# Properties

 Label 1849.2.a.g.1.2 Level $1849$ Weight $2$ Character 1849.1 Self dual yes Analytic conductor $14.764$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7643393337$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 1849.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.44949 q^{2} -2.44949 q^{3} +4.00000 q^{4} -2.44949 q^{5} -6.00000 q^{6} +2.44949 q^{7} +4.89898 q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+2.44949 q^{2} -2.44949 q^{3} +4.00000 q^{4} -2.44949 q^{5} -6.00000 q^{6} +2.44949 q^{7} +4.89898 q^{8} +3.00000 q^{9} -6.00000 q^{10} -1.00000 q^{11} -9.79796 q^{12} -3.00000 q^{13} +6.00000 q^{14} +6.00000 q^{15} +4.00000 q^{16} -7.00000 q^{17} +7.34847 q^{18} +4.89898 q^{19} -9.79796 q^{20} -6.00000 q^{21} -2.44949 q^{22} +1.00000 q^{23} -12.0000 q^{24} +1.00000 q^{25} -7.34847 q^{26} +9.79796 q^{28} -2.44949 q^{29} +14.6969 q^{30} -3.00000 q^{31} +2.44949 q^{33} -17.1464 q^{34} -6.00000 q^{35} +12.0000 q^{36} -4.89898 q^{37} +12.0000 q^{38} +7.34847 q^{39} -12.0000 q^{40} -5.00000 q^{41} -14.6969 q^{42} -4.00000 q^{44} -7.34847 q^{45} +2.44949 q^{46} -10.0000 q^{47} -9.79796 q^{48} -1.00000 q^{49} +2.44949 q^{50} +17.1464 q^{51} -12.0000 q^{52} -1.00000 q^{53} +2.44949 q^{55} +12.0000 q^{56} -12.0000 q^{57} -6.00000 q^{58} -10.0000 q^{59} +24.0000 q^{60} +7.34847 q^{61} -7.34847 q^{62} +7.34847 q^{63} -8.00000 q^{64} +7.34847 q^{65} +6.00000 q^{66} +9.00000 q^{67} -28.0000 q^{68} -2.44949 q^{69} -14.6969 q^{70} +4.89898 q^{71} +14.6969 q^{72} -12.2474 q^{73} -12.0000 q^{74} -2.44949 q^{75} +19.5959 q^{76} -2.44949 q^{77} +18.0000 q^{78} -6.00000 q^{79} -9.79796 q^{80} -9.00000 q^{81} -12.2474 q^{82} +1.00000 q^{83} -24.0000 q^{84} +17.1464 q^{85} +6.00000 q^{87} -4.89898 q^{88} -17.1464 q^{89} -18.0000 q^{90} -7.34847 q^{91} +4.00000 q^{92} +7.34847 q^{93} -24.4949 q^{94} -12.0000 q^{95} +11.0000 q^{97} -2.44949 q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{4} - 12q^{6} + 6q^{9} + O(q^{10})$$ $$2q + 8q^{4} - 12q^{6} + 6q^{9} - 12q^{10} - 2q^{11} - 6q^{13} + 12q^{14} + 12q^{15} + 8q^{16} - 14q^{17} - 12q^{21} + 2q^{23} - 24q^{24} + 2q^{25} - 6q^{31} - 12q^{35} + 24q^{36} + 24q^{38} - 24q^{40} - 10q^{41} - 8q^{44} - 20q^{47} - 2q^{49} - 24q^{52} - 2q^{53} + 24q^{56} - 24q^{57} - 12q^{58} - 20q^{59} + 48q^{60} - 16q^{64} + 12q^{66} + 18q^{67} - 56q^{68} - 24q^{74} + 36q^{78} - 12q^{79} - 18q^{81} + 2q^{83} - 48q^{84} + 12q^{87} - 36q^{90} + 8q^{92} - 24q^{95} + 22q^{97} - 6q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.44949 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$3$$ −2.44949 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$4$$ 4.00000 2.00000
$$5$$ −2.44949 −1.09545 −0.547723 0.836660i $$-0.684505\pi$$
−0.547723 + 0.836660i $$0.684505\pi$$
$$6$$ −6.00000 −2.44949
$$7$$ 2.44949 0.925820 0.462910 0.886405i $$-0.346805\pi$$
0.462910 + 0.886405i $$0.346805\pi$$
$$8$$ 4.89898 1.73205
$$9$$ 3.00000 1.00000
$$10$$ −6.00000 −1.89737
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ −9.79796 −2.82843
$$13$$ −3.00000 −0.832050 −0.416025 0.909353i $$-0.636577\pi$$
−0.416025 + 0.909353i $$0.636577\pi$$
$$14$$ 6.00000 1.60357
$$15$$ 6.00000 1.54919
$$16$$ 4.00000 1.00000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 7.34847 1.73205
$$19$$ 4.89898 1.12390 0.561951 0.827170i $$-0.310051\pi$$
0.561951 + 0.827170i $$0.310051\pi$$
$$20$$ −9.79796 −2.19089
$$21$$ −6.00000 −1.30931
$$22$$ −2.44949 −0.522233
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ −12.0000 −2.44949
$$25$$ 1.00000 0.200000
$$26$$ −7.34847 −1.44115
$$27$$ 0 0
$$28$$ 9.79796 1.85164
$$29$$ −2.44949 −0.454859 −0.227429 0.973795i $$-0.573032\pi$$
−0.227429 + 0.973795i $$0.573032\pi$$
$$30$$ 14.6969 2.68328
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ 2.44949 0.426401
$$34$$ −17.1464 −2.94059
$$35$$ −6.00000 −1.01419
$$36$$ 12.0000 2.00000
$$37$$ −4.89898 −0.805387 −0.402694 0.915335i $$-0.631926\pi$$
−0.402694 + 0.915335i $$0.631926\pi$$
$$38$$ 12.0000 1.94666
$$39$$ 7.34847 1.17670
$$40$$ −12.0000 −1.89737
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ −14.6969 −2.26779
$$43$$ 0 0
$$44$$ −4.00000 −0.603023
$$45$$ −7.34847 −1.09545
$$46$$ 2.44949 0.361158
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ −9.79796 −1.41421
$$49$$ −1.00000 −0.142857
$$50$$ 2.44949 0.346410
$$51$$ 17.1464 2.40098
$$52$$ −12.0000 −1.66410
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 0 0
$$55$$ 2.44949 0.330289
$$56$$ 12.0000 1.60357
$$57$$ −12.0000 −1.58944
$$58$$ −6.00000 −0.787839
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 24.0000 3.09839
$$61$$ 7.34847 0.940875 0.470438 0.882433i $$-0.344096\pi$$
0.470438 + 0.882433i $$0.344096\pi$$
$$62$$ −7.34847 −0.933257
$$63$$ 7.34847 0.925820
$$64$$ −8.00000 −1.00000
$$65$$ 7.34847 0.911465
$$66$$ 6.00000 0.738549
