# Properties

 Label 1617.2.a.t.1.2 Level $1617$ Weight $2$ Character 1617.1 Self dual yes Analytic conductor $12.912$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9118100068$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ 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 $$\chi$$ $$=$$ 1617.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.46260 q^{2} -1.00000 q^{3} +0.139194 q^{4} -2.39821 q^{5} -1.46260 q^{6} -2.72161 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.46260 q^{2} -1.00000 q^{3} +0.139194 q^{4} -2.39821 q^{5} -1.46260 q^{6} -2.72161 q^{8} +1.00000 q^{9} -3.50761 q^{10} -1.00000 q^{11} -0.139194 q^{12} -5.04502 q^{13} +2.39821 q^{15} -4.25901 q^{16} +6.36842 q^{17} +1.46260 q^{18} +5.32340 q^{19} -0.333816 q^{20} -1.46260 q^{22} +4.92520 q^{23} +2.72161 q^{24} +0.751399 q^{25} -7.37883 q^{26} -1.00000 q^{27} +5.04502 q^{29} +3.50761 q^{30} +7.57201 q^{31} -0.786003 q^{32} +1.00000 q^{33} +9.31444 q^{34} +0.139194 q^{36} +4.24860 q^{37} +7.78600 q^{38} +5.04502 q^{39} +6.52699 q^{40} +0.646809 q^{41} -10.5180 q^{43} -0.139194 q^{44} -2.39821 q^{45} +7.20359 q^{46} -0.526989 q^{47} +4.25901 q^{48} +1.09899 q^{50} -6.36842 q^{51} -0.702237 q^{52} +3.72161 q^{53} -1.46260 q^{54} +2.39821 q^{55} -5.32340 q^{57} +7.37883 q^{58} -7.97021 q^{59} +0.333816 q^{60} +2.00000 q^{61} +11.0748 q^{62} +7.36842 q^{64} +12.0990 q^{65} +1.46260 q^{66} +8.76663 q^{67} +0.886447 q^{68} -4.92520 q^{69} -11.4432 q^{71} -2.72161 q^{72} +13.0450 q^{73} +6.21400 q^{74} -0.751399 q^{75} +0.740987 q^{76} +7.37883 q^{78} +11.4432 q^{79} +10.2140 q^{80} +1.00000 q^{81} +0.946021 q^{82} -13.1648 q^{83} -15.2728 q^{85} -15.3836 q^{86} -5.04502 q^{87} +2.72161 q^{88} -11.8504 q^{89} -3.50761 q^{90} +0.685559 q^{92} -7.57201 q^{93} -0.770774 q^{94} -12.7666 q^{95} +0.786003 q^{96} +1.87122 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 - 3 * q^3 + 6 * q^4 - 4 * q^5 - 2 * q^6 + 3 * q^8 + 3 * q^9 $$3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + 11 q^{10} - 3 q^{11} - 6 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{19} + 3 q^{20} - 2 q^{22} + 10 q^{23} - 3 q^{24} + 15 q^{25} + q^{26} - 3 q^{27} - 4 q^{29} - 11 q^{30} + 2 q^{31} + 8 q^{32} + 3 q^{33} + 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 18 q^{40} - 14 q^{41} - 14 q^{43} - 6 q^{44} - 4 q^{45} + 28 q^{46} + 4 q^{48} - 19 q^{50} + 8 q^{51} + 29 q^{52} - 2 q^{54} + 4 q^{55} - 8 q^{57} - q^{58} - 3 q^{60} + 6 q^{61} + 38 q^{62} - 5 q^{64} + 14 q^{65} + 2 q^{66} - 4 q^{67} - 42 q^{68} - 10 q^{69} - 12 q^{71} + 3 q^{72} + 20 q^{73} + 29 q^{74} - 15 q^{75} + 11 q^{76} - q^{78} + 12 q^{79} + 41 q^{80} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{85} + 24 q^{86} + 4 q^{87} - 3 q^{88} - 26 q^{89} + 11 q^{90} + 26 q^{92} - 2 q^{93} - 35 q^{94} - 8 q^{95} - 8 q^{96} + 4 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 - 3 * q^3 + 6 * q^4 - 4 * q^5 - 2 * q^6 + 3 * q^8 + 3 * q^9 + 11 * q^10 - 3 * q^11 - 6 * q^12 + 4 * q^13 + 4 * q^15 - 4 * q^16 - 8 * q^17 + 2 * q^18 + 8 * q^19 + 3 * q^20 - 2 * q^22 + 10 * q^23 - 3 * q^24 + 15 * q^25 + q^26 - 3 * q^27 - 4 * q^29 - 11 * q^30 + 2 * q^31 + 8 * q^32 + 3 * q^33 + 4 * q^34 + 6 * q^36 + 13 * q^38 - 4 * q^39 + 18 * q^40 - 14 * q^41 - 14 * q^43 - 6 * q^44 - 4 * q^45 + 28 * q^46 + 4 * q^48 - 19 * q^50 + 8 * q^51 + 29 * q^52 - 2 * q^54 + 4 * q^55 - 8 * q^57 - q^58 - 3 * q^60 + 6 * q^61 + 38 * q^62 - 5 * q^64 + 14 * q^65 + 2 * q^66 - 4 * q^67 - 42 * q^68 - 10 * q^69 - 12 * q^71 + 3 * q^72 + 20 * q^73 + 29 * q^74 - 15 * q^75 + 11 * q^76 - q^78 + 12 * q^79 + 41 * q^80 + 3 * q^81 + 6 * q^82 - 6 * q^83 - 6 * q^85 + 24 * q^86 + 4 * q^87 - 3 * q^88 - 26 * q^89 + 11 * q^90 + 26 * q^92 - 2 * q^93 - 35 * q^94 - 8 * q^95 - 8 * q^96 + 4 * q^97 - 3 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.46260 1.03421 0.517107 0.855921i $$-0.327009\pi$$
0.517107 + 0.855921i $$0.327009\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.139194 0.0695971
$$5$$ −2.39821 −1.07251 −0.536255 0.844056i $$-0.680162\pi$$
−0.536255 + 0.844056i $$0.680162\pi$$
$$6$$ −1.46260 −0.597103
$$7$$ 0 0
$$8$$ −2.72161 −0.962235
$$9$$ 1.00000 0.333333
$$10$$ −3.50761 −1.10921
$$11$$ −1.00000 −0.301511
$$12$$ −0.139194 −0.0401819
$$13$$ −5.04502 −1.39924 −0.699618 0.714517i $$-0.746646\pi$$
−0.699618 + 0.714517i $$0.746646\pi$$
$$14$$ 0 0
$$15$$ 2.39821 0.619214
$$16$$ −4.25901 −1.06475
$$17$$ 6.36842 1.54457 0.772284 0.635277i $$-0.219114\pi$$
0.772284 + 0.635277i $$0.219114\pi$$
$$18$$ 1.46260 0.344738
$$19$$ 5.32340 1.22127 0.610636 0.791911i $$-0.290914\pi$$
0.610636 + 0.791911i $$0.290914\pi$$
$$20$$ −0.333816 −0.0746436
