# Properties

 Label 1850.2.a.be.1.2 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 1.2 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 $$-0.602316\pi$$
−0.315930 + 0.948783i $$0.602316\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.09441 −0.446792
$$7$$ −3.20984 −1.21320 −0.606602 0.795006i $$-0.707468\pi$$
−0.606602 + 0.795006i $$0.707468\pi$$
$$8$$ 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 $$-0.493496\pi$$
0.0204313 + 0.999791i $$0.493496\pi$$
$$14$$ −3.20984 −0.857865
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.978989 0.237440 0.118720 0.992928i $$-0.462121\pi$$
0.118720 + 0.992928i $$0.462121\pi$$
$$18$$ −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 $$-0.578316\pi$$
−0.243561 + 0.969886i $$0.578316\pi$$
$$24$$ −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 $$-0.612361\pi$$
−0.345706 + 0.938343i $$0.612361\pi$$
$$44$$ 3.82327 0.576380
$$45$$ 0 0
$$46$$ −2.33616 −0.344448
$$47$$ 6.23085 0.908863 0.454431 0.890782i $$-0.349843\pi$$
0.454431 + 0.890782i $$0.349843\pi$$
$$48$$ −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 $$-0.217276\pi$$
0.775939 + 0.630807i $$0.217276\pi$$
$$54$$ 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 $$-0.445103\pi$$
0.171609 + 0.985165i $$0.445103\pi$$
$$68$$ 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 $$-0.805521\pi$$
−0.819090 + 0.573665i $$0.805521\pi$$
$$74$$ 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 $$-0.233429\pi$$
0.742944 + 0.669354i $$0.233429\pi$$
$$84$$ 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 $$-0.549824\pi$$
−0.155887 + 0.987775i $$0.549824\pi$$
$$98$$ 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 $$-0.539649\pi$$
−0.124240 + 0.992252i $$0.539649\pi$$
$$104$$ 0.147332 0.0144471
$$105$$ 0 0
$$106$$ 11.2978 1.09734
$$107$$ 5.70291 0.551321 0.275661 0.961255i $$-0.411103\pi$$
0.275661 + 0.961255i $$0.411103\pi$$
$$108$$ 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 $$-0.532703\pi$$
−0.102559 + 0.994727i $$0.532703\pi$$
$$114$$ −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 $$-0.576923\pi$$
−0.239316 + 0.970942i $$0.576923\pi$$
$$128$$ 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 $$-0.536843\pi$$
−0.115489 + 0.993309i $$0.536843\pi$$
$$138$$ 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 $$-0.234834\pi$$
0.739982 + 0.672627i $$0.234834\pi$$
$$158$$ 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 $$-0.821510\pi$$
−0.846860 + 0.531817i $$0.821510\pi$$
$$164$$ 11.8265 0.923491
$$165$$ 0 0
$$166$$ 13.5371 1.05068
$$167$$ −11.7729 −0.911012 −0.455506 0.890233i $$-0.650542\pi$$
−0.455506 + 0.890233i $$0.650542\pi$$
$$168$$ 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 $$-0.321139\pi$$
0.532801 + 0.846240i $$0.321139\pi$$
$$174$$ −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 $$-0.668649\pi$$
−0.505383 + 0.862895i $$0.668649\pi$$
$$194$$ −3.07063 −0.220458
$$195$$ 0 0
$$196$$ 3.30305 0.235932
$$197$$ 3.36608 0.239823 0.119912 0.992785i $$-0.461739\pi$$
0.119912 + 0.992785i $$0.461739\pi$$
$$198$$ −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 $$-0.672998\pi$$
−0.517125 + 0.855910i $$0.672998\pi$$
$$224$$ −3.20984 −0.214466
$$225$$ 0 0
$$226$$ −2.18044 −0.145041
$$227$$ 6.46090 0.428825 0.214413 0.976743i $$-0.431216\pi$$
0.214413 + 0.976743i $$0.431216\pi$$
$$228$$ −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 $$-0.189238\pi$$
0.828423 + 0.560103i $$0.189238\pi$$
$$234$$ −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 $$-0.554448\pi$$
−0.170222 + 0.985406i $$0.554448\pi$$
$$258$$ 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 $$-0.651800\pi$$
−0.459022 + 0.888425i $$0.651800\pi$$
$$264$$ −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 $$-0.411647\pi$$