$$67$$ 9.00000 1.09952 0.549762 0.835321i $$-0.314718\pi$$
0.549762 + 0.835321i $$0.314718\pi$$
$$68$$ −28.0000 −3.39550
$$69$$ −2.44949 −0.294884
$$70$$ −14.6969 −1.75662
$$71$$ 4.89898 0.581402 0.290701 0.956814i $$-0.406112\pi$$
0.290701 + 0.956814i $$0.406112\pi$$
$$72$$ 14.6969 1.73205
$$73$$ −12.2474 −1.43346 −0.716728 0.697353i $$-0.754361\pi$$
−0.716728 + 0.697353i $$0.754361\pi$$
$$74$$ −12.0000 −1.39497
$$75$$ −2.44949 −0.282843
$$76$$ 19.5959 2.24781
$$77$$ −2.44949 −0.279145
$$78$$ 18.0000 2.03810
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ −9.79796 −1.09545
$$81$$ −9.00000 −1.00000
$$82$$ −12.2474 −1.35250
$$83$$ 1.00000 0.109764 0.0548821 0.998493i $$-0.482522\pi$$
0.0548821 + 0.998493i $$0.482522\pi$$
$$84$$ −24.0000 −2.61861
$$85$$ 17.1464 1.85979
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ −4.89898 −0.522233
$$89$$ −17.1464 −1.81752 −0.908759 0.417322i $$-0.862969\pi$$
−0.908759 + 0.417322i $$0.862969\pi$$
$$90$$ −18.0000 −1.89737
$$91$$ −7.34847 −0.770329
$$92$$ 4.00000 0.417029
$$93$$ 7.34847 0.762001
$$94$$ −24.4949 −2.52646
$$95$$ −12.0000 −1.23117
$$96$$ 0 0
$$97$$ 11.0000 1.11688 0.558440 0.829545i $$-0.311400\pi$$
0.558440 + 0.829545i $$0.311400\pi$$
$$98$$ −2.44949 −0.247436
$$99$$ −3.00000 −0.301511
$$100$$ 4.00000 0.400000
$$101$$ −5.00000 −0.497519 −0.248759 0.968565i $$-0.580023\pi$$
−0.248759 + 0.968565i $$0.580023\pi$$
$$102$$ 42.0000 4.15862
$$103$$ 17.0000 1.67506 0.837530 0.546392i $$-0.183999\pi$$
0.837530 + 0.546392i $$0.183999\pi$$
$$104$$ −14.6969 −1.44115
$$105$$ 14.6969 1.43427
$$106$$ −2.44949 −0.237915
$$107$$ 14.0000 1.35343 0.676716 0.736245i $$-0.263403\pi$$
0.676716 + 0.736245i $$0.263403\pi$$
$$108$$ 0 0
$$109$$ 13.0000 1.24517 0.622587 0.782551i $$-0.286082\pi$$
0.622587 + 0.782551i $$0.286082\pi$$
$$110$$ 6.00000 0.572078
$$111$$ 12.0000 1.13899
$$112$$ 9.79796 0.925820
$$113$$ 14.6969 1.38257 0.691286 0.722581i $$-0.257045\pi$$
0.691286 + 0.722581i $$0.257045\pi$$
$$114$$ −29.3939 −2.75299
$$115$$ −2.44949 −0.228416
$$116$$ −9.79796 −0.909718
$$117$$ −9.00000 −0.832050
$$118$$ −24.4949 −2.25494
$$119$$ −17.1464 −1.57181
$$120$$ 29.3939 2.68328
$$121$$ −10.0000 −0.909091
$$122$$ 18.0000 1.62964
$$123$$ 12.2474 1.10432
$$124$$ −12.0000 −1.07763
$$125$$ 9.79796 0.876356
$$126$$ 18.0000 1.60357
$$127$$ 1.00000 0.0887357 0.0443678 0.999015i $$-0.485873\pi$$
0.0443678 + 0.999015i $$0.485873\pi$$
$$128$$ −19.5959 −1.73205
$$129$$ 0 0
$$130$$ 18.0000 1.57870
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 9.79796 0.852803
$$133$$ 12.0000 1.04053
$$134$$ 22.0454 1.90443
$$135$$ 0 0
$$136$$ −34.2929 −2.94059
$$137$$ −4.89898 −0.418548 −0.209274 0.977857i $$-0.567110\pi$$
−0.209274 + 0.977857i $$0.567110\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ −24.0000 −2.02837
$$141$$ 24.4949 2.06284
$$142$$ 12.0000 1.00702
$$143$$ 3.00000 0.250873
$$144$$ 12.0000 1.00000
$$145$$ 6.00000 0.498273
$$146$$ −30.0000 −2.48282
$$147$$ 2.44949 0.202031
$$148$$ −19.5959 −1.61077
$$149$$ 4.89898 0.401340 0.200670 0.979659i $$-0.435688\pi$$
0.200670 + 0.979659i $$0.435688\pi$$
$$150$$ −6.00000 −0.489898
$$151$$ −17.1464 −1.39536 −0.697678 0.716411i $$-0.745783\pi$$
−0.697678 + 0.716411i $$0.745783\pi$$
$$152$$ 24.0000 1.94666
$$153$$ −21.0000 −1.69775
$$154$$ −6.00000 −0.483494
$$155$$ 7.34847 0.590243
$$156$$ 29.3939 2.35339
$$157$$ −9.79796 −0.781962 −0.390981 0.920399i $$-0.627864\pi$$
−0.390981 + 0.920399i $$0.627864\pi$$
$$158$$ −14.6969 −1.16923
$$159$$ 2.44949 0.194257
$$160$$ 0 0
$$161$$ 2.44949 0.193047
$$162$$ −22.0454 −1.73205
$$163$$ 2.44949 0.191859 0.0959294 0.995388i $$-0.469418\pi$$
0.0959294 + 0.995388i $$0.469418\pi$$
$$164$$ −20.0000 −1.56174
$$165$$ −6.00000 −0.467099
$$166$$ 2.44949 0.190117
$$167$$ 5.00000 0.386912 0.193456 0.981109i $$-0.438030\pi$$
0.193456 + 0.981109i $$0.438030\pi$$
$$168$$ −29.3939 −2.26779
$$169$$ −4.00000 −0.307692
$$170$$ 42.0000 3.22125
$$171$$ 14.6969 1.12390
$$172$$ 0 0
$$173$$ −10.0000 −0.760286 −0.380143 0.924928i $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ 14.6969 1.11417
$$175$$ 2.44949 0.185164
$$176$$ −4.00000 −0.301511
$$177$$ 24.4949 1.84115
$$178$$ −42.0000 −3.14803
$$179$$ −2.44949 −0.183083 −0.0915417 0.995801i $$-0.529179\pi$$
−0.0915417 + 0.995801i $$0.529179\pi$$
$$180$$ −29.3939 −2.19089
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ −18.0000 −1.33425
$$183$$ −18.0000 −1.33060
$$184$$ 4.89898 0.361158
$$185$$ 12.0000 0.882258
$$186$$ 18.0000 1.31982
$$187$$ 7.00000 0.511891
$$188$$ −40.0000 −2.91730
$$189$$ 0 0
$$190$$ −29.3939 −2.13246
$$191$$ 14.6969 1.06343 0.531717 0.846922i $$-0.321547\pi$$
0.531717 + 0.846922i $$0.321547\pi$$
$$192$$ 19.5959 1.41421
$$193$$ 15.0000 1.07972 0.539862 0.841754i $$-0.318476\pi$$
0.539862 + 0.841754i $$0.318476\pi$$
$$194$$ 26.9444 1.93449
$$195$$ −18.0000 −1.28901
$$196$$ −4.00000 −0.285714
