# Properties

 Label 1850.2.a.t.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} -1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} -1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{11} -0.618034 q^{12} -3.85410 q^{13} +1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.61803 q^{18} -4.47214 q^{19} +0.763932 q^{21} +3.61803 q^{22} +3.85410 q^{23} +0.618034 q^{24} +3.85410 q^{26} +3.47214 q^{27} -1.23607 q^{28} +6.32624 q^{29} +9.61803 q^{31} -1.00000 q^{32} +2.23607 q^{33} +4.47214 q^{34} -2.61803 q^{36} +1.00000 q^{37} +4.47214 q^{38} +2.38197 q^{39} +7.38197 q^{41} -0.763932 q^{42} +0.763932 q^{43} -3.61803 q^{44} -3.85410 q^{46} -3.23607 q^{47} -0.618034 q^{48} -5.47214 q^{49} +2.76393 q^{51} -3.85410 q^{52} +8.47214 q^{53} -3.47214 q^{54} +1.23607 q^{56} +2.76393 q^{57} -6.32624 q^{58} -9.23607 q^{59} +8.38197 q^{61} -9.61803 q^{62} +3.23607 q^{63} +1.00000 q^{64} -2.23607 q^{66} +10.0902 q^{67} -4.47214 q^{68} -2.38197 q^{69} -14.9443 q^{71} +2.61803 q^{72} +4.09017 q^{73} -1.00000 q^{74} -4.47214 q^{76} +4.47214 q^{77} -2.38197 q^{78} +11.5623 q^{79} +5.70820 q^{81} -7.38197 q^{82} +5.52786 q^{83} +0.763932 q^{84} -0.763932 q^{86} -3.90983 q^{87} +3.61803 q^{88} -10.4721 q^{89} +4.76393 q^{91} +3.85410 q^{92} -5.94427 q^{93} +3.23607 q^{94} +0.618034 q^{96} -8.47214 q^{97} +5.47214 q^{98} +9.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} + 17 q^{31} - 2 q^{32} - 3 q^{36} + 2 q^{37} + 7 q^{39} + 17 q^{41} - 6 q^{42} + 6 q^{43} - 5 q^{44} - q^{46} - 2 q^{47} + q^{48} - 2 q^{49} + 10 q^{51} - q^{52} + 8 q^{53} + 2 q^{54} - 2 q^{56} + 10 q^{57} + 3 q^{58} - 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{63} + 2 q^{64} + 9 q^{67} - 7 q^{69} - 12 q^{71} + 3 q^{72} - 3 q^{73} - 2 q^{74} - 7 q^{78} + 3 q^{79} - 2 q^{81} - 17 q^{82} + 20 q^{83} + 6 q^{84} - 6 q^{86} - 19 q^{87} + 5 q^{88} - 12 q^{89} + 14 q^{91} + q^{92} + 6 q^{93} + 2 q^{94} - q^{96} - 8 q^{97} + 2 q^{98} + 10 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −0.618034 −0.356822 −0.178411 0.983956i $$-0.557096\pi$$
−0.178411 + 0.983956i $$0.557096\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0.618034 0.252311
$$7$$ −1.23607 −0.467190 −0.233595 0.972334i $$-0.575049\pi$$
−0.233595 + 0.972334i $$0.575049\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.61803 −0.872678
$$10$$ 0 0
$$11$$ −3.61803 −1.09088 −0.545439 0.838150i $$-0.683637\pi$$
−0.545439 + 0.838150i $$0.683637\pi$$
$$12$$ −0.618034 −0.178411
$$13$$ −3.85410 −1.06894 −0.534468 0.845189i $$-0.679488\pi$$
−0.534468 + 0.845189i $$0.679488\pi$$
$$14$$ 1.23607 0.330353
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.47214 −1.08465 −0.542326 0.840168i $$-0.682456\pi$$
−0.542326 + 0.840168i $$0.682456\pi$$
$$18$$ 2.61803 0.617077
$$19$$ −4.47214 −1.02598 −0.512989 0.858395i $$-0.671462\pi$$
−0.512989 + 0.858395i $$0.671462\pi$$
$$20$$ 0 0
$$21$$ 0.763932 0.166704
$$22$$ 3.61803 0.771367
$$23$$ 3.85410 0.803636 0.401818 0.915720i $$-0.368378\pi$$
0.401818 + 0.915720i $$0.368378\pi$$
$$24$$ 0.618034 0.126156
$$25$$ 0 0
$$26$$ 3.85410 0.755852
$$27$$ 3.47214 0.668213
$$28$$ −1.23607 −0.233595
$$29$$ 6.32624 1.17475 0.587376 0.809314i $$-0.300161\pi$$
0.587376 + 0.809314i $$0.300161\pi$$
$$30$$ 0 0
$$31$$ 9.61803 1.72745 0.863725 0.503964i $$-0.168125\pi$$
0.863725 + 0.503964i $$0.168125\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 2.23607 0.389249
$$34$$ 4.47214 0.766965
$$35$$ 0 0
$$36$$ −2.61803 −0.436339
$$37$$ 1.00000 0.164399
$$38$$ 4.47214 0.725476
$$39$$ 2.38197 0.381420
$$40$$ 0 0
$$41$$ 7.38197 1.15287 0.576435 0.817143i $$-0.304444\pi$$
0.576435 + 0.817143i $$0.304444\pi$$
$$42$$ −0.763932 −0.117877
$$43$$ 0.763932 0.116499 0.0582493 0.998302i $$-0.481448\pi$$
0.0582493 + 0.998302i $$0.481448\pi$$
$$44$$ −3.61803 −0.545439
$$45$$ 0 0
$$46$$ −3.85410 −0.568256
$$47$$ −3.23607 −0.472029 −0.236015 0.971750i $$-0.575841\pi$$
−0.236015 + 0.971750i $$0.575841\pi$$
$$48$$ −0.618034 −0.0892055
$$49$$ −5.47214 −0.781734
$$50$$ 0 0
$$51$$ 2.76393 0.387028
$$52$$ −3.85410 −0.534468
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ −3.47214 −0.472498
$$55$$ 0 0
$$56$$ 1.23607 0.165177
$$57$$ 2.76393 0.366092
$$58$$ −6.32624 −0.830676
$$59$$ −9.23607 −1.20243 −0.601217 0.799086i $$-0.705317\pi$$
−0.601217 + 0.799086i $$0.705317\pi$$
$$60$$ 0 0
$$61$$ 8.38197 1.07320 0.536600 0.843836i $$-0.319708\pi$$
0.536600 + 0.843836i $$0.319708\pi$$
$$62$$ −9.61803 −1.22149
$$63$$ 3.23607 0.407706
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.23607 −0.275241
$$67$$ 10.0902 1.23271 0.616355 0.787468i $$-0.288609\pi$$
0.616355 + 0.787468i $$0.288609\pi$$
$$68$$ −4.47214 −0.542326
$$69$$ −2.38197 −0.286755
$$70$$ 0 0
$$71$$ −14.9443 −1.77356 −0.886779 0.462193i $$-0.847063\pi$$
