# Properties

 Label 693.2.a.l.1.2 Level $693$ Weight $2$ Character 693.1 Self dual yes Analytic conductor $5.534$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 693.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.53363286007$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.86081$$ of defining polynomial Character $$\chi$$ $$=$$ 693.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +O(q^{10})$$ $$q-1.46260 q^{2} +0.139194 q^{4} -2.39821 q^{5} -1.00000 q^{7} +2.72161 q^{8} +3.50761 q^{10} +1.00000 q^{11} +5.04502 q^{13} +1.46260 q^{14} -4.25901 q^{16} +6.36842 q^{17} -5.32340 q^{19} -0.333816 q^{20} -1.46260 q^{22} -4.92520 q^{23} +0.751399 q^{25} -7.37883 q^{26} -0.139194 q^{28} -5.04502 q^{29} -7.57201 q^{31} +0.786003 q^{32} -9.31444 q^{34} +2.39821 q^{35} +4.24860 q^{37} +7.78600 q^{38} -6.52699 q^{40} +0.646809 q^{41} -10.5180 q^{43} +0.139194 q^{44} +7.20359 q^{46} -0.526989 q^{47} +1.00000 q^{49} -1.09899 q^{50} +0.702237 q^{52} -3.72161 q^{53} -2.39821 q^{55} -2.72161 q^{56} +7.37883 q^{58} -7.97021 q^{59} -2.00000 q^{61} +11.0748 q^{62} +7.36842 q^{64} -12.0990 q^{65} +8.76663 q^{67} +0.886447 q^{68} -3.50761 q^{70} +11.4432 q^{71} -13.0450 q^{73} -6.21400 q^{74} -0.740987 q^{76} -1.00000 q^{77} +11.4432 q^{79} +10.2140 q^{80} -0.946021 q^{82} -13.1648 q^{83} -15.2728 q^{85} +15.3836 q^{86} +2.72161 q^{88} -11.8504 q^{89} -5.04502 q^{91} -0.685559 q^{92} +0.770774 q^{94} +12.7666 q^{95} -1.87122 q^{97} -1.46260 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + O(q^{10})$$ $$3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.46260 −1.03421 −0.517107 0.855921i $$-0.672991\pi$$
−0.517107 + 0.855921i $$0.672991\pi$$
$$3$$ 0 0
$$4$$ 0.139194 0.0695971
$$5$$ −2.39821 −1.07251 −0.536255 0.844056i $$-0.680162\pi$$
−0.536255 + 0.844056i $$0.680162\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 2.72161 0.962235
$$9$$ 0 0
$$10$$ 3.50761 1.10921
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 5.04502 1.39924 0.699618 0.714517i $$-0.253354\pi$$
0.699618 + 0.714517i $$0.253354\pi$$
$$14$$ 1.46260 0.390896
$$15$$ 0 0
$$16$$ −4.25901 −1.06475
$$17$$ 6.36842 1.54457 0.772284 0.635277i $$-0.219114\pi$$
0.772284 + 0.635277i $$0.219114\pi$$
$$18$$ 0 0
$$19$$ −5.32340 −1.22127 −0.610636 0.791911i $$-0.709086\pi$$
−0.610636 + 0.791911i $$0.709086\pi$$
$$20$$ −0.333816 −0.0746436
$$21$$ 0 0
$$22$$ −1.46260 −0.311827
$$23$$ −4.92520 −1.02697 −0.513487 0.858097i $$-0.671647\pi$$
−0.513487 + 0.858097i $$0.671647\pi$$
$$24$$ 0 0
$$25$$ 0.751399 0.150280
$$26$$ −7.37883 −1.44711
$$27$$ 0 0
$$28$$ −0.139194 −0.0263052
$$29$$ −5.04502 −0.936836 −0.468418 0.883507i $$-0.655176\pi$$
−0.468418 + 0.883507i $$0.655176\pi$$
$$30$$ 0 0
$$31$$ −7.57201 −1.35997 −0.679986 0.733225i $$-0.738014\pi$$
−0.679986 + 0.733225i $$0.738014\pi$$
$$32$$ 0.786003 0.138947
$$33$$ 0 0
$$34$$ −9.31444 −1.59741
$$35$$ 2.39821 0.405371
$$36$$ 0 0
$$37$$ 4.24860 0.698466 0.349233 0.937036i $$-0.386442\pi$$
0.349233 + 0.937036i $$0.386442\pi$$
$$38$$ 7.78600 1.26306
$$39$$ 0 0
$$40$$ −6.52699 −1.03201
$$41$$ 0.646809 0.101015 0.0505073 0.998724i $$-0.483916\pi$$
0.0505073 + 0.998724i $$0.483916\pi$$
$$42$$ 0 0
$$43$$ −10.5180 −1.60398 −0.801992 0.597335i $$-0.796226\pi$$
−0.801992 + 0.597335i $$0.796226\pi$$
$$44$$ 0.139194 0.0209843
$$45$$ 0 0
$$46$$ 7.20359 1.06211
$$47$$ −0.526989 −0.0768693 −0.0384347 0.999261i $$-0.512237\pi$$
−0.0384347 + 0.999261i $$0.512237\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −1.09899 −0.155421
$$51$$ 0 0
$$52$$ 0.702237 0.0973827
$$53$$ −3.72161 −0.511203 −0.255601 0.966782i $$-0.582273\pi$$
−0.255601 + 0.966782i $$0.582273\pi$$
$$54$$ 0 0
$$55$$ −2.39821 −0.323374
$$56$$ −2.72161 −0.363691
$$57$$ 0 0
$$58$$ 7.37883 0.968888
$$59$$ −7.97021 −1.03763 −0.518817 0.854886i $$-0.673627\pi$$
−0.518817 + 0.854886i $$0.673627\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 11.0748 1.40650
$$63$$ 0 0
$$64$$ 7.36842 0.921053
$$65$$ −12.0990 −1.50070
$$66$$ 0 0
$$67$$ 8.76663 1.07101 0.535507 0.844531i $$-0.320121\pi$$
0.535507 + 0.844531i $$0.320121\pi$$
$$68$$ 0.886447 0.107497
$$69$$ 0 0
$$70$$ −3.50761 −0.419240
$$71$$ 11.4432 1.35806 0.679030 0.734110i $$-0.262400\pi$$
0.679030 + 0.734110i $$0.262400\pi$$
$$72$$ 0 0
$$73$$ −13.0450 −1.52680 −0.763402 0.645924i $$-0.776472\pi$$
−0.763402 + 0.645924i $$0.776472\pi$$
$$74$$ −6.21400 −0.722363
$$75$$ 0 0
$$76$$ −0.740987 −0.0849970