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ −7.34847 −0.522233
$$199$$ −19.5959 −1.38912 −0.694559 0.719436i $$-0.744400\pi$$
−0.694559 + 0.719436i $$0.744400\pi$$
$$200$$ 4.89898 0.346410
$$201$$ −22.0454 −1.55496
$$202$$ −12.2474 −0.861727
$$203$$ −6.00000 −0.421117
$$204$$ 68.5857 4.80196
$$205$$ 12.2474 0.855399
$$206$$ 41.6413 2.90129
$$207$$ 3.00000 0.208514
$$208$$ −12.0000 −0.832050
$$209$$ −4.89898 −0.338869
$$210$$ 36.0000 2.48424
$$211$$ 14.6969 1.01178 0.505889 0.862598i $$-0.331164\pi$$
0.505889 + 0.862598i $$0.331164\pi$$
$$212$$ −4.00000 −0.274721
$$213$$ −12.0000 −0.822226
$$214$$ 34.2929 2.34421
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −7.34847 −0.498847
$$218$$ 31.8434 2.15670
$$219$$ 30.0000 2.02721
$$220$$ 9.79796 0.660578
$$221$$ 21.0000 1.41261
$$222$$ 29.3939 1.97279
$$223$$ 17.1464 1.14821 0.574105 0.818782i $$-0.305350\pi$$
0.574105 + 0.818782i $$0.305350\pi$$
$$224$$ 0 0
$$225$$ 3.00000 0.200000
$$226$$ 36.0000 2.39468
$$227$$ 9.79796 0.650313 0.325157 0.945660i $$-0.394583\pi$$
0.325157 + 0.945660i $$0.394583\pi$$
$$228$$ −48.0000 −3.17888
$$229$$ 7.00000 0.462573 0.231287 0.972886i $$-0.425707\pi$$
0.231287 + 0.972886i $$0.425707\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 6.00000 0.394771
$$232$$ −12.0000 −0.787839
$$233$$ 4.89898 0.320943 0.160471 0.987040i $$-0.448699\pi$$
0.160471 + 0.987040i $$0.448699\pi$$
$$234$$ −22.0454 −1.44115
$$235$$ 24.4949 1.59787
$$236$$ −40.0000 −2.60378
$$237$$ 14.6969 0.954669
$$238$$ −42.0000 −2.72246
$$239$$ −26.0000 −1.68180 −0.840900 0.541190i $$-0.817974\pi$$
−0.840900 + 0.541190i $$0.817974\pi$$
$$240$$ 24.0000 1.54919
$$241$$ 19.5959 1.26228 0.631142 0.775667i $$-0.282587\pi$$
0.631142 + 0.775667i $$0.282587\pi$$
$$242$$ −24.4949 −1.57459
$$243$$ 22.0454 1.41421
$$244$$ 29.3939 1.88175
$$245$$ 2.44949 0.156492
$$246$$ 30.0000 1.91273
$$247$$ −14.6969 −0.935144
$$248$$ −14.6969 −0.933257
$$249$$ −2.44949 −0.155230
$$250$$ 24.0000 1.51789
$$251$$ 5.00000 0.315597 0.157799 0.987471i $$-0.449560\pi$$
0.157799 + 0.987471i $$0.449560\pi$$
$$252$$ 29.3939 1.85164
$$253$$ −1.00000 −0.0628695
$$254$$ 2.44949 0.153695
$$255$$ −42.0000 −2.63014
$$256$$ −32.0000 −2.00000
$$257$$ 2.44949 0.152795 0.0763975 0.997077i $$-0.475658\pi$$
0.0763975 + 0.997077i $$0.475658\pi$$
$$258$$ 0 0
$$259$$ −12.0000 −0.745644
$$260$$ 29.3939 1.82293
$$261$$ −7.34847 −0.454859
$$262$$ 0 0
$$263$$ 14.6969 0.906252 0.453126 0.891446i $$-0.350309\pi$$
0.453126 + 0.891446i $$0.350309\pi$$
$$264$$ 12.0000 0.738549
$$265$$ 2.44949 0.150471
$$266$$ 29.3939 1.80225
$$267$$ 42.0000 2.57036
$$268$$ 36.0000 2.19905
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 0 0
$$271$$ 3.00000 0.182237 0.0911185 0.995840i $$-0.470956\pi$$
0.0911185 + 0.995840i $$0.470956\pi$$
$$272$$ −28.0000 −1.69775
$$273$$ 18.0000 1.08941
$$274$$ −12.0000 −0.724947
$$275$$ −1.00000 −0.0603023
$$276$$ −9.79796 −0.589768
$$277$$ −17.1464 −1.03023 −0.515115 0.857121i $$-0.672251\pi$$
−0.515115 + 0.857121i $$0.672251\pi$$
$$278$$ −17.1464 −1.02837
$$279$$ −9.00000 −0.538816
$$280$$ −29.3939 −1.75662
$$281$$ 1.00000 0.0596550 0.0298275 0.999555i $$-0.490504\pi$$
0.0298275 + 0.999555i $$0.490504\pi$$
$$282$$ 60.0000 3.57295
$$283$$ −21.0000 −1.24832 −0.624160 0.781296i $$-0.714559\pi$$
−0.624160 + 0.781296i $$0.714559\pi$$
$$284$$ 19.5959 1.16280
$$285$$ 29.3939 1.74114
$$286$$ 7.34847 0.434524
$$287$$ −12.2474 −0.722944
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 14.6969 0.863034
$$291$$ −26.9444 −1.57951
$$292$$ −48.9898 −2.86691
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 24.4949 1.42615
$$296$$ −24.0000 −1.39497
$$297$$ 0 0
$$298$$ 12.0000 0.695141
$$299$$ −3.00000 −0.173494
$$300$$ −9.79796 −0.565685
$$301$$ 0 0
$$302$$ −42.0000 −2.41683
$$303$$ 12.2474 0.703598
$$304$$ 19.5959 1.12390
$$305$$ −18.0000 −1.03068
$$306$$ −51.4393 −2.94059
$$307$$ 23.0000 1.31268 0.656340 0.754466i $$-0.272104\pi$$
0.656340 + 0.754466i $$0.272104\pi$$
$$308$$ −9.79796 −0.558291
$$309$$ −41.6413 −2.36889
$$310$$ 18.0000 1.02233
$$311$$ −25.0000 −1.41762 −0.708810 0.705399i $$-0.750768\pi$$
−0.708810 + 0.705399i $$0.750768\pi$$
$$312$$ 36.0000 2.03810
$$313$$ 7.34847 0.415360 0.207680 0.978197i $$-0.433409\pi$$
0.207680 + 0.978197i $$0.433409\pi$$
$$314$$ −24.0000 −1.35440
$$315$$ −18.0000 −1.01419
$$316$$ −24.0000 −1.35011
$$317$$ −17.0000 −0.954815 −0.477408 0.878682i $$-0.658423\pi$$
−0.477408 + 0.878682i $$0.658423\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 2.44949 0.137145
$$320$$ 19.5959 1.09545
$$321$$ −34.2929 −1.91404
$$322$$ 6.00000 0.334367
$$323$$ −34.2929 −1.90811
$$324$$ −36.0000 −2.00000
$$325$$ −3.00000 −0.166410
$$326$$ 6.00000 0.332309
$$327$$ −31.8434 −1.76094
$$328$$ −24.4949 −1.35250
$$329$$ −24.4949 −1.35045
$$330$$ −14.6969 −0.809040
$$331$$ −2.44949 −0.134636 −0.0673181 0.997732i $$-0.521444\pi$$