−0.886779 + 0.462193i $$0.847063\pi$$
$$72$$ 2.61803 0.308538
$$73$$ 4.09017 0.478718 0.239359 0.970931i $$-0.423063\pi$$
0.239359 + 0.970931i $$0.423063\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −4.47214 −0.512989
$$77$$ 4.47214 0.509647
$$78$$ −2.38197 −0.269705
$$79$$ 11.5623 1.30086 0.650431 0.759566i $$-0.274589\pi$$
0.650431 + 0.759566i $$0.274589\pi$$
$$80$$ 0 0
$$81$$ 5.70820 0.634245
$$82$$ −7.38197 −0.815202
$$83$$ 5.52786 0.606762 0.303381 0.952869i $$-0.401885\pi$$
0.303381 + 0.952869i $$0.401885\pi$$
$$84$$ 0.763932 0.0833518
$$85$$ 0 0
$$86$$ −0.763932 −0.0823769
$$87$$ −3.90983 −0.419178
$$88$$ 3.61803 0.385684
$$89$$ −10.4721 −1.11004 −0.555022 0.831836i $$-0.687290\pi$$
−0.555022 + 0.831836i $$0.687290\pi$$
$$90$$ 0 0
$$91$$ 4.76393 0.499396
$$92$$ 3.85410 0.401818
$$93$$ −5.94427 −0.616392
$$94$$ 3.23607 0.333775
$$95$$ 0 0
$$96$$ 0.618034 0.0630778
$$97$$ −8.47214 −0.860215 −0.430108 0.902778i $$-0.641524\pi$$
−0.430108 + 0.902778i $$0.641524\pi$$
$$98$$ 5.47214 0.552769
$$99$$ 9.47214 0.951985
$$100$$ 0 0
$$101$$ 12.4721 1.24102 0.620512 0.784197i $$-0.286925\pi$$
0.620512 + 0.784197i $$0.286925\pi$$
$$102$$ −2.76393 −0.273670
$$103$$ 16.2705 1.60318 0.801590 0.597873i $$-0.203987\pi$$
0.801590 + 0.597873i $$0.203987\pi$$
$$104$$ 3.85410 0.377926
$$105$$ 0 0
$$106$$ −8.47214 −0.822887
$$107$$ −8.32624 −0.804928 −0.402464 0.915436i $$-0.631846\pi$$
−0.402464 + 0.915436i $$0.631846\pi$$
$$108$$ 3.47214 0.334106
$$109$$ −14.9443 −1.43140 −0.715701 0.698407i $$-0.753893\pi$$
−0.715701 + 0.698407i $$0.753893\pi$$
$$110$$ 0 0
$$111$$ −0.618034 −0.0586612
$$112$$ −1.23607 −0.116797
$$113$$ −10.9443 −1.02955 −0.514775 0.857325i $$-0.672125\pi$$
−0.514775 + 0.857325i $$0.672125\pi$$
$$114$$ −2.76393 −0.258866
$$115$$ 0 0
$$116$$ 6.32624 0.587376
$$117$$ 10.0902 0.932837
$$118$$ 9.23607 0.850249
$$119$$ 5.52786 0.506738
$$120$$ 0 0
$$121$$ 2.09017 0.190015
$$122$$ −8.38197 −0.758868
$$123$$ −4.56231 −0.411369
$$124$$ 9.61803 0.863725
$$125$$ 0 0
$$126$$ −3.23607 −0.288292
$$127$$ −0.472136 −0.0418953 −0.0209476 0.999781i $$-0.506668\pi$$
−0.0209476 + 0.999781i $$0.506668\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −0.472136 −0.0415693
$$130$$ 0 0
$$131$$ 8.65248 0.755970 0.377985 0.925812i $$-0.376617\pi$$
0.377985 + 0.925812i $$0.376617\pi$$
$$132$$ 2.23607 0.194625
$$133$$ 5.52786 0.479327
$$134$$ −10.0902 −0.871658
$$135$$ 0 0
$$136$$ 4.47214 0.383482
$$137$$ −19.3262 −1.65115 −0.825576 0.564291i $$-0.809150\pi$$
−0.825576 + 0.564291i $$0.809150\pi$$
$$138$$ 2.38197 0.202766
$$139$$ −1.85410 −0.157263 −0.0786314 0.996904i $$-0.525055\pi$$
−0.0786314 + 0.996904i $$0.525055\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ 14.9443 1.25410
$$143$$ 13.9443 1.16608
$$144$$ −2.61803 −0.218169
$$145$$ 0 0
$$146$$ −4.09017 −0.338505
$$147$$ 3.38197 0.278940
$$148$$ 1.00000 0.0821995
$$149$$ −6.18034 −0.506313 −0.253157 0.967425i $$-0.581469\pi$$
−0.253157 + 0.967425i $$0.581469\pi$$
$$150$$ 0 0
$$151$$ 17.7082 1.44107 0.720537 0.693417i $$-0.243896\pi$$
0.720537 + 0.693417i $$0.243896\pi$$
$$152$$ 4.47214 0.362738
$$153$$ 11.7082 0.946552
$$154$$ −4.47214 −0.360375
$$155$$ 0 0
$$156$$ 2.38197 0.190710
$$157$$ 7.52786 0.600789 0.300394 0.953815i $$-0.402882\pi$$
0.300394 + 0.953815i $$0.402882\pi$$
$$158$$ −11.5623 −0.919848
$$159$$ −5.23607 −0.415247
$$160$$ 0 0
$$161$$ −4.76393 −0.375450
$$162$$ −5.70820 −0.448479
$$163$$ 12.4721 0.976893 0.488447 0.872594i $$-0.337564\pi$$
0.488447 + 0.872594i $$0.337564\pi$$
$$164$$ 7.38197 0.576435
$$165$$ 0 0
$$166$$ −5.52786 −0.429045
$$167$$ −7.14590 −0.552966 −0.276483 0.961019i $$-0.589169\pi$$
−0.276483 + 0.961019i $$0.589169\pi$$
$$168$$ −0.763932 −0.0589386
$$169$$ 1.85410 0.142623
$$170$$ 0 0
$$171$$ 11.7082 0.895349
$$172$$ 0.763932 0.0582493
$$173$$ −8.47214 −0.644125 −0.322062 0.946718i $$-0.604376\pi$$
−0.322062 + 0.946718i $$0.604376\pi$$
$$174$$ 3.90983 0.296403
$$175$$ 0 0
$$176$$ −3.61803 −0.272720
$$177$$ 5.70820 0.429055
$$178$$ 10.4721 0.784920
$$179$$ 18.6525 1.39415 0.697076 0.716997i $$-0.254484\pi$$
0.697076 + 0.716997i $$0.254484\pi$$
$$180$$ 0 0
$$181$$ 5.52786 0.410883 0.205441 0.978669i $$-0.434137\pi$$
0.205441 + 0.978669i $$0.434137\pi$$
$$182$$ −4.76393 −0.353126
$$183$$ −5.18034 −0.382942
$$184$$ −3.85410 −0.284128
$$185$$ 0 0
$$186$$ 5.94427 0.435855
$$187$$ 16.1803 1.18322
$$188$$ −3.23607 −0.236015
$$189$$ −4.29180 −0.312182
$$190$$ 0 0
$$191$$ 4.09017 0.295954 0.147977 0.988991i $$-0.452724\pi$$
0.147977 + 0.988991i $$0.452724\pi$$
$$192$$ −0.618034 −0.0446028
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 8.47214 0.608264
$$195$$ 0 0
$$196$$ −5.47214 −0.390867
$$197$$ 16.4721 1.17359 0.586796 0.809735i $$-0.300389\pi$$
