.0383412
$$82$$ −11.8265 −1.30601
$$83$$ −13.5371 −1.48589 −0.742944 0.669354i $$-0.766571\pi$$
−0.742944 + 0.669354i $$0.766571\pi$$
$$84$$ 3.51289 0.383287
$$85$$ 0 0
$$86$$ −4.53390 −0.488903
$$87$$ 6.89945 0.739699
$$88$$ −3.82327 −0.407562
$$89$$ −6.46929 −0.685743 −0.342872 0.939382i $$-0.611400\pi$$
−0.342872 + 0.939382i $$0.611400\pi$$
$$90$$ 0 0
$$91$$ −0.472913 −0.0495747
$$92$$ 2.33616 0.243561
$$93$$ −3.96585 −0.411239
$$94$$ 6.23085 0.642663
$$95$$ 0 0
$$96$$ −1.09441 −0.111698
$$97$$ 3.07063 0.311775 0.155887 0.987775i $$-0.450176\pi$$
0.155887 + 0.987775i $$0.450176\pi$$
$$98$$ −3.30305 −0.333658
$$99$$ −6.89054 −0.692525
$$100$$ 0 0
$$101$$ −6.57513 −0.654250 −0.327125 0.944981i $$-0.606080\pi$$
−0.327125 + 0.944981i $$0.606080\pi$$
$$102$$ 1.07142 0.106086
$$103$$ 2.52180 0.248480 0.124240 0.992252i $$-0.460351\pi$$
0.124240 + 0.992252i $$0.460351\pi$$
$$104$$ 0.147332 0.0144471
$$105$$ 0 0
$$106$$ 11.2978 1.09734
$$107$$ −5.70291 −0.551321 −0.275661 0.961255i $$-0.588897\pi$$
−0.275661 + 0.961255i $$0.588897\pi$$
$$108$$ −5.25565 −0.505726
$$109$$ 11.9318 1.14286 0.571428 0.820652i $$-0.306390\pi$$
0.571428 + 0.820652i $$0.306390\pi$$
$$110$$ 0 0
$$111$$ −1.09441 −0.103877
$$112$$ 3.20984 0.303301
$$113$$ 2.18044 0.205119 0.102559 0.994727i $$-0.467297\pi$$
0.102559 + 0.994727i $$0.467297\pi$$
$$114$$ −2.92858 −0.274287
$$115$$ 0 0
$$116$$ 6.30425 0.585335
$$117$$ 0.265531 0.0245484
$$118$$ −6.92858 −0.637828
$$119$$ −3.14239 −0.288063
$$120$$ 0 0
$$121$$ 3.61741 0.328856
$$122$$ −10.4885 −0.949583
$$123$$ 12.9430 1.16703
$$124$$ −3.62372 −0.325420
$$125$$ 0 0
$$126$$ 5.78496 0.515365
$$127$$ 5.39390 0.478631 0.239316 0.970942i $$-0.423077\pi$$
0.239316 + 0.970942i $$0.423077\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.96195 0.436876
$$130$$ 0 0
$$131$$ −8.97060 −0.783765 −0.391883 0.920015i $$-0.628176\pi$$
−0.391883 + 0.920015i $$0.628176\pi$$
$$132$$ 4.18424 0.364191
$$133$$ 8.58933 0.744789
$$134$$ 2.80936 0.242692
$$135$$ 0 0
$$136$$ 0.978989 0.0839476
$$137$$ 2.70352 0.230978 0.115489 0.993309i $$-0.463157\pi$$
0.115489 + 0.993309i $$0.463157\pi$$
$$138$$ −2.55672 −0.217642
$$139$$ 4.30358 0.365025 0.182512 0.983204i $$-0.441577\pi$$
0.182512 + 0.983204i $$0.441577\pi$$
$$140$$ 0 0
$$141$$ −6.81912 −0.574273
$$142$$ 12.3189 1.03378
$$143$$ −0.563292 −0.0471048
$$144$$ −1.80226 −0.150188
$$145$$ 0 0
$$146$$ −13.9966 −1.15837
$$147$$ 3.61490 0.298152
$$148$$ −1.00000 −0.0821995
$$149$$ 21.6949 1.77732 0.888659 0.458568i $$-0.151638\pi$$
0.888659 + 0.458568i $$0.151638\pi$$
$$150$$ 0 0
$$151$$ 2.06744 0.168246 0.0841230 0.996455i $$-0.473191\pi$$
0.0841230 + 0.996455i $$0.473191\pi$$
$$152$$ −2.67594 −0.217047
$$153$$ 1.76439 0.142643
$$154$$ −12.2721 −0.988912
$$155$$ 0 0
$$156$$ −0.161242 −0.0129097
$$157$$ −18.5439 −1.47996 −0.739982 0.672627i $$-0.765166\pi$$
−0.739982 + 0.672627i $$0.765166\pi$$
$$158$$ −15.6057 −1.24152
$$159$$ −12.3645 −0.980569
$$160$$ 0 0
$$161$$ 7.49868 0.590979
$$162$$ 0.345071 0.0271113
$$163$$ 21.6240 1.69372 0.846860 0.531817i $$-0.178490\pi$$
0.846860 + 0.531817i $$0.178490\pi$$
$$164$$ 11.8265 0.923491
$$165$$ 0 0
$$166$$ 13.5371 1.05068
$$167$$ 11.7729 0.911012 0.455506 0.890233i $$-0.349458\pi$$
0.455506 + 0.890233i $$0.349458\pi$$
$$168$$ −3.51289 −0.271025
$$169$$ −12.9783 −0.998330
$$170$$ 0 0
$$171$$ −4.82274 −0.368804
$$172$$ 4.53390 0.345706
$$173$$ −14.0158 −1.06560 −0.532801 0.846240i $$-0.678861\pi$$
−0.532801 + 0.846240i $$0.678861\pi$$
$$174$$ −6.89945 −0.523046
$$175$$ 0 0
$$176$$ 3.82327 0.288190
$$177$$ 7.58273 0.569953
$$178$$ 6.46929 0.484894
$$179$$ −12.4160 −0.928019 −0.464009 0.885830i $$-0.653590\pi$$
−0.464009 + 0.885830i $$0.653590\pi$$
$$180$$ 0 0
$$181$$ 8.19748 0.609314 0.304657 0.952462i $$-0.401458\pi$$
0.304657 + 0.952462i $$0.401458\pi$$
$$182$$ 0.472913 0.0350546
$$183$$ 11.4787 0.848532
$$184$$ −2.33616 −0.172224
$$185$$ 0 0
$$186$$ 3.96585 0.290790
$$187$$ −3.74294 −0.273711
$$188$$ −6.23085 −0.454431
$$189$$ −16.8698 −1.22710
$$190$$ 0 0
$$191$$ 8.00137 0.578959 0.289479 0.957184i $$-0.406518\pi$$