−0.0673181 + 0.997732i $$0.521444\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −14.6969 −0.805387
$$334$$ 12.2474 0.670151
$$335$$ −22.0454 −1.20447
$$336$$ −24.0000 −1.30931
$$337$$ −1.00000 −0.0544735 −0.0272367 0.999629i $$-0.508671\pi$$
−0.0272367 + 0.999629i $$0.508671\pi$$
$$338$$ −9.79796 −0.532939
$$339$$ −36.0000 −1.95525
$$340$$ 68.5857 3.71958
$$341$$ 3.00000 0.162459
$$342$$ 36.0000 1.94666
$$343$$ −19.5959 −1.05808
$$344$$ 0 0
$$345$$ 6.00000 0.323029
$$346$$ −24.4949 −1.31685
$$347$$ 17.1464 0.920468 0.460234 0.887798i $$-0.347765\pi$$
0.460234 + 0.887798i $$0.347765\pi$$
$$348$$ 24.0000 1.28654
$$349$$ −2.44949 −0.131118 −0.0655591 0.997849i $$-0.520883\pi$$
−0.0655591 + 0.997849i $$0.520883\pi$$
$$350$$ 6.00000 0.320713
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −29.0000 −1.54351 −0.771757 0.635917i $$-0.780622\pi$$
−0.771757 + 0.635917i $$0.780622\pi$$
$$354$$ 60.0000 3.18896
$$355$$ −12.0000 −0.636894
$$356$$ −68.5857 −3.63504
$$357$$ 42.0000 2.22288
$$358$$ −6.00000 −0.317110
$$359$$ −11.0000 −0.580558 −0.290279 0.956942i $$-0.593748\pi$$
−0.290279 + 0.956942i $$0.593748\pi$$
$$360$$ −36.0000 −1.89737
$$361$$ 5.00000 0.263158
$$362$$ 29.3939 1.54491
$$363$$ 24.4949 1.28565
$$364$$ −29.3939 −1.54066
$$365$$ 30.0000 1.57027
$$366$$ −44.0908 −2.30466
$$367$$ 6.00000 0.313197 0.156599 0.987662i $$-0.449947\pi$$
0.156599 + 0.987662i $$0.449947\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −15.0000 −0.780869
$$370$$ 29.3939 1.52811
$$371$$ −2.44949 −0.127171
$$372$$ 29.3939 1.52400
$$373$$ 34.2929 1.77562 0.887808 0.460213i $$-0.152227\pi$$
0.887808 + 0.460213i $$0.152227\pi$$
$$374$$ 17.1464 0.886621
$$375$$ −24.0000 −1.23935
$$376$$ −48.9898 −2.52646
$$377$$ 7.34847 0.378465
$$378$$ 0 0
$$379$$ −21.0000 −1.07870 −0.539349 0.842082i $$-0.681330\pi$$
−0.539349 + 0.842082i $$0.681330\pi$$
$$380$$ −48.0000 −2.46235
$$381$$ −2.44949 −0.125491
$$382$$ 36.0000 1.84192
$$383$$ −24.4949 −1.25163 −0.625815 0.779971i $$-0.715234\pi$$
−0.625815 + 0.779971i $$0.715234\pi$$
$$384$$ 48.0000 2.44949
$$385$$ 6.00000 0.305788
$$386$$ 36.7423 1.87014
$$387$$ 0 0
$$388$$ 44.0000 2.23376
$$389$$ −9.79796 −0.496776 −0.248388 0.968661i $$-0.579901\pi$$
−0.248388 + 0.968661i $$0.579901\pi$$
$$390$$ −44.0908 −2.23263
$$391$$ −7.00000 −0.354005
$$392$$ −4.89898 −0.247436
$$393$$ 0 0
$$394$$ 19.5959 0.987228
$$395$$ 14.6969 0.739483
$$396$$ −12.0000 −0.603023
$$397$$ −12.0000 −0.602263 −0.301131 0.953583i $$-0.597364\pi$$
−0.301131 + 0.953583i $$0.597364\pi$$
$$398$$ −48.0000 −2.40602
$$399$$ −29.3939 −1.47153
$$400$$ 4.00000 0.200000
$$401$$ −5.00000 −0.249688 −0.124844 0.992176i $$-0.539843\pi$$
−0.124844 + 0.992176i $$0.539843\pi$$
$$402$$ −54.0000 −2.69328
$$403$$ 9.00000 0.448322
$$404$$ −20.0000 −0.995037
$$405$$ 22.0454 1.09545
$$406$$ −14.6969 −0.729397
$$407$$ 4.89898 0.242833
$$408$$ 84.0000 4.15862
$$409$$ 36.7423 1.81679 0.908396 0.418111i $$-0.137308\pi$$
0.908396 + 0.418111i $$0.137308\pi$$
$$410$$ 30.0000 1.48159
$$411$$ 12.0000 0.591916
$$412$$ 68.0000 3.35012
$$413$$ −24.4949 −1.20532
$$414$$ 7.34847 0.361158
$$415$$ −2.44949 −0.120241
$$416$$ 0 0
$$417$$ 17.1464 0.839664
$$418$$ −12.0000 −0.586939
$$419$$ −2.44949 −0.119665 −0.0598327 0.998208i $$-0.519057\pi$$
−0.0598327 + 0.998208i $$0.519057\pi$$
$$420$$ 58.7878 2.86855
$$421$$ −9.79796 −0.477523 −0.238762 0.971078i $$-0.576741\pi$$
−0.238762 + 0.971078i $$0.576741\pi$$
$$422$$ 36.0000 1.75245
$$423$$ −30.0000 −1.45865
$$424$$ −4.89898 −0.237915
$$425$$ −7.00000 −0.339550
$$426$$ −29.3939 −1.42414
$$427$$ 18.0000 0.871081
$$428$$ 56.0000 2.70686
$$429$$ −7.34847 −0.354787
$$430$$ 0 0
$$431$$ −17.0000 −0.818861 −0.409431 0.912341i $$-0.634273\pi$$
−0.409431 + 0.912341i $$0.634273\pi$$
$$432$$ 0 0
$$433$$ −12.2474 −0.588575 −0.294287 0.955717i $$-0.595082\pi$$
−0.294287 + 0.955717i $$0.595082\pi$$
$$434$$ −18.0000 −0.864028
$$435$$ −14.6969 −0.704664
$$436$$ 52.0000 2.49035
$$437$$ 4.89898 0.234350
$$438$$ 73.4847 3.51123
$$439$$ −29.0000 −1.38409 −0.692047 0.721852i $$-0.743291\pi$$
−0.692047 + 0.721852i $$0.743291\pi$$
$$440$$ 12.0000 0.572078
$$441$$ −3.00000 −0.142857
$$442$$ 51.4393 2.44672
$$443$$ −34.0000 −1.61539 −0.807694 0.589601i $$-0.799285\pi$$
−0.807694 + 0.589601i $$0.799285\pi$$
$$444$$ 48.0000 2.27798
$$445$$ 42.0000 1.99099
$$446$$ 42.0000 1.98876
$$447$$ −12.0000 −0.567581
$$448$$ −19.5959 −0.925820
$$449$$ −26.9444 −1.27158 −0.635792 0.771860i $$-0.719326\pi$$
−0.635792 + 0.771860i $$0.719326\pi$$
$$450$$ 7.34847 0.346410
$$451$$ 5.00000 0.235441
$$452$$ 58.7878 2.76514
$$453$$ 42.0000 1.97333
$$454$$ 24.0000 1.12638
$$455$$ 18.0000 0.843853
$$456$$ −58.7878 −2.75299
$$457$$ −31.8434 −1.48957 −0.744785 0.667305i $$-0.767448\pi$$
−0.744785 + 0.667305i $$0.767448\pi$$
$$458$$ 17.1464 0.801200
$$459$$ 0 0
$$460$$ −9.79796 −0.456832