0.586796 + 0.809735i $$0.300389\pi$$
$$198$$ −9.47214 −0.673155
$$199$$ 20.9443 1.48470 0.742350 0.670012i $$-0.233711\pi$$
0.742350 + 0.670012i $$0.233711\pi$$
$$200$$ 0 0
$$201$$ −6.23607 −0.439858
$$202$$ −12.4721 −0.877536
$$203$$ −7.81966 −0.548833
$$204$$ 2.76393 0.193514
$$205$$ 0 0
$$206$$ −16.2705 −1.13362
$$207$$ −10.0902 −0.701315
$$208$$ −3.85410 −0.267234
$$209$$ 16.1803 1.11922
$$210$$ 0 0
$$211$$ −22.2705 −1.53317 −0.766583 0.642146i $$-0.778044\pi$$
−0.766583 + 0.642146i $$0.778044\pi$$
$$212$$ 8.47214 0.581869
$$213$$ 9.23607 0.632845
$$214$$ 8.32624 0.569170
$$215$$ 0 0
$$216$$ −3.47214 −0.236249
$$217$$ −11.8885 −0.807047
$$218$$ 14.9443 1.01215
$$219$$ −2.52786 −0.170817
$$220$$ 0 0
$$221$$ 17.2361 1.15942
$$222$$ 0.618034 0.0414797
$$223$$ 8.18034 0.547796 0.273898 0.961759i $$-0.411687\pi$$
0.273898 + 0.961759i $$0.411687\pi$$
$$224$$ 1.23607 0.0825883
$$225$$ 0 0
$$226$$ 10.9443 0.728002
$$227$$ 17.7082 1.17533 0.587667 0.809103i $$-0.300046\pi$$
0.587667 + 0.809103i $$0.300046\pi$$
$$228$$ 2.76393 0.183046
$$229$$ 17.1246 1.13163 0.565813 0.824534i $$-0.308562\pi$$
0.565813 + 0.824534i $$0.308562\pi$$
$$230$$ 0 0
$$231$$ −2.76393 −0.181853
$$232$$ −6.32624 −0.415338
$$233$$ −13.5623 −0.888496 −0.444248 0.895904i $$-0.646529\pi$$
−0.444248 + 0.895904i $$0.646529\pi$$
$$234$$ −10.0902 −0.659615
$$235$$ 0 0
$$236$$ −9.23607 −0.601217
$$237$$ −7.14590 −0.464176
$$238$$ −5.52786 −0.358318
$$239$$ 3.14590 0.203491 0.101746 0.994810i $$-0.467557\pi$$
0.101746 + 0.994810i $$0.467557\pi$$
$$240$$ 0 0
$$241$$ −10.4721 −0.674570 −0.337285 0.941403i $$-0.609509\pi$$
−0.337285 + 0.941403i $$0.609509\pi$$
$$242$$ −2.09017 −0.134361
$$243$$ −13.9443 −0.894525
$$244$$ 8.38197 0.536600
$$245$$ 0 0
$$246$$ 4.56231 0.290882
$$247$$ 17.2361 1.09670
$$248$$ −9.61803 −0.610746
$$249$$ −3.41641 −0.216506
$$250$$ 0 0
$$251$$ −3.05573 −0.192876 −0.0964379 0.995339i $$-0.530745\pi$$
−0.0964379 + 0.995339i $$0.530745\pi$$
$$252$$ 3.23607 0.203853
$$253$$ −13.9443 −0.876669
$$254$$ 0.472136 0.0296244
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.9443 1.18171 0.590856 0.806777i $$-0.298790\pi$$
0.590856 + 0.806777i $$0.298790\pi$$
$$258$$ 0.472136 0.0293939
$$259$$ −1.23607 −0.0768055
$$260$$ 0 0
$$261$$ −16.5623 −1.02518
$$262$$ −8.65248 −0.534552
$$263$$ 8.76393 0.540407 0.270204 0.962803i $$-0.412909\pi$$
0.270204 + 0.962803i $$0.412909\pi$$
$$264$$ −2.23607 −0.137620
$$265$$ 0 0
$$266$$ −5.52786 −0.338935
$$267$$ 6.47214 0.396088
$$268$$ 10.0902 0.616355
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ −4.94427 −0.300343 −0.150172 0.988660i $$-0.547983\pi$$
−0.150172 + 0.988660i $$0.547983\pi$$
$$272$$ −4.47214 −0.271163
$$273$$ −2.94427 −0.178195
$$274$$ 19.3262 1.16754
$$275$$ 0 0
$$276$$ −2.38197 −0.143378
$$277$$ 7.79837 0.468559 0.234279 0.972169i $$-0.424727\pi$$
0.234279 + 0.972169i $$0.424727\pi$$
$$278$$ 1.85410 0.111202
$$279$$ −25.1803 −1.50751
$$280$$ 0 0
$$281$$ −5.88854 −0.351281 −0.175641 0.984454i $$-0.556200\pi$$
−0.175641 + 0.984454i $$0.556200\pi$$
$$282$$ −2.00000 −0.119098
$$283$$ −11.2361 −0.667915 −0.333957 0.942588i $$-0.608384\pi$$
−0.333957 + 0.942588i $$0.608384\pi$$
$$284$$ −14.9443 −0.886779
$$285$$ 0 0
$$286$$ −13.9443 −0.824542
$$287$$ −9.12461 −0.538609
$$288$$ 2.61803 0.154269
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 5.23607 0.306944
$$292$$ 4.09017 0.239359
$$293$$ −18.6525 −1.08969 −0.544845 0.838537i $$-0.683411\pi$$
−0.544845 + 0.838537i $$0.683411\pi$$
$$294$$ −3.38197 −0.197240
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ −12.5623 −0.728939
$$298$$ 6.18034 0.358017
$$299$$ −14.8541 −0.859035
$$300$$ 0 0
$$301$$ −0.944272 −0.0544269
$$302$$ −17.7082 −1.01899
$$303$$ −7.70820 −0.442825
$$304$$ −4.47214 −0.256495
$$305$$ 0 0
$$306$$ −11.7082 −0.669313
$$307$$ 6.14590 0.350765 0.175382 0.984500i $$-0.443884\pi$$
0.175382 + 0.984500i $$0.443884\pi$$
$$308$$ 4.47214 0.254824
$$309$$ −10.0557 −0.572050
$$310$$ 0 0
$$311$$ 2.03444 0.115363 0.0576813 0.998335i $$-0.481629\pi$$
0.0576813 + 0.998335i $$0.481629\pi$$
$$312$$ −2.38197 −0.134852
$$313$$ −5.81966 −0.328947 −0.164473 0.986382i $$-0.552592\pi$$
−0.164473 + 0.986382i $$0.552592\pi$$
$$314$$ −7.52786 −0.424822
$$315$$ 0 0
$$316$$ 11.5623 0.650431
$$317$$ −3.05573 −0.171627 −0.0858134 0.996311i $$-0.527349\pi$$
−0.0858134 + 0.996311i $$0.527349\pi$$
$$318$$ 5.23607 0.293624
$$319$$ −22.8885 −1.28151
$$320$$ 0 0
$$321$$ 5.14590 0.287216
$$322$$ 4.76393 0.265484
$$323$$ 20.0000 1.11283
$$324$$ 5.70820 0.317122
$$325$$ 0 0
$$326$$ −12.4721 −0.690768
$$327$$ 9.23607 0.510756
$$328$$ −7.38197 −0.407601
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 5.52786 0.303381
$$333$$ −2.61803 −0.143467
$$334$$ 7.14590 0.391006