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ 11.4432 1.28746 0.643732 0.765251i $$-0.277385\pi$$
0.643732 + 0.765251i $$0.277385\pi$$
$$80$$ 10.2140 1.14196
$$81$$ 0 0
$$82$$ −0.946021 −0.104471
$$83$$ −13.1648 −1.44503 −0.722514 0.691356i $$-0.757014\pi$$
−0.722514 + 0.691356i $$0.757014\pi$$
$$84$$ 0 0
$$85$$ −15.2728 −1.65657
$$86$$ 15.3836 1.65886
$$87$$ 0 0
$$88$$ 2.72161 0.290125
$$89$$ −11.8504 −1.25614 −0.628070 0.778157i $$-0.716155\pi$$
−0.628070 + 0.778157i $$0.716155\pi$$
$$90$$ 0 0
$$91$$ −5.04502 −0.528861
$$92$$ −0.685559 −0.0714744
$$93$$ 0 0
$$94$$ 0.770774 0.0794993
$$95$$ 12.7666 1.30983
$$96$$ 0 0
$$97$$ −1.87122 −0.189993 −0.0949967 0.995478i $$-0.530284\pi$$
−0.0949967 + 0.995478i $$0.530284\pi$$
$$98$$ −1.46260 −0.147745
$$99$$ 0 0
$$100$$ 0.104590 0.0104590
$$101$$ −4.51803 −0.449560 −0.224780 0.974409i $$-0.572166\pi$$
−0.224780 + 0.974409i $$0.572166\pi$$
$$102$$ 0 0
$$103$$ −10.6468 −1.04906 −0.524531 0.851392i $$-0.675759\pi$$
−0.524531 + 0.851392i $$0.675759\pi$$
$$104$$ 13.7306 1.34639
$$105$$ 0 0
$$106$$ 5.44322 0.528693
$$107$$ −15.9702 −1.54390 −0.771949 0.635684i $$-0.780718\pi$$
−0.771949 + 0.635684i $$0.780718\pi$$
$$108$$ 0 0
$$109$$ 12.7756 1.22368 0.611840 0.790982i $$-0.290430\pi$$
0.611840 + 0.790982i $$0.290430\pi$$
$$110$$ 3.50761 0.334438
$$111$$ 0 0
$$112$$ 4.25901 0.402439
$$113$$ −18.7368 −1.76261 −0.881307 0.472544i $$-0.843336\pi$$
−0.881307 + 0.472544i $$0.843336\pi$$
$$114$$ 0 0
$$115$$ 11.8116 1.10144
$$116$$ −0.702237 −0.0652010
$$117$$ 0 0
$$118$$ 11.6572 1.07313
$$119$$ −6.36842 −0.583792
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.92520 0.264835
$$123$$ 0 0
$$124$$ −1.05398 −0.0946501
$$125$$ 10.1890 0.911334
$$126$$ 0 0
$$127$$ −2.27839 −0.202174 −0.101087 0.994878i $$-0.532232\pi$$
−0.101087 + 0.994878i $$0.532232\pi$$
$$128$$ −12.3490 −1.09151
$$129$$ 0 0
$$130$$ 17.6960 1.55204
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 5.32340 0.461598
$$134$$ −12.8221 −1.10766
$$135$$ 0 0
$$136$$ 17.3324 1.48624
$$137$$ 4.77559 0.408006 0.204003 0.978970i $$-0.434605\pi$$
0.204003 + 0.978970i $$0.434605\pi$$
$$138$$ 0 0
$$139$$ −15.4432 −1.30988 −0.654939 0.755682i $$-0.727306\pi$$
−0.654939 + 0.755682i $$0.727306\pi$$
$$140$$ 0.333816 0.0282126
$$141$$ 0 0
$$142$$ −16.7368 −1.40452
$$143$$ 5.04502 0.421885
$$144$$ 0 0
$$145$$ 12.0990 1.00477
$$146$$ 19.0796 1.57904
$$147$$ 0 0
$$148$$ 0.591380 0.0486112
$$149$$ 9.84143 0.806241 0.403121 0.915147i $$-0.367925\pi$$
0.403121 + 0.915147i $$0.367925\pi$$
$$150$$ 0 0
$$151$$ −4.12878 −0.335996 −0.167998 0.985787i $$-0.553730\pi$$
−0.167998 + 0.985787i $$0.553730\pi$$
$$152$$ −14.4882 −1.17515
$$153$$ 0 0
$$154$$ 1.46260 0.117860
$$155$$ 18.1592 1.45859
$$156$$ 0 0
$$157$$ −0.946021 −0.0755007 −0.0377504 0.999287i $$-0.512019\pi$$
−0.0377504 + 0.999287i $$0.512019\pi$$
$$158$$ −16.7368 −1.33151
$$159$$ 0 0
$$160$$ −1.88500 −0.149022
$$161$$ 4.92520 0.388160
$$162$$ 0 0
$$163$$ 8.76663 0.686655 0.343328 0.939216i $$-0.388446\pi$$
0.343328 + 0.939216i $$0.388446\pi$$
$$164$$ 0.0900320 0.00703032
$$165$$ 0 0
$$166$$ 19.2549 1.49447
$$167$$ 24.3684 1.88568 0.942842 0.333239i $$-0.108142\pi$$
0.942842 + 0.333239i $$0.108142\pi$$
$$168$$ 0 0
$$169$$ 12.4522 0.957860
$$170$$ 22.3380 1.71324
$$171$$ 0 0
$$172$$ −1.46405 −0.111633
$$173$$ −12.3476 −0.938770 −0.469385 0.882994i $$-0.655524\pi$$
−0.469385 + 0.882994i $$0.655524\pi$$
$$174$$ 0 0
$$175$$ −0.751399 −0.0568004
$$176$$ −4.25901 −0.321035
$$177$$ 0 0
$$178$$ 17.3324 1.29912
$$179$$ 5.59283 0.418028 0.209014 0.977913i $$-0.432975\pi$$
0.209014 + 0.977913i $$0.432975\pi$$
$$180$$ 0 0
$$181$$ 13.5720 1.00880 0.504400 0.863470i $$-0.331714\pi$$
0.504400 + 0.863470i $$0.331714\pi$$
$$182$$ 7.37883 0.546955
$$183$$ 0 0
$$184$$ −13.4045 −0.988191
$$185$$ −10.1890 −0.749112
$$186$$ 0 0
$$187$$ 6.36842 0.465705
$$188$$ −0.0733538 −0.00534988
$$189$$ 0 0
$$190$$ −18.6724 −1.35464
$$191$$ 9.42240 0.681781 0.340890 0.940103i $$-0.389271\pi$$
0.340890 + 0.940103i $$0.389271\pi$$
$$192$$ 0 0
$$193$$ −10.1288 −0.729086 −0.364543 0.931187i $$-0.618775\pi$$
−0.364543 + 0.931187i $$0.618775\pi$$
$$194$$ 2.73684 0.196494
$$195$$ 0 0
$$196$$ 0.139194 0.00994244
$$197$$ 2.25756 0.160845 0.0804224 0.996761i $$-0.474373\pi$$
0.0804224 + 0.996761i $$0.474373\pi$$
$$198$$ 0 0
$$199$$ 3.07480 0.217967 0.108984 0.994044i $$-0.465240\pi$$
0.108984 + 0.994044i $$0.465240\pi$$
$$200$$ 2.04502 0.144604
$$201$$ 0 0
