# Properties

 Label 2541.2.a.bg.1.2 Level $2541$ Weight $2$ Character 2541.1 Self dual yes Analytic conductor $20.290$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2541 = 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2541.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2899871536$$ 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 $$\chi$$ $$=$$ 2541.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} -1.46260 q^{6} +1.00000 q^{7} +2.72161 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} -1.46260 q^{6} +1.00000 q^{7} +2.72161 q^{8} +1.00000 q^{9} -3.50761 q^{10} +0.139194 q^{12} -5.04502 q^{13} -1.46260 q^{14} +2.39821 q^{15} -4.25901 q^{16} +6.36842 q^{17} -1.46260 q^{18} +5.32340 q^{19} +0.333816 q^{20} +1.00000 q^{21} +4.92520 q^{23} +2.72161 q^{24} +0.751399 q^{25} +7.37883 q^{26} +1.00000 q^{27} +0.139194 q^{28} -5.04502 q^{29} -3.50761 q^{30} -7.57201 q^{31} +0.786003 q^{32} -9.31444 q^{34} +2.39821 q^{35} +0.139194 q^{36} +4.24860 q^{37} -7.78600 q^{38} -5.04502 q^{39} +6.52699 q^{40} +0.646809 q^{41} -1.46260 q^{42} +10.5180 q^{43} +2.39821 q^{45} -7.20359 q^{46} +0.526989 q^{47} -4.25901 q^{48} +1.00000 q^{49} -1.09899 q^{50} +6.36842 q^{51} -0.702237 q^{52} +3.72161 q^{53} -1.46260 q^{54} +2.72161 q^{56} +5.32340 q^{57} +7.37883 q^{58} +7.97021 q^{59} +0.333816 q^{60} +2.00000 q^{61} +11.0748 q^{62} +1.00000 q^{63} +7.36842 q^{64} -12.0990 q^{65} +8.76663 q^{67} +0.886447 q^{68} +4.92520 q^{69} -3.50761 q^{70} -11.4432 q^{71} +2.72161 q^{72} +13.0450 q^{73} -6.21400 q^{74} +0.751399 q^{75} +0.740987 q^{76} +7.37883 q^{78} -11.4432 q^{79} -10.2140 q^{80} +1.00000 q^{81} -0.946021 q^{82} -13.1648 q^{83} +0.139194 q^{84} +15.2728 q^{85} -15.3836 q^{86} -5.04502 q^{87} +11.8504 q^{89} -3.50761 q^{90} -5.04502 q^{91} +0.685559 q^{92} -7.57201 q^{93} -0.770774 q^{94} +12.7666 q^{95} +0.786003 q^{96} -1.87122 q^{97} -1.46260 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 3 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 11 q^{10} + 6 q^{12} + 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} - 8 q^{17} - 2 q^{18} + 8 q^{19} - 3 q^{20} + 3 q^{21} + 10 q^{23} - 3 q^{24} + 15 q^{25} - q^{26} + 3 q^{27} + 6 q^{28} + 4 q^{29} + 11 q^{30} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 6 q^{36} - 13 q^{38} + 4 q^{39} + 18 q^{40} - 14 q^{41} - 2 q^{42} + 14 q^{43} + 4 q^{45} - 28 q^{46} - 4 q^{48} + 3 q^{49} + 19 q^{50} - 8 q^{51} + 29 q^{52} - 2 q^{54} - 3 q^{56} + 8 q^{57} - q^{58} - 3 q^{60} + 6 q^{61} + 38 q^{62} + 3 q^{63} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 10 q^{69} + 11 q^{70} - 12 q^{71} - 3 q^{72} + 20 q^{73} - 29 q^{74} + 15 q^{75} + 11 q^{76} - q^{78} - 12 q^{79} - 41 q^{80} + 3 q^{81} - 6 q^{82} - 6 q^{83} + 6 q^{84} + 6 q^{85} + 24 q^{86} + 4 q^{87} + 26 q^{89} + 11 q^{90} + 4 q^{91} + 26 q^{92} - 2 q^{93} - 35 q^{94} + 8 q^{95} - 8 q^{96} - 4 q^{97} - 2 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 + 3 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 11 * q^10 + 6 * q^12 + 4 * q^13 - 2 * q^14 + 4 * q^15 - 4 * q^16 - 8 * q^17 - 2 * q^18 + 8 * q^19 - 3 * q^20 + 3 * q^21 + 10 * q^23 - 3 * q^24 + 15 * q^25 - q^26 + 3 * q^27 + 6 * q^28 + 4 * q^29 + 11 * q^30 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 6 * q^36 - 13 * q^38 + 4 * q^39 + 18 * q^40 - 14 * q^41 - 2 * q^42 + 14 * q^43 + 4 * q^45 - 28 * q^46 - 4 * q^48 + 3 * q^49 + 19 * q^50 - 8 * q^51 + 29 * q^52 - 2 * q^54 - 3 * q^56 + 8 * q^57 - q^58 - 3 * q^60 + 6 * q^61 + 38 * q^62 + 3 * q^63 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 10 * q^69 + 11 * q^70 - 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^84 + 6 * q^85 + 24 * q^86 + 4 * q^87 + 26 * q^89 + 11 * q^90 + 4 * q^91 + 26 * q^92 - 2 * q^93 - 35 * q^94 + 8 * q^95 - 8 * q^96 - 4 * q^97 - 2 * q^98

## 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 $$0.672991\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 0.139194 0.0695971
$$5$$ 2.39821 1.07251 0.536255 0.844056i $$-0.319838\pi$$
0.536255 + 0.844056i $$0.319838\pi$$
$$6$$ −1.46260 −0.597103
$$7$$ 1.00000 0.377964
$$8$$ 2.72161 0.962235
$$9$$ 1.00000 0.333333
$$10$$ −3.50761 −1.10921
$$11$$ 0 0
$$12$$ 0.139194 0.0401819
$$13$$ −5.04502 −1.39924 −0.699618 0.714517i $$-0.746646\pi$$
−0.699618 + 0.714517i $$0.746646\pi$$
$$14$$ −1.46260 −0.390896
$$15$$ 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
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 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
$$28$$ 0.139194 0.0263052
$$29$$ −5.04502 −0.936836 −0.468418 0.883507i $$-0.655176\pi$$
−0.468418 + 0.883507i $$0.655176\pi$$
$$30$$ −3.50761 −0.640400
$$31$$ −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
$$36$$ 0.139194 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 $$0.483916\pi$$
$$42$$ −1.46260 −0.225684
$$43$$ 10.5180 1.60398 0.801992 0.597335i $$-0.203774\pi$$
0.801992 + 0.597335i $$0.203774\pi$$
$$44$$ 0 0
$$45$$ 2.39821 0.357504
$$46$$ −7.20359 −1.06211
$$47$$ 0.526989 0.0768693 0.0384347 0.999261i $$-0.487763\pi$$
0.0384347 + 0.999261i $$0.487763\pi$$
$$48$$ −4.25901 −0.614736
$$49$$ 1.00000 0.142857
$$50$$ −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
$$55$$ 0 0
$$56$$ 2.72161 0.363691
$$57$$ 5.32340 0.705102
$$58$$ 7.37883 0.968888
$$59$$ 7.97021 1.03763 0.518817 0.854886i $$-0.326373\pi$$
0.518817 + 0.854886i $$0.326373\pi$$
$$60$$ 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
$$63$$ 1.00000 0.125988
$$64$$ 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
$$69$$ 4.92520 0.592924
$$70$$ −3.50761 −0.419240
$$71$$ −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 $$-0.722615\pi$$
−0.643732 + 0.765251i $$0.722615\pi$$
$$80$$ −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 $$0.757014\pi$$
$$84$$ 0.139194 0.0151873
$$85$$ 15.2728 1.65657
$$86$$ −15.3836 −1.65886
$$87$$ −5.04502 −0.540882
$$88$$ 0 0
$$89$$ 11.8504 1.25614 0.628070 0.778157i $$-0.283845\pi$$
0.628070 + 0.778157i $$0.283845\pi$$
$$90$$ −3.50761 −0.369735
$$91$$ −5.04502 −0.528861
$$92$$ 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 $$-0.530284\pi$$