$$335$$ 0 0
$$336$$ 0.763932 0.0416759
$$337$$ 17.0344 0.927925 0.463963 0.885855i $$-0.346427\pi$$
0.463963 + 0.885855i $$0.346427\pi$$
$$338$$ −1.85410 −0.100850
$$339$$ 6.76393 0.367366
$$340$$ 0 0
$$341$$ −34.7984 −1.88444
$$342$$ −11.7082 −0.633107
$$343$$ 15.4164 0.832408
$$344$$ −0.763932 −0.0411885
$$345$$ 0 0
$$346$$ 8.47214 0.455465
$$347$$ −12.7639 −0.685204 −0.342602 0.939481i $$-0.611308\pi$$
−0.342602 + 0.939481i $$0.611308\pi$$
$$348$$ −3.90983 −0.209589
$$349$$ 12.1803 0.651999 0.325999 0.945370i $$-0.394299\pi$$
0.325999 + 0.945370i $$0.394299\pi$$
$$350$$ 0 0
$$351$$ −13.3820 −0.714277
$$352$$ 3.61803 0.192842
$$353$$ −29.7082 −1.58121 −0.790604 0.612328i $$-0.790233\pi$$
−0.790604 + 0.612328i $$0.790233\pi$$
$$354$$ −5.70820 −0.303388
$$355$$ 0 0
$$356$$ −10.4721 −0.555022
$$357$$ −3.41641 −0.180815
$$358$$ −18.6525 −0.985814
$$359$$ 4.47214 0.236030 0.118015 0.993012i $$-0.462347\pi$$
0.118015 + 0.993012i $$0.462347\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −5.52786 −0.290538
$$363$$ −1.29180 −0.0678017
$$364$$ 4.76393 0.249698
$$365$$ 0 0
$$366$$ 5.18034 0.270781
$$367$$ 27.1246 1.41589 0.707947 0.706266i $$-0.249622\pi$$
0.707947 + 0.706266i $$0.249622\pi$$
$$368$$ 3.85410 0.200909
$$369$$ −19.3262 −1.00608
$$370$$ 0 0
$$371$$ −10.4721 −0.543686
$$372$$ −5.94427 −0.308196
$$373$$ −14.2918 −0.740001 −0.370001 0.929032i $$-0.620643\pi$$
−0.370001 + 0.929032i $$0.620643\pi$$
$$374$$ −16.1803 −0.836665
$$375$$ 0 0
$$376$$ 3.23607 0.166887
$$377$$ −24.3820 −1.25574
$$378$$ 4.29180 0.220746
$$379$$ 16.9098 0.868600 0.434300 0.900768i $$-0.356996\pi$$
0.434300 + 0.900768i $$0.356996\pi$$
$$380$$ 0 0
$$381$$ 0.291796 0.0149492
$$382$$ −4.09017 −0.209271
$$383$$ 17.8885 0.914062 0.457031 0.889451i $$-0.348913\pi$$
0.457031 + 0.889451i $$0.348913\pi$$
$$384$$ 0.618034 0.0315389
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −2.00000 −0.101666
$$388$$ −8.47214 −0.430108
$$389$$ −0.145898 −0.00739732 −0.00369866 0.999993i $$-0.501177\pi$$
−0.00369866 + 0.999993i $$0.501177\pi$$
$$390$$ 0 0
$$391$$ −17.2361 −0.871665
$$392$$ 5.47214 0.276385
$$393$$ −5.34752 −0.269747
$$394$$ −16.4721 −0.829854
$$395$$ 0 0
$$396$$ 9.47214 0.475993
$$397$$ 10.6525 0.534632 0.267316 0.963609i $$-0.413863\pi$$
0.267316 + 0.963609i $$0.413863\pi$$
$$398$$ −20.9443 −1.04984
$$399$$ −3.41641 −0.171034
$$400$$ 0 0
$$401$$ −9.23607 −0.461227 −0.230614 0.973045i $$-0.574073\pi$$
−0.230614 + 0.973045i $$0.574073\pi$$
$$402$$ 6.23607 0.311027
$$403$$ −37.0689 −1.84653
$$404$$ 12.4721 0.620512
$$405$$ 0 0
$$406$$ 7.81966 0.388083
$$407$$ −3.61803 −0.179339
$$408$$ −2.76393 −0.136835
$$409$$ −3.81966 −0.188870 −0.0944350 0.995531i $$-0.530104\pi$$
−0.0944350 + 0.995531i $$0.530104\pi$$
$$410$$ 0 0
$$411$$ 11.9443 0.589167
$$412$$ 16.2705 0.801590
$$413$$ 11.4164 0.561765
$$414$$ 10.0902 0.495905
$$415$$ 0 0
$$416$$ 3.85410 0.188963
$$417$$ 1.14590 0.0561149
$$418$$ −16.1803 −0.791406
$$419$$ 9.56231 0.467149 0.233575 0.972339i $$-0.424958\pi$$
0.233575 + 0.972339i $$0.424958\pi$$
$$420$$ 0 0
$$421$$ −1.96556 −0.0957954 −0.0478977 0.998852i $$-0.515252\pi$$
−0.0478977 + 0.998852i $$0.515252\pi$$
$$422$$ 22.2705 1.08411
$$423$$ 8.47214 0.411929
$$424$$ −8.47214 −0.411443
$$425$$ 0 0
$$426$$ −9.23607 −0.447489
$$427$$ −10.3607 −0.501388
$$428$$ −8.32624 −0.402464
$$429$$ −8.61803 −0.416083
$$430$$ 0 0
$$431$$ −32.3607 −1.55876 −0.779380 0.626552i $$-0.784466\pi$$
−0.779380 + 0.626552i $$0.784466\pi$$
$$432$$ 3.47214 0.167053
$$433$$ 36.3262 1.74573 0.872864 0.487964i $$-0.162260\pi$$
0.872864 + 0.487964i $$0.162260\pi$$
$$434$$ 11.8885 0.570668
$$435$$ 0 0
$$436$$ −14.9443 −0.715701
$$437$$ −17.2361 −0.824513
$$438$$ 2.52786 0.120786
$$439$$ −16.7984 −0.801743 −0.400871 0.916134i $$-0.631293\pi$$
−0.400871 + 0.916134i $$0.631293\pi$$
$$440$$ 0 0
$$441$$ 14.3262 0.682202
$$442$$ −17.2361 −0.819836
$$443$$ 18.2705 0.868058 0.434029 0.900899i $$-0.357092\pi$$
0.434029 + 0.900899i $$0.357092\pi$$
$$444$$ −0.618034 −0.0293306
$$445$$ 0 0
$$446$$ −8.18034 −0.387350
$$447$$ 3.81966 0.180664
$$448$$ −1.23607 −0.0583987
$$449$$ −19.5279 −0.921577 −0.460788 0.887510i $$-0.652433\pi$$
−0.460788 + 0.887510i $$0.652433\pi$$
$$450$$ 0 0
$$451$$ −26.7082 −1.25764
$$452$$ −10.9443 −0.514775
$$453$$ −10.9443 −0.514207
$$454$$ −17.7082 −0.831087
$$455$$ 0 0
$$456$$ −2.76393 −0.129433
$$457$$ 29.2361 1.36761 0.683803 0.729667i $$-0.260325\pi$$
0.683803 + 0.729667i $$0.260325\pi$$
$$458$$ −17.1246 −0.800181
$$459$$ −15.5279 −0.724779
$$460$$ 0 0
$$461$$ −21.0557 −0.980663 −0.490332 0.871536i $$-0.663124\pi$$
−0.490332 + 0.871536i $$0.663124\pi$$
$$462$$ 2.76393 0.128590
$$463$$ −15.5623 −0.723242 −0.361621 0.932325i $$-0.617777\pi$$
−0.361621 + 0.932325i $$0.617777\pi$$