$$202$$ 6.60806 0.464941
$$203$$ 5.04502 0.354091
$$204$$ 0 0
$$205$$ −1.55118 −0.108339
$$206$$ 15.5720 1.08495
$$207$$ 0 0
$$208$$ −21.4868 −1.48984
$$209$$ −5.32340 −0.368228
$$210$$ 0 0
$$211$$ −14.6468 −1.00833 −0.504164 0.863608i $$-0.668199\pi$$
−0.504164 + 0.863608i $$0.668199\pi$$
$$212$$ −0.518027 −0.0355782
$$213$$ 0 0
$$214$$ 23.3580 1.59672
$$215$$ 25.2244 1.72029
$$216$$ 0 0
$$217$$ 7.57201 0.514021
$$218$$ −18.6856 −1.26555
$$219$$ 0 0
$$220$$ −0.333816 −0.0225059
$$221$$ 32.1288 2.16122
$$222$$ 0 0
$$223$$ −1.90997 −0.127901 −0.0639505 0.997953i $$-0.520370\pi$$
−0.0639505 + 0.997953i $$0.520370\pi$$
$$224$$ −0.786003 −0.0525170
$$225$$ 0 0
$$226$$ 27.4045 1.82292
$$227$$ 3.20359 0.212629 0.106315 0.994333i $$-0.466095\pi$$
0.106315 + 0.994333i $$0.466095\pi$$
$$228$$ 0 0
$$229$$ −18.3088 −1.20988 −0.604941 0.796270i $$-0.706803\pi$$
−0.604941 + 0.796270i $$0.706803\pi$$
$$230$$ −17.2757 −1.13913
$$231$$ 0 0
$$232$$ −13.7306 −0.901456
$$233$$ 16.5872 1.08667 0.543333 0.839517i $$-0.317162\pi$$
0.543333 + 0.839517i $$0.317162\pi$$
$$234$$ 0 0
$$235$$ 1.26383 0.0824432
$$236$$ −1.10941 −0.0722162
$$237$$ 0 0
$$238$$ 9.31444 0.603766
$$239$$ −2.91623 −0.188635 −0.0943177 0.995542i $$-0.530067\pi$$
−0.0943177 + 0.995542i $$0.530067\pi$$
$$240$$ 0 0
$$241$$ −6.09899 −0.392871 −0.196435 0.980517i $$-0.562937\pi$$
−0.196435 + 0.980517i $$0.562937\pi$$
$$242$$ −1.46260 −0.0940194
$$243$$ 0 0
$$244$$ −0.278388 −0.0178220
$$245$$ −2.39821 −0.153216
$$246$$ 0 0
$$247$$ −26.8567 −1.70885
$$248$$ −20.6081 −1.30861
$$249$$ 0 0
$$250$$ −14.9025 −0.942514
$$251$$ −1.62262 −0.102419 −0.0512093 0.998688i $$-0.516308\pi$$
−0.0512093 + 0.998688i $$0.516308\pi$$
$$252$$ 0 0
$$253$$ −4.92520 −0.309644
$$254$$ 3.33237 0.209091
$$255$$ 0 0
$$256$$ 3.32485 0.207803
$$257$$ 6.89541 0.430124 0.215062 0.976600i $$-0.431005\pi$$
0.215062 + 0.976600i $$0.431005\pi$$
$$258$$ 0 0
$$259$$ −4.24860 −0.263995
$$260$$ −1.68411 −0.104444
$$261$$ 0 0
$$262$$ 5.85039 0.361439
$$263$$ −5.08377 −0.313478 −0.156739 0.987640i $$-0.550098\pi$$
−0.156739 + 0.987640i $$0.550098\pi$$
$$264$$ 0 0
$$265$$ 8.92520 0.548270
$$266$$ −7.78600 −0.477390
$$267$$ 0 0
$$268$$ 1.22026 0.0745394
$$269$$ 0.886447 0.0540476 0.0270238 0.999635i $$-0.491397\pi$$
0.0270238 + 0.999635i $$0.491397\pi$$
$$270$$ 0 0
$$271$$ −25.3234 −1.53829 −0.769144 0.639076i $$-0.779317\pi$$
−0.769144 + 0.639076i $$0.779317\pi$$
$$272$$ −27.1232 −1.64458
$$273$$ 0 0
$$274$$ −6.98477 −0.421965
$$275$$ 0.751399 0.0453111
$$276$$ 0 0
$$277$$ −24.8269 −1.49170 −0.745851 0.666113i $$-0.767957\pi$$
−0.745851 + 0.666113i $$0.767957\pi$$
$$278$$ 22.5872 1.35469
$$279$$ 0 0
$$280$$ 6.52699 0.390062
$$281$$ −1.90101 −0.113404 −0.0567022 0.998391i $$-0.518059\pi$$
−0.0567022 + 0.998391i $$0.518059\pi$$
$$282$$ 0 0
$$283$$ −22.3178 −1.32666 −0.663328 0.748329i $$-0.730857\pi$$
−0.663328 + 0.748329i $$0.730857\pi$$
$$284$$ 1.59283 0.0945171
$$285$$ 0 0
$$286$$ −7.37883 −0.436320
$$287$$ −0.646809 −0.0381799
$$288$$ 0 0
$$289$$ 23.5568 1.38569
$$290$$ −17.6960 −1.03914
$$291$$ 0 0
$$292$$ −1.81579 −0.106261
$$293$$ −12.0900 −0.706307 −0.353154 0.935565i $$-0.614891\pi$$
−0.353154 + 0.935565i $$0.614891\pi$$
$$294$$ 0 0
$$295$$ 19.1142 1.11287
$$296$$ 11.5630 0.672088
$$297$$ 0 0
$$298$$ −14.3941 −0.833826
$$299$$ −24.8477 −1.43698
$$300$$ 0 0
$$301$$ 10.5180 0.606249
$$302$$ 6.03875 0.347491
$$303$$ 0 0
$$304$$ 22.6724 1.30035
$$305$$ 4.79641 0.274642
$$306$$ 0 0
$$307$$ −13.5928 −0.775784 −0.387892 0.921705i $$-0.626797\pi$$
−0.387892 + 0.921705i $$0.626797\pi$$
$$308$$ −0.139194 −0.00793132
$$309$$ 0 0
$$310$$ −26.5597 −1.50849
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 14.9252 0.843622 0.421811 0.906684i $$-0.361395\pi$$
0.421811 + 0.906684i $$0.361395\pi$$
$$314$$ 1.38365 0.0780838
$$315$$ 0 0
$$316$$ 1.59283 0.0896037
$$317$$ −3.97918 −0.223493 −0.111746 0.993737i $$-0.535644\pi$$
−0.111746 + 0.993737i $$0.535644\pi$$
$$318$$ 0 0
$$319$$ −5.04502 −0.282467
$$320$$ −17.6710 −0.987839
$$321$$ 0 0
$$322$$ −7.20359 −0.401440
$$323$$ −33.9017 −1.88634
$$324$$ 0 0
$$325$$ 3.79082 0.210277
$$326$$ −12.8221 −0.710148
$$327$$ 0 0
$$328$$ 1.76036 0.0971997
$$329$$ 0.526989 0.0290539
$$330$$ 0 0
$$331$$ 23.4432 1.28856 0.644278 0.764791i $$-0.277158\pi$$
0.644278 + 0.764791i $$0.277158\pi$$
$$332$$ −1.83247 −0.100570
$$333$$ 0 0
$$334$$ −35.6412 −1.95020
$$335$$ −21.0242 −1.14867
$$336$$ 0 0