$$21$$ 0 0
$$22$$ −1.46260 −0.311827
$$23$$ 4.92520 1.02697 0.513487 0.858097i $$-0.328353\pi$$
0.513487 + 0.858097i $$0.328353\pi$$
$$24$$ 2.72161 0.555547
$$25$$ 0.751399 0.150280
$$26$$ −7.37883 −1.44711
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.04502 0.936836 0.468418 0.883507i $$-0.344824\pi$$
0.468418 + 0.883507i $$0.344824\pi$$
$$30$$ 3.50761 0.640400
$$31$$ 7.57201 1.35997 0.679986 0.733225i $$-0.261986\pi$$
0.679986 + 0.733225i $$0.261986\pi$$
$$32$$ −0.786003 −0.138947
$$33$$ 1.00000 0.174078
$$34$$ 9.31444 1.59741
$$35$$ 0 0
$$36$$ 0.139194 0.0231990
$$37$$ 4.24860 0.698466 0.349233 0.937036i $$-0.386442\pi$$
0.349233 + 0.937036i $$0.386442\pi$$
$$38$$ 7.78600 1.26306
$$39$$ 5.04502 0.807849
$$40$$ 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
$$45$$ −2.39821 −0.357504
$$46$$ 7.20359 1.06211
$$47$$ −0.526989 −0.0768693 −0.0384347 0.999261i $$-0.512237\pi$$
−0.0384347 + 0.999261i $$0.512237\pi$$
$$48$$ 4.25901 0.614736
$$49$$ 0 0
$$50$$ 1.09899 0.155421
$$51$$ −6.36842 −0.891757
$$52$$ −0.702237 −0.0973827
$$53$$ 3.72161 0.511203 0.255601 0.966782i $$-0.417727\pi$$
0.255601 + 0.966782i $$0.417727\pi$$
$$54$$ −1.46260 −0.199034
$$55$$ 2.39821 0.323374
$$56$$ 0 0
$$57$$ −5.32340 −0.705102
$$58$$ 7.37883 0.968888
$$59$$ −7.97021 −1.03763 −0.518817 0.854886i $$-0.673627\pi$$
−0.518817 + 0.854886i $$0.673627\pi$$
$$60$$ 0.333816 0.0430955
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 11.0748 1.40650
$$63$$ 0 0
$$64$$ 7.36842 0.921053
$$65$$ 12.0990 1.50070
$$66$$ 1.46260 0.180033
$$67$$ 8.76663 1.07101 0.535507 0.844531i $$-0.320121\pi$$
0.535507 + 0.844531i $$0.320121\pi$$
$$68$$ 0.886447 0.107497
$$69$$ −4.92520 −0.592924
$$70$$ 0 0
$$71$$ −11.4432 −1.35806 −0.679030 0.734110i $$-0.737600\pi$$
−0.679030 + 0.734110i $$0.737600\pi$$
$$72$$ −2.72161 −0.320745
$$73$$ 13.0450 1.52680 0.763402 0.645924i $$-0.223528\pi$$
0.763402 + 0.645924i $$0.223528\pi$$
$$74$$ 6.21400 0.722363
$$75$$ −0.751399 −0.0867641
$$76$$ 0.740987 0.0849970
$$77$$ 0 0
$$78$$ 7.37883 0.835488
$$79$$ 11.4432 1.28746 0.643732 0.765251i $$-0.277385\pi$$
0.643732 + 0.765251i $$0.277385\pi$$
$$80$$ 10.2140 1.14196
$$81$$ 1.00000 0.111111
$$82$$ 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
$$87$$ −5.04502 −0.540882
$$88$$ 2.72161 0.290125
$$89$$ −11.8504 −1.25614 −0.628070 0.778157i $$-0.716155\pi$$
−0.628070 + 0.778157i $$0.716155\pi$$
$$90$$ −3.50761 −0.369735
$$91$$ 0 0
$$92$$ 0.685559 0.0714744
$$93$$ −7.57201 −0.785180
$$94$$ −0.770774 −0.0794993
$$95$$ −12.7666 −1.30983
$$96$$ 0.786003 0.0802211
$$97$$ 1.87122 0.189993 0.0949967 0.995478i $$-0.469716\pi$$
0.0949967 + 0.995478i $$0.469716\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0.104590 0.0104590
$$101$$ −4.51803 −0.449560 −0.224780 0.974409i $$-0.572166\pi$$
−0.224780 + 0.974409i $$0.572166\pi$$
$$102$$ −9.31444 −0.922267
$$103$$ 10.6468 1.04906 0.524531 0.851392i $$-0.324241\pi$$
0.524531 + 0.851392i $$0.324241\pi$$
$$104$$ 13.7306 1.34639
$$105$$ 0 0
$$106$$ 5.44322 0.528693
$$107$$ 15.9702 1.54390 0.771949 0.635684i $$-0.219282\pi$$
0.771949 + 0.635684i $$0.219282\pi$$
$$108$$ −0.139194 −0.0133940
$$109$$ 12.7756 1.22368 0.611840 0.790982i $$-0.290430\pi$$
0.611840 + 0.790982i $$0.290430\pi$$
$$110$$ 3.50761 0.334438
$$111$$ −4.24860 −0.403259
$$112$$ 0 0
$$113$$ 18.7368 1.76261 0.881307 0.472544i $$-0.156664\pi$$
0.881307 + 0.472544i $$0.156664\pi$$
$$114$$ −7.78600 −0.729226
$$115$$ −11.8116 −1.10144
$$116$$ 0.702237 0.0652010
$$117$$ −5.04502 −0.466412
$$118$$ −11.6572 −1.07313
$$119$$ 0 0
$$120$$ −6.52699 −0.595830
$$121$$ 1.00000 0.0909091
$$122$$ 2.92520 0.264835
$$123$$ −0.646809 −0.0583208
$$124$$ 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
$$129$$ 10.5180 0.926061
$$130$$ 17.6960 1.55204
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0.139194 0.0121153
$$133$$ 0 0
$$134$$ 12.8221 1.10766
$$135$$ 2.39821 0.206405
$$136$$ −17.3324 −1.48624
$$137$$ −4.77559 −0.408006 −0.204003 0.978970i $$-0.565395\pi$$
−0.204003 + 0.978970i $$0.565395\pi$$
$$138$$ −7.20359 −0.613210
$$139$$ 15.4432 1.30988 0.654939 0.755682i $$-0.272694\pi$$
0.654939 + 0.755682i $$0.272694\pi$$
$$140$$ 0 0
$$141$$ 0.526989 0.0443805
$$142$$ −16.7368 −1.40452
$$143$$ 5.04502 0.421885
$$144$$ −4.25901 −0.354918
$$145$$ −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 $$-0.632075\pi$$
−0.403121 + 0.915147i $$0.632075\pi$$
$$150$$ −1.09899 −0.0897326
$$151$$ −4.12878 −0.335996 −0.167998 0.985787i $$-0.553730\pi$$
−0.167998 + 0.985787i $$0.553730\pi$$
$$152$$ −14.4882 −1.17515
$$153$$ 6.36842 0.514856
$$154$$ 0 0
$$155$$ −18.1592 −1.45859