$$464$$ 6.32624 0.293688
$$465$$ 0 0
$$466$$ 13.5623 0.628262
$$467$$ −20.3607 −0.942180 −0.471090 0.882085i $$-0.656139\pi$$
−0.471090 + 0.882085i $$0.656139\pi$$
$$468$$ 10.0902 0.466418
$$469$$ −12.4721 −0.575910
$$470$$ 0 0
$$471$$ −4.65248 −0.214375
$$472$$ 9.23607 0.425124
$$473$$ −2.76393 −0.127086
$$474$$ 7.14590 0.328222
$$475$$ 0 0
$$476$$ 5.52786 0.253369
$$477$$ −22.1803 −1.01557
$$478$$ −3.14590 −0.143890
$$479$$ 16.4377 0.751057 0.375529 0.926811i $$-0.377461\pi$$
0.375529 + 0.926811i $$0.377461\pi$$
$$480$$ 0 0
$$481$$ −3.85410 −0.175732
$$482$$ 10.4721 0.476993
$$483$$ 2.94427 0.133969
$$484$$ 2.09017 0.0950077
$$485$$ 0 0
$$486$$ 13.9443 0.632525
$$487$$ 25.3050 1.14668 0.573338 0.819319i $$-0.305648\pi$$
0.573338 + 0.819319i $$0.305648\pi$$
$$488$$ −8.38197 −0.379434
$$489$$ −7.70820 −0.348577
$$490$$ 0 0
$$491$$ 27.4508 1.23884 0.619420 0.785060i $$-0.287368\pi$$
0.619420 + 0.785060i $$0.287368\pi$$
$$492$$ −4.56231 −0.205685
$$493$$ −28.2918 −1.27420
$$494$$ −17.2361 −0.775487
$$495$$ 0 0
$$496$$ 9.61803 0.431862
$$497$$ 18.4721 0.828589
$$498$$ 3.41641 0.153093
$$499$$ 23.7082 1.06132 0.530662 0.847583i $$-0.321943\pi$$
0.530662 + 0.847583i $$0.321943\pi$$
$$500$$ 0 0
$$501$$ 4.41641 0.197311
$$502$$ 3.05573 0.136384
$$503$$ −7.90983 −0.352682 −0.176341 0.984329i $$-0.556426\pi$$
−0.176341 + 0.984329i $$0.556426\pi$$
$$504$$ −3.23607 −0.144146
$$505$$ 0 0
$$506$$ 13.9443 0.619898
$$507$$ −1.14590 −0.0508911
$$508$$ −0.472136 −0.0209476
$$509$$ −4.29180 −0.190231 −0.0951153 0.995466i $$-0.530322\pi$$
−0.0951153 + 0.995466i $$0.530322\pi$$
$$510$$ 0 0
$$511$$ −5.05573 −0.223652
$$512$$ −1.00000 −0.0441942
$$513$$ −15.5279 −0.685572
$$514$$ −18.9443 −0.835596
$$515$$ 0 0
$$516$$ −0.472136 −0.0207846
$$517$$ 11.7082 0.514926
$$518$$ 1.23607 0.0543097
$$519$$ 5.23607 0.229838
$$520$$ 0 0
$$521$$ 25.4164 1.11351 0.556757 0.830676i $$-0.312046\pi$$
0.556757 + 0.830676i $$0.312046\pi$$
$$522$$ 16.5623 0.724912
$$523$$ −34.1803 −1.49460 −0.747301 0.664486i $$-0.768651\pi$$
−0.747301 + 0.664486i $$0.768651\pi$$
$$524$$ 8.65248 0.377985
$$525$$ 0 0
$$526$$ −8.76393 −0.382126
$$527$$ −43.0132 −1.87368
$$528$$ 2.23607 0.0973124
$$529$$ −8.14590 −0.354169
$$530$$ 0 0
$$531$$ 24.1803 1.04934
$$532$$ 5.52786 0.239663
$$533$$ −28.4508 −1.23234
$$534$$ −6.47214 −0.280077
$$535$$ 0 0
$$536$$ −10.0902 −0.435829
$$537$$ −11.5279 −0.497464
$$538$$ −4.00000 −0.172452
$$539$$ 19.7984 0.852776
$$540$$ 0 0
$$541$$ 5.32624 0.228993 0.114496 0.993424i $$-0.463475\pi$$
0.114496 + 0.993424i $$0.463475\pi$$
$$542$$ 4.94427 0.212375
$$543$$ −3.41641 −0.146612
$$544$$ 4.47214 0.191741
$$545$$ 0 0
$$546$$ 2.94427 0.126003
$$547$$ −42.0689 −1.79874 −0.899368 0.437193i $$-0.855973\pi$$
−0.899368 + 0.437193i $$0.855973\pi$$
$$548$$ −19.3262 −0.825576
$$549$$ −21.9443 −0.936559
$$550$$ 0 0
$$551$$ −28.2918 −1.20527
$$552$$ 2.38197 0.101383
$$553$$ −14.2918 −0.607749
$$554$$ −7.79837 −0.331321
$$555$$ 0 0
$$556$$ −1.85410 −0.0786314
$$557$$ 0.562306 0.0238257 0.0119128 0.999929i $$-0.496208\pi$$
0.0119128 + 0.999929i $$0.496208\pi$$
$$558$$ 25.1803 1.06597
$$559$$ −2.94427 −0.124529
$$560$$ 0 0
$$561$$ −10.0000 −0.422200
$$562$$ 5.88854 0.248393
$$563$$ 27.8885 1.17536 0.587681 0.809093i $$-0.300041\pi$$
0.587681 + 0.809093i $$0.300041\pi$$
$$564$$ 2.00000 0.0842152
$$565$$ 0 0
$$566$$ 11.2361 0.472287
$$567$$ −7.05573 −0.296313
$$568$$ 14.9443 0.627048
$$569$$ −21.8885 −0.917615 −0.458808 0.888536i $$-0.651723\pi$$
−0.458808 + 0.888536i $$0.651723\pi$$
$$570$$ 0 0
$$571$$ 9.56231 0.400170 0.200085 0.979779i $$-0.435878\pi$$
0.200085 + 0.979779i $$0.435878\pi$$
$$572$$ 13.9443 0.583039
$$573$$ −2.52786 −0.105603
$$574$$ 9.12461 0.380854
$$575$$ 0 0
$$576$$ −2.61803 −0.109085
$$577$$ 20.6525 0.859774 0.429887 0.902883i $$-0.358553\pi$$
0.429887 + 0.902883i $$0.358553\pi$$
$$578$$ −3.00000 −0.124784
$$579$$ 2.47214 0.102738
$$580$$ 0 0
$$581$$ −6.83282 −0.283473
$$582$$ −5.23607 −0.217042
$$583$$ −30.6525 −1.26950
$$584$$ −4.09017 −0.169252
$$585$$ 0 0
$$586$$ 18.6525 0.770527
$$587$$ 30.9443 1.27721 0.638603 0.769536i $$-0.279512\pi$$
0.638603 + 0.769536i $$0.279512\pi$$
$$588$$ 3.38197 0.139470
$$589$$ −43.0132 −1.77233
$$590$$ 0 0
$$591$$ −10.1803 −0.418763
$$592$$ 1.00000 0.0410997
$$593$$ 2.56231 0.105221 0.0526106 0.998615i $$-0.483246\pi$$
0.0526106 + 0.998615i $$0.483246\pi$$
$$594$$ 12.5623 0.515438
$$595$$ 0 0
$$596$$ −6.18034 −0.253157
$$597$$ −12.9443 −0.529774
$$598$$ 14.8541 0.607429
$$599$$ 38.3607 1.56737 0.783687 0.621155i $$-0.213336\pi$$
0.783687 + 0.621155i $$0.213336\pi$$
$$600$$ 0 0
$$601$$ 35.6869 1.45570 0.727850 0.685737i $$-0.240520\pi$$
0.727850 + 0.685737i $$0.240520\pi$$
$$602$$ 0.944272 0.0384856
$$603$$ −26.4164 −1.07576