0.289479 + 0.957184i $$0.406518\pi$$
$$192$$ 1.09441 0.0789824
$$193$$ 14.0420 1.01077 0.505383 0.862895i $$-0.331351\pi$$
0.505383 + 0.862895i $$0.331351\pi$$
$$194$$ −3.07063 −0.220458
$$195$$ 0 0
$$196$$ 3.30305 0.235932
$$197$$ −3.36608 −0.239823 −0.119912 0.992785i $$-0.538261\pi$$
−0.119912 + 0.992785i $$0.538261\pi$$
$$198$$ 6.89054 0.489689
$$199$$ −14.8890 −1.05545 −0.527725 0.849415i $$-0.676955\pi$$
−0.527725 + 0.849415i $$0.676955\pi$$
$$200$$ 0 0
$$201$$ −3.07460 −0.216866
$$202$$ 6.57513 0.462624
$$203$$ 20.2356 1.42026
$$204$$ −1.07142 −0.0750142
$$205$$ 0 0
$$206$$ −2.52180 −0.175702
$$207$$ −4.21037 −0.292641
$$208$$ −0.147332 −0.0102157
$$209$$ 10.2308 0.707683
$$210$$ 0 0
$$211$$ −9.80544 −0.675035 −0.337517 0.941319i $$-0.609587\pi$$
−0.337517 + 0.941319i $$0.609587\pi$$
$$212$$ −11.2978 −0.775939
$$213$$ −13.4819 −0.923766
$$214$$ 5.70291 0.389843
$$215$$ 0 0
$$216$$ 5.25565 0.357602
$$217$$ −11.6316 −0.789601
$$218$$ −11.9318 −0.808121
$$219$$ 15.3181 1.03510
$$220$$ 0 0
$$221$$ 0.144237 0.00970241
$$222$$ 1.09441 0.0734522
$$223$$ 15.4447 1.03425 0.517125 0.855910i $$-0.327002\pi$$
0.517125 + 0.855910i $$0.327002\pi$$
$$224$$ −3.20984 −0.214466
$$225$$ 0 0
$$226$$ −2.18044 −0.145041
$$227$$ −6.46090 −0.428825 −0.214413 0.976743i $$-0.568784\pi$$
−0.214413 + 0.976743i $$0.568784\pi$$
$$228$$ 2.92858 0.193950
$$229$$ 7.97458 0.526975 0.263488 0.964663i $$-0.415127\pi$$
0.263488 + 0.964663i $$0.415127\pi$$
$$230$$ 0 0
$$231$$ 13.4307 0.883676
$$232$$ −6.30425 −0.413894
$$233$$ −25.2907 −1.65685 −0.828423 0.560103i $$-0.810762\pi$$
−0.828423 + 0.560103i $$0.810762\pi$$
$$234$$ −0.265531 −0.0173583
$$235$$ 0 0
$$236$$ 6.92858 0.451012
$$237$$ 17.0791 1.10941
$$238$$ 3.14239 0.203691
$$239$$ 5.11481 0.330850 0.165425 0.986222i $$-0.447100\pi$$
0.165425 + 0.986222i $$0.447100\pi$$
$$240$$ 0 0
$$241$$ 16.4415 1.05909 0.529544 0.848282i $$-0.322363\pi$$
0.529544 + 0.848282i $$0.322363\pi$$
$$242$$ −3.61741 −0.232536
$$243$$ 15.3893 0.987225
$$244$$ 10.4885 0.671457
$$245$$ 0 0
$$246$$ −12.9430 −0.825217
$$247$$ −0.394252 −0.0250857
$$248$$ 3.62372 0.230107
$$249$$ −14.8152 −0.938872
$$250$$ 0 0
$$251$$ 6.35586 0.401178 0.200589 0.979675i $$-0.435714\pi$$
0.200589 + 0.979675i $$0.435714\pi$$
$$252$$ −5.78496 −0.364418
$$253$$ 8.93177 0.561535
$$254$$ −5.39390 −0.338444
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 5.45772 0.340443 0.170222 0.985406i $$-0.445552\pi$$
0.170222 + 0.985406i $$0.445552\pi$$
$$258$$ −4.96195 −0.308918
$$259$$ −3.20984 −0.199450
$$260$$ 0 0
$$261$$ −11.3619 −0.703284
$$262$$ 8.97060 0.554206
$$263$$ 14.8882 0.918044 0.459022 0.888425i $$-0.348200\pi$$
0.459022 + 0.888425i $$0.348200\pi$$
$$264$$ −4.18424 −0.257522
$$265$$ 0 0
$$266$$ −8.58933 −0.526646
$$267$$ −7.08007 −0.433293
$$268$$ −2.80936 −0.171609
$$269$$ 15.5573 0.948546 0.474273 0.880378i $$-0.342711\pi$$
0.474273 + 0.880378i $$0.342711\pi$$
$$270$$ 0 0
$$271$$ 0.462920 0.0281204 0.0140602 0.999901i $$-0.495524\pi$$
0.0140602 + 0.999901i $$0.495524\pi$$
$$272$$ −0.978989 −0.0593599
$$273$$ −0.517562 −0.0313242
$$274$$ −2.70352 −0.163326
$$275$$ 0 0
$$276$$ 2.55672 0.153896
$$277$$ −9.12112 −0.548035 −0.274018 0.961725i $$-0.588353\pi$$
−0.274018 + 0.961725i $$0.588353\pi$$
$$278$$ −4.30358 −0.258111
$$279$$ 6.53089 0.390994
$$280$$ 0 0
$$281$$ −24.8734 −1.48382 −0.741912 0.670497i $$-0.766081\pi$$
−0.741912 + 0.670497i $$0.766081\pi$$
$$282$$ 6.81912 0.406073
$$283$$ −19.7259 −1.17258 −0.586292 0.810100i $$-0.699413\pi$$
−0.586292 + 0.810100i $$0.699413\pi$$
$$284$$ −12.3189 −0.730990
$$285$$ 0 0
$$286$$ 0.563292 0.0333081
$$287$$ 37.9610 2.24077
$$288$$ 1.80226 0.106199
$$289$$ −16.0416 −0.943622
$$290$$ 0 0
$$291$$ 3.36053 0.196998
$$292$$ 13.9966 0.819090
$$293$$ 2.65528 0.155123 0.0775615 0.996988i $$-0.475287\pi$$
0.0775615 + 0.996988i $$0.475287\pi$$
$$294$$ −3.61490 −0.210825
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ −20.0938 −1.16596
$$298$$ −21.6949 −1.25675
$$299$$ −0.344191 −0.0199051
$$300$$ 0 0
$$301$$ 14.5531 0.838825
$$302$$ −2.06744 −0.118968
$$303$$ −7.19590 −0.413394
$$304$$ 2.67594 0.153476