$$461$$ 16.0000 0.745194 0.372597 0.927993i $$-0.378467\pi$$
0.372597 + 0.927993i $$0.378467\pi$$
$$462$$ 14.6969 0.683763
$$463$$ 12.2474 0.569187 0.284594 0.958648i $$-0.408141\pi$$
0.284594 + 0.958648i $$0.408141\pi$$
$$464$$ −9.79796 −0.454859
$$465$$ −18.0000 −0.834730
$$466$$ 12.0000 0.555889
$$467$$ −19.5959 −0.906791 −0.453395 0.891309i $$-0.649787\pi$$
−0.453395 + 0.891309i $$0.649787\pi$$
$$468$$ −36.0000 −1.66410
$$469$$ 22.0454 1.01796
$$470$$ 60.0000 2.76759
$$471$$ 24.0000 1.10586
$$472$$ −48.9898 −2.25494
$$473$$ 0 0
$$474$$ 36.0000 1.65353
$$475$$ 4.89898 0.224781
$$476$$ −68.5857 −3.14362
$$477$$ −3.00000 −0.137361
$$478$$ −63.6867 −2.91296
$$479$$ −1.00000 −0.0456912 −0.0228456 0.999739i $$-0.507273\pi$$
−0.0228456 + 0.999739i $$0.507273\pi$$
$$480$$ 0 0
$$481$$ 14.6969 0.670123
$$482$$ 48.0000 2.18634
$$483$$ −6.00000 −0.273009
$$484$$ −40.0000 −1.81818
$$485$$ −26.9444 −1.22348
$$486$$ 54.0000 2.44949
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ 36.0000 1.62964
$$489$$ −6.00000 −0.271329
$$490$$ 6.00000 0.271052
$$491$$ −22.0454 −0.994895 −0.497448 0.867494i $$-0.665729\pi$$
−0.497448 + 0.867494i $$0.665729\pi$$
$$492$$ 48.9898 2.20863
$$493$$ 17.1464 0.772236
$$494$$ −36.0000 −1.61972
$$495$$ 7.34847 0.330289
$$496$$ −12.0000 −0.538816
$$497$$ 12.0000 0.538274
$$498$$ −6.00000 −0.268866
$$499$$ −7.34847 −0.328963 −0.164481 0.986380i $$-0.552595\pi$$
−0.164481 + 0.986380i $$0.552595\pi$$
$$500$$ 39.1918 1.75271
$$501$$ −12.2474 −0.547176
$$502$$ 12.2474 0.546630
$$503$$ 14.6969 0.655304 0.327652 0.944798i $$-0.393743\pi$$
0.327652 + 0.944798i $$0.393743\pi$$
$$504$$ 36.0000 1.60357
$$505$$ 12.2474 0.545004
$$506$$ −2.44949 −0.108893
$$507$$ 9.79796 0.435143
$$508$$ 4.00000 0.177471
$$509$$ −19.0000 −0.842160 −0.421080 0.907023i $$-0.638349\pi$$
−0.421080 + 0.907023i $$0.638349\pi$$
$$510$$ −102.879 −4.55554
$$511$$ −30.0000 −1.32712
$$512$$ −39.1918 −1.73205
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ −41.6413 −1.83494
$$516$$ 0 0
$$517$$ 10.0000 0.439799
$$518$$ −29.3939 −1.29149
$$519$$ 24.4949 1.07521
$$520$$ 36.0000 1.57870
$$521$$ 22.0454 0.965827 0.482913 0.875668i $$-0.339579\pi$$
0.482913 + 0.875668i $$0.339579\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ −2.44949 −0.107109 −0.0535544 0.998565i $$-0.517055\pi$$
−0.0535544 + 0.998565i $$0.517055\pi$$
$$524$$ 0 0
$$525$$ −6.00000 −0.261861
$$526$$ 36.0000 1.56967
$$527$$ 21.0000 0.914774
$$528$$ 9.79796 0.426401
$$529$$ −22.0000 −0.956522
$$530$$ 6.00000 0.260623
$$531$$ −30.0000 −1.30189
$$532$$ 48.0000 2.08106
$$533$$ 15.0000 0.649722
$$534$$ 102.879 4.45199
$$535$$ −34.2929 −1.48261
$$536$$ 44.0908 1.90443
$$537$$ 6.00000 0.258919
$$538$$ 12.2474 0.528025
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 33.0000 1.41878 0.709390 0.704816i $$-0.248970\pi$$
0.709390 + 0.704816i $$0.248970\pi$$
$$542$$ 7.34847 0.315644
$$543$$ −29.3939 −1.26141
$$544$$ 0 0
$$545$$ −31.8434 −1.36402
$$546$$ 44.0908 1.88691
$$547$$ −27.0000 −1.15444 −0.577218 0.816590i $$-0.695862\pi$$
−0.577218 + 0.816590i $$0.695862\pi$$
$$548$$ −19.5959 −0.837096
$$549$$ 22.0454 0.940875
$$550$$ −2.44949 −0.104447
$$551$$ −12.0000 −0.511217
$$552$$ −12.0000 −0.510754
$$553$$ −14.6969 −0.624977
$$554$$ −42.0000 −1.78441
$$555$$ −29.3939 −1.24770
$$556$$ −28.0000 −1.18746
$$557$$ −41.0000 −1.73723 −0.868613 0.495491i $$-0.834988\pi$$
−0.868613 + 0.495491i $$0.834988\pi$$
$$558$$ −22.0454 −0.933257
$$559$$ 0 0
$$560$$ −24.0000 −1.01419
$$561$$ −17.1464 −0.723923
$$562$$ 2.44949 0.103325
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 97.9796 4.12568
$$565$$ −36.0000 −1.51453
$$566$$ −51.4393 −2.16215
$$567$$ −22.0454 −0.925820
$$568$$ 24.0000 1.00702
$$569$$ 23.0000 0.964210 0.482105 0.876113i $$-0.339872\pi$$
0.482105 + 0.876113i $$0.339872\pi$$
$$570$$ 72.0000 3.01575
$$571$$ −7.34847 −0.307524 −0.153762 0.988108i $$-0.549139\pi$$
−0.153762 + 0.988108i $$0.549139\pi$$
$$572$$ 12.0000 0.501745
$$573$$ −36.0000 −1.50392
$$574$$ −30.0000 −1.25218
$$575$$ 1.00000 0.0417029
$$576$$ −24.0000 −1.00000
$$577$$ −12.2474 −0.509868 −0.254934 0.966958i $$-0.582054\pi$$
−0.254934 + 0.966958i $$0.582054\pi$$
$$578$$ 78.3837 3.26033
$$579$$ −36.7423 −1.52696
$$580$$ 24.0000 0.996546
$$581$$ 2.44949 0.101622
$$582$$ −66.0000 −2.73579
$$583$$ 1.00000 0.0414158
$$584$$ −60.0000 −2.48282
$$585$$ 22.0454 0.911465
$$586$$ 39.1918 1.61900
$$587$$ 24.4949 1.01101 0.505506 0.862823i $$-0.331306\pi$$
0.505506 + 0.862823i $$0.331306\pi$$
$$588$$ 9.79796 0.404061
$$589$$ −14.6969 −0.605577
$$590$$ 60.0000 2.47016
$$591$$ −19.5959 −0.806068
$$592$$ −19.5959 −0.805387
$$593$$ −7.34847 −0.301765 −0.150883 0.988552i $$-0.548212\pi$$
−0.150883 + 0.988552i $$0.548212\pi$$
$$594$$ 0 0
$$595$$ 42.0000 1.72183
$$596$$ 19.5959 0.802680
$$597$$ 48.0000 1.96451
$$598$$ −7.34847 −0.300501