$$337$$ 11.1648 0.608187 0.304094 0.952642i $$-0.401646\pi$$
0.304094 + 0.952642i $$0.401646\pi$$
$$338$$ −18.2125 −0.990632
$$339$$ 0 0
$$340$$ −2.12588 −0.115292
$$341$$ −7.57201 −0.410047
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −28.6260 −1.54341
$$345$$ 0 0
$$346$$ 18.0596 0.970889
$$347$$ 22.5872 1.21255 0.606273 0.795256i $$-0.292664\pi$$
0.606273 + 0.795256i $$0.292664\pi$$
$$348$$ 0 0
$$349$$ 27.9315 1.49514 0.747568 0.664185i $$-0.231221\pi$$
0.747568 + 0.664185i $$0.231221\pi$$
$$350$$ 1.09899 0.0587437
$$351$$ 0 0
$$352$$ 0.786003 0.0418941
$$353$$ 16.5478 0.880751 0.440376 0.897814i $$-0.354845\pi$$
0.440376 + 0.897814i $$0.354845\pi$$
$$354$$ 0 0
$$355$$ −27.4432 −1.45654
$$356$$ −1.64951 −0.0874236
$$357$$ 0 0
$$358$$ −8.18006 −0.432330
$$359$$ −22.0305 −1.16272 −0.581362 0.813645i $$-0.697480\pi$$
−0.581362 + 0.813645i $$0.697480\pi$$
$$360$$ 0 0
$$361$$ 9.33863 0.491507
$$362$$ −19.8504 −1.04331
$$363$$ 0 0
$$364$$ −0.702237 −0.0368072
$$365$$ 31.2847 1.63751
$$366$$ 0 0
$$367$$ −19.3836 −1.01182 −0.505909 0.862587i $$-0.668843\pi$$
−0.505909 + 0.862587i $$0.668843\pi$$
$$368$$ 20.9765 1.09347
$$369$$ 0 0
$$370$$ 14.9025 0.774742
$$371$$ 3.72161 0.193216
$$372$$ 0 0
$$373$$ 29.2549 1.51476 0.757380 0.652975i $$-0.226479\pi$$
0.757380 + 0.652975i $$0.226479\pi$$
$$374$$ −9.31444 −0.481638
$$375$$ 0 0
$$376$$ −1.43426 −0.0739663
$$377$$ −25.4522 −1.31085
$$378$$ 0 0
$$379$$ 12.5270 0.643468 0.321734 0.946830i $$-0.395734\pi$$
0.321734 + 0.946830i $$0.395734\pi$$
$$380$$ 1.77704 0.0911602
$$381$$ 0 0
$$382$$ −13.7812 −0.705107
$$383$$ 17.5928 0.898952 0.449476 0.893293i $$-0.351611\pi$$
0.449476 + 0.893293i $$0.351611\pi$$
$$384$$ 0 0
$$385$$ 2.39821 0.122224
$$386$$ 14.8143 0.754030
$$387$$ 0 0
$$388$$ −0.260463 −0.0132230
$$389$$ −20.0900 −1.01861 −0.509303 0.860588i $$-0.670097\pi$$
−0.509303 + 0.860588i $$0.670097\pi$$
$$390$$ 0 0
$$391$$ −31.3657 −1.58623
$$392$$ 2.72161 0.137462
$$393$$ 0 0
$$394$$ −3.30191 −0.166348
$$395$$ −27.4432 −1.38082
$$396$$ 0 0
$$397$$ −35.1053 −1.76188 −0.880941 0.473226i $$-0.843090\pi$$
−0.880941 + 0.473226i $$0.843090\pi$$
$$398$$ −4.49720 −0.225424
$$399$$ 0 0
$$400$$ −3.20022 −0.160011
$$401$$ −9.57201 −0.478003 −0.239002 0.971019i $$-0.576820\pi$$
−0.239002 + 0.971019i $$0.576820\pi$$
$$402$$ 0 0
$$403$$ −38.2009 −1.90292
$$404$$ −0.628883 −0.0312881
$$405$$ 0 0
$$406$$ −7.37883 −0.366205
$$407$$ 4.24860 0.210595
$$408$$ 0 0
$$409$$ 38.1801 1.88788 0.943941 0.330113i $$-0.107087\pi$$
0.943941 + 0.330113i $$0.107087\pi$$
$$410$$ 2.26875 0.112046
$$411$$ 0 0
$$412$$ −1.48197 −0.0730116
$$413$$ 7.97021 0.392189
$$414$$ 0 0
$$415$$ 31.5720 1.54981
$$416$$ 3.96540 0.194420
$$417$$ 0 0
$$418$$ 7.78600 0.380826
$$419$$ −7.17380 −0.350463 −0.175231 0.984527i $$-0.556067\pi$$
−0.175231 + 0.984527i $$0.556067\pi$$
$$420$$ 0 0
$$421$$ 15.1530 0.738511 0.369255 0.929328i $$-0.379613\pi$$
0.369255 + 0.929328i $$0.379613\pi$$
$$422$$ 21.4224 1.04283
$$423$$ 0 0
$$424$$ −10.1288 −0.491897
$$425$$ 4.78522 0.232117
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ −2.22296 −0.107451
$$429$$ 0 0
$$430$$ −36.8932 −1.77915
$$431$$ −5.56304 −0.267962 −0.133981 0.990984i $$-0.542776\pi$$
−0.133981 + 0.990984i $$0.542776\pi$$
$$432$$ 0 0
$$433$$ −25.6412 −1.23224 −0.616119 0.787653i $$-0.711296\pi$$
−0.616119 + 0.787653i $$0.711296\pi$$
$$434$$ −11.0748 −0.531608
$$435$$ 0 0
$$436$$ 1.77829 0.0851645
$$437$$ 26.2188 1.25422
$$438$$ 0 0
$$439$$ 23.6710 1.12976 0.564878 0.825175i $$-0.308923\pi$$
0.564878 + 0.825175i $$0.308923\pi$$
$$440$$ −6.52699 −0.311162
$$441$$ 0 0
$$442$$ −46.9915 −2.23516
$$443$$ 18.0305 0.856653 0.428326 0.903624i $$-0.359103\pi$$
0.428326 + 0.903624i $$0.359103\pi$$
$$444$$ 0 0
$$445$$ 28.4197 1.34722
$$446$$ 2.79352 0.132277
$$447$$ 0 0
$$448$$ −7.36842 −0.348125
$$449$$ 34.9765 1.65064 0.825321 0.564664i $$-0.190994\pi$$
0.825321 + 0.564664i $$0.190994\pi$$
$$450$$ 0 0
$$451$$ 0.646809 0.0304570
$$452$$ −2.60806 −0.122673
$$453$$ 0 0
$$454$$ −4.68556 −0.219904
$$455$$ 12.0990 0.567210
$$456$$ 0 0
$$457$$ 4.53595 0.212183 0.106091 0.994356i $$-0.466166\pi$$
0.106091 + 0.994356i $$0.466166\pi$$
$$458$$ 26.7785 1.25128
$$459$$ 0 0
$$460$$ 1.64411 0.0766571
$$461$$ 2.79641 0.130242 0.0651210 0.997877i $$-0.479257\pi$$
0.0651210 + 0.997877i $$0.479257\pi$$
$$462$$ 0 0
$$463$$ 38.3595 1.78272 0.891358 0.453301i $$-0.149754\pi$$
0.891358 + 0.453301i $$0.149754\pi$$
$$464$$ 21.4868 0.997499