$$305$$ 0 0
$$306$$ −1.76439 −0.100864
$$307$$ 25.3545 1.44706 0.723528 0.690295i $$-0.242519\pi$$
0.723528 + 0.690295i $$0.242519\pi$$
$$308$$ 12.2721 0.699267
$$309$$ 2.75989 0.157005
$$310$$ 0 0
$$311$$ 11.1234 0.630748 0.315374 0.948967i $$-0.397870\pi$$
0.315374 + 0.948967i $$0.397870\pi$$
$$312$$ 0.161242 0.00912855
$$313$$ −3.49109 −0.197328 −0.0986640 0.995121i $$-0.531457\pi$$
−0.0986640 + 0.995121i $$0.531457\pi$$
$$314$$ 18.5439 1.04649
$$315$$ 0 0
$$316$$ 15.6057 0.877890
$$317$$ −18.3656 −1.03152 −0.515759 0.856734i $$-0.672490\pi$$
−0.515759 + 0.856734i $$0.672490\pi$$
$$318$$ 12.3645 0.693367
$$319$$ 24.1029 1.34950
$$320$$ 0 0
$$321$$ −6.24134 −0.348357
$$322$$ −7.49868 −0.417885
$$323$$ −2.61972 −0.145765
$$324$$ −0.345071 −0.0191706
$$325$$ 0 0
$$326$$ −21.6240 −1.19764
$$327$$ 13.0583 0.722124
$$328$$ −11.8265 −0.653007
$$329$$ −20.0000 −1.10264
$$330$$ 0 0
$$331$$ −21.9365 −1.20574 −0.602870 0.797839i $$-0.705976\pi$$
−0.602870 + 0.797839i $$0.705976\pi$$
$$332$$ −13.5371 −0.742944
$$333$$ 1.80226 0.0987633
$$334$$ −11.7729 −0.644183
$$335$$ 0 0
$$336$$ 3.51289 0.191644
$$337$$ 8.65472 0.471453 0.235726 0.971819i $$-0.424253\pi$$
0.235726 + 0.971819i $$0.424253\pi$$
$$338$$ 12.9783 0.705926
$$339$$ 2.38630 0.129606
$$340$$ 0 0
$$341$$ −13.8545 −0.750262
$$342$$ 4.82274 0.260784
$$343$$ −11.8666 −0.640737
$$344$$ −4.53390 −0.244451
$$345$$ 0 0
$$346$$ 14.0158 0.753495
$$347$$ 21.3265 1.14486 0.572432 0.819952i $$-0.306000\pi$$
0.572432 + 0.819952i $$0.306000\pi$$
$$348$$ 6.89945 0.369849
$$349$$ −1.16305 −0.0622569 −0.0311285 0.999515i $$-0.509910\pi$$
−0.0311285 + 0.999515i $$0.509910\pi$$
$$350$$ 0 0
$$351$$ 0.774328 0.0413306
$$352$$ −3.82327 −0.203781
$$353$$ 0.470079 0.0250198 0.0125099 0.999922i $$-0.496018\pi$$
0.0125099 + 0.999922i $$0.496018\pi$$
$$354$$ −7.58273 −0.403017
$$355$$ 0 0
$$356$$ −6.46929 −0.342872
$$357$$ −3.43908 −0.182015
$$358$$ 12.4160 0.656208
$$359$$ −33.3086 −1.75796 −0.878981 0.476856i $$-0.841776\pi$$
−0.878981 + 0.476856i $$0.841776\pi$$
$$360$$ 0 0
$$361$$ −11.8393 −0.623123
$$362$$ −8.19748 −0.430850
$$363$$ 3.95894 0.207790
$$364$$ −0.472913 −0.0247874
$$365$$ 0 0
$$366$$ −11.4787 −0.600003
$$367$$ 11.1884 0.584029 0.292014 0.956414i $$-0.405674\pi$$
0.292014 + 0.956414i $$0.405674\pi$$
$$368$$ 2.33616 0.121781
$$369$$ −21.3144 −1.10958
$$370$$ 0 0
$$371$$ −36.2642 −1.88275
$$372$$ −3.96585 −0.205620
$$373$$ −23.1634 −1.19936 −0.599678 0.800242i $$-0.704705\pi$$
−0.599678 + 0.800242i $$0.704705\pi$$
$$374$$ 3.74294 0.193543
$$375$$ 0 0
$$376$$ 6.23085 0.321331
$$377$$ −0.928820 −0.0478366
$$378$$ 16.8698 0.867688
$$379$$ −26.7721 −1.37519 −0.687595 0.726095i $$-0.741333\pi$$
−0.687595 + 0.726095i $$0.741333\pi$$
$$380$$ 0 0
$$381$$ 5.90315 0.302428
$$382$$ −8.00137 −0.409386
$$383$$ −22.3189 −1.14044 −0.570220 0.821492i $$-0.693142\pi$$
−0.570220 + 0.821492i $$0.693142\pi$$
$$384$$ −1.09441 −0.0558490
$$385$$ 0 0
$$386$$ −14.0420 −0.714720
$$387$$ −8.17127 −0.415369
$$388$$ 3.07063 0.155887
$$389$$ −24.3011 −1.23211 −0.616057 0.787701i $$-0.711271\pi$$
−0.616057 + 0.787701i $$0.711271\pi$$
$$390$$ 0 0
$$391$$ −2.28707 −0.115662
$$392$$ −3.30305 −0.166829
$$393$$ −9.81754 −0.495229
$$394$$ 3.36608 0.169581
$$395$$ 0 0
$$396$$ −6.89054 −0.346262
$$397$$ −28.1888 −1.41476 −0.707378 0.706835i $$-0.750122\pi$$
−0.707378 + 0.706835i $$0.750122\pi$$
$$398$$ 14.8890 0.746316
$$399$$ 9.40027 0.470602
$$400$$ 0 0
$$401$$ −15.9528 −0.796644 −0.398322 0.917246i $$-0.630407\pi$$
−0.398322 + 0.917246i $$0.630407\pi$$
$$402$$ 3.07460 0.153347
$$403$$ 0.533891 0.0265950
$$404$$ −6.57513 −0.327125
$$405$$ 0 0
$$406$$ −20.2356 −1.00428
$$407$$ −3.82327 −0.189513
$$408$$ 1.07142 0.0530431
$$409$$ 31.6014 1.56259 0.781294 0.624164i $$-0.214560\pi$$
0.781294 + 0.624164i $$0.214560\pi$$
$$410$$ 0 0
$$411$$ 2.95877 0.145945
$$412$$ 2.52180 0.124240
$$413$$ 22.2396 1.09434
$$414$$ 4.21037 0.206928
$$415$$ 0 0
$$416$$ 0.147332 0.00722356
$$417$$ 4.70989 0.230644
$$418$$ −10.2308 −0.500407
$$419$$ −18.4266 −0.900197 −0.450098 0.892979i $$-0.648611\pi$$