$$604$$ 17.7082 0.720537
$$605$$ 0 0
$$606$$ 7.70820 0.313124
$$607$$ −5.96556 −0.242135 −0.121067 0.992644i $$-0.538632\pi$$
−0.121067 + 0.992644i $$0.538632\pi$$
$$608$$ 4.47214 0.181369
$$609$$ 4.83282 0.195836
$$610$$ 0 0
$$611$$ 12.4721 0.504569
$$612$$ 11.7082 0.473276
$$613$$ 36.1803 1.46131 0.730655 0.682747i $$-0.239215\pi$$
0.730655 + 0.682747i $$0.239215\pi$$
$$614$$ −6.14590 −0.248028
$$615$$ 0 0
$$616$$ −4.47214 −0.180187
$$617$$ 11.0902 0.446473 0.223237 0.974764i $$-0.428338\pi$$
0.223237 + 0.974764i $$0.428338\pi$$
$$618$$ 10.0557 0.404501
$$619$$ 18.2705 0.734354 0.367177 0.930151i $$-0.380324\pi$$
0.367177 + 0.930151i $$0.380324\pi$$
$$620$$ 0 0
$$621$$ 13.3820 0.537000
$$622$$ −2.03444 −0.0815737
$$623$$ 12.9443 0.518601
$$624$$ 2.38197 0.0953550
$$625$$ 0 0
$$626$$ 5.81966 0.232600
$$627$$ −10.0000 −0.399362
$$628$$ 7.52786 0.300394
$$629$$ −4.47214 −0.178316
$$630$$ 0 0
$$631$$ 26.3951 1.05077 0.525387 0.850864i $$-0.323921\pi$$
0.525387 + 0.850864i $$0.323921\pi$$
$$632$$ −11.5623 −0.459924
$$633$$ 13.7639 0.547067
$$634$$ 3.05573 0.121358
$$635$$ 0 0
$$636$$ −5.23607 −0.207624
$$637$$ 21.0902 0.835623
$$638$$ 22.8885 0.906166
$$639$$ 39.1246 1.54775
$$640$$ 0 0
$$641$$ −22.5066 −0.888956 −0.444478 0.895790i $$-0.646611\pi$$
−0.444478 + 0.895790i $$0.646611\pi$$
$$642$$ −5.14590 −0.203092
$$643$$ 33.2361 1.31070 0.655351 0.755324i $$-0.272521\pi$$
0.655351 + 0.755324i $$0.272521\pi$$
$$644$$ −4.76393 −0.187725
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ −18.9098 −0.743422 −0.371711 0.928348i $$-0.621229\pi$$
−0.371711 + 0.928348i $$0.621229\pi$$
$$648$$ −5.70820 −0.224239
$$649$$ 33.4164 1.31171
$$650$$ 0 0
$$651$$ 7.34752 0.287972
$$652$$ 12.4721 0.488447
$$653$$ 4.72949 0.185079 0.0925396 0.995709i $$-0.470502\pi$$
0.0925396 + 0.995709i $$0.470502\pi$$
$$654$$ −9.23607 −0.361159
$$655$$ 0 0
$$656$$ 7.38197 0.288217
$$657$$ −10.7082 −0.417767
$$658$$ −4.00000 −0.155936
$$659$$ −15.4508 −0.601880 −0.300940 0.953643i $$-0.597300\pi$$
−0.300940 + 0.953643i $$0.597300\pi$$
$$660$$ 0 0
$$661$$ 1.67376 0.0651018 0.0325509 0.999470i $$-0.489637\pi$$
0.0325509 + 0.999470i $$0.489637\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ −10.6525 −0.413708
$$664$$ −5.52786 −0.214523
$$665$$ 0 0
$$666$$ 2.61803 0.101447
$$667$$ 24.3820 0.944073
$$668$$ −7.14590 −0.276483
$$669$$ −5.05573 −0.195466
$$670$$ 0 0
$$671$$ −30.3262 −1.17073
$$672$$ −0.763932 −0.0294693
$$673$$ −17.8541 −0.688225 −0.344113 0.938928i $$-0.611820\pi$$
−0.344113 + 0.938928i $$0.611820\pi$$
$$674$$ −17.0344 −0.656142
$$675$$ 0 0
$$676$$ 1.85410 0.0713116
$$677$$ −3.34752 −0.128656 −0.0643279 0.997929i $$-0.520490\pi$$
−0.0643279 + 0.997929i $$0.520490\pi$$
$$678$$ −6.76393 −0.259767
$$679$$ 10.4721 0.401884
$$680$$ 0 0
$$681$$ −10.9443 −0.419385
$$682$$ 34.7984 1.33250
$$683$$ 35.4164 1.35517 0.677586 0.735444i $$-0.263026\pi$$
0.677586 + 0.735444i $$0.263026\pi$$
$$684$$ 11.7082 0.447674
$$685$$ 0 0
$$686$$ −15.4164 −0.588601
$$687$$ −10.5836 −0.403789
$$688$$ 0.763932 0.0291246
$$689$$ −32.6525 −1.24396
$$690$$ 0 0
$$691$$ −35.7771 −1.36102 −0.680512 0.732737i $$-0.738243\pi$$
−0.680512 + 0.732737i $$0.738243\pi$$
$$692$$ −8.47214 −0.322062
$$693$$ −11.7082 −0.444758
$$694$$ 12.7639 0.484512
$$695$$ 0 0
$$696$$ 3.90983 0.148202
$$697$$ −33.0132 −1.25046
$$698$$ −12.1803 −0.461033
$$699$$ 8.38197 0.317035
$$700$$ 0 0
$$701$$ 3.97871 0.150274 0.0751370 0.997173i $$-0.476061\pi$$
0.0751370 + 0.997173i $$0.476061\pi$$
$$702$$ 13.3820 0.505070
$$703$$ −4.47214 −0.168670
$$704$$ −3.61803 −0.136360
$$705$$ 0 0
$$706$$ 29.7082 1.11808
$$707$$ −15.4164 −0.579794
$$708$$ 5.70820 0.214527
$$709$$ 43.2148 1.62297 0.811483 0.584377i $$-0.198661\pi$$
0.811483 + 0.584377i $$0.198661\pi$$
$$710$$ 0 0
$$711$$ −30.2705 −1.13523
$$712$$ 10.4721 0.392460
$$713$$ 37.0689 1.38824
$$714$$ 3.41641 0.127856
$$715$$ 0 0
$$716$$ 18.6525 0.697076
$$717$$ −1.94427 −0.0726102
$$718$$ −4.47214 −0.166899
$$719$$ 0.583592 0.0217643 0.0108822 0.999941i $$-0.496536\pi$$
0.0108822 + 0.999941i $$0.496536\pi$$
$$720$$ 0 0
$$721$$ −20.1115 −0.748990
$$722$$ −1.00000 −0.0372161
$$723$$ 6.47214 0.240701
$$724$$ 5.52786 0.205441
$$725$$ 0 0
$$726$$ 1.29180 0.0479430
$$727$$ 23.1459 0.858434 0.429217 0.903201i $$-0.358790\pi$$
0.429217 + 0.903201i $$0.358790\pi$$
$$728$$ −4.76393 −0.176563
$$729$$ −8.50658 −0.315058
$$730$$ 0 0
$$731$$ −3.41641 −0.126360
$$732$$ −5.18034 −0.191471
$$733$$ −36.4721 −1.34713 −0.673565 0.739128i $$-0.735238\pi$$
−0.673565 + 0.739128i $$0.735238\pi$$
$$734$$ −27.1246 −1.00119
$$735$$ 0 0
$$736$$ −3.85410 −0.142064
$$737$$ −36.5066 −1.34474
$$738$$ 19.3262 0.711409
$$739$$ −22.0902 −0.812600 −0.406300 0.913740i $$-0.633181\pi$$
−0.406300 + 0.913740i $$0.633181\pi$$