$$465$$ 0 0
$$466$$ −24.2605 −1.12384
$$467$$ 20.4674 0.947119 0.473560 0.880762i $$-0.342969\pi$$
0.473560 + 0.880762i $$0.342969\pi$$
$$468$$ 0 0
$$469$$ −8.76663 −0.404805
$$470$$ −1.84848 −0.0852638
$$471$$ 0 0
$$472$$ −21.6918 −0.998447
$$473$$ −10.5180 −0.483619
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ −0.886447 −0.0406302
$$477$$ 0 0
$$478$$ 4.26528 0.195089
$$479$$ 11.6137 0.530641 0.265321 0.964160i $$-0.414522\pi$$
0.265321 + 0.964160i $$0.414522\pi$$
$$480$$ 0 0
$$481$$ 21.4343 0.977318
$$482$$ 8.92038 0.406312
$$483$$ 0 0
$$484$$ 0.139194 0.00632701
$$485$$ 4.48757 0.203770
$$486$$ 0 0
$$487$$ 32.4793 1.47178 0.735888 0.677103i $$-0.236765\pi$$
0.735888 + 0.677103i $$0.236765\pi$$
$$488$$ −5.44322 −0.246403
$$489$$ 0 0
$$490$$ 3.50761 0.158458
$$491$$ −26.6766 −1.20390 −0.601949 0.798535i $$-0.705609\pi$$
−0.601949 + 0.798535i $$0.705609\pi$$
$$492$$ 0 0
$$493$$ −32.1288 −1.44701
$$494$$ 39.2805 1.76731
$$495$$ 0 0
$$496$$ 32.2493 1.44804
$$497$$ −11.4432 −0.513299
$$498$$ 0 0
$$499$$ −41.2459 −1.84642 −0.923210 0.384296i $$-0.874444\pi$$
−0.923210 + 0.384296i $$0.874444\pi$$
$$500$$ 1.41825 0.0634262
$$501$$ 0 0
$$502$$ 2.37324 0.105923
$$503$$ −30.5180 −1.36073 −0.680366 0.732873i $$-0.738179\pi$$
−0.680366 + 0.732873i $$0.738179\pi$$
$$504$$ 0 0
$$505$$ 10.8352 0.482159
$$506$$ 7.20359 0.320238
$$507$$ 0 0
$$508$$ −0.317138 −0.0140707
$$509$$ 18.9944 0.841912 0.420956 0.907081i $$-0.361695\pi$$
0.420956 + 0.907081i $$0.361695\pi$$
$$510$$ 0 0
$$511$$ 13.0450 0.577078
$$512$$ 19.8352 0.876599
$$513$$ 0 0
$$514$$ −10.0852 −0.444840
$$515$$ 25.5333 1.12513
$$516$$ 0 0
$$517$$ −0.526989 −0.0231770
$$518$$ 6.21400 0.273027
$$519$$ 0 0
$$520$$ −32.9288 −1.44402
$$521$$ −25.2430 −1.10592 −0.552958 0.833209i $$-0.686501\pi$$
−0.552958 + 0.833209i $$0.686501\pi$$
$$522$$ 0 0
$$523$$ −2.93416 −0.128302 −0.0641509 0.997940i $$-0.520434\pi$$
−0.0641509 + 0.997940i $$0.520434\pi$$
$$524$$ −0.556777 −0.0243229
$$525$$ 0 0
$$526$$ 7.43551 0.324204
$$527$$ −48.2217 −2.10057
$$528$$ 0 0
$$529$$ 1.25756 0.0546767
$$530$$ −13.0540 −0.567029
$$531$$ 0 0
$$532$$ 0.740987 0.0321258
$$533$$ 3.26316 0.141343
$$534$$ 0 0
$$535$$ 38.2999 1.65585
$$536$$ 23.8594 1.03057
$$537$$ 0 0
$$538$$ −1.29652 −0.0558968
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 8.90437 0.382829 0.191414 0.981509i $$-0.438693\pi$$
0.191414 + 0.981509i $$0.438693\pi$$
$$542$$ 37.0380 1.59092
$$543$$ 0 0
$$544$$ 5.00560 0.214613
$$545$$ −30.6385 −1.31241
$$546$$ 0 0
$$547$$ −29.4737 −1.26020 −0.630102 0.776513i $$-0.716987\pi$$
−0.630102 + 0.776513i $$0.716987\pi$$
$$548$$ 0.664734 0.0283960
$$549$$ 0 0
$$550$$ −1.09899 −0.0468613
$$551$$ 26.8567 1.14413
$$552$$ 0 0
$$553$$ −11.4432 −0.486615
$$554$$ 36.3117 1.54274
$$555$$ 0 0
$$556$$ −2.14961 −0.0911636
$$557$$ −14.8954 −0.631139 −0.315569 0.948903i $$-0.602196\pi$$
−0.315569 + 0.948903i $$0.602196\pi$$
$$558$$ 0 0
$$559$$ −53.0636 −2.24435
$$560$$ −10.2140 −0.431620
$$561$$ 0 0
$$562$$ 2.78041 0.117284
$$563$$ 7.81164 0.329222 0.164611 0.986359i $$-0.447363\pi$$
0.164611 + 0.986359i $$0.447363\pi$$
$$564$$ 0 0
$$565$$ 44.9348 1.89042
$$566$$ 32.6420 1.37205
$$567$$ 0 0
$$568$$ 31.1440 1.30677
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 25.5512 1.06928 0.534642 0.845079i $$-0.320447\pi$$
0.534642 + 0.845079i $$0.320447\pi$$
$$572$$ 0.702237 0.0293620
$$573$$ 0 0
$$574$$ 0.946021 0.0394862
$$575$$ −3.70079 −0.154334
$$576$$ 0 0
$$577$$ −29.5124 −1.22862 −0.614309 0.789065i $$-0.710565\pi$$
−0.614309 + 0.789065i $$0.710565\pi$$
$$578$$ −34.4541 −1.43310
$$579$$ 0 0
$$580$$ 1.68411 0.0699288
$$581$$ 13.1648 0.546169
$$582$$ 0 0
$$583$$ −3.72161 −0.154133
$$584$$ −35.5035 −1.46914
$$585$$ 0 0
$$586$$ 17.6829 0.730472
$$587$$ −31.3955 −1.29583 −0.647916 0.761712i $$-0.724359\pi$$
−0.647916 + 0.761712i $$0.724359\pi$$
$$588$$ 0 0
$$589$$ 40.3088 1.66090
$$590$$ −27.9564 −1.15095
$$591$$ 0 0
$$592$$ −18.0948 −0.743694
$$593$$ 7.90997 0.324823 0.162412 0.986723i $$-0.448073\pi$$
0.162412 + 0.986723i $$0.448073\pi$$
$$594$$ 0 0
$$595$$ 15.2728 0.626123
$$596$$ 1.36987 0.0561120
$$597$$ 0 0
$$598$$ 36.3422 1.48614
$$599$$ −27.4432 −1.12130 −0.560650 0.828053i $$-0.689449\pi$$
−0.560650 + 0.828053i $$0.689449\pi$$
$$600$$ 0 0
$$601$$ 31.9910 1.30494 0.652471 0.757814i $$-0.273732\pi$$
0.652471 + 0.757814i $$0.273732\pi$$
$$602$$ −15.3836 −0.626991
$$603$$ 0 0
$$604$$ −0.574702 −0.0233843