$$156$$ 0.702237 0.0562239
$$157$$ 0.946021 0.0755007 0.0377504 0.999287i $$-0.487981\pi$$
0.0377504 + 0.999287i $$0.487981\pi$$
$$158$$ 16.7368 1.33151
$$159$$ −3.72161 −0.295143
$$160$$ 1.88500 0.149022
$$161$$ 0 0
$$162$$ 1.46260 0.114913
$$163$$ 8.76663 0.686655 0.343328 0.939216i $$-0.388446\pi$$
0.343328 + 0.939216i $$0.388446\pi$$
$$164$$ 0.0900320 0.00703032
$$165$$ −2.39821 −0.186700
$$166$$ −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
$$171$$ 5.32340 0.407091
$$172$$ −1.46405 −0.111633
$$173$$ −12.3476 −0.938770 −0.469385 0.882994i $$-0.655524\pi$$
−0.469385 + 0.882994i $$0.655524\pi$$
$$174$$ −7.37883 −0.559388
$$175$$ 0 0
$$176$$ 4.25901 0.321035
$$177$$ 7.97021 0.599078
$$178$$ −17.3324 −1.29912
$$179$$ −5.59283 −0.418028 −0.209014 0.977913i $$-0.567025\pi$$
−0.209014 + 0.977913i $$0.567025\pi$$
$$180$$ −0.333816 −0.0248812
$$181$$ −13.5720 −1.00880 −0.504400 0.863470i $$-0.668286\pi$$
−0.504400 + 0.863470i $$0.668286\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ −13.4045 −0.988191
$$185$$ −10.1890 −0.749112
$$186$$ −11.0748 −0.812044
$$187$$ −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 $$-0.610729\pi$$
−0.340890 + 0.940103i $$0.610729\pi$$
$$192$$ −7.36842 −0.531770
$$193$$ −10.1288 −0.729086 −0.364543 0.931187i $$-0.618775\pi$$
−0.364543 + 0.931187i $$0.618775\pi$$
$$194$$ 2.73684 0.196494
$$195$$ −12.0990 −0.866427
$$196$$ 0 0
$$197$$ −2.25756 −0.160845 −0.0804224 0.996761i $$-0.525627\pi$$
−0.0804224 + 0.996761i $$0.525627\pi$$
$$198$$ −1.46260 −0.103942
$$199$$ −3.07480 −0.217967 −0.108984 0.994044i $$-0.534760\pi$$
−0.108984 + 0.994044i $$0.534760\pi$$
$$200$$ −2.04502 −0.144604
$$201$$ −8.76663 −0.618350
$$202$$ −6.60806 −0.464941
$$203$$ 0 0
$$204$$ −0.886447 −0.0620637
$$205$$ −1.55118 −0.108339
$$206$$ 15.5720 1.08495
$$207$$ 4.92520 0.342325
$$208$$ 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
$$213$$ 11.4432 0.784077
$$214$$ 23.3580 1.59672
$$215$$ 25.2244 1.72029
$$216$$ 2.72161 0.185182
$$217$$ 0 0
$$218$$ 18.6856 1.26555
$$219$$ −13.0450 −0.881500
$$220$$ 0.333816 0.0225059
$$221$$ −32.1288 −2.16122
$$222$$ −6.21400 −0.417056
$$223$$ 1.90997 0.127901 0.0639505 0.997953i $$-0.479630\pi$$
0.0639505 + 0.997953i $$0.479630\pi$$
$$224$$ 0 0
$$225$$ 0.751399 0.0500933
$$226$$ 27.4045 1.82292
$$227$$ 3.20359 0.212629 0.106315 0.994333i $$-0.466095\pi$$
0.106315 + 0.994333i $$0.466095\pi$$
$$228$$ −0.740987 −0.0490730
$$229$$ 18.3088 1.20988 0.604941 0.796270i $$-0.293197\pi$$
0.604941 + 0.796270i $$0.293197\pi$$
$$230$$ −17.2757 −1.13913
$$231$$ 0 0
$$232$$ −13.7306 −0.901456
$$233$$ −16.5872 −1.08667 −0.543333 0.839517i $$-0.682838\pi$$
−0.543333 + 0.839517i $$0.682838\pi$$
$$234$$ −7.37883 −0.482369
$$235$$ 1.26383 0.0824432
$$236$$ −1.10941 −0.0722162
$$237$$ −11.4432 −0.743317
$$238$$ 0 0
$$239$$ 2.91623 0.188635 0.0943177 0.995542i $$-0.469933\pi$$
0.0943177 + 0.995542i $$0.469933\pi$$
$$240$$ −10.2140 −0.659311
$$241$$ 6.09899 0.392871 0.196435 0.980517i $$-0.437063\pi$$
0.196435 + 0.980517i $$0.437063\pi$$
$$242$$ 1.46260 0.0940194
$$243$$ −1.00000 −0.0641500
$$244$$ 0.278388 0.0178220
$$245$$ 0 0
$$246$$ −0.946021 −0.0603161
$$247$$ −26.8567 −1.70885
$$248$$ −20.6081 −1.30861
$$249$$ 13.1648 0.834288
$$250$$ 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
$$255$$ 15.2728 0.956419
$$256$$ 3.32485 0.207803
$$257$$ 6.89541 0.430124 0.215062 0.976600i $$-0.431005\pi$$
0.215062 + 0.976600i $$0.431005\pi$$
$$258$$ 15.3836 0.957744
$$259$$ 0 0
$$260$$ 1.68411 0.104444
$$261$$ 5.04502 0.312279
$$262$$ −5.85039 −0.361439
$$263$$ 5.08377 0.313478 0.156739 0.987640i $$-0.449902\pi$$
0.156739 + 0.987640i $$0.449902\pi$$
$$264$$ −2.72161 −0.167504
$$265$$ −8.92520 −0.548270
$$266$$ 0 0
$$267$$ 11.8504 0.725232
$$268$$ 1.22026 0.0745394
$$269$$ 0.886447 0.0540476 0.0270238 0.999635i $$-0.491397\pi$$
0.0270238 + 0.999635i $$0.491397\pi$$
$$270$$ 3.50761 0.213467
$$271$$ 25.3234 1.53829 0.769144 0.639076i $$-0.220683\pi$$
0.769144 + 0.639076i $$0.220683\pi$$
$$272$$ −27.1232 −1.64458
$$273$$ 0 0
$$274$$ −6.98477 −0.421965
$$275$$ −0.751399 −0.0453111
$$276$$ −0.685559 −0.0412658
$$277$$ −24.8269 −1.49170 −0.745851 0.666113i $$-0.767957\pi$$
−0.745851 + 0.666113i $$0.767957\pi$$
$$278$$ 22.5872 1.35469
$$279$$ 7.57201 0.453324
$$280$$ 0 0
$$281$$ 1.90101 0.113404 0.0567022 0.998391i $$-0.481941\pi$$
0.0567022 + 0.998391i $$0.481941\pi$$
$$282$$ 0.770774 0.0458989
$$283$$ 22.3178 1.32666 0.663328 0.748329i $$-0.269143\pi$$
0.663328 + 0.748329i $$0.269143\pi$$
$$284$$ −1.59283 −0.0945171
$$285$$ 12.7666 0.756230
$$286$$ 7.37883 0.436320
$$287$$ 0 0
$$288$$ −0.786003 −0.0463157