$$740$$ 0 0
$$741$$ −10.6525 −0.391328
$$742$$ 10.4721 0.384444
$$743$$ −10.0689 −0.369392 −0.184696 0.982796i $$-0.559130\pi$$
−0.184696 + 0.982796i $$0.559130\pi$$
$$744$$ 5.94427 0.217928
$$745$$ 0 0
$$746$$ 14.2918 0.523260
$$747$$ −14.4721 −0.529508
$$748$$ 16.1803 0.591612
$$749$$ 10.2918 0.376054
$$750$$ 0 0
$$751$$ −18.9443 −0.691286 −0.345643 0.938366i $$-0.612339\pi$$
−0.345643 + 0.938366i $$0.612339\pi$$
$$752$$ −3.23607 −0.118007
$$753$$ 1.88854 0.0688224
$$754$$ 24.3820 0.887939
$$755$$ 0 0
$$756$$ −4.29180 −0.156091
$$757$$ 10.8541 0.394499 0.197250 0.980353i $$-0.436799\pi$$
0.197250 + 0.980353i $$0.436799\pi$$
$$758$$ −16.9098 −0.614193
$$759$$ 8.61803 0.312815
$$760$$ 0 0
$$761$$ −25.8541 −0.937210 −0.468605 0.883408i $$-0.655243\pi$$
−0.468605 + 0.883408i $$0.655243\pi$$
$$762$$ −0.291796 −0.0105707
$$763$$ 18.4721 0.668736
$$764$$ 4.09017 0.147977
$$765$$ 0 0
$$766$$ −17.8885 −0.646339
$$767$$ 35.5967 1.28532
$$768$$ −0.618034 −0.0223014
$$769$$ −11.8885 −0.428712 −0.214356 0.976756i $$-0.568765\pi$$
−0.214356 + 0.976756i $$0.568765\pi$$
$$770$$ 0 0
$$771$$ −11.7082 −0.421661
$$772$$ −4.00000 −0.143963
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 8.47214 0.304132
$$777$$ 0.763932 0.0274059
$$778$$ 0.145898 0.00523070
$$779$$ −33.0132 −1.18282
$$780$$ 0 0
$$781$$ 54.0689 1.93474
$$782$$ 17.2361 0.616361
$$783$$ 21.9656 0.784985
$$784$$ −5.47214 −0.195433
$$785$$ 0 0
$$786$$ 5.34752 0.190740
$$787$$ 34.4721 1.22880 0.614399 0.788995i $$-0.289398\pi$$
0.614399 + 0.788995i $$0.289398\pi$$
$$788$$ 16.4721 0.586796
$$789$$ −5.41641 −0.192829
$$790$$ 0 0
$$791$$ 13.5279 0.480995
$$792$$ −9.47214 −0.336578
$$793$$ −32.3050 −1.14718
$$794$$ −10.6525 −0.378042
$$795$$ 0 0
$$796$$ 20.9443 0.742350
$$797$$ −12.7295 −0.450902 −0.225451 0.974255i $$-0.572386\pi$$
−0.225451 + 0.974255i $$0.572386\pi$$
$$798$$ 3.41641 0.120940
$$799$$ 14.4721 0.511987
$$800$$ 0 0
$$801$$ 27.4164 0.968711
$$802$$ 9.23607 0.326137
$$803$$ −14.7984 −0.522223
$$804$$ −6.23607 −0.219929
$$805$$ 0 0
$$806$$ 37.0689 1.30570
$$807$$ −2.47214 −0.0870233
$$808$$ −12.4721 −0.438768
$$809$$ 27.1246 0.953651 0.476825 0.878998i $$-0.341787\pi$$
0.476825 + 0.878998i $$0.341787\pi$$
$$810$$ 0 0
$$811$$ 53.1033 1.86471 0.932355 0.361544i $$-0.117750\pi$$
0.932355 + 0.361544i $$0.117750\pi$$
$$812$$ −7.81966 −0.274416
$$813$$ 3.05573 0.107169
$$814$$ 3.61803 0.126812
$$815$$ 0 0
$$816$$ 2.76393 0.0967570
$$817$$ −3.41641 −0.119525
$$818$$ 3.81966 0.133551
$$819$$ −12.4721 −0.435812
$$820$$ 0 0
$$821$$ 21.4164 0.747438 0.373719 0.927542i $$-0.378082\pi$$
0.373719 + 0.927542i $$0.378082\pi$$
$$822$$ −11.9443 −0.416604
$$823$$ 33.8885 1.18128 0.590640 0.806935i $$-0.298875\pi$$
0.590640 + 0.806935i $$0.298875\pi$$
$$824$$ −16.2705 −0.566810
$$825$$ 0 0
$$826$$ −11.4164 −0.397228
$$827$$ −56.0689 −1.94971 −0.974853 0.222849i $$-0.928464\pi$$
−0.974853 + 0.222849i $$0.928464\pi$$
$$828$$ −10.0902 −0.350658
$$829$$ 31.2016 1.08368 0.541839 0.840483i $$-0.317728\pi$$
0.541839 + 0.840483i $$0.317728\pi$$
$$830$$ 0 0
$$831$$ −4.81966 −0.167192
$$832$$ −3.85410 −0.133617
$$833$$ 24.4721 0.847909
$$834$$ −1.14590 −0.0396792
$$835$$ 0 0
$$836$$ 16.1803 0.559609
$$837$$ 33.3951 1.15430
$$838$$ −9.56231 −0.330324
$$839$$ −5.34752 −0.184617 −0.0923085 0.995730i $$-0.529425\pi$$
−0.0923085 + 0.995730i $$0.529425\pi$$
$$840$$ 0 0
$$841$$ 11.0213 0.380044
$$842$$ 1.96556 0.0677376
$$843$$ 3.63932 0.125345
$$844$$ −22.2705 −0.766583
$$845$$ 0 0
$$846$$ −8.47214 −0.291278
$$847$$ −2.58359 −0.0887733
$$848$$ 8.47214 0.290934
$$849$$ 6.94427 0.238327
$$850$$ 0 0
$$851$$ 3.85410 0.132117
$$852$$ 9.23607 0.316422
$$853$$ 12.7426 0.436300 0.218150 0.975915i $$-0.429998\pi$$
0.218150 + 0.975915i $$0.429998\pi$$
$$854$$ 10.3607 0.354535
$$855$$ 0 0
$$856$$ 8.32624 0.284585
$$857$$ 26.9443 0.920399 0.460199 0.887816i $$-0.347778\pi$$
0.460199 + 0.887816i $$0.347778\pi$$
$$858$$ 8.61803 0.294215
$$859$$ 26.5836 0.907020 0.453510 0.891251i $$-0.350172\pi$$
0.453510 + 0.891251i $$0.350172\pi$$
$$860$$ 0 0
$$861$$ 5.63932 0.192188
$$862$$ 32.3607 1.10221
$$863$$ 7.41641 0.252457 0.126229 0.992001i $$-0.459713\pi$$
0.126229 + 0.992001i $$0.459713\pi$$
$$864$$ −3.47214 −0.118124
$$865$$ 0 0
$$866$$ −36.3262 −1.23442
$$867$$ −1.85410 −0.0629686
$$868$$ −11.8885 −0.403523
$$869$$ −41.8328 −1.41908
$$870$$ 0 0
$$871$$ −38.8885 −1.31769
$$872$$ 14.9443 0.506077
$$873$$ 22.1803 0.750691
$$874$$ 17.2361 0.583019
$$875$$ 0 0
$$876$$ −2.52786 −0.0854086
$$877$$ −36.8328 −1.24376 −0.621878 0.783114i $$-0.713630\pi$$
−0.621878 + 0.783114i $$0.713630\pi$$
$$878$$ 16.7984 0.566918
$$879$$ 11.5279 0.388825
$$880$$ 0 0
$$881$$ −22.7426 −0.766219 −0.383110 0.923703i $$-0.625147\pi$$