−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 $$0.572166\pi$$
$$102$$ −9.31444 −0.922267
$$103$$ −10.6468 −1.04906 −0.524531 0.851392i $$-0.675759\pi$$
−0.524531 + 0.851392i $$0.675759\pi$$
$$104$$ −13.7306 −1.34639
$$105$$ 2.39821 0.234041
$$106$$ −5.44322 −0.528693
$$107$$ −15.9702 −1.54390 −0.771949 0.635684i $$-0.780718\pi$$
−0.771949 + 0.635684i $$0.780718\pi$$
$$108$$ 0.139194 0.0133940
$$109$$ −12.7756 −1.22368 −0.611840 0.790982i $$-0.709570\pi$$
−0.611840 + 0.790982i $$0.709570\pi$$
$$110$$ 0 0
$$111$$ 4.24860 0.403259
$$112$$ −4.25901 −0.402439
$$113$$ 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
$$119$$ 6.36842 0.583792
$$120$$ 6.52699 0.595830
$$121$$ 0 0
$$122$$ −2.92520 −0.264835
$$123$$ 0.646809 0.0583208
$$124$$ −1.05398 −0.0946501
$$125$$ −10.1890 −0.911334
$$126$$ −1.46260 −0.130299
$$127$$ 2.27839 0.202174 0.101087 0.994878i $$-0.467768\pi$$
0.101087 + 0.994878i $$0.467768\pi$$
$$128$$ −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 0
$$133$$ 5.32340 0.461598
$$134$$ −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 $$0.272694\pi$$
$$140$$ 0.333816 0.0282126
$$141$$ 0.526989 0.0443805
$$142$$ 16.7368 1.40452
$$143$$ 0 0
$$144$$ −4.25901 −0.354918
$$145$$ −12.0990 −1.00477
$$146$$ −19.0796 −1.57904
$$147$$ 1.00000 0.0824786
$$148$$ 0.591380 0.0486112
$$149$$ 9.84143 0.806241 0.403121 0.915147i $$-0.367925\pi$$
0.403121 + 0.915147i $$0.367925\pi$$
$$150$$ −1.09899 −0.0897326
$$151$$ 4.12878 0.335996 0.167998 0.985787i $$-0.446270\pi$$
0.167998 + 0.985787i $$0.446270\pi$$
$$152$$ 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 $$-0.512019\pi$$
−0.0377504 + 0.999287i $$0.512019\pi$$
$$158$$ 16.7368 1.33151
$$159$$ 3.72161 0.295143
$$160$$ 1.88500 0.149022
$$161$$ 4.92520 0.388160
$$162$$ −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
$$165$$ 0 0
$$166$$ 19.2549 1.49447
$$167$$ 24.3684 1.88568 0.942842 0.333239i $$-0.108142\pi$$
0.942842 + 0.333239i $$0.108142\pi$$
$$168$$ 2.72161 0.209977
$$169$$ 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
$$175$$ 0.751399 0.0568004
$$176$$ 0 0
$$177$$ 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 $$-0.331714\pi$$
0.504400 + 0.863470i $$0.331714\pi$$
$$182$$ 7.37883 0.546955
$$183$$ 2.00000 0.147844
$$184$$ 13.4045 0.988191
$$185$$ 10.1890 0.749112
$$186$$ 11.0748 0.812044
$$187$$ 0 0
$$188$$ 0.0733538 0.00534988
$$189$$ 1.00000 0.0727393
$$190$$ −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 $$-0.381225\pi$$
0.364543 + 0.931187i $$0.381225\pi$$
$$194$$ 2.73684 0.196494
$$195$$ −12.0990 −0.866427
$$196$$ 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
$$201$$ 8.76663 0.618350
$$202$$ 6.60806 0.464941
$$203$$ −5.04502 −0.354091
$$204$$ 0.886447 0.0620637
$$205$$ 1.55118 0.108339
$$206$$ 15.5720 1.08495
$$207$$ 4.92520 0.342325
$$208$$ 21.4868 1.48984
$$209$$ 0 0
$$210$$ −3.50761 −0.242048
$$211$$ 14.6468 1.00833 0.504164 0.863608i $$-0.331801\pi$$
0.504164 + 0.863608i $$0.331801\pi$$
$$212$$ 0.518027 0.0355782