$$605$$ −2.39821 −0.0975010
$$606$$ 0 0
$$607$$ 7.41344 0.300902 0.150451 0.988617i $$-0.451927\pi$$
0.150451 + 0.988617i $$0.451927\pi$$
$$608$$ −4.18421 −0.169692
$$609$$ 0 0
$$610$$ −7.01523 −0.284038
$$611$$ −2.65867 −0.107558
$$612$$ 0 0
$$613$$ −33.9917 −1.37291 −0.686456 0.727171i $$-0.740835\pi$$
−0.686456 + 0.727171i $$0.740835\pi$$
$$614$$ 19.8809 0.802326
$$615$$ 0 0
$$616$$ −2.72161 −0.109657
$$617$$ 44.0305 1.77260 0.886300 0.463112i $$-0.153267\pi$$
0.886300 + 0.463112i $$0.153267\pi$$
$$618$$ 0 0
$$619$$ 40.0096 1.60812 0.804061 0.594546i $$-0.202668\pi$$
0.804061 + 0.594546i $$0.202668\pi$$
$$620$$ 2.52766 0.101513
$$621$$ 0 0
$$622$$ 11.7008 0.469159
$$623$$ 11.8504 0.474776
$$624$$ 0 0
$$625$$ −28.1924 −1.12770
$$626$$ −21.8296 −0.872485
$$627$$ 0 0
$$628$$ −0.131681 −0.00525463
$$629$$ 27.0569 1.07883
$$630$$ 0 0
$$631$$ 28.5568 1.13683 0.568414 0.822743i $$-0.307557\pi$$
0.568414 + 0.822743i $$0.307557\pi$$
$$632$$ 31.1440 1.23884
$$633$$ 0 0
$$634$$ 5.81994 0.231139
$$635$$ 5.46405 0.216834
$$636$$ 0 0
$$637$$ 5.04502 0.199891
$$638$$ 7.37883 0.292131
$$639$$ 0 0
$$640$$ 29.6156 1.17066
$$641$$ 31.1053 1.22858 0.614292 0.789079i $$-0.289442\pi$$
0.614292 + 0.789079i $$0.289442\pi$$
$$642$$ 0 0
$$643$$ −5.48197 −0.216188 −0.108094 0.994141i $$-0.534475\pi$$
−0.108094 + 0.994141i $$0.534475\pi$$
$$644$$ 0.685559 0.0270148
$$645$$ 0 0
$$646$$ 49.5845 1.95088
$$647$$ 9.26383 0.364199 0.182099 0.983280i $$-0.441711\pi$$
0.182099 + 0.983280i $$0.441711\pi$$
$$648$$ 0 0
$$649$$ −7.97021 −0.312858
$$650$$ −5.54445 −0.217471
$$651$$ 0 0
$$652$$ 1.22026 0.0477892
$$653$$ 29.9821 1.17329 0.586645 0.809844i $$-0.300449\pi$$
0.586645 + 0.809844i $$0.300449\pi$$
$$654$$ 0 0
$$655$$ 9.59283 0.374823
$$656$$ −2.75477 −0.107556
$$657$$ 0 0
$$658$$ −0.770774 −0.0300479
$$659$$ −23.9702 −0.933747 −0.466873 0.884324i $$-0.654620\pi$$
−0.466873 + 0.884324i $$0.654620\pi$$
$$660$$ 0 0
$$661$$ −40.4585 −1.57365 −0.786826 0.617175i $$-0.788277\pi$$
−0.786826 + 0.617175i $$0.788277\pi$$
$$662$$ −34.2880 −1.33264
$$663$$ 0 0
$$664$$ −35.8296 −1.39046
$$665$$ −12.7666 −0.495069
$$666$$ 0 0
$$667$$ 24.8477 0.962107
$$668$$ 3.39194 0.131238
$$669$$ 0 0
$$670$$ 30.7499 1.18797
$$671$$ −2.00000 −0.0772091
$$672$$ 0 0
$$673$$ 21.8712 0.843073 0.421537 0.906811i $$-0.361491\pi$$
0.421537 + 0.906811i $$0.361491\pi$$
$$674$$ −16.3297 −0.628995
$$675$$ 0 0
$$676$$ 1.73327 0.0666643
$$677$$ 1.26316 0.0485472 0.0242736 0.999705i $$-0.492273\pi$$
0.0242736 + 0.999705i $$0.492273\pi$$
$$678$$ 0 0
$$679$$ 1.87122 0.0718108
$$680$$ −41.5666 −1.59401
$$681$$ 0 0
$$682$$ 11.0748 0.424076
$$683$$ −37.6441 −1.44041 −0.720206 0.693760i $$-0.755953\pi$$
−0.720206 + 0.693760i $$0.755953\pi$$
$$684$$ 0 0
$$685$$ −11.4529 −0.437591
$$686$$ 1.46260 0.0558423
$$687$$ 0 0
$$688$$ 44.7964 1.70785
$$689$$ −18.7756 −0.715293
$$690$$ 0 0
$$691$$ 14.3892 0.547393 0.273696 0.961816i $$-0.411754\pi$$
0.273696 + 0.961816i $$0.411754\pi$$
$$692$$ −1.71871 −0.0653357
$$693$$ 0 0
$$694$$ −33.0361 −1.25403
$$695$$ 37.0361 1.40486
$$696$$ 0 0
$$697$$ 4.11915 0.156024
$$698$$ −40.8525 −1.54629
$$699$$ 0 0
$$700$$ −0.104590 −0.00395314
$$701$$ 39.2936 1.48410 0.742050 0.670345i $$-0.233854\pi$$
0.742050 + 0.670345i $$0.233854\pi$$
$$702$$ 0 0
$$703$$ −22.6170 −0.853017
$$704$$ 7.36842 0.277708
$$705$$ 0 0
$$706$$ −24.2028 −0.910885
$$707$$ 4.51803 0.169918
$$708$$ 0 0
$$709$$ −49.2430 −1.84936 −0.924680 0.380745i $$-0.875667\pi$$
−0.924680 + 0.380745i $$0.875667\pi$$
$$710$$ 40.1384 1.50637
$$711$$ 0 0
$$712$$ −32.2522 −1.20870
$$713$$ 37.2936 1.39666
$$714$$ 0 0
$$715$$ −12.0990 −0.452477
$$716$$ 0.778489 0.0290935
$$717$$ 0 0
$$718$$ 32.2217 1.20250
$$719$$ 7.41344 0.276475 0.138237 0.990399i $$-0.455856\pi$$
0.138237 + 0.990399i $$0.455856\pi$$
$$720$$ 0 0
$$721$$ 10.6468 0.396508
$$722$$ −13.6587 −0.508323
$$723$$ 0 0
$$724$$ 1.88914 0.0702095
$$725$$ −3.79082 −0.140787
$$726$$ 0 0
$$727$$ 18.9557 0.703026 0.351513 0.936183i $$-0.385667\pi$$
0.351513 + 0.936183i $$0.385667\pi$$
$$728$$ −13.7306 −0.508889
$$729$$ 0 0
$$730$$ −45.7569 −1.69354
$$731$$ −66.9832 −2.47746
$$732$$ 0 0
$$733$$ −3.59283 −0.132704 −0.0663521 0.997796i $$-0.521136\pi$$
−0.0663521 + 0.997796i $$0.521136\pi$$
$$734$$ 28.3505 1.04644
$$735$$ 0 0
$$736$$ −3.87122 −0.142695
$$737$$ 8.76663 0.322923
$$738$$ 0 0
$$739$$ −26.7756 −0.984956 −0.492478 0.870325i $$-0.663909\pi$$
−0.492478 + 0.870325i $$0.663909\pi$$