−0.450098 + 0.892979i $$0.648611\pi$$
$$420$$ 0 0
$$421$$ −19.1570 −0.933655 −0.466828 0.884348i $$-0.654603\pi$$
−0.466828 + 0.884348i $$0.654603\pi$$
$$422$$ 9.80544 0.477322
$$423$$ 11.2296 0.546003
$$424$$ 11.2978 0.548672
$$425$$ 0 0
$$426$$ 13.4819 0.653201
$$427$$ 33.6663 1.62923
$$428$$ −5.70291 −0.275661
$$429$$ −0.616473 −0.0297636
$$430$$ 0 0
$$431$$ −10.3947 −0.500694 −0.250347 0.968156i $$-0.580545\pi$$
−0.250347 + 0.968156i $$0.580545\pi$$
$$432$$ −5.25565 −0.252863
$$433$$ −3.36622 −0.161770 −0.0808852 0.996723i $$-0.525775\pi$$
−0.0808852 + 0.996723i $$0.525775\pi$$
$$434$$ 11.6316 0.558332
$$435$$ 0 0
$$436$$ 11.9318 0.571428
$$437$$ 6.25142 0.299046
$$438$$ −15.3181 −0.731926
$$439$$ −32.5045 −1.55135 −0.775677 0.631130i $$-0.782591\pi$$
−0.775677 + 0.631130i $$0.782591\pi$$
$$440$$ 0 0
$$441$$ −5.95296 −0.283474
$$442$$ −0.144237 −0.00686064
$$443$$ 23.5479 1.11879 0.559397 0.828900i $$-0.311033\pi$$
0.559397 + 0.828900i $$0.311033\pi$$
$$444$$ −1.09441 −0.0519385
$$445$$ 0 0
$$446$$ −15.4447 −0.731325
$$447$$ 23.7432 1.12302
$$448$$ 3.20984 0.151651
$$449$$ 12.5629 0.592878 0.296439 0.955052i $$-0.404201\pi$$
0.296439 + 0.955052i $$0.404201\pi$$
$$450$$ 0 0
$$451$$ 45.2158 2.12913
$$452$$ 2.18044 0.102559
$$453$$ 2.26263 0.106308
$$454$$ 6.46090 0.303225
$$455$$ 0 0
$$456$$ −2.92858 −0.137143
$$457$$ −6.53390 −0.305643 −0.152821 0.988254i $$-0.548836\pi$$
−0.152821 + 0.988254i $$0.548836\pi$$
$$458$$ −7.97458 −0.372628
$$459$$ 5.14523 0.240159
$$460$$ 0 0
$$461$$ 24.7616 1.15326 0.576631 0.817005i $$-0.304367\pi$$
0.576631 + 0.817005i $$0.304367\pi$$
$$462$$ −13.4307 −0.624853
$$463$$ −25.9239 −1.20479 −0.602393 0.798200i $$-0.705786\pi$$
−0.602393 + 0.798200i $$0.705786\pi$$
$$464$$ 6.30425 0.292667
$$465$$ 0 0
$$466$$ 25.2907 1.17157
$$467$$ −28.8182 −1.33355 −0.666773 0.745261i $$-0.732325\pi$$
−0.666773 + 0.745261i $$0.732325\pi$$
$$468$$ 0.265531 0.0122742
$$469$$ −9.01759 −0.416394
$$470$$ 0 0
$$471$$ −20.2947 −0.935129
$$472$$ −6.92858 −0.318914
$$473$$ 17.3343 0.797033
$$474$$ −17.0791 −0.784469
$$475$$ 0 0
$$476$$ −3.14239 −0.144031
$$477$$ 20.3617 0.932297
$$478$$ −5.11481 −0.233946
$$479$$ −41.7873 −1.90931 −0.954655 0.297714i $$-0.903776\pi$$
−0.954655 + 0.297714i $$0.903776\pi$$
$$480$$ 0 0
$$481$$ 0.147332 0.00671778
$$482$$ −16.4415 −0.748888
$$483$$ 8.20665 0.373416
$$484$$ 3.61741 0.164428
$$485$$ 0 0
$$486$$ −15.3893 −0.698073
$$487$$ −9.34463 −0.423446 −0.211723 0.977330i $$-0.567907\pi$$
−0.211723 + 0.977330i $$0.567907\pi$$
$$488$$ −10.4885 −0.474791
$$489$$ 23.6655 1.07019
$$490$$ 0 0
$$491$$ −10.7558 −0.485404 −0.242702 0.970101i $$-0.578034\pi$$
−0.242702 + 0.970101i $$0.578034\pi$$
$$492$$ 12.9430 0.583516
$$493$$ −6.17179 −0.277963
$$494$$ 0.394252 0.0177383
$$495$$ 0 0
$$496$$ −3.62372 −0.162710
$$497$$ −39.5415 −1.77368
$$498$$ 14.8152 0.663883
$$499$$ 29.0702 1.30136 0.650680 0.759352i $$-0.274484\pi$$
0.650680 + 0.759352i $$0.274484\pi$$
$$500$$ 0 0
$$501$$ 12.8844 0.575631
$$502$$ −6.35586 −0.283676
$$503$$ −36.2683 −1.61712 −0.808561 0.588412i $$-0.799753\pi$$
−0.808561 + 0.588412i $$0.799753\pi$$
$$504$$ 5.78496 0.257683
$$505$$ 0 0
$$506$$ −8.93177 −0.397065
$$507$$ −14.2036 −0.630804
$$508$$ 5.39390 0.239316
$$509$$ −22.2170 −0.984751 −0.492375 0.870383i $$-0.663871\pi$$
−0.492375 + 0.870383i $$0.663871\pi$$
$$510$$ 0 0
$$511$$ 44.9268 1.98745
$$512$$ −1.00000 −0.0441942
$$513$$ −14.0638 −0.620933
$$514$$ −5.45772 −0.240730
$$515$$ 0 0
$$516$$ 4.96195 0.218438
$$517$$ −23.8222 −1.04770
$$518$$ 3.20984 0.141032
$$519$$ −15.3391 −0.673311
$$520$$ 0 0
$$521$$ 41.2732 1.80821 0.904107 0.427306i $$-0.140537\pi$$
0.904107 + 0.427306i $$0.140537\pi$$
$$522$$ 11.3619 0.497297
$$523$$ −11.0354 −0.482545 −0.241273 0.970457i $$-0.577565\pi$$
−0.241273 + 0.970457i $$0.577565\pi$$
$$524$$ −8.97060 −0.391883
$$525$$ 0 0
$$526$$ −14.8882 −0.649155
$$527$$ 3.54758 0.154535
$$528$$ 4.18424 0.182096
$$529$$ −17.5424 −0.762712
$$530$$ 0 0
$$531$$ −12.4871 −0.541895
$$532$$ 8.58933 0.372395
$$533$$ −1.74242 −0.0754726
$$534$$ 7.08007 0.306385
$$535$$ 0 0
$$536$$ 2.80936 0.121346