0.274018 + 0.961725i $$0.411647\pi$$
$$278$$ 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 $$-0.300587\pi$$
0.586292 + 0.810100i $$0.300587\pi$$
$$284$$ −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 $$-0.524713\pi$$
−0.0775615 + 0.996988i $$0.524713\pi$$
$$294$$ −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 $$-0.757481\pi$$
−0.723528 + 0.690295i $$0.757481\pi$$
$$308$$ −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 $$-0.468543\pi$$
0.0986640 + 0.995121i $$0.468543\pi$$
$$314$$ 18.5439 1.04649
$$315$$ 0 0
$$316$$ 15.6057 0.877890
$$317$$ 18.3656 1.03152 0.515759 0.856734i $$-0.327510\pi$$
0.515759 + 0.856734i $$0.327510\pi$$
$$318$$ −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 $$-0.575747\pi$$
−0.235726 + 0.971819i $$0.575747\pi$$
$$338$$ −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 $$-0.694000\pi$$
−0.572432 + 0.819952i $$0.694000\pi$$
$$348$$ −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 $$-0.503982\pi$$
−0.0125099 + 0.999922i $$0.503982\pi$$
$$354$$ −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 $$-0.594326\pi$$
−0.292014 + 0.956414i $$0.594326\pi$$
$$368$$ −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 $$-0.295295\pi$$
0.599678 + 0.800242i $$0.295295\pi$$
$$374$$ 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 $$-0.306858\pi$$
0.570220 + 0.821492i $$0.306858\pi$$
$$384$$ −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 $$-0.249878\pi$$
0.707378 + 0.706835i $$0.249878\pi$$
$$398$$ −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 $$-0.474225\pi$$
0.0808852 + 0.996723i $$0.474225\pi$$
$$434$$ 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 $$-0.688967\pi$$
−0.559397 + 0.828900i $$0.688967\pi$$
$$444$$ −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 $$-0.451164\pi$$
0.152821 + 0.988254i $$0.451164\pi$$
$$458$$ 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 $$-0.294214\pi$$
0.602393 + 0.798200i $$0.294214\pi$$
$$464$$ 6.30425 0.292667
$$465$$ 0 0
$$466$$ 25.2907 1.17157
$$467$$ 28.8182 1.33355 0.666773 0.745261i $$-0.267675\pi$$
0.666773 + 0.745261i $$0.267675\pi$$
$$468$$ −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 $$-0.432093\pi$$
0.211723 + 0.977330i $$0.432093\pi$$
$$488$$ 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 $$-0.200247\pi$$
0.808561 + 0.588412i $$0.200247\pi$$
$$504$$ 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 $$-0.422435\pi$$
0.241273 + 0.970457i $$0.422435\pi$$
$$524$$ −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 $$-0.395921\pi$$
0.321177 + 0.947019i $$0.395921\pi$$
$$548$$ −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 $$-0.188943\pi$$
0.828942 + 0.559334i $$0.188943\pi$$
$$558$$ 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 $$-0.694263\pi$$
−0.573109 + 0.819479i $$0.694263\pi$$
$$564$$ −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 $$-0.253812\pi$$
0.698587 + 0.715525i $$0.253812\pi$$
$$578$$ −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 $$-0.531820\pi$$
−0.0997976 + 0.995008i $$0.531820\pi$$
$$588$$ −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 $$-0.153234\pi$$
0.886348 + 0.463020i $$0.153234\pi$$
$$594$$ 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 $$-0.359505\pi$$
0.427185 + 0.904164i $$0.359505\pi$$
$$608$$ 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 $$-0.534056\pi$$
−0.106787 + 0.994282i $$0.534056\pi$$
$$614$$ −25.3545 −1.02322
$$615$$ 0 0
$$616$$ −12.2721 −0.494456
$$617$$ −34.8545 −1.40319 −0.701595 0.712576i $$-0.747528\pi$$
−0.701595 + 0.712576i $$0.747528\pi$$
$$618$$ 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 $$-0.300261\pi$$
0.587121 + 0.809499i $$0.300261\pi$$
$$644$$ 7.49868 0.295489
$$645$$ 0 0
$$646$$ 2.61972 0.103071