$$289$$ 23.5568 1.38569
$$290$$ −17.6960 −1.03914
$$291$$ −1.87122 −0.109693
$$292$$ 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
$$297$$ 1.00000 0.0580259
$$298$$ −14.3941 −0.833826
$$299$$ −24.8477 −1.43698
$$300$$ −0.104590 −0.00603853
$$301$$ 0 0
$$302$$ −6.03875 −0.347491
$$303$$ 4.51803 0.259554
$$304$$ −22.6724 −1.30035
$$305$$ −4.79641 −0.274642
$$306$$ 9.31444 0.532471
$$307$$ 13.5928 0.775784 0.387892 0.921705i $$-0.373203\pi$$
0.387892 + 0.921705i $$0.373203\pi$$
$$308$$ 0 0
$$309$$ −10.6468 −0.605676
$$310$$ −26.5597 −1.50849
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ −13.7306 −0.777341
$$313$$ −14.9252 −0.843622 −0.421811 0.906684i $$-0.638605\pi$$
−0.421811 + 0.906684i $$0.638605\pi$$
$$314$$ 1.38365 0.0780838
$$315$$ 0 0
$$316$$ 1.59283 0.0896037
$$317$$ 3.97918 0.223493 0.111746 0.993737i $$-0.464356\pi$$
0.111746 + 0.993737i $$0.464356\pi$$
$$318$$ −5.44322 −0.305241
$$319$$ −5.04502 −0.282467
$$320$$ −17.6710 −0.987839
$$321$$ −15.9702 −0.891370
$$322$$ 0 0
$$323$$ 33.9017 1.88634
$$324$$ 0.139194 0.00773301
$$325$$ −3.79082 −0.210277
$$326$$ 12.8221 0.710148
$$327$$ −12.7756 −0.706492
$$328$$ −1.76036 −0.0971997
$$329$$ 0 0
$$330$$ −3.50761 −0.193088
$$331$$ 23.4432 1.28856 0.644278 0.764791i $$-0.277158\pi$$
0.644278 + 0.764791i $$0.277158\pi$$
$$332$$ −1.83247 −0.100570
$$333$$ 4.24860 0.232822
$$334$$ 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
$$339$$ −18.7368 −1.01765
$$340$$ −2.12588 −0.115292
$$341$$ −7.57201 −0.410047
$$342$$ 7.78600 0.421019
$$343$$ 0 0
$$344$$ 28.6260 1.54341
$$345$$ 11.8116 0.635918
$$346$$ −18.0596 −0.970889
$$347$$ −22.5872 −1.21255 −0.606273 0.795256i $$-0.707336\pi$$
−0.606273 + 0.795256i $$0.707336\pi$$
$$348$$ −0.702237 −0.0376438
$$349$$ −27.9315 −1.49514 −0.747568 0.664185i $$-0.768779\pi$$
−0.747568 + 0.664185i $$0.768779\pi$$
$$350$$ 0 0
$$351$$ 5.04502 0.269283
$$352$$ 0.786003 0.0418941
$$353$$ 16.5478 0.880751 0.440376 0.897814i $$-0.354845\pi$$
0.440376 + 0.897814i $$0.354845\pi$$
$$354$$ 11.6572 0.619574
$$355$$ 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 $$-0.302520\pi$$
0.581362 + 0.813645i $$0.302520\pi$$
$$360$$ 6.52699 0.344003
$$361$$ 9.33863 0.491507
$$362$$ −19.8504 −1.04331
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −31.2847 −1.63751
$$366$$ −2.92520 −0.152902
$$367$$ 19.3836 1.01182 0.505909 0.862587i $$-0.331157\pi$$
0.505909 + 0.862587i $$0.331157\pi$$
$$368$$ −20.9765 −1.09347
$$369$$ 0.646809 0.0336715
$$370$$ −14.9025 −0.774742
$$371$$ 0 0
$$372$$ −1.05398 −0.0546463
$$373$$ 29.2549 1.51476 0.757380 0.652975i $$-0.226479\pi$$
0.757380 + 0.652975i $$0.226479\pi$$
$$374$$ −9.31444 −0.481638
$$375$$ −10.1890 −0.526159
$$376$$ 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
$$381$$ 2.27839 0.116725
$$382$$ −13.7812 −0.705107
$$383$$ 17.5928 0.898952 0.449476 0.893293i $$-0.351611\pi$$
0.449476 + 0.893293i $$0.351611\pi$$
$$384$$ −12.3490 −0.630185
$$385$$ 0 0
$$386$$ −14.8143 −0.754030
$$387$$ −10.5180 −0.534661
$$388$$ 0.260463 0.0132230
$$389$$ 20.0900 1.01861 0.509303 0.860588i $$-0.329903\pi$$
0.509303 + 0.860588i $$0.329903\pi$$
$$390$$ −17.6960 −0.896070
$$391$$ 31.3657 1.58623
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ −3.30191 −0.166348
$$395$$ −27.4432 −1.38082
$$396$$ −0.139194 −0.00699477
$$397$$ 35.1053 1.76188 0.880941 0.473226i $$-0.156910\pi$$
0.880941 + 0.473226i $$0.156910\pi$$
$$398$$ −4.49720 −0.225424
$$399$$ 0 0
$$400$$ −3.20022 −0.160011
$$401$$ 9.57201 0.478003 0.239002 0.971019i $$-0.423180\pi$$
0.239002 + 0.971019i $$0.423180\pi$$
$$402$$ −12.8221 −0.639506
$$403$$ −38.2009 −1.90292
$$404$$ −0.628883 −0.0312881
$$405$$ −2.39821 −0.119168
$$406$$ 0 0
$$407$$ −4.24860 −0.210595
$$408$$ 17.3324 0.858080
$$409$$ −38.1801 −1.88788 −0.943941 0.330113i $$-0.892913\pi$$
−0.943941 + 0.330113i $$0.892913\pi$$
$$410$$ −2.26875 −0.112046
$$411$$ 4.77559 0.235563
$$412$$ 1.48197 0.0730116
$$413$$ 0 0
$$414$$ 7.20359 0.354037
$$415$$ 31.5720 1.54981
$$416$$ 3.96540 0.194420
$$417$$ −15.4432 −0.756258
$$418$$ −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
$$423$$ −0.526989 −0.0256231
$$424$$ −10.1288 −0.491897
$$425$$ 4.78522 0.232117
$$426$$ 16.7368 0.810903
$$427$$ 0 0
$$428$$ 2.22296 0.107451
$$429$$ −5.04502 −0.243576
$$430$$ 36.8932 1.77915
$$431$$ 5.56304 0.267962 0.133981 0.990984i $$-0.457224\pi$$
0.133981 + 0.990984i $$0.457224\pi$$
$$432$$ 4.25901 0.204912
$$433$$ 25.6412 1.23224 0.616119 0.787653i $$-0.288704\pi$$
0.616119 + 0.787653i $$0.288704\pi$$
$$434$$ 0 0
$$435$$ 12.0990 0.580102
$$436$$ 1.77829 0.0851645
$$437$$ 26.2188 1.25422