$$537$$ −13.5883 −0.586377
$$538$$ −15.5573 −0.670723
$$539$$ 12.6285 0.543946
$$540$$ 0 0
$$541$$ −32.2338 −1.38584 −0.692920 0.721014i $$-0.743676\pi$$
−0.692920 + 0.721014i $$0.743676\pi$$
$$542$$ −0.462920 −0.0198841
$$543$$ 8.97142 0.385001
$$544$$ 0.978989 0.0419738
$$545$$ 0 0
$$546$$ 0.517562 0.0221496
$$547$$ −15.0234 −0.642355 −0.321177 0.947019i $$-0.604079\pi$$
−0.321177 + 0.947019i $$0.604079\pi$$
$$548$$ 2.70352 0.115489
$$549$$ −18.9030 −0.806760
$$550$$ 0 0
$$551$$ 16.8698 0.718677
$$552$$ −2.55672 −0.108821
$$553$$ 50.0918 2.13012
$$554$$ 9.12112 0.387519
$$555$$ 0 0
$$556$$ 4.30358 0.182512
$$557$$ −39.1275 −1.65788 −0.828942 0.559334i $$-0.811057\pi$$
−0.828942 + 0.559334i $$0.811057\pi$$
$$558$$ −6.53089 −0.276475
$$559$$ −0.667989 −0.0282529
$$560$$ 0 0
$$561$$ −4.09632 −0.172947
$$562$$ 24.8734 1.04922
$$563$$ 27.1970 1.14622 0.573109 0.819479i $$-0.305737\pi$$
0.573109 + 0.819479i $$0.305737\pi$$
$$564$$ −6.81912 −0.287137
$$565$$ 0 0
$$566$$ 19.7259 0.829142
$$567$$ −1.10762 −0.0465157
$$568$$ 12.3189 0.516888
$$569$$ −33.4499 −1.40229 −0.701146 0.713018i $$-0.747328\pi$$
−0.701146 + 0.713018i $$0.747328\pi$$
$$570$$ 0 0
$$571$$ 28.9145 1.21003 0.605017 0.796213i $$-0.293166\pi$$
0.605017 + 0.796213i $$0.293166\pi$$
$$572$$ −0.563292 −0.0235524
$$573$$ 8.75680 0.365821
$$574$$ −37.9610 −1.58446
$$575$$ 0 0
$$576$$ −1.80226 −0.0750942
$$577$$ −33.5613 −1.39717 −0.698587 0.715525i $$-0.746188\pi$$
−0.698587 + 0.715525i $$0.746188\pi$$
$$578$$ 16.0416 0.667242
$$579$$ 15.3678 0.638663
$$580$$ 0 0
$$581$$ −43.4518 −1.80268
$$582$$ −3.36053 −0.139299
$$583$$ −43.1948 −1.78894
$$584$$ −13.9966 −0.579184
$$585$$ 0 0
$$586$$ −2.65528 −0.109689
$$587$$ 4.83581 0.199595 0.0997976 0.995008i $$-0.468180\pi$$
0.0997976 + 0.995008i $$0.468180\pi$$
$$588$$ 3.61490 0.149076
$$589$$ −9.69686 −0.399552
$$590$$ 0 0
$$591$$ −3.68388 −0.151535
$$592$$ −1.00000 −0.0410997
$$593$$ −43.1680 −1.77270 −0.886348 0.463020i $$-0.846766\pi$$
−0.886348 + 0.463020i $$0.846766\pi$$
$$594$$ 20.0938 0.824459
$$595$$ 0 0
$$596$$ 21.6949 0.888659
$$597$$ −16.2947 −0.666896
$$598$$ 0.344191 0.0140750
$$599$$ −38.7714 −1.58416 −0.792078 0.610420i $$-0.791001\pi$$
−0.792078 + 0.610420i $$0.791001\pi$$
$$600$$ 0 0
$$601$$ 1.56064 0.0636597 0.0318299 0.999493i $$-0.489867\pi$$
0.0318299 + 0.999493i $$0.489867\pi$$
$$602$$ −14.5531 −0.593139
$$603$$ 5.06320 0.206190
$$604$$ 2.06744 0.0841230
$$605$$ 0 0
$$606$$ 7.19590 0.292314
$$607$$ −21.0494 −0.854370 −0.427185 0.904164i $$-0.640495\pi$$
−0.427185 + 0.904164i $$0.640495\pi$$
$$608$$ −2.67594 −0.108524
$$609$$ 22.1461 0.897405
$$610$$ 0 0
$$611$$ 0.918005 0.0371385
$$612$$ 1.76439 0.0713214
$$613$$ 5.28786 0.213574 0.106787 0.994282i $$-0.465944\pi$$
0.106787 + 0.994282i $$0.465944\pi$$
$$614$$ −25.3545 −1.02322
$$615$$ 0 0
$$616$$ −12.2721 −0.494456
$$617$$ 34.8545 1.40319 0.701595 0.712576i $$-0.252472\pi$$
0.701595 + 0.712576i $$0.252472\pi$$
$$618$$ −2.75989 −0.111019
$$619$$ −7.30122 −0.293461 −0.146730 0.989177i $$-0.546875\pi$$
−0.146730 + 0.989177i $$0.546875\pi$$
$$620$$ 0 0
$$621$$ −12.2780 −0.492701
$$622$$ −11.1234 −0.446006
$$623$$ −20.7654 −0.831946
$$624$$ −0.161242 −0.00645486
$$625$$ 0 0
$$626$$ 3.49109 0.139532
$$627$$ 11.1968 0.447156
$$628$$ −18.5439 −0.739982
$$629$$ 0.978989 0.0390348
$$630$$ 0 0
$$631$$ 24.2752 0.966381 0.483190 0.875515i $$-0.339478\pi$$
0.483190 + 0.875515i $$0.339478\pi$$
$$632$$ −15.6057 −0.620762
$$633$$ −10.7312 −0.426527
$$634$$ 18.3656 0.729393
$$635$$ 0 0
$$636$$ −12.3645 −0.490285
$$637$$ −0.486646 −0.0192816
$$638$$ −24.1029 −0.954241
$$639$$ 22.2018 0.878290
$$640$$ 0 0
$$641$$ −8.22087 −0.324705 −0.162353 0.986733i $$-0.551908\pi$$
−0.162353 + 0.986733i $$0.551908\pi$$
$$642$$ 6.24134 0.246326
$$643$$ −29.7758 −1.17424 −0.587121 0.809499i $$-0.699739\pi$$
−0.587121 + 0.809499i $$0.699739\pi$$
$$644$$ 7.49868 0.295489
$$645$$ 0 0
$$646$$ 2.61972 0.103071
$$647$$ 35.6356 1.40098 0.700490 0.713662i $$-0.252965\pi$$
0.700490 + 0.713662i $$0.252965\pi$$
$$648$$ 0.345071 0.0135557
$$649$$ 26.4899 1.03982
$$650$$ 0 0
$$651$$ −12.7297 −0.498917