$$647$$ −35.6356 −1.40098 −0.700490 0.713662i $$-0.747035\pi$$
−0.700490 + 0.713662i $$0.747035\pi$$
$$648$$ −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 $$-0.236736\pi$$
0.735951 + 0.677035i $$0.236736\pi$$
$$654$$ −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 $$-0.552931\pi$$
−0.165523 + 0.986206i $$0.552931\pi$$
$$674$$ −8.65472 −0.333368
$$675$$ 0 0
$$676$$ −12.9783 −0.499165
$$677$$ −9.13523 −0.351096 −0.175548 0.984471i $$-0.556170\pi$$
−0.175548 + 0.984471i $$0.556170\pi$$
$$678$$ 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 $$-0.784508\pi$$
−0.779463 + 0.626449i $$0.784508\pi$$
$$684$$ −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 $$-0.210885\pi$$
0.788448 + 0.615101i $$0.210885\pi$$
$$728$$ −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 $$-0.439610\pi$$
0.188584 + 0.982057i $$0.439610\pi$$
$$734$$ −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 $$-0.198197\pi$$
0.812334 + 0.583193i $$0.198197\pi$$
$$744$$ 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 $$-0.461858\pi$$
0.119541 + 0.992829i $$0.461858\pi$$
$$758$$ −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 $$-0.843001\pi$$
−0.880810 + 0.473470i $$0.843001\pi$$
$$774$$ 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 $$-0.521465\pi$$
−0.0673842 + 0.997727i $$0.521465\pi$$
$$788$$ 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 $$-0.675312\pi$$
−0.523334 + 0.852128i $$0.675312\pi$$
$$798$$ 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 $$-0.216492\pi$$
0.777491 + 0.628893i $$0.216492\pi$$
$$824$$ −2.52180 −0.0878510
$$825$$ 0 0
$$826$$ −22.2396 −0.773815
$$827$$ −55.1277 −1.91698 −0.958488 0.285131i $$-0.907963\pi$$
−0.958488 + 0.285131i $$0.907963\pi$$
$$828$$ 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 $$-0.847744\pi$$
−0.887766 + 0.460294i $$0.847744\pi$$
$$854$$ −33.6663 −1.15204
$$855$$ 0 0
$$856$$ 5.70291 0.194921
$$857$$ −4.63553 −0.158347 −0.0791733 0.996861i $$-0.525228\pi$$
−0.0791733 + 0.996861i $$0.525228\pi$$
$$858$$ −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 $$-0.438693\pi$$
0.191414 + 0.981509i $$0.438693\pi$$
$$864$$ 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 $$-0.597978\pi$$
−0.302969 + 0.953000i $$0.597978\pi$$
$$878$$ −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 $$-0.339114\pi$$
0.484191 + 0.874962i $$0.339114\pi$$
$$884$$ 0.144237 0.00485121
$$885$$ 0 0
$$886$$ −23.5479 −0.791107
$$887$$ −23.6957 −0.795625 −0.397812 0.917467i $$-0.630230\pi$$
−0.397812 + 0.917467i $$0.630230\pi$$
$$888$$ −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 $$-0.529454\pi$$
−0.0923994 + 0.995722i $$0.529454\pi$$
$$908$$ 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 $$-0.664208\pi$$
−0.493296 + 0.869862i $$0.664208\pi$$
$$938$$ −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 $$-0.785631\pi$$
−0.781668 + 0.623694i $$0.785631\pi$$
$$948$$ −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 $$-0.544854\pi$$
−0.140449 + 0.990088i $$0.544854\pi$$
$$954$$ −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 $$-0.596646\pi$$
−0.298978 + 0.954260i $$0.596646\pi$$
$$968$$ 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 $$-0.351114\pi$$
0.450868 + 0.892590i $$0.351114\pi$$
$$978$$ 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 $$-0.632666\pi$$
−0.404819 + 0.914397i $$0.632666\pi$$
$$984$$ −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 $$-0.287489\pi$$
0.619120 + 0.785296i $$0.287489\pi$$
$$998$$ 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.be.1.2 5
5.2 odd 4 370.2.b.d.149.9 yes 10
5.3 odd 4 370.2.b.d.149.2 10
5.4 even 2 1850.2.a.bd.1.4 5
15.2 even 4 3330.2.d.p.1999.2 10
15.8 even 4 3330.2.d.p.1999.7 10

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