$$438$$ −19.0796 −0.911659
$$439$$ −23.6710 −1.12976 −0.564878 0.825175i $$-0.691077\pi$$
−0.564878 + 0.825175i $$0.691077\pi$$
$$440$$ −6.52699 −0.311162
$$441$$ 0 0
$$442$$ −46.9915 −2.23516
$$443$$ −18.0305 −0.856653 −0.428326 0.903624i $$-0.640897\pi$$
−0.428326 + 0.903624i $$0.640897\pi$$
$$444$$ −0.591380 −0.0280657
$$445$$ 28.4197 1.34722
$$446$$ 2.79352 0.132277
$$447$$ 9.84143 0.465484
$$448$$ 0 0
$$449$$ −34.9765 −1.65064 −0.825321 0.564664i $$-0.809006\pi$$
−0.825321 + 0.564664i $$0.809006\pi$$
$$450$$ 1.09899 0.0518071
$$451$$ −0.646809 −0.0304570
$$452$$ 2.60806 0.122673
$$453$$ 4.12878 0.193987
$$454$$ 4.68556 0.219904
$$455$$ 0 0
$$456$$ 14.4882 0.678474
$$457$$ 4.53595 0.212183 0.106091 0.994356i $$-0.466166\pi$$
0.106091 + 0.994356i $$0.466166\pi$$
$$458$$ 26.7785 1.25128
$$459$$ −6.36842 −0.297252
$$460$$ −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
$$465$$ 18.1592 0.842115
$$466$$ −24.2605 −1.12384
$$467$$ 20.4674 0.947119 0.473560 0.880762i $$-0.342969\pi$$
0.473560 + 0.880762i $$0.342969\pi$$
$$468$$ −0.702237 −0.0324609
$$469$$ 0 0
$$470$$ 1.84848 0.0852638
$$471$$ −0.946021 −0.0435904
$$472$$ 21.6918 0.998447
$$473$$ 10.5180 0.483619
$$474$$ −16.7368 −0.768749
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 3.72161 0.170401
$$478$$ 4.26528 0.195089
$$479$$ 11.6137 0.530641 0.265321 0.964160i $$-0.414522\pi$$
0.265321 + 0.964160i $$0.414522\pi$$
$$480$$ −1.88500 −0.0860380
$$481$$ −21.4343 −0.977318
$$482$$ 8.92038 0.406312
$$483$$ 0 0
$$484$$ 0.139194 0.00632701
$$485$$ −4.48757 −0.203770
$$486$$ −1.46260 −0.0663448
$$487$$ 32.4793 1.47178 0.735888 0.677103i $$-0.236765\pi$$
0.735888 + 0.677103i $$0.236765\pi$$
$$488$$ −5.44322 −0.246403
$$489$$ −8.76663 −0.396441
$$490$$ 0 0
$$491$$ 26.6766 1.20390 0.601949 0.798535i $$-0.294391\pi$$
0.601949 + 0.798535i $$0.294391\pi$$
$$492$$ −0.0900320 −0.00405895
$$493$$ 32.1288 1.44701
$$494$$ −39.2805 −1.76731
$$495$$ 2.39821 0.107791
$$496$$ −32.2493 −1.44804
$$497$$ 0 0
$$498$$ 19.2549 0.862831
$$499$$ −41.2459 −1.84642 −0.923210 0.384296i $$-0.874444\pi$$
−0.923210 + 0.384296i $$0.874444\pi$$
$$500$$ 1.41825 0.0634262
$$501$$ −24.3684 −1.08870
$$502$$ −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
$$507$$ −12.4522 −0.553021
$$508$$ −0.317138 −0.0140707
$$509$$ 18.9944 0.841912 0.420956 0.907081i $$-0.361695\pi$$
0.420956 + 0.907081i $$0.361695\pi$$
$$510$$ 22.3380 0.989142
$$511$$ 0 0
$$512$$ −19.8352 −0.876599
$$513$$ −5.32340 −0.235034
$$514$$ 10.0852 0.444840
$$515$$ −25.5333 −1.12513
$$516$$ 1.46405 0.0644511
$$517$$ 0.526989 0.0231770
$$518$$ 0 0
$$519$$ 12.3476 0.541999
$$520$$ −32.9288 −1.44402
$$521$$ −25.2430 −1.10592 −0.552958 0.833209i $$-0.686501\pi$$
−0.552958 + 0.833209i $$0.686501\pi$$
$$522$$ 7.37883 0.322963
$$523$$ 2.93416 0.128302 0.0641509 0.997940i $$-0.479566\pi$$
0.0641509 + 0.997940i $$0.479566\pi$$
$$524$$ −0.556777 −0.0243229
$$525$$ 0 0
$$526$$ 7.43551 0.324204
$$527$$ 48.2217 2.10057
$$528$$ −4.25901 −0.185350
$$529$$ 1.25756 0.0546767
$$530$$ −13.0540 −0.567029
$$531$$ −7.97021 −0.345878
$$532$$ 0 0
$$533$$ −3.26316 −0.141343
$$534$$ 17.3324 0.750045
$$535$$ −38.2999 −1.65585
$$536$$ −23.8594 −1.03057
$$537$$ 5.59283 0.241348
$$538$$ 1.29652 0.0558968
$$539$$ 0 0
$$540$$ 0.333816 0.0143652
$$541$$ 8.90437 0.382829 0.191414 0.981509i $$-0.438693\pi$$
0.191414 + 0.981509i $$0.438693\pi$$
$$542$$ 37.0380 1.59092
$$543$$ 13.5720 0.582430
$$544$$ −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
$$549$$ 2.00000 0.0853579
$$550$$ −1.09899 −0.0468613
$$551$$ 26.8567 1.14413
$$552$$ 13.4045 0.570532
$$553$$ 0 0
$$554$$ −36.3117 −1.54274
$$555$$ 10.1890 0.432500
$$556$$ 2.14961 0.0911636
$$557$$ 14.8954 0.631139 0.315569 0.948903i $$-0.397804\pi$$
0.315569 + 0.948903i $$0.397804\pi$$
$$558$$ 11.0748 0.468834
$$559$$ 53.0636 2.24435
$$560$$ 0 0
$$561$$ 6.36842 0.268875
$$562$$ 2.78041 0.117284
$$563$$ 7.81164 0.329222 0.164611 0.986359i $$-0.447363\pi$$
0.164611 + 0.986359i $$0.447363\pi$$
$$564$$ 0.0733538 0.00308875
$$565$$ −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 $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 18.6724 0.782103
$$571$$ 25.5512 1.06928 0.534642 0.845079i $$-0.320447\pi$$
0.534642 + 0.845079i $$0.320447\pi$$
$$572$$ 0.702237 0.0293620
$$573$$ 9.42240 0.393626
$$574$$ 0 0
$$575$$ 3.70079 0.154334
$$576$$ 7.36842 0.307018
$$577$$ 29.5124 1.22862 0.614309 0.789065i $$-0.289435\pi$$
0.614309 + 0.789065i $$0.289435\pi$$
$$578$$ 34.4541 1.43310
$$579$$ 10.1288 0.420938
$$580$$ −1.68411 −0.0699288
$$581$$ 0 0
$$582$$ −2.73684 −0.113446
$$583$$ −3.72161 −0.154133
$$584$$ −35.5035 −1.46914