−0.383110 + 0.923703i $$0.625147\pi$$
$$882$$ −14.3262 −0.482390
$$883$$ −29.3050 −0.986190 −0.493095 0.869975i $$-0.664135\pi$$
−0.493095 + 0.869975i $$0.664135\pi$$
$$884$$ 17.2361 0.579712
$$885$$ 0 0
$$886$$ −18.2705 −0.613810
$$887$$ 7.88854 0.264871 0.132436 0.991192i $$-0.457720\pi$$
0.132436 + 0.991192i $$0.457720\pi$$
$$888$$ 0.618034 0.0207399
$$889$$ 0.583592 0.0195731
$$890$$ 0 0
$$891$$ −20.6525 −0.691884
$$892$$ 8.18034 0.273898
$$893$$ 14.4721 0.484292
$$894$$ −3.81966 −0.127749
$$895$$ 0 0
$$896$$ 1.23607 0.0412941
$$897$$ 9.18034 0.306523
$$898$$ 19.5279 0.651653
$$899$$ 60.8460 2.02933
$$900$$ 0 0
$$901$$ −37.8885 −1.26225
$$902$$ 26.7082 0.889286
$$903$$ 0.583592 0.0194207
$$904$$ 10.9443 0.364001
$$905$$ 0 0
$$906$$ 10.9443 0.363599
$$907$$ 20.1115 0.667790 0.333895 0.942610i $$-0.391637\pi$$
0.333895 + 0.942610i $$0.391637\pi$$
$$908$$ 17.7082 0.587667
$$909$$ −32.6525 −1.08301
$$910$$ 0 0
$$911$$ −21.8885 −0.725200 −0.362600 0.931945i $$-0.618111\pi$$
−0.362600 + 0.931945i $$0.618111\pi$$
$$912$$ 2.76393 0.0915229
$$913$$ −20.0000 −0.661903
$$914$$ −29.2361 −0.967043
$$915$$ 0 0
$$916$$ 17.1246 0.565813
$$917$$ −10.6950 −0.353182
$$918$$ 15.5279 0.512496
$$919$$ 29.8885 0.985932 0.492966 0.870049i $$-0.335913\pi$$
0.492966 + 0.870049i $$0.335913\pi$$
$$920$$ 0 0
$$921$$ −3.79837 −0.125161
$$922$$ 21.0557 0.693433
$$923$$ 57.5967 1.89582
$$924$$ −2.76393 −0.0909267
$$925$$ 0 0
$$926$$ 15.5623 0.511409
$$927$$ −42.5967 −1.39906
$$928$$ −6.32624 −0.207669
$$929$$ 28.4508 0.933442 0.466721 0.884405i $$-0.345435\pi$$
0.466721 + 0.884405i $$0.345435\pi$$
$$930$$ 0 0
$$931$$ 24.4721 0.802042
$$932$$ −13.5623 −0.444248
$$933$$ −1.25735 −0.0411639
$$934$$ 20.3607 0.666222
$$935$$ 0 0
$$936$$ −10.0902 −0.329808
$$937$$ −57.0476 −1.86366 −0.931832 0.362890i $$-0.881790\pi$$
−0.931832 + 0.362890i $$0.881790\pi$$
$$938$$ 12.4721 0.407230
$$939$$ 3.59675 0.117375
$$940$$ 0 0
$$941$$ −3.81966 −0.124517 −0.0622587 0.998060i $$-0.519830\pi$$
−0.0622587 + 0.998060i $$0.519830\pi$$
$$942$$ 4.65248 0.151586
$$943$$ 28.4508 0.926487
$$944$$ −9.23607 −0.300608
$$945$$ 0 0
$$946$$ 2.76393 0.0898632
$$947$$ 34.8328 1.13191 0.565957 0.824435i $$-0.308507\pi$$
0.565957 + 0.824435i $$0.308507\pi$$
$$948$$ −7.14590 −0.232088
$$949$$ −15.7639 −0.511719
$$950$$ 0 0
$$951$$ 1.88854 0.0612402
$$952$$ −5.52786 −0.179159
$$953$$ 44.4508 1.43990 0.719952 0.694024i $$-0.244164\pi$$
0.719952 + 0.694024i $$0.244164\pi$$
$$954$$ 22.1803 0.718115
$$955$$ 0 0
$$956$$ 3.14590 0.101746
$$957$$ 14.1459 0.457272
$$958$$ −16.4377 −0.531078
$$959$$ 23.8885 0.771401
$$960$$ 0 0
$$961$$ 61.5066 1.98408
$$962$$ 3.85410 0.124261
$$963$$ 21.7984 0.702443
$$964$$ −10.4721 −0.337285
$$965$$ 0 0
$$966$$ −2.94427 −0.0947304
$$967$$ −11.7295 −0.377195 −0.188597 0.982054i $$-0.560394\pi$$
−0.188597 + 0.982054i $$0.560394\pi$$
$$968$$ −2.09017 −0.0671806
$$969$$ −12.3607 −0.397082
$$970$$ 0 0
$$971$$ 26.6738 0.856002 0.428001 0.903778i $$-0.359218\pi$$
0.428001 + 0.903778i $$0.359218\pi$$
$$972$$ −13.9443 −0.447263
$$973$$ 2.29180 0.0734716
$$974$$ −25.3050 −0.810823
$$975$$ 0 0
$$976$$ 8.38197 0.268300
$$977$$ −52.4721 −1.67873 −0.839366 0.543566i $$-0.817074\pi$$
−0.839366 + 0.543566i $$0.817074\pi$$
$$978$$ 7.70820 0.246481
$$979$$ 37.8885 1.21092
$$980$$ 0 0
$$981$$ 39.1246 1.24915
$$982$$ −27.4508 −0.875992
$$983$$ −39.7771 −1.26869 −0.634346 0.773049i $$-0.718731\pi$$
−0.634346 + 0.773049i $$0.718731\pi$$
$$984$$ 4.56231 0.145441
$$985$$ 0 0
$$986$$ 28.2918 0.900994
$$987$$ −2.47214 −0.0786890
$$988$$ 17.2361 0.548352
$$989$$ 2.94427 0.0936224
$$990$$ 0 0
$$991$$ −54.1033 −1.71865 −0.859324 0.511431i $$-0.829116\pi$$
−0.859324 + 0.511431i $$0.829116\pi$$
$$992$$ −9.61803 −0.305373
$$993$$ −17.3050 −0.549156
$$994$$ −18.4721 −0.585901
$$995$$ 0 0
$$996$$ −3.41641 −0.108253
$$997$$ −53.7771 −1.70314 −0.851569 0.524243i $$-0.824348\pi$$
−0.851569 + 0.524243i $$0.824348\pi$$
$$998$$ −23.7082 −0.750470
$$999$$ 3.47214 0.109854
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.t.1.1 2
5.2 odd 4 1850.2.b.j.149.2 4
5.3 odd 4 1850.2.b.j.149.3 4
5.4 even 2 74.2.a.b.1.2 2
15.14 odd 2 666.2.a.i.1.2 2
20.19 odd 2 592.2.a.g.1.1 2
35.34 odd 2 3626.2.a.s.1.1 2
40.19 odd 2 2368.2.a.u.1.2 2
40.29 even 2 2368.2.a.y.1.1 2
55.54 odd 2 8954.2.a.j.1.2 2
60.59 even 2 5328.2.a.bc.1.2 2
185.184 even 2 2738.2.a.g.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 2 5.4 even 2
592.2.a.g.1.1 2 20.19 odd 2
666.2.a.i.1.2 2 15.14 odd 2
1850.2.a.t.1.1 2 1.1 even 1 trivial
1850.2.b.j.149.2 4 5.2 odd 4
1850.2.b.j.149.3 4 5.3 odd 4
2368.2.a.u.1.2 2 40.19 odd 2
2368.2.a.y.1.1 2 40.29 even 2
2738.2.a.g.1.2 2 185.184 even 2
3626.2.a.s.1.1 2 35.34 odd 2
5328.2.a.bc.1.2 2 60.59 even 2
8954.2.a.j.1.2 2 55.54 odd 2