$$740$$ −1.41825 −0.0521360
$$741$$ 0 0
$$742$$ −5.44322 −0.199827
$$743$$ 33.8027 1.24010 0.620050 0.784562i $$-0.287112\pi$$
0.620050 + 0.784562i $$0.287112\pi$$
$$744$$ 0 0
$$745$$ −23.6018 −0.864703
$$746$$ −42.7881 −1.56658
$$747$$ 0 0
$$748$$ 0.886447 0.0324117
$$749$$ 15.9702 0.583539
$$750$$ 0 0
$$751$$ 35.3955 1.29160 0.645800 0.763506i $$-0.276524\pi$$
0.645800 + 0.763506i $$0.276524\pi$$
$$752$$ 2.24445 0.0818468
$$753$$ 0 0
$$754$$ 37.2263 1.35570
$$755$$ 9.90168 0.360359
$$756$$ 0 0
$$757$$ −29.3442 −1.06653 −0.533267 0.845947i $$-0.679036\pi$$
−0.533267 + 0.845947i $$0.679036\pi$$
$$758$$ −18.3220 −0.665483
$$759$$ 0 0
$$760$$ 34.7458 1.26036
$$761$$ −12.9044 −0.467783 −0.233892 0.972263i $$-0.575146\pi$$
−0.233892 + 0.972263i $$0.575146\pi$$
$$762$$ 0 0
$$763$$ −12.7756 −0.462507
$$764$$ 1.31154 0.0474500
$$765$$ 0 0
$$766$$ −25.7312 −0.929708
$$767$$ −40.2099 −1.45189
$$768$$ 0 0
$$769$$ 9.78186 0.352743 0.176371 0.984324i $$-0.443564\pi$$
0.176371 + 0.984324i $$0.443564\pi$$
$$770$$ −3.50761 −0.126406
$$771$$ 0 0
$$772$$ −1.40987 −0.0507422
$$773$$ −29.7223 −1.06904 −0.534518 0.845157i $$-0.679507\pi$$
−0.534518 + 0.845157i $$0.679507\pi$$
$$774$$ 0 0
$$775$$ −5.68960 −0.204376
$$776$$ −5.09273 −0.182818
$$777$$ 0 0
$$778$$ 29.3836 1.05345
$$779$$ −3.44322 −0.123366
$$780$$ 0 0
$$781$$ 11.4432 0.409471
$$782$$ 45.8755 1.64050
$$783$$ 0 0
$$784$$ −4.25901 −0.152108
$$785$$ 2.26875 0.0809753
$$786$$ 0 0
$$787$$ 17.0242 0.606847 0.303423 0.952856i $$-0.401870\pi$$
0.303423 + 0.952856i $$0.401870\pi$$
$$788$$ 0.314240 0.0111943
$$789$$ 0 0
$$790$$ 40.1384 1.42806
$$791$$ 18.7368 0.666205
$$792$$ 0 0
$$793$$ −10.0900 −0.358308
$$794$$ 51.3449 1.82216
$$795$$ 0 0
$$796$$ 0.427995 0.0151699
$$797$$ 36.4287 1.29037 0.645185 0.764027i $$-0.276780\pi$$
0.645185 + 0.764027i $$0.276780\pi$$
$$798$$ 0 0
$$799$$ −3.35609 −0.118730
$$800$$ 0.590602 0.0208809
$$801$$ 0 0
$$802$$ 14.0000 0.494357
$$803$$ −13.0450 −0.460349
$$804$$ 0 0
$$805$$ −11.8116 −0.416306
$$806$$ 55.8726 1.96803
$$807$$ 0 0
$$808$$ −12.2963 −0.432583
$$809$$ −44.4882 −1.56412 −0.782062 0.623201i $$-0.785832\pi$$
−0.782062 + 0.623201i $$0.785832\pi$$
$$810$$ 0 0
$$811$$ 7.65307 0.268736 0.134368 0.990932i $$-0.457100\pi$$
0.134368 + 0.990932i $$0.457100\pi$$
$$812$$ 0.702237 0.0246437
$$813$$ 0 0
$$814$$ −6.21400 −0.217800
$$815$$ −21.0242 −0.736445
$$816$$ 0 0
$$817$$ 55.9917 1.95890
$$818$$ −55.8421 −1.95247
$$819$$ 0 0
$$820$$ −0.215915 −0.00754009
$$821$$ 44.3691 1.54849 0.774246 0.632885i $$-0.218129\pi$$
0.774246 + 0.632885i $$0.218129\pi$$
$$822$$ 0 0
$$823$$ 6.61702 0.230655 0.115327 0.993328i $$-0.463208\pi$$
0.115327 + 0.993328i $$0.463208\pi$$
$$824$$ −28.9765 −1.00944
$$825$$ 0 0
$$826$$ −11.6572 −0.405607
$$827$$ −39.7126 −1.38094 −0.690472 0.723359i $$-0.742597\pi$$
−0.690472 + 0.723359i $$0.742597\pi$$
$$828$$ 0 0
$$829$$ −3.90997 −0.135799 −0.0678994 0.997692i $$-0.521630\pi$$
−0.0678994 + 0.997692i $$0.521630\pi$$
$$830$$ −46.1772 −1.60283
$$831$$ 0 0
$$832$$ 37.1738 1.28877
$$833$$ 6.36842 0.220653
$$834$$ 0 0
$$835$$ −58.4405 −2.02242
$$836$$ −0.740987 −0.0256276
$$837$$ 0 0
$$838$$ 10.4924 0.362453
$$839$$ 9.58097 0.330772 0.165386 0.986229i $$-0.447113\pi$$
0.165386 + 0.986229i $$0.447113\pi$$
$$840$$ 0 0
$$841$$ −3.54781 −0.122338
$$842$$ −22.1627 −0.763778
$$843$$ 0 0
$$844$$ −2.03875 −0.0701767
$$845$$ −29.8629 −1.02732
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 15.8504 0.544305
$$849$$ 0 0
$$850$$ −6.99886 −0.240059
$$851$$ −20.9252 −0.717307
$$852$$ 0 0
$$853$$ −14.5568 −0.498415 −0.249207 0.968450i $$-0.580170\pi$$
−0.249207 + 0.968450i $$0.580170\pi$$
$$854$$ −2.92520 −0.100098
$$855$$ 0 0
$$856$$ −43.4647 −1.48559
$$857$$ −10.4793 −0.357965 −0.178983 0.983852i $$-0.557281\pi$$
−0.178983 + 0.983852i $$0.557281\pi$$
$$858$$ 0 0
$$859$$ 2.88645 0.0984843 0.0492421 0.998787i $$-0.484319\pi$$
0.0492421 + 0.998787i $$0.484319\pi$$
$$860$$ 3.51109 0.119727
$$861$$ 0 0
$$862$$ 8.13650 0.277130
$$863$$ 20.5485 0.699479 0.349739 0.936847i $$-0.386270\pi$$
0.349739 + 0.936847i $$0.386270\pi$$
$$864$$ 0 0
$$865$$ 29.6121 1.00684
$$866$$ 37.5028 1.27440
$$867$$ 0 0
$$868$$ 1.05398 0.0357744
$$869$$ 11.4432 0.388185
$$870$$ 0 0
$$871$$ 44.2278 1.49860
$$872$$ 34.7702 1.17747
$$873$$ 0 0
$$874$$ −38.3476 −1.29713
$$875$$ −10.1890 −0.344452
$$876$$ 0 0
$$877$$ 59.1149 1.99617 0.998084 0.0618724i $$-0.0197072\pi$$
0.998084 + 0.0618724i $$0.0197072\pi$$