$$585$$ 12.0990 0.500232
$$586$$ −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
$$591$$ 2.25756 0.0928638
$$592$$ −18.0948 −0.743694
$$593$$ 7.90997 0.324823 0.162412 0.986723i $$-0.448073\pi$$
0.162412 + 0.986723i $$0.448073\pi$$
$$594$$ 1.46260 0.0600111
$$595$$ 0 0
$$596$$ −1.36987 −0.0561120
$$597$$ 3.07480 0.125843
$$598$$ −36.3422 −1.48614
$$599$$ 27.4432 1.12130 0.560650 0.828053i $$-0.310551\pi$$
0.560650 + 0.828053i $$0.310551\pi$$
$$600$$ 2.04502 0.0834874
$$601$$ −31.9910 −1.30494 −0.652471 0.757814i $$-0.726268\pi$$
−0.652471 + 0.757814i $$0.726268\pi$$
$$602$$ 0 0
$$603$$ 8.76663 0.357005
$$604$$ −0.574702 −0.0233843
$$605$$ −2.39821 −0.0975010
$$606$$ 6.60806 0.268434
$$607$$ −7.41344 −0.300902 −0.150451 0.988617i $$-0.548073\pi$$
−0.150451 + 0.988617i $$0.548073\pi$$
$$608$$ −4.18421 −0.169692
$$609$$ 0 0
$$610$$ −7.01523 −0.284038
$$611$$ 2.65867 0.107558
$$612$$ 0.886447 0.0358325
$$613$$ −33.9917 −1.37291 −0.686456 0.727171i $$-0.740835\pi$$
−0.686456 + 0.727171i $$0.740835\pi$$
$$614$$ 19.8809 0.802326
$$615$$ 1.55118 0.0625497
$$616$$ 0 0
$$617$$ −44.0305 −1.77260 −0.886300 0.463112i $$-0.846733\pi$$
−0.886300 + 0.463112i $$0.846733\pi$$
$$618$$ −15.5720 −0.626398
$$619$$ −40.0096 −1.60812 −0.804061 0.594546i $$-0.797332\pi$$
−0.804061 + 0.594546i $$0.797332\pi$$
$$620$$ −2.52766 −0.101513
$$621$$ −4.92520 −0.197641
$$622$$ −11.7008 −0.469159
$$623$$ 0 0
$$624$$ −21.4868 −0.860160
$$625$$ −28.1924 −1.12770
$$626$$ −21.8296 −0.872485
$$627$$ 5.32340 0.212596
$$628$$ 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
$$633$$ 14.6468 0.582158
$$634$$ 5.81994 0.231139
$$635$$ 5.46405 0.216834
$$636$$ −0.518027 −0.0205411
$$637$$ 0 0
$$638$$ −7.37883 −0.292131
$$639$$ −11.4432 −0.452687
$$640$$ −29.6156 −1.17066
$$641$$ −31.1053 −1.22858 −0.614292 0.789079i $$-0.710558\pi$$
−0.614292 + 0.789079i $$0.710558\pi$$
$$642$$ −23.3580 −0.921867
$$643$$ 5.48197 0.216188 0.108094 0.994141i $$-0.465525\pi$$
0.108094 + 0.994141i $$0.465525\pi$$
$$644$$ 0 0
$$645$$ −25.2244 −0.993210
$$646$$ 49.5845 1.95088
$$647$$ 9.26383 0.364199 0.182099 0.983280i $$-0.441711\pi$$
0.182099 + 0.983280i $$0.441711\pi$$
$$648$$ −2.72161 −0.106915
$$649$$ 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 $$-0.699551\pi$$
−0.586645 + 0.809844i $$0.699551\pi$$
$$654$$ −18.6856 −0.730663
$$655$$ 9.59283 0.374823
$$656$$ −2.75477 −0.107556
$$657$$ 13.0450 0.508935
$$658$$ 0 0
$$659$$ 23.9702 0.933747 0.466873 0.884324i $$-0.345380\pi$$
0.466873 + 0.884324i $$0.345380\pi$$
$$660$$ −0.333816 −0.0129938
$$661$$ 40.4585 1.57365 0.786826 0.617175i $$-0.211723\pi$$
0.786826 + 0.617175i $$0.211723\pi$$
$$662$$ 34.2880 1.33264
$$663$$ 32.1288 1.24778
$$664$$ 35.8296 1.39046
$$665$$ 0 0
$$666$$ 6.21400 0.240788
$$667$$ 24.8477 0.962107
$$668$$ 3.39194 0.131238
$$669$$ −1.90997 −0.0738436
$$670$$ −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
$$675$$ −0.751399 −0.0289214
$$676$$ 1.73327 0.0666643
$$677$$ 1.26316 0.0485472 0.0242736 0.999705i $$-0.492273\pi$$
0.0242736 + 0.999705i $$0.492273\pi$$
$$678$$ −27.4045 −1.05246
$$679$$ 0 0
$$680$$ 41.5666 1.59401
$$681$$ −3.20359 −0.122762
$$682$$ −11.0748 −0.424076
$$683$$ 37.6441 1.44041 0.720206 0.693760i $$-0.244047\pi$$
0.720206 + 0.693760i $$0.244047\pi$$
$$684$$ 0.740987 0.0283323
$$685$$ 11.4529 0.437591
$$686$$ 0 0
$$687$$ −18.3088 −0.698526
$$688$$ 44.7964 1.70785
$$689$$ −18.7756 −0.715293
$$690$$ 17.2757 0.657674
$$691$$ −14.3892 −0.547393 −0.273696 0.961816i $$-0.588246\pi$$
−0.273696 + 0.961816i $$0.588246\pi$$
$$692$$ −1.71871 −0.0653357
$$693$$ 0 0
$$694$$ −33.0361 −1.25403
$$695$$ −37.0361 −1.40486
$$696$$ 13.7306 0.520456
$$697$$ 4.11915 0.156024
$$698$$ −40.8525 −1.54629
$$699$$ 16.5872 0.627387
$$700$$ 0 0
$$701$$ −39.2936 −1.48410 −0.742050 0.670345i $$-0.766146\pi$$
−0.742050 + 0.670345i $$0.766146\pi$$
$$702$$ 7.37883 0.278496
$$703$$ 22.6170 0.853017
$$704$$ −7.36842 −0.277708
$$705$$ −1.26383 −0.0475986
$$706$$ 24.2028 0.910885
$$707$$ 0 0
$$708$$ 1.10941 0.0416941
$$709$$ −49.2430 −1.84936 −0.924680 0.380745i $$-0.875667\pi$$
−0.924680 + 0.380745i $$0.875667\pi$$
$$710$$ 40.1384 1.50637
$$711$$ 11.4432 0.429154
$$712$$ 32.2522 1.20870
$$713$$ 37.2936 1.39666
$$714$$ 0 0
$$715$$ −12.0990 −0.452477
$$716$$ −0.778489 −0.0290935
$$717$$ −2.91623 −0.108909
$$718$$ 32.2217 1.20250
$$719$$ 7.41344 0.276475 0.138237 0.990399i $$-0.455856\pi$$
0.138237 + 0.990399i $$0.455856\pi$$
$$720$$ 10.2140 0.380653
$$721$$ 0 0
$$722$$ 13.6587 0.508323
$$723$$ −6.09899 −0.226824
$$724$$ −1.88914 −0.0702095
$$725$$ 3.79082 0.140787
$$726$$ −1.46260 −0.0542821