$$652$$ 21.6240 0.846860
$$653$$ −37.6128 −1.47190 −0.735951 0.677035i $$-0.763264\pi$$
−0.735951 + 0.677035i $$0.763264\pi$$
$$654$$ −13.0583 −0.510619
$$655$$ 0 0
$$656$$ 11.8265 0.461746
$$657$$ −25.2256 −0.984142
$$658$$ 20.0000 0.779681
$$659$$ 45.0553 1.75510 0.877552 0.479481i $$-0.159175\pi$$
0.877552 + 0.479481i $$0.159175\pi$$
$$660$$ 0 0
$$661$$ 7.32365 0.284857 0.142429 0.989805i $$-0.454509\pi$$
0.142429 + 0.989805i $$0.454509\pi$$
$$662$$ 21.9365 0.852587
$$663$$ 0.157854 0.00613056
$$664$$ 13.5371 0.525341
$$665$$ 0 0
$$666$$ −1.80226 −0.0698362
$$667$$ 14.7277 0.570260
$$668$$ 11.7729 0.455506
$$669$$ 16.9028 0.653501
$$670$$ 0 0
$$671$$ 40.1003 1.54806
$$672$$ −3.51289 −0.135512
$$673$$ 8.58807 0.331046 0.165523 0.986206i $$-0.447069\pi$$
0.165523 + 0.986206i $$0.447069\pi$$
$$674$$ −8.65472 −0.333368
$$675$$ 0 0
$$676$$ −12.9783 −0.499165
$$677$$ 9.13523 0.351096 0.175548 0.984471i $$-0.443830\pi$$
0.175548 + 0.984471i $$0.443830\pi$$
$$678$$ −2.38630 −0.0916454
$$679$$ 9.85621 0.378247
$$680$$ 0 0
$$681$$ −7.07089 −0.270957
$$682$$ 13.8545 0.530515
$$683$$ 40.7414 1.55893 0.779463 0.626449i $$-0.215492\pi$$
0.779463 + 0.626449i $$0.215492\pi$$
$$684$$ −4.82274 −0.184402
$$685$$ 0 0
$$686$$ 11.8666 0.453069
$$687$$ 8.72748 0.332974
$$688$$ 4.53390 0.172853
$$689$$ 1.66454 0.0634139
$$690$$ 0 0
$$691$$ −28.7543 −1.09387 −0.546933 0.837176i $$-0.684205\pi$$
−0.546933 + 0.837176i $$0.684205\pi$$
$$692$$ −14.0158 −0.532801
$$693$$ −22.1175 −0.840174
$$694$$ −21.3265 −0.809541
$$695$$ 0 0
$$696$$ −6.89945 −0.261523
$$697$$ −11.5780 −0.438547
$$698$$ 1.16305 0.0440223
$$699$$ −27.6784 −1.04689
$$700$$ 0 0
$$701$$ −7.87020 −0.297253 −0.148627 0.988893i $$-0.547485\pi$$
−0.148627 + 0.988893i $$0.547485\pi$$
$$702$$ −0.774328 −0.0292251
$$703$$ −2.67594 −0.100925
$$704$$ 3.82327 0.144095
$$705$$ 0 0
$$706$$ −0.470079 −0.0176917
$$707$$ −21.1051 −0.793738
$$708$$ 7.58273 0.284976
$$709$$ 33.0458 1.24106 0.620530 0.784182i $$-0.286917\pi$$
0.620530 + 0.784182i $$0.286917\pi$$
$$710$$ 0 0
$$711$$ −28.1256 −1.05479
$$712$$ 6.46929 0.242447
$$713$$ −8.46558 −0.317039
$$714$$ 3.43908 0.128704
$$715$$ 0 0
$$716$$ −12.4160 −0.464009
$$717$$ 5.59771 0.209050
$$718$$ 33.3086 1.24307
$$719$$ 4.94138 0.184282 0.0921412 0.995746i $$-0.470629\pi$$
0.0921412 + 0.995746i $$0.470629\pi$$
$$720$$ 0 0
$$721$$ 8.09456 0.301457
$$722$$ 11.8393 0.440615
$$723$$ 17.9937 0.669195
$$724$$ 8.19748 0.304657
$$725$$ 0 0
$$726$$ −3.95894 −0.146930
$$727$$ −42.5178 −1.57690 −0.788448 0.615101i $$-0.789115\pi$$
−0.788448 + 0.615101i $$0.789115\pi$$
$$728$$ 0.472913 0.0175273
$$729$$ 17.8775 0.662129
$$730$$ 0 0
$$731$$ −4.43863 −0.164169
$$732$$ 11.4787 0.424266
$$733$$ −10.2114 −0.377167 −0.188584 0.982057i $$-0.560390\pi$$
−0.188584 + 0.982057i $$0.560390\pi$$
$$734$$ −11.1884 −0.412971
$$735$$ 0 0
$$736$$ −2.33616 −0.0861119
$$737$$ −10.7410 −0.395648
$$738$$ 21.3144 0.784593
$$739$$ 25.5528 0.939975 0.469988 0.882673i $$-0.344258\pi$$
0.469988 + 0.882673i $$0.344258\pi$$
$$740$$ 0 0
$$741$$ −0.431475 −0.0158506
$$742$$ 36.2642 1.33130
$$743$$ −44.2852 −1.62467 −0.812334 0.583193i $$-0.801803\pi$$
−0.812334 + 0.583193i $$0.801803\pi$$
$$744$$ 3.96585 0.145395
$$745$$ 0 0
$$746$$ 23.1634 0.848072
$$747$$ 24.3974 0.892652
$$748$$ −3.74294 −0.136855
$$749$$ −18.3054 −0.668865
$$750$$ 0 0
$$751$$ 41.2897 1.50668 0.753342 0.657629i $$-0.228440\pi$$
0.753342 + 0.657629i $$0.228440\pi$$
$$752$$ −6.23085 −0.227216
$$753$$ 6.95593 0.253488
$$754$$ 0.928820 0.0338256
$$755$$ 0 0
$$756$$ −16.8698 −0.613548
$$757$$ −6.57802 −0.239082 −0.119541 0.992829i $$-0.538142\pi$$
−0.119541 + 0.992829i $$0.538142\pi$$
$$758$$ 26.7721 0.972406
$$759$$ 9.77504 0.354811
$$760$$ 0 0
$$761$$ 20.3166 0.736475 0.368237 0.929732i $$-0.379961\pi$$
0.368237 + 0.929732i $$0.379961\pi$$
$$762$$ −5.90315 −0.213849
$$763$$ 38.2990 1.38652
$$764$$ 8.00137 0.289479
$$765$$ 0 0
$$766$$ 22.3189 0.806413
$$767$$ −1.02080 −0.0368591
$$768$$ 1.09441 0.0394912
$$769$$ 25.2435 0.910303 0.455151 0.890414i $$-0.349585\pi$$
0.455151 + 0.890414i $$0.349585\pi$$
$$770$$ 0 0