$$213$$ −11.4432 −0.784077
$$214$$ 23.3580 1.59672
$$215$$ 25.2244 1.72029
$$216$$ 2.72161 0.185182
$$217$$ −7.57201 −0.514021
$$218$$ 18.6856 1.26555
$$219$$ 13.0450 0.881500
$$220$$ 0 0
$$221$$ −32.1288 −2.16122
$$222$$ −6.21400 −0.417056
$$223$$ −1.90997 −0.127901 −0.0639505 0.997953i $$-0.520370\pi$$
−0.0639505 + 0.997953i $$0.520370\pi$$
$$224$$ 0.786003 0.0525170
$$225$$ 0.751399 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 $$-0.706803\pi$$
−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 $$0.317162\pi$$
$$234$$ 7.37883 0.482369
$$235$$ 1.26383 0.0824432
$$236$$ 1.10941 0.0722162
$$237$$ −11.4432 −0.743317
$$238$$ −9.31444 −0.603766
$$239$$ −2.91623 −0.188635 −0.0943177 0.995542i $$-0.530067\pi$$
−0.0943177 + 0.995542i $$0.530067\pi$$
$$240$$ −10.2140 −0.659311
$$241$$ 6.09899 0.392871 0.196435 0.980517i $$-0.437063\pi$$
0.196435 + 0.980517i $$0.437063\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0.278388 0.0178220
$$245$$ 2.39821 0.153216
$$246$$ −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 $$-0.483692\pi$$
0.0512093 + 0.998688i $$0.483692\pi$$
$$252$$ 0.139194 0.00876841
$$253$$ 0 0
$$254$$ −3.33237 −0.209091
$$255$$ 15.2728 0.956419
$$256$$ 3.32485 0.207803
$$257$$ −6.89541 −0.430124 −0.215062 0.976600i $$-0.568995\pi$$
−0.215062 + 0.976600i $$0.568995\pi$$
$$258$$ −15.3836 −0.957744
$$259$$ 4.24860 0.263995
$$260$$ −1.68411 −0.104444
$$261$$ −5.04502 −0.312279
$$262$$ 5.85039 0.361439
$$263$$ −5.08377 −0.313478 −0.156739 0.987640i $$-0.550098\pi$$
−0.156739 + 0.987640i $$0.550098\pi$$
$$264$$ 0 0
$$265$$ 8.92520 0.548270
$$266$$ −7.78600 −0.477390
$$267$$ 11.8504 0.725232
$$268$$ 1.22026 0.0745394
$$269$$ −0.886447 −0.0540476 −0.0270238 0.999635i $$-0.508603\pi$$
−0.0270238 + 0.999635i $$0.508603\pi$$
$$270$$ −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
$$273$$ −5.04502 −0.305338
$$274$$ 6.98477 0.421965
$$275$$ 0 0
$$276$$ 0.685559 0.0412658
$$277$$ 24.8269 1.49170 0.745851 0.666113i $$-0.232043\pi$$
0.745851 + 0.666113i $$0.232043\pi$$
$$278$$ −22.5872 −1.35469
$$279$$ −7.57201 −0.453324
$$280$$ 6.52699 0.390062
$$281$$ −1.90101 −0.113404 −0.0567022 0.998391i $$-0.518059\pi$$
−0.0567022 + 0.998391i $$0.518059\pi$$
$$282$$ −0.770774 −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
$$286$$ 0 0
$$287$$ 0.646809 0.0381799
$$288$$ 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 $$0.614891\pi$$
$$294$$ −1.46260 −0.0853005
$$295$$ 19.1142 1.11287
$$296$$ 11.5630 0.672088
$$297$$ 0 0
$$298$$ −14.3941 −0.833826
$$299$$ −24.8477 −1.43698
$$300$$ 0.104590 0.00603853
$$301$$ 10.5180 0.606249
$$302$$ −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 $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ −13.7306 −0.777341
$$313$$ 14.9252 0.843622 0.421811 0.906684i $$-0.361395\pi$$
0.421811 + 0.906684i $$0.361395\pi$$
$$314$$ 1.38365 0.0780838
$$315$$ 2.39821 0.135124
$$316$$ −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
$$319$$ 0 0
$$320$$ 17.6710 0.987839
$$321$$ −15.9702 −0.891370
$$322$$ −7.20359 