$$599$$ −17.0000 −0.694601 −0.347301 0.937754i $$-0.612902\pi$$
−0.347301 + 0.937754i $$0.612902\pi$$
$$600$$ −12.0000 −0.489898
$$601$$ −19.5959 −0.799334 −0.399667 0.916660i $$-0.630874\pi$$
−0.399667 + 0.916660i $$0.630874\pi$$
$$602$$ 0 0
$$603$$ 27.0000 1.09952
$$604$$ −68.5857 −2.79071
$$605$$ 24.4949 0.995859
$$606$$ 30.0000 1.21867
$$607$$ −39.1918 −1.59075 −0.795374 0.606119i $$-0.792725\pi$$
−0.795374 + 0.606119i $$0.792725\pi$$
$$608$$ 0 0
$$609$$ 14.6969 0.595550
$$610$$ −44.0908 −1.78518
$$611$$ 30.0000 1.21367
$$612$$ −84.0000 −3.39550
$$613$$ 12.0000 0.484675 0.242338 0.970192i $$-0.422086\pi$$
0.242338 + 0.970192i $$0.422086\pi$$
$$614$$ 56.3383 2.27363
$$615$$ −30.0000 −1.20972
$$616$$ −12.0000 −0.483494
$$617$$ 43.0000 1.73111 0.865557 0.500810i $$-0.166964\pi$$
0.865557 + 0.500810i $$0.166964\pi$$
$$618$$ −102.000 −4.10304
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ 29.3939 1.18049
$$621$$ 0 0
$$622$$ −61.2372 −2.45539
$$623$$ −42.0000 −1.68269
$$624$$ 29.3939 1.17670
$$625$$ −29.0000 −1.16000
$$626$$ 18.0000 0.719425
$$627$$ 12.0000 0.479234
$$628$$ −39.1918 −1.56392
$$629$$ 34.2929 1.36735
$$630$$ −44.0908 −1.75662
$$631$$ 14.6969 0.585076 0.292538 0.956254i $$-0.405500\pi$$
0.292538 + 0.956254i $$0.405500\pi$$
$$632$$ −29.3939 −1.16923
$$633$$ −36.0000 −1.43087
$$634$$ −41.6413 −1.65379
$$635$$ −2.44949 −0.0972050
$$636$$ 9.79796 0.388514
$$637$$ 3.00000 0.118864
$$638$$ 6.00000 0.237542
$$639$$ 14.6969 0.581402
$$640$$ 48.0000 1.89737
$$641$$ −22.0454 −0.870741 −0.435371 0.900251i $$-0.643383\pi$$
−0.435371 + 0.900251i $$0.643383\pi$$
$$642$$ −84.0000 −3.31522
$$643$$ −10.0000 −0.394362 −0.197181 0.980367i $$-0.563179\pi$$
−0.197181 + 0.980367i $$0.563179\pi$$
$$644$$ 9.79796 0.386094
$$645$$ 0 0
$$646$$ −84.0000 −3.30494
$$647$$ 44.0908 1.73339 0.866694 0.498839i $$-0.166240\pi$$
0.866694 + 0.498839i $$0.166240\pi$$
$$648$$ −44.0908 −1.73205
$$649$$ 10.0000 0.392534
$$650$$ −7.34847 −0.288231
$$651$$ 18.0000 0.705476
$$652$$ 9.79796 0.383718
$$653$$ −34.2929 −1.34198 −0.670992 0.741465i $$-0.734131\pi$$
−0.670992 + 0.741465i $$0.734131\pi$$
$$654$$ −78.0000 −3.05004
$$655$$ 0 0
$$656$$ −20.0000 −0.780869
$$657$$ −36.7423 −1.43346
$$658$$ −60.0000 −2.33904
$$659$$ 1.00000 0.0389545 0.0194772 0.999810i $$-0.493800\pi$$
0.0194772 + 0.999810i $$0.493800\pi$$
$$660$$ −24.0000 −0.934199
$$661$$ −45.0000 −1.75030 −0.875149 0.483854i $$-0.839236\pi$$
−0.875149 + 0.483854i $$0.839236\pi$$
$$662$$ −6.00000 −0.233197
$$663$$ −51.4393 −1.99774
$$664$$ 4.89898 0.190117
$$665$$ −29.3939 −1.13985
$$666$$ −36.0000 −1.39497
$$667$$ −2.44949 −0.0948446
$$668$$ 20.0000 0.773823
$$669$$ −42.0000 −1.62381
$$670$$ −54.0000 −2.08620
$$671$$ −7.34847 −0.283685
$$672$$ 0 0
$$673$$ −26.9444 −1.03863 −0.519315 0.854583i $$-0.673813\pi$$
−0.519315 + 0.854583i $$0.673813\pi$$
$$674$$ −2.44949 −0.0943508
$$675$$ 0 0
$$676$$ −16.0000 −0.615385
$$677$$ 24.4949 0.941415 0.470708 0.882289i $$-0.343999\pi$$
0.470708 + 0.882289i $$0.343999\pi$$
$$678$$ −88.1816 −3.38660
$$679$$ 26.9444 1.03403
$$680$$ 84.0000 3.22125
$$681$$ −24.0000 −0.919682
$$682$$ 7.34847 0.281387
$$683$$ 23.0000 0.880071 0.440035 0.897980i $$-0.354966\pi$$
0.440035 + 0.897980i $$0.354966\pi$$
$$684$$ 58.7878 2.24781
$$685$$ 12.0000 0.458496
$$686$$ −48.0000 −1.83265
$$687$$ −17.1464 −0.654177
$$688$$ 0 0
$$689$$ 3.00000 0.114291
$$690$$ 14.6969 0.559503
$$691$$ −26.9444 −1.02501 −0.512506 0.858683i $$-0.671283\pi$$
−0.512506 + 0.858683i $$0.671283\pi$$
$$692$$ −40.0000 −1.52057
$$693$$ −7.34847 −0.279145
$$694$$ 42.0000 1.59430
$$695$$ 17.1464 0.650401
$$696$$ 29.3939 1.11417
$$697$$ 35.0000 1.32572
$$698$$ −6.00000 −0.227103
$$699$$ −12.0000 −0.453882
$$700$$ 9.79796 0.370328
$$701$$ −8.00000 −0.302156 −0.151078 0.988522i $$-0.548274\pi$$
−0.151078 + 0.988522i $$0.548274\pi$$
$$702$$ 0 0
$$703$$ −24.0000 −0.905177
$$704$$ 8.00000 0.301511
$$705$$ −60.0000 −2.25973
$$706$$ −71.0352 −2.67345
$$707$$ −12.2474 −0.460613
$$708$$ 97.9796 3.68230
$$709$$ −15.0000 −0.563337 −0.281668 0.959512i $$-0.590888\pi$$
−0.281668 + 0.959512i $$0.590888\pi$$
$$710$$ −29.3939 −1.10313
$$711$$ −18.0000 −0.675053
$$712$$ −84.0000 −3.14803
$$713$$ −3.00000 −0.112351
$$714$$ 102.879 3.85013
$$715$$ −7.34847 −0.274817
$$716$$ −9.79796 −0.366167
$$717$$ 63.6867 2.37842
$$718$$ −26.9444 −1.00556
$$719$$ −4.00000 −0.149175 −0.0745874 0.997214i $$-0.523764\pi$$
−0.0745874 + 0.997214i $$0.523764\pi$$
$$720$$ −29.3939 −1.09545
$$721$$ 41.6413 1.55080
$$722$$ 12.2474 0.455803
$$723$$ −48.0000 −1.78514
$$724$$ 48.0000 1.78391
$$725$$ −2.44949 −0.0909718
$$726$$ 60.0000 2.22681
$$727$$ −29.3939 −1.09016 −0.545079 0.838385i $$-0.683500\pi$$
−0.545079 + 0.838385i $$0.683500\pi$$
$$728$$ −36.0000 −1.33425
$$729$$ −27.0000 −1.00000
$$730$$ 73.4847 2.71979
$$731$$ 0 0
$$732$$ −72.0000 −2.66120