$$727$$ −18.9557 −0.703026 −0.351513 0.936183i $$-0.614333\pi$$
−0.351513 + 0.936183i $$0.614333\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −45.7569 −1.69354
$$731$$ −66.9832 −2.47746
$$732$$ −0.278388 −0.0102895
$$733$$ 3.59283 0.132704 0.0663521 0.997796i $$-0.478864\pi$$
0.0663521 + 0.997796i $$0.478864\pi$$
$$734$$ 28.3505 1.04644
$$735$$ 0 0
$$736$$ −3.87122 −0.142695
$$737$$ −8.76663 −0.322923
$$738$$ 0.946021 0.0348235
$$739$$ −26.7756 −0.984956 −0.492478 0.870325i $$-0.663909\pi$$
−0.492478 + 0.870325i $$0.663909\pi$$
$$740$$ −1.41825 −0.0521360
$$741$$ 26.8567 0.986604
$$742$$ 0 0
$$743$$ −33.8027 −1.24010 −0.620050 0.784562i $$-0.712888\pi$$
−0.620050 + 0.784562i $$0.712888\pi$$
$$744$$ 20.6081 0.755528
$$745$$ 23.6018 0.864703
$$746$$ 42.7881 1.56658
$$747$$ −13.1648 −0.481676
$$748$$ −0.886447 −0.0324117
$$749$$ 0 0
$$750$$ −14.9025 −0.544161
$$751$$ 35.3955 1.29160 0.645800 0.763506i $$-0.276524\pi$$
0.645800 + 0.763506i $$0.276524\pi$$
$$752$$ 2.24445 0.0818468
$$753$$ 1.62262 0.0591314
$$754$$ −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
$$759$$ 4.92520 0.178773
$$760$$ 34.7458 1.26036
$$761$$ −12.9044 −0.467783 −0.233892 0.972263i $$-0.575146\pi$$
−0.233892 + 0.972263i $$0.575146\pi$$
$$762$$ 3.33237 0.120719
$$763$$ 0 0
$$764$$ −1.31154 −0.0474500
$$765$$ −15.2728 −0.552189
$$766$$ 25.7312 0.929708
$$767$$ 40.2099 1.45189
$$768$$ −3.32485 −0.119975
$$769$$ −9.78186 −0.352743 −0.176371 0.984324i $$-0.556436\pi$$
−0.176371 + 0.984324i $$0.556436\pi$$
$$770$$ 0 0
$$771$$ −6.89541 −0.248332
$$772$$ −1.40987 −0.0507422
$$773$$ −29.7223 −1.06904 −0.534518 0.845157i $$-0.679507\pi$$
−0.534518 + 0.845157i $$0.679507\pi$$
$$774$$ −15.3836 −0.552954
$$775$$ 5.68960 0.204376
$$776$$ −5.09273 −0.182818
$$777$$ 0 0
$$778$$ 29.3836 1.05345
$$779$$ 3.44322 0.123366
$$780$$ −1.68411 −0.0603008
$$781$$ 11.4432 0.409471
$$782$$ 45.8755 1.64050
$$783$$ −5.04502 −0.180294
$$784$$ 0 0
$$785$$ −2.26875 −0.0809753
$$786$$ 5.85039 0.208677
$$787$$ −17.0242 −0.606847 −0.303423 0.952856i $$-0.598130\pi$$
−0.303423 + 0.952856i $$0.598130\pi$$
$$788$$ −0.314240 −0.0111943
$$789$$ −5.08377 −0.180987
$$790$$ −40.1384 −1.42806
$$791$$ 0 0
$$792$$ 2.72161 0.0967083
$$793$$ −10.0900 −0.358308
$$794$$ 51.3449 1.82216
$$795$$ 8.92520 0.316544
$$796$$ −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
$$801$$ −11.8504 −0.418713
$$802$$ 14.0000 0.494357
$$803$$ −13.0450 −0.460349
$$804$$ −1.22026 −0.0430354
$$805$$ 0 0
$$806$$ −55.8726 −1.96803
$$807$$ −0.886447 −0.0312044
$$808$$ 12.2963 0.432583
$$809$$ 44.4882 1.56412 0.782062 0.623201i $$-0.214168\pi$$
0.782062 + 0.623201i $$0.214168\pi$$
$$810$$ −3.50761 −0.123245
$$811$$ −7.65307 −0.268736 −0.134368 0.990932i $$-0.542900\pi$$
−0.134368 + 0.990932i $$0.542900\pi$$
$$812$$ 0 0
$$813$$ −25.3234 −0.888131
$$814$$ −6.21400 −0.217800
$$815$$ −21.0242 −0.736445
$$816$$ 27.1232 0.949501
$$817$$ −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 $$-0.781871\pi$$
−0.774246 + 0.632885i $$0.781871\pi$$
$$822$$ 6.98477 0.243622
$$823$$ 6.61702 0.230655 0.115327 0.993328i $$-0.463208\pi$$
0.115327 + 0.993328i $$0.463208\pi$$
$$824$$ −28.9765 −1.00944
$$825$$ 0.751399 0.0261604
$$826$$ 0 0
$$827$$ 39.7126 1.38094 0.690472 0.723359i $$-0.257403\pi$$
0.690472 + 0.723359i $$0.257403\pi$$
$$828$$ 0.685559 0.0238248
$$829$$ 3.90997 0.135799 0.0678994 0.997692i $$-0.478370\pi$$
0.0678994 + 0.997692i $$0.478370\pi$$
$$830$$ 46.1772 1.60283
$$831$$ 24.8269 0.861235
$$832$$ −37.1738 −1.28877
$$833$$ 0 0
$$834$$ −22.5872 −0.782132
$$835$$ −58.4405 −2.02242
$$836$$ −0.740987 −0.0256276
$$837$$ −7.57201 −0.261727
$$838$$ −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
$$843$$ −1.90101 −0.0654741
$$844$$ −2.03875 −0.0701767
$$845$$ −29.8629 −1.02732
$$846$$ −0.770774 −0.0264998
$$847$$ 0 0
$$848$$ −15.8504 −0.544305
$$849$$ −22.3178 −0.765945
$$850$$ 6.99886 0.240059
$$851$$ 20.9252 0.717307
$$852$$ 1.59283 0.0545694
$$853$$ 14.5568 0.498415 0.249207 0.968450i $$-0.419830\pi$$
0.249207 + 0.968450i $$0.419830\pi$$
$$854$$ 0 0
$$855$$ −12.7666 −0.436609
$$856$$ −43.4647 −1.48559
$$857$$ −10.4793 −0.357965 −0.178983 0.983852i $$-0.557281\pi$$
−0.178983 + 0.983852i $$0.557281\pi$$
$$858$$ −7.37883 −0.251909
$$859$$ −2.88645 −0.0984843 −0.0492421 0.998787i $$-0.515681\pi$$
−0.0492421 + 0.998787i $$0.515681\pi$$
$$860$$ 3.51109 0.119727
$$861$$ 0 0
$$862$$ 8.13650 0.277130
$$863$$ −20.5485 −0.699479 −0.349739 0.936847i $$-0.613730\pi$$
−0.349739 + 0.936847i $$0.613730\pi$$
$$864$$ 0.786003 0.0267404
$$865$$ 29.6121 1.00684
$$866$$ 37.5028 1.27440
$$867$$ −23.5568 −0.800030
$$868$$ 0 0