$$771$$ 5.97300 0.215112
$$772$$ 14.0420 0.505383
$$773$$ 48.9781 1.76162 0.880810 0.473470i $$-0.156999\pi$$
0.880810 + 0.473470i $$0.156999\pi$$
$$774$$ 8.17127 0.293710
$$775$$ 0 0
$$776$$ −3.07063 −0.110229
$$777$$ −3.51289 −0.126024
$$778$$ 24.3011 0.871237
$$779$$ 31.6469 1.13387
$$780$$ 0 0
$$781$$ −47.0984 −1.68531
$$782$$ 2.28707 0.0817855
$$783$$ −33.1330 −1.18408
$$784$$ 3.30305 0.117966
$$785$$ 0 0
$$786$$ 9.81754 0.350180
$$787$$ 3.78073 0.134768 0.0673842 0.997727i $$-0.478535\pi$$
0.0673842 + 0.997727i $$0.478535\pi$$
$$788$$ −3.36608 −0.119912
$$789$$ 16.2938 0.580075
$$790$$ 0 0
$$791$$ 6.99886 0.248851
$$792$$ 6.89054 0.244845
$$793$$ −1.54529 −0.0548750
$$794$$ 28.1888 1.00038
$$795$$ 0 0
$$796$$ −14.8890 −0.527725
$$797$$ 29.5487 1.04667 0.523334 0.852128i $$-0.324688\pi$$
0.523334 + 0.852128i $$0.324688\pi$$
$$798$$ −9.40027 −0.332766
$$799$$ 6.09993 0.215800
$$800$$ 0 0
$$801$$ 11.6593 0.411963
$$802$$ 15.9528 0.563312
$$803$$ 53.5129 1.88843
$$804$$ −3.07460 −0.108433
$$805$$ 0 0
$$806$$ −0.533891 −0.0188055
$$807$$ 17.0261 0.599348
$$808$$ 6.57513 0.231312
$$809$$ 47.4231 1.66731 0.833654 0.552287i $$-0.186245\pi$$
0.833654 + 0.552287i $$0.186245\pi$$
$$810$$ 0 0
$$811$$ 25.6962 0.902317 0.451159 0.892444i $$-0.351011\pi$$
0.451159 + 0.892444i $$0.351011\pi$$
$$812$$ 20.2356 0.710131
$$813$$ 0.506626 0.0177681
$$814$$ 3.82327 0.134006
$$815$$ 0 0
$$816$$ −1.07142 −0.0375071
$$817$$ 12.1324 0.424460
$$818$$ −31.6014 −1.10492
$$819$$ 0.852312 0.0297822
$$820$$ 0 0
$$821$$ 0.286013 0.00998192 0.00499096 0.999988i $$-0.498411\pi$$
0.00499096 + 0.999988i $$0.498411\pi$$
$$822$$ −2.95877 −0.103199
$$823$$ −44.6093 −1.55498 −0.777491 0.628893i $$-0.783508\pi$$
−0.777491 + 0.628893i $$0.783508\pi$$
$$824$$ −2.52180 −0.0878510
$$825$$ 0 0
$$826$$ −22.2396 −0.773815
$$827$$ 55.1277 1.91698 0.958488 0.285131i $$-0.0920373\pi$$
0.958488 + 0.285131i $$0.0920373\pi$$
$$828$$ −4.21037 −0.146320
$$829$$ −17.2029 −0.597482 −0.298741 0.954334i $$-0.596567\pi$$
−0.298741 + 0.954334i $$0.596567\pi$$
$$830$$ 0 0
$$831$$ −9.98227 −0.346281
$$832$$ −0.147332 −0.00510783
$$833$$ −3.23365 −0.112039
$$834$$ −4.70989 −0.163090
$$835$$ 0 0
$$836$$ 10.2308 0.353841
$$837$$ 19.0450 0.658292
$$838$$ 18.4266 0.636535
$$839$$ 27.3527 0.944320 0.472160 0.881513i $$-0.343474\pi$$
0.472160 + 0.881513i $$0.343474\pi$$
$$840$$ 0 0
$$841$$ 10.7436 0.370467
$$842$$ 19.1570 0.660194
$$843$$ −27.2218 −0.937568
$$844$$ −9.80544 −0.337517
$$845$$ 0 0
$$846$$ −11.2296 −0.386082
$$847$$ 11.6113 0.398969
$$848$$ −11.2978 −0.387970
$$849$$ −21.5883 −0.740908
$$850$$ 0 0
$$851$$ −2.33616 −0.0800824
$$852$$ −13.4819 −0.461883
$$853$$ 51.8565 1.77553 0.887766 0.460294i $$-0.152256\pi$$
0.887766 + 0.460294i $$0.152256\pi$$
$$854$$ −33.6663 −1.15204
$$855$$ 0 0
$$856$$ 5.70291 0.194921
$$857$$ 4.63553 0.158347 0.0791733 0.996861i $$-0.474772\pi$$
0.0791733 + 0.996861i $$0.474772\pi$$
$$858$$ 0.616473 0.0210461
$$859$$ 43.1006 1.47057 0.735287 0.677755i $$-0.237047\pi$$
0.735287 + 0.677755i $$0.237047\pi$$
$$860$$ 0 0
$$861$$ 41.5450 1.41585
$$862$$ 10.3947 0.354044
$$863$$ −11.2463 −0.382829 −0.191414 0.981509i $$-0.561307\pi$$
−0.191414 + 0.981509i $$0.561307\pi$$
$$864$$ 5.25565 0.178801
$$865$$ 0 0
$$866$$ 3.36622 0.114389
$$867$$ −17.5561 −0.596237
$$868$$ −11.6316 −0.394801
$$869$$ 59.6649 2.02399
$$870$$ 0 0
$$871$$ 0.413910 0.0140248
$$872$$ −11.9318 −0.404061
$$873$$ −5.53407 −0.187300
$$874$$ −6.25142 −0.211457
$$875$$ 0 0
$$876$$ 15.3181 0.517550
$$877$$ 17.9444 0.605939 0.302969 0.953000i $$-0.402022\pi$$
0.302969 + 0.953000i $$0.402022\pi$$
$$878$$ 32.5045 1.09697
$$879$$ 2.90597 0.0980160
$$880$$ 0 0
$$881$$ −16.6495 −0.560935 −0.280468 0.959864i $$-0.590489\pi$$
−0.280468 + 0.959864i $$0.590489\pi$$
$$882$$ 5.95296 0.200446
$$883$$ −28.7758 −0.968383 −0.484191 0.874962i $$-0.660886\pi$$
−0.484191 + 0.874962i $$0.660886\pi$$
$$884$$ 0.144237 0.00485121
$$885$$ 0 0
$$886$$ −23.5479 −0.791107
$$887$$ 23.6957 0.795625 0.397812 0.917467i $$-0.369770\pi$$
0.397812 + 0.917467i $$0.369770\pi$$
$$888$$ 1.09441 0.0367261