$$878$$ −34.6212 −1.16841
$$879$$ 0 0
$$880$$ 10.2140 0.344314
$$881$$ −30.3982 −1.02414 −0.512071 0.858943i $$-0.671121\pi$$
−0.512071 + 0.858943i $$0.671121\pi$$
$$882$$ 0 0
$$883$$ −35.6114 −1.19842 −0.599210 0.800592i $$-0.704519\pi$$
−0.599210 + 0.800592i $$0.704519\pi$$
$$884$$ 4.47214 0.150414
$$885$$ 0 0
$$886$$ −26.3713 −0.885962
$$887$$ −22.9736 −0.771377 −0.385689 0.922629i $$-0.626036\pi$$
−0.385689 + 0.922629i $$0.626036\pi$$
$$888$$ 0 0
$$889$$ 2.27839 0.0764147
$$890$$ −41.5666 −1.39332
$$891$$ 0 0
$$892$$ −0.265856 −0.00890153
$$893$$ 2.80538 0.0938784
$$894$$ 0 0
$$895$$ −13.4128 −0.448339
$$896$$ 12.3490 0.412553
$$897$$ 0 0
$$898$$ −51.1565 −1.70712
$$899$$ 38.2009 1.27407
$$900$$ 0 0
$$901$$ −23.7008 −0.789588
$$902$$ −0.946021 −0.0314991
$$903$$ 0 0
$$904$$ −50.9944 −1.69605
$$905$$ −32.5485 −1.08195
$$906$$ 0 0
$$907$$ −57.1745 −1.89845 −0.949224 0.314602i $$-0.898129\pi$$
−0.949224 + 0.314602i $$0.898129\pi$$
$$908$$ 0.445920 0.0147984
$$909$$ 0 0
$$910$$ −17.6960 −0.586616
$$911$$ −6.82687 −0.226184 −0.113092 0.993584i $$-0.536076\pi$$
−0.113092 + 0.993584i $$0.536076\pi$$
$$912$$ 0 0
$$913$$ −13.1648 −0.435692
$$914$$ −6.63428 −0.219442
$$915$$ 0 0
$$916$$ −2.54848 −0.0842043
$$917$$ 4.00000 0.132092
$$918$$ 0 0
$$919$$ 12.0692 0.398126 0.199063 0.979987i $$-0.436210\pi$$
0.199063 + 0.979987i $$0.436210\pi$$
$$920$$ 32.1467 1.05985
$$921$$ 0 0
$$922$$ −4.09003 −0.134698
$$923$$ 57.7312 1.90025
$$924$$ 0 0
$$925$$ 3.19239 0.104965
$$926$$ −56.1045 −1.84371
$$927$$ 0 0
$$928$$ −3.96540 −0.130171
$$929$$ −26.8954 −0.882410 −0.441205 0.897406i $$-0.645449\pi$$
−0.441205 + 0.897406i $$0.645449\pi$$
$$930$$ 0 0
$$931$$ −5.32340 −0.174468
$$932$$ 2.30885 0.0756288
$$933$$ 0 0
$$934$$ −29.9356 −0.979523
$$935$$ −15.2728 −0.499474
$$936$$ 0 0
$$937$$ −14.9944 −0.489846 −0.244923 0.969543i $$-0.578763\pi$$
−0.244923 + 0.969543i $$0.578763\pi$$
$$938$$ 12.8221 0.418655
$$939$$ 0 0
$$940$$ 0.175918 0.00573780
$$941$$ 30.1205 0.981900 0.490950 0.871188i $$-0.336650\pi$$
0.490950 + 0.871188i $$0.336650\pi$$
$$942$$ 0 0
$$943$$ −3.18566 −0.103739
$$944$$ 33.9452 1.10482
$$945$$ 0 0
$$946$$ 15.3836 0.500166
$$947$$ 17.3532 0.563903 0.281951 0.959429i $$-0.409018\pi$$
0.281951 + 0.959429i $$0.409018\pi$$
$$948$$ 0 0
$$949$$ −65.8123 −2.13636
$$950$$ 5.85039 0.189812
$$951$$ 0 0
$$952$$ −17.3324 −0.561745
$$953$$ 2.14064 0.0693422 0.0346711 0.999399i $$-0.488962\pi$$
0.0346711 + 0.999399i $$0.488962\pi$$
$$954$$ 0 0
$$955$$ −22.5969 −0.731217
$$956$$ −0.405923 −0.0131285
$$957$$ 0 0
$$958$$ −16.9861 −0.548796
$$959$$ −4.77559 −0.154212
$$960$$ 0 0
$$961$$ 26.3353 0.849525
$$962$$ −31.3497 −1.01076
$$963$$ 0 0
$$964$$ −0.848944 −0.0273427
$$965$$ 24.2909 0.781952
$$966$$ 0 0
$$967$$ −1.53326 −0.0493062 −0.0246531 0.999696i $$-0.507848\pi$$
−0.0246531 + 0.999696i $$0.507848\pi$$
$$968$$ 2.72161 0.0874759
$$969$$ 0 0
$$970$$ −6.56351 −0.210742
$$971$$ 26.5574 0.852269 0.426135 0.904660i $$-0.359875\pi$$
0.426135 + 0.904660i $$0.359875\pi$$
$$972$$ 0 0
$$973$$ 15.4432 0.495087
$$974$$ −47.5041 −1.52213
$$975$$ 0 0
$$976$$ 8.51803 0.272655
$$977$$ 55.9017 1.78845 0.894227 0.447615i $$-0.147726\pi$$
0.894227 + 0.447615i $$0.147726\pi$$
$$978$$ 0 0
$$979$$ −11.8504 −0.378740
$$980$$ −0.333816 −0.0106634
$$981$$ 0 0
$$982$$ 39.0171 1.24509
$$983$$ 53.0361 1.69159 0.845794 0.533510i $$-0.179127\pi$$
0.845794 + 0.533510i $$0.179127\pi$$
$$984$$ 0 0
$$985$$ −5.41411 −0.172508
$$986$$ 46.9915 1.49651
$$987$$ 0 0
$$988$$ −3.73829 −0.118931
$$989$$ 51.8034 1.64725
$$990$$ 0 0
$$991$$ 14.7362 0.468110 0.234055 0.972223i $$-0.424800\pi$$
0.234055 + 0.972223i $$0.424800\pi$$
$$992$$ −5.95162 −0.188964
$$993$$ 0 0
$$994$$ 16.7368 0.530860
$$995$$ −7.37402 −0.233772
$$996$$ 0 0
$$997$$ −45.0665 −1.42727 −0.713635 0.700517i $$-0.752953\pi$$
−0.713635 + 0.700517i $$0.752953\pi$$
$$998$$ 60.3262 1.90959
$$999$$ 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 693.2.a.l.1.2 3
3.2 odd 2 231.2.a.e.1.2 3
7.6 odd 2 4851.2.a.bi.1.2 3
11.10 odd 2 7623.2.a.cd.1.2 3
12.11 even 2 3696.2.a.bo.1.2 3
15.14 odd 2 5775.2.a.bp.1.2 3
21.20 even 2 1617.2.a.t.1.2 3
33.32 even 2 2541.2.a.bg.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 3.2 odd 2
693.2.a.l.1.2 3 1.1 even 1 trivial
1617.2.a.t.1.2 3 21.20 even 2
2541.2.a.bg.1.2 3 33.32 even 2
3696.2.a.bo.1.2 3 12.11 even 2
4851.2.a.bi.1.2 3 7.6 odd 2
5775.2.a.bp.1.2 3 15.14 odd 2
7623.2.a.cd.1.2 3 11.10 odd 2