$$733$$ 48.9898 1.80948 0.904740 0.425965i $$-0.140065\pi$$
0.904740 + 0.425965i $$0.140065\pi$$
$$734$$ 14.6969 0.542474
$$735$$ −6.00000 −0.221313
$$736$$ 0 0
$$737$$ −9.00000 −0.331519
$$738$$ −36.7423 −1.35250
$$739$$ 4.89898 0.180212 0.0901059 0.995932i $$-0.471279\pi$$
0.0901059 + 0.995932i $$0.471279\pi$$
$$740$$ 48.0000 1.76452
$$741$$ 36.0000 1.32249
$$742$$ −6.00000 −0.220267
$$743$$ 14.6969 0.539178 0.269589 0.962975i $$-0.413112\pi$$
0.269589 + 0.962975i $$0.413112\pi$$
$$744$$ 36.0000 1.31982
$$745$$ −12.0000 −0.439646
$$746$$ 84.0000 3.07546
$$747$$ 3.00000 0.109764
$$748$$ 28.0000 1.02378
$$749$$ 34.2929 1.25303
$$750$$ −58.7878 −2.14663
$$751$$ 51.4393 1.87705 0.938523 0.345217i $$-0.112195\pi$$
0.938523 + 0.345217i $$0.112195\pi$$
$$752$$ −40.0000 −1.45865
$$753$$ −12.2474 −0.446322
$$754$$ 18.0000 0.655521
$$755$$ 42.0000 1.52854
$$756$$ 0 0
$$757$$ 4.89898 0.178056 0.0890282 0.996029i $$-0.471624\pi$$
0.0890282 + 0.996029i $$0.471624\pi$$
$$758$$ −51.4393 −1.86836
$$759$$ 2.44949 0.0889108
$$760$$ −58.7878 −2.13246
$$761$$ −4.89898 −0.177588 −0.0887939 0.996050i $$-0.528301\pi$$
−0.0887939 + 0.996050i $$0.528301\pi$$
$$762$$ −6.00000 −0.217357
$$763$$ 31.8434 1.15281
$$764$$ 58.7878 2.12687
$$765$$ 51.4393 1.85979
$$766$$ −60.0000 −2.16789
$$767$$ 30.0000 1.08324
$$768$$ 78.3837 2.82843
$$769$$ 24.0000 0.865462 0.432731 0.901523i $$-0.357550\pi$$
0.432731 + 0.901523i $$0.357550\pi$$
$$770$$ 14.6969 0.529641
$$771$$ −6.00000 −0.216085
$$772$$ 60.0000 2.15945
$$773$$ 26.9444 0.969122 0.484561 0.874757i $$-0.338979\pi$$
0.484561 + 0.874757i $$0.338979\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 53.8888 1.93449
$$777$$ 29.3939 1.05450
$$778$$ −24.0000 −0.860442
$$779$$ −24.4949 −0.877621
$$780$$ −72.0000 −2.57801
$$781$$ −4.89898 −0.175299
$$782$$ −17.1464 −0.613155
$$783$$ 0 0
$$784$$ −4.00000 −0.142857
$$785$$ 24.0000 0.856597
$$786$$ 0 0
$$787$$ 38.0000 1.35455 0.677277 0.735728i $$-0.263160\pi$$
0.677277 + 0.735728i $$0.263160\pi$$
$$788$$ 32.0000 1.13995
$$789$$ −36.0000 −1.28163
$$790$$ 36.0000 1.28082
$$791$$ 36.0000 1.28001
$$792$$ −14.6969 −0.522233
$$793$$ −22.0454 −0.782855
$$794$$ −29.3939 −1.04315
$$795$$ −6.00000 −0.212798
$$796$$ −78.3837 −2.77824
$$797$$ −28.0000 −0.991811 −0.495905 0.868377i $$-0.665164\pi$$
−0.495905 + 0.868377i $$0.665164\pi$$
$$798$$ −72.0000 −2.54877
$$799$$ 70.0000 2.47642
$$800$$ 0 0
$$801$$ −51.4393 −1.81752
$$802$$ −12.2474 −0.432472
$$803$$ 12.2474 0.432203
$$804$$ −88.1816 −3.10993
$$805$$ −6.00000 −0.211472
$$806$$ 22.0454 0.776516
$$807$$ −12.2474 −0.431131
$$808$$ −24.4949 −0.861727
$$809$$ 28.0000 0.984428 0.492214 0.870474i $$-0.336188\pi$$
0.492214 + 0.870474i $$0.336188\pi$$
$$810$$ 54.0000 1.89737
$$811$$ −7.34847 −0.258040 −0.129020 0.991642i $$-0.541183\pi$$
−0.129020 + 0.991642i $$0.541183\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ −7.34847 −0.257722
$$814$$ 12.0000 0.420600
$$815$$ −6.00000 −0.210171
$$816$$ 68.5857 2.40098
$$817$$ 0 0
$$818$$ 90.0000 3.14678
$$819$$ −22.0454 −0.770329
$$820$$ 48.9898 1.71080
$$821$$ 31.0000 1.08191 0.540954 0.841052i $$-0.318063\pi$$
0.540954 + 0.841052i $$0.318063\pi$$
$$822$$ 29.3939 1.02523
$$823$$ −9.00000 −0.313720 −0.156860 0.987621i $$-0.550137\pi$$
−0.156860 + 0.987621i $$0.550137\pi$$
$$824$$ 83.2827 2.90129
$$825$$ 2.44949 0.0852803
$$826$$ −60.0000 −2.08767
$$827$$ −32.0000 −1.11275 −0.556375 0.830932i $$-0.687808\pi$$
−0.556375 + 0.830932i $$0.687808\pi$$
$$828$$ 12.0000 0.417029
$$829$$ −44.0908 −1.53134 −0.765669 0.643235i $$-0.777592\pi$$
−0.765669 + 0.643235i $$0.777592\pi$$
$$830$$ −6.00000 −0.208263
$$831$$ 42.0000 1.45696
$$832$$ 24.0000 0.832050
$$833$$ 7.00000 0.242536
$$834$$ 42.0000 1.45434
$$835$$ −12.2474 −0.423840
$$836$$ −19.5959 −0.677739
$$837$$ 0 0
$$838$$ −6.00000 −0.207267
$$839$$ 26.9444 0.930224 0.465112 0.885252i $$-0.346014\pi$$
0.465112 + 0.885252i $$0.346014\pi$$
$$840$$ 72.0000 2.48424
$$841$$ −23.0000 −0.793103
$$842$$ −24.0000 −0.827095
$$843$$ −2.44949 −0.0843649
$$844$$ 58.7878 2.02356
$$845$$ 9.79796 0.337060
$$846$$ −73.4847 −2.52646
$$847$$ −24.4949 −0.841655
$$848$$ −4.00000 −0.137361
$$849$$ 51.4393 1.76539
$$850$$ −17.1464 −0.588118
$$851$$ −4.89898 −0.167935
$$852$$ −48.0000 −1.64445
$$853$$ −5.00000 −0.171197 −0.0855984 0.996330i $$-0.527280\pi$$
−0.0855984 + 0.996330i $$0.527280\pi$$
$$854$$ 44.0908 1.50876
$$855$$ −36.0000 −1.23117
$$856$$ 68.5857 2.34421
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ −18.0000 −0.614510
$$859$$ 48.9898 1.67151 0.835755 0.549102i $$-0.185030\pi$$
0.835755 + 0.549102i $$0.185030\pi$$
$$860$$ 0 0
$$861$$ 30.0000 1.02240
$$862$$ −41.6413 −1.41831
$$863$$ 9.79796 0.333526 0.166763 0.985997i $$-0.446668\pi$$
0.166763 + 0.985997i $$0.446668\pi$$
$$864$$ 0 0
$$865$$ 24.4949 0.832851
$$866$$ −30.0000 −1.01944
$$867$$ −78.3837 −2.66205