$$889$$ 17.3135 0.580678
$$890$$ 0 0
$$891$$ −1.31930 −0.0441982
$$892$$ 15.4447 0.517125
$$893$$ −16.6734 −0.557953
$$894$$ −23.7432 −0.794092
$$895$$ 0 0
$$896$$ −3.20984 −0.107233
$$897$$ −0.376687 −0.0125772
$$898$$ −12.5629 −0.419228
$$899$$ −22.8448 −0.761918
$$900$$ 0 0
$$901$$ 11.0605 0.368478
$$902$$ −45.2158 −1.50552
$$903$$ 15.9271 0.530019
$$904$$ −2.18044 −0.0725204
$$905$$ 0 0
$$906$$ −2.26263 −0.0751710
$$907$$ 5.56548 0.184799 0.0923994 0.995722i $$-0.470546\pi$$
0.0923994 + 0.995722i $$0.470546\pi$$
$$908$$ −6.46090 −0.214413
$$909$$ 11.8501 0.393043
$$910$$ 0 0
$$911$$ −17.2514 −0.571565 −0.285782 0.958295i $$-0.592253\pi$$
−0.285782 + 0.958295i $$0.592253\pi$$
$$912$$ 2.92858 0.0969751
$$913$$ −51.7559 −1.71287
$$914$$ 6.53390 0.216122
$$915$$ 0 0
$$916$$ 7.97458 0.263488
$$917$$ −28.7942 −0.950867
$$918$$ −5.14523 −0.169818
$$919$$ −8.34007 −0.275114 −0.137557 0.990494i $$-0.543925\pi$$
−0.137557 + 0.990494i $$0.543925\pi$$
$$920$$ 0 0
$$921$$ 27.7483 0.914336
$$922$$ −24.7616 −0.815480
$$923$$ 1.81497 0.0597403
$$924$$ 13.4307 0.441838
$$925$$ 0 0
$$926$$ 25.9239 0.851913
$$927$$ −4.54494 −0.149275
$$928$$ −6.30425 −0.206947
$$929$$ −48.2529 −1.58313 −0.791564 0.611087i $$-0.790733\pi$$
−0.791564 + 0.611087i $$0.790733\pi$$
$$930$$ 0 0
$$931$$ 8.83876 0.289679
$$932$$ −25.2907 −0.828423
$$933$$ 12.1736 0.398544
$$934$$ 28.8182 0.942959
$$935$$ 0 0
$$936$$ −0.265531 −0.00867916
$$937$$ 30.2000 0.986592 0.493296 0.869862i $$-0.335792\pi$$
0.493296 + 0.869862i $$0.335792\pi$$
$$938$$ 9.01759 0.294435
$$939$$ −3.82069 −0.124684
$$940$$ 0 0
$$941$$ 14.9206 0.486399 0.243199 0.969976i $$-0.421803\pi$$
0.243199 + 0.969976i $$0.421803\pi$$
$$942$$ 20.2947 0.661236
$$943$$ 27.6285 0.899707
$$944$$ 6.92858 0.225506
$$945$$ 0 0
$$946$$ −17.3343 −0.563587
$$947$$ 48.1091 1.56334 0.781668 0.623694i $$-0.214369\pi$$
0.781668 + 0.623694i $$0.214369\pi$$
$$948$$ 17.0791 0.554703
$$949$$ −2.06215 −0.0669403
$$950$$ 0 0
$$951$$ −20.0996 −0.651774
$$952$$ 3.14239 0.101846
$$953$$ 8.67149 0.280897 0.140449 0.990088i $$-0.455146\pi$$
0.140449 + 0.990088i $$0.455146\pi$$
$$954$$ −20.3617 −0.659234
$$955$$ 0 0
$$956$$ 5.11481 0.165425
$$957$$ 26.3785 0.852695
$$958$$ 41.7873 1.35009
$$959$$ 8.67787 0.280223
$$960$$ 0 0
$$961$$ −17.8686 −0.576408
$$962$$ −0.147332 −0.00475018
$$963$$ 10.2781 0.331208
$$964$$ 16.4415 0.529544
$$965$$ 0 0
$$966$$ −8.20665 −0.264045
$$967$$ 18.5944 0.597955 0.298978 0.954260i $$-0.403354\pi$$
0.298978 + 0.954260i $$0.403354\pi$$
$$968$$ −3.61741 −0.116268
$$969$$ −2.86705 −0.0921029
$$970$$ 0 0
$$971$$ −32.6679 −1.04836 −0.524182 0.851606i $$-0.675629\pi$$
−0.524182 + 0.851606i $$0.675629\pi$$
$$972$$ 15.3893 0.493612
$$973$$ 13.8138 0.442850
$$974$$ 9.34463 0.299421
$$975$$ 0 0
$$976$$ 10.4885 0.335728
$$977$$ −28.1856 −0.901737 −0.450868 0.892590i $$-0.648886\pi$$
−0.450868 + 0.892590i $$0.648886\pi$$
$$978$$ −23.6655 −0.756740
$$979$$ −24.7338 −0.790497
$$980$$ 0 0
$$981$$ −21.5042 −0.686575
$$982$$ 10.7558 0.343232
$$983$$ 25.3845 0.809639 0.404819 0.914397i $$-0.367334\pi$$
0.404819 + 0.914397i $$0.367334\pi$$
$$984$$ −12.9430 −0.412608
$$985$$ 0 0
$$986$$ 6.17179 0.196550
$$987$$ −21.8883 −0.696711
$$988$$ −0.394252 −0.0125428
$$989$$ 10.5919 0.336803
$$990$$ 0 0
$$991$$ −21.1083 −0.670527 −0.335264 0.942124i $$-0.608825\pi$$
−0.335264 + 0.942124i $$0.608825\pi$$
$$992$$ 3.62372 0.115053
$$993$$ −24.0076 −0.761858
$$994$$ 39.5415 1.25418
$$995$$ 0 0
$$996$$ −14.8152 −0.469436
$$997$$ −39.0978 −1.23824 −0.619120 0.785296i $$-0.712511\pi$$
−0.619120 + 0.785296i $$0.712511\pi$$
$$998$$ −29.0702 −0.920201
$$999$$ 5.25565 0.166282
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 1850.2.a.bd.1.4 5
5.2 odd 4 370.2.b.d.149.2 10
5.3 odd 4 370.2.b.d.149.9 yes 10
5.4 even 2 1850.2.a.be.1.2 5
15.2 even 4 3330.2.d.p.1999.7 10
15.8 even 4 3330.2.d.p.1999.2 10

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.2 10 5.2 odd 4
370.2.b.d.149.9 yes 10 5.3 odd 4
1850.2.a.bd.1.4 5 1.1 even 1 trivial
1850.2.a.be.1.2 5 5.4 even 2
3330.2.d.p.1999.2 10 15.8 even 4
3330.2.d.p.1999.7 10 15.2 even 4