$$869$$ −11.4432 −0.388185
$$870$$ 17.6960 0.599950
$$871$$ −44.2278 −1.49860
$$872$$ −34.7702 −1.17747
$$873$$ 1.87122 0.0633311
$$874$$ 38.3476 1.29713
$$875$$ 0 0
$$876$$ −1.81579 −0.0613499
$$877$$ 59.1149 1.99617 0.998084 0.0618724i $$-0.0197072\pi$$
0.998084 + 0.0618724i $$0.0197072\pi$$
$$878$$ −34.6212 −1.16841
$$879$$ 12.0900 0.407787
$$880$$ −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
$$885$$ −19.1142 −0.642518
$$886$$ −26.3713 −0.885962
$$887$$ −22.9736 −0.771377 −0.385689 0.922629i $$-0.626036\pi$$
−0.385689 + 0.922629i $$0.626036\pi$$
$$888$$ 11.5630 0.388030
$$889$$ 0 0
$$890$$ 41.5666 1.39332
$$891$$ −1.00000 −0.0335013
$$892$$ 0.265856 0.00890153
$$893$$ −2.80538 −0.0938784
$$894$$ 14.3941 0.481409
$$895$$ 13.4128 0.448339
$$896$$ 0 0
$$897$$ 24.8477 0.829640
$$898$$ −51.1565 −1.70712
$$899$$ 38.2009 1.27407
$$900$$ 0.104590 0.00348634
$$901$$ 23.7008 0.789588
$$902$$ −0.946021 −0.0314991
$$903$$ 0 0
$$904$$ −50.9944 −1.69605
$$905$$ 32.5485 1.08195
$$906$$ 6.03875 0.200624
$$907$$ −57.1745 −1.89845 −0.949224 0.314602i $$-0.898129\pi$$
−0.949224 + 0.314602i $$0.898129\pi$$
$$908$$ 0.445920 0.0147984
$$909$$ −4.51803 −0.149853
$$910$$ 0 0
$$911$$ 6.82687 0.226184 0.113092 0.993584i $$-0.463924\pi$$
0.113092 + 0.993584i $$0.463924\pi$$
$$912$$ 22.6724 0.750760
$$913$$ 13.1648 0.435692
$$914$$ 6.63428 0.219442
$$915$$ 4.79641 0.158565
$$916$$ 2.54848 0.0842043
$$917$$ 0 0
$$918$$ −9.31444 −0.307422
$$919$$ 12.0692 0.398126 0.199063 0.979987i $$-0.436210\pi$$
0.199063 + 0.979987i $$0.436210\pi$$
$$920$$ 32.1467 1.05985
$$921$$ −13.5928 −0.447899
$$922$$ 4.09003 0.134698
$$923$$ 57.7312 1.90025
$$924$$ 0 0
$$925$$ 3.19239 0.104965
$$926$$ 56.1045 1.84371
$$927$$ 10.6468 0.349687
$$928$$ −3.96540 −0.130171
$$929$$ −26.8954 −0.882410 −0.441205 0.897406i $$-0.645449\pi$$
−0.441205 + 0.897406i $$0.645449\pi$$
$$930$$ 26.5597 0.870926
$$931$$ 0 0
$$932$$ −2.30885 −0.0756288
$$933$$ 8.00000 0.261908
$$934$$ 29.9356 0.979523
$$935$$ 15.2728 0.499474
$$936$$ 13.7306 0.448798
$$937$$ 14.9944 0.489846 0.244923 0.969543i $$-0.421237\pi$$
0.244923 + 0.969543i $$0.421237\pi$$
$$938$$ 0 0
$$939$$ 14.9252 0.487065
$$940$$ 0.175918 0.00573780
$$941$$ 30.1205 0.981900 0.490950 0.871188i $$-0.336650\pi$$
0.490950 + 0.871188i $$0.336650\pi$$
$$942$$ −1.38365 −0.0450817
$$943$$ 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 $$-0.590982\pi$$
−0.281951 + 0.959429i $$0.590982\pi$$
$$948$$ −1.59283 −0.0517327
$$949$$ −65.8123 −2.13636
$$950$$ 5.85039 0.189812
$$951$$ −3.97918 −0.129034
$$952$$ 0 0
$$953$$ −2.14064 −0.0693422 −0.0346711 0.999399i $$-0.511038\pi$$
−0.0346711 + 0.999399i $$0.511038\pi$$
$$954$$ 5.44322 0.176231
$$955$$ 22.5969 0.731217
$$956$$ 0.405923 0.0131285
$$957$$ 5.04502 0.163082
$$958$$ 16.9861 0.548796
$$959$$ 0 0
$$960$$ 17.6710 0.570329
$$961$$ 26.3353 0.849525
$$962$$ −31.3497 −1.01076
$$963$$ 15.9702 0.514633
$$964$$ 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
$$969$$ −33.9017 −1.08908
$$970$$ −6.56351 −0.210742
$$971$$ 26.5574 0.852269 0.426135 0.904660i $$-0.359875\pi$$
0.426135 + 0.904660i $$0.359875\pi$$
$$972$$ −0.139194 −0.00446465
$$973$$ 0 0
$$974$$ 47.5041 1.52213
$$975$$ 3.79082 0.121403
$$976$$ −8.51803 −0.272655
$$977$$ −55.9017 −1.78845 −0.894227 0.447615i $$-0.852274\pi$$
−0.894227 + 0.447615i $$0.852274\pi$$
$$978$$ −12.8221 −0.410004
$$979$$ 11.8504 0.378740
$$980$$ 0 0
$$981$$ 12.7756 0.407893
$$982$$ 39.0171 1.24509
$$983$$ 53.0361 1.69159 0.845794 0.533510i $$-0.179127\pi$$
0.845794 + 0.533510i $$0.179127\pi$$
$$984$$ 1.76036 0.0561183
$$985$$ 5.41411 0.172508
$$986$$ 46.9915 1.49651
$$987$$ 0 0
$$988$$ −3.73829 −0.118931
$$989$$ −51.8034 −1.64725
$$990$$ 3.50761 0.111479
$$991$$ 14.7362 0.468110 0.234055 0.972223i $$-0.424800\pi$$
0.234055 + 0.972223i $$0.424800\pi$$
$$992$$ −5.95162 −0.188964
$$993$$ −23.4432 −0.743948
$$994$$ 0 0
$$995$$ 7.37402 0.233772
$$996$$ 1.83247 0.0580640
$$997$$ 45.0665 1.42727 0.713635 0.700517i $$-0.247047\pi$$
0.713635 + 0.700517i $$0.247047\pi$$
$$998$$ −60.3262 −1.90959
$$999$$ −4.24860 −0.134420
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 1617.2.a.t.1.2 3
3.2 odd 2 4851.2.a.bi.1.2 3
7.6 odd 2 231.2.a.e.1.2 3
21.20 even 2 693.2.a.l.1.2 3
28.27 even 2 3696.2.a.bo.1.2 3
35.34 odd 2 5775.2.a.bp.1.2 3
77.76 even 2 2541.2.a.bg.1.2 3
231.230 odd 2 7623.2.a.cd.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 7.6 odd 2
693.2.a.l.1.2 3 21.20 even 2
1617.2.a.t.1.2 3 1.1 even 1 trivial
2541.2.a.bg.1.2 3 77.76 even 2
3696.2.a.bo.1.2 3 28.27 even 2
4851.2.a.bi.1.2 3 3.2 odd 2
5775.2.a.bp.1.2 3 35.34 odd 2
7623.2.a.cd.1.2 3 231.230 odd 2