$$868$$ −29.3939 −0.997693
$$869$$ 6.00000 0.203536
$$870$$ −36.0000 −1.22051
$$871$$ −27.0000 −0.914860
$$872$$ 63.6867 2.15670
$$873$$ 33.0000 1.11688
$$874$$ 12.0000 0.405906
$$875$$ 24.0000 0.811348
$$876$$ 120.000 4.05442
$$877$$ −55.0000 −1.85722 −0.928609 0.371060i $$-0.878995\pi$$
−0.928609 + 0.371060i $$0.878995\pi$$
$$878$$ −71.0352 −2.39732
$$879$$ −39.1918 −1.32191
$$880$$ 9.79796 0.330289
$$881$$ 7.00000 0.235836 0.117918 0.993023i $$-0.462378\pi$$
0.117918 + 0.993023i $$0.462378\pi$$
$$882$$ −7.34847 −0.247436
$$883$$ 31.0000 1.04323 0.521617 0.853180i $$-0.325329\pi$$
0.521617 + 0.853180i $$0.325329\pi$$
$$884$$ 84.0000 2.82523
$$885$$ −60.0000 −2.01688
$$886$$ −83.2827 −2.79794
$$887$$ 29.3939 0.986950 0.493475 0.869760i $$-0.335726\pi$$
0.493475 + 0.869760i $$0.335726\pi$$
$$888$$ 58.7878 1.97279
$$889$$ 2.44949 0.0821532
$$890$$ 102.879 3.44850
$$891$$ 9.00000 0.301511
$$892$$ 68.5857 2.29642
$$893$$ −48.9898 −1.63938
$$894$$ −29.3939 −0.983078
$$895$$ 6.00000 0.200558
$$896$$ −48.0000 −1.60357
$$897$$ 7.34847 0.245358
$$898$$ −66.0000 −2.20245
$$899$$ 7.34847 0.245085
$$900$$ 12.0000 0.400000
$$901$$ 7.00000 0.233204
$$902$$ 12.2474 0.407795
$$903$$ 0 0
$$904$$ 72.0000 2.39468
$$905$$ −29.3939 −0.977086
$$906$$ 102.879 3.41791
$$907$$ −27.0000 −0.896520 −0.448260 0.893903i $$-0.647956\pi$$
−0.448260 + 0.893903i $$0.647956\pi$$
$$908$$ 39.1918 1.30063
$$909$$ −15.0000 −0.497519
$$910$$ 44.0908 1.46160
$$911$$ −22.0454 −0.730397 −0.365198 0.930930i $$-0.618999\pi$$
−0.365198 + 0.930930i $$0.618999\pi$$
$$912$$ −48.0000 −1.58944
$$913$$ −1.00000 −0.0330952
$$914$$ −78.0000 −2.58001
$$915$$ 44.0908 1.45760
$$916$$ 28.0000 0.925146
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 3.00000 0.0989609 0.0494804 0.998775i $$-0.484243\pi$$
0.0494804 + 0.998775i $$0.484243\pi$$
$$920$$ −12.0000 −0.395628
$$921$$ −56.3383 −1.85641
$$922$$ 39.1918 1.29071
$$923$$ −14.6969 −0.483756
$$924$$ 24.0000 0.789542
$$925$$ −4.89898 −0.161077
$$926$$ 30.0000 0.985861
$$927$$ 51.0000 1.67506
$$928$$ 0 0
$$929$$ 53.8888 1.76803 0.884017 0.467455i $$-0.154829\pi$$
0.884017 + 0.467455i $$0.154829\pi$$
$$930$$ −44.0908 −1.44579
$$931$$ −4.89898 −0.160558
$$932$$ 19.5959 0.641886
$$933$$ 61.2372 2.00482
$$934$$ −48.0000 −1.57061
$$935$$ −17.1464 −0.560748
$$936$$ −44.0908 −1.44115
$$937$$ 2.44949 0.0800213 0.0400107 0.999199i $$-0.487261\pi$$
0.0400107 + 0.999199i $$0.487261\pi$$
$$938$$ 54.0000 1.76316
$$939$$ −18.0000 −0.587408
$$940$$ 97.9796 3.19574
$$941$$ −29.0000 −0.945373 −0.472686 0.881231i $$-0.656716\pi$$
−0.472686 + 0.881231i $$0.656716\pi$$
$$942$$ 58.7878 1.91541
$$943$$ −5.00000 −0.162822
$$944$$ −40.0000 −1.30189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 25.0000 0.812391 0.406195 0.913786i $$-0.366855\pi$$
0.406195 + 0.913786i $$0.366855\pi$$
$$948$$ 58.7878 1.90934
$$949$$ 36.7423 1.19271
$$950$$ 12.0000 0.389331
$$951$$ 41.6413 1.35031
$$952$$ −84.0000 −2.72246
$$953$$ −56.3383 −1.82498 −0.912488 0.409104i $$-0.865841\pi$$
−0.912488 + 0.409104i $$0.865841\pi$$
$$954$$ −7.34847 −0.237915
$$955$$ −36.0000 −1.16493
$$956$$ −104.000 −3.36360
$$957$$ −6.00000 −0.193952
$$958$$ −2.44949 −0.0791394
$$959$$ −12.0000 −0.387500
$$960$$ −48.0000 −1.54919
$$961$$ −22.0000 −0.709677
$$962$$ 36.0000 1.16069
$$963$$ 42.0000 1.35343
$$964$$ 78.3837 2.52457
$$965$$ −36.7423 −1.18278
$$966$$ −14.6969 −0.472866
$$967$$ 49.0000 1.57573 0.787867 0.615846i $$-0.211185\pi$$
0.787867 + 0.615846i $$0.211185\pi$$
$$968$$ −48.9898 −1.57459
$$969$$ 84.0000 2.69847
$$970$$ −66.0000 −2.11913
$$971$$ −41.0000 −1.31575 −0.657876 0.753126i $$-0.728545\pi$$
−0.657876 + 0.753126i $$0.728545\pi$$
$$972$$ 88.1816 2.82843
$$973$$ −17.1464 −0.549689
$$974$$ −9.79796 −0.313947
$$975$$ 7.34847 0.235339
$$976$$ 29.3939 0.940875
$$977$$ 28.0000 0.895799 0.447900 0.894084i $$-0.352172\pi$$
0.447900 + 0.894084i $$0.352172\pi$$
$$978$$ −14.6969 −0.469956
$$979$$ 17.1464 0.548002
$$980$$ 9.79796 0.312984
$$981$$ 39.0000 1.24517
$$982$$ −54.0000 −1.72321
$$983$$ −12.2474 −0.390633 −0.195316 0.980740i $$-0.562573\pi$$
−0.195316 + 0.980740i $$0.562573\pi$$
$$984$$ 60.0000 1.91273
$$985$$ −19.5959 −0.624378
$$986$$ 42.0000 1.33755
$$987$$ 60.0000 1.90982
$$988$$ −58.7878 −1.87029
$$989$$ 0 0
$$990$$ 18.0000 0.572078
$$991$$ 31.8434 1.01154 0.505769 0.862669i $$-0.331209\pi$$
0.505769 + 0.862669i $$0.331209\pi$$
$$992$$ 0 0
$$993$$ 6.00000 0.190404
$$994$$ 29.3939 0.932317
$$995$$ 48.0000 1.52170
$$996$$ −9.79796 −0.310460
$$997$$ 31.8434 1.00849 0.504245 0.863561i $$-0.331771\pi$$
0.504245 + 0.863561i $$0.331771\pi$$
$$998$$ −18.0000 −0.569780
$$999$$ 0 0
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 1849.2.a.g.1.2 yes 2
43.42 odd 2 inner 1849.2.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1849.2.a.g.1.1 2 43.42 odd 2 inner
1849.2.a.g.1.2 yes 2 1.1 even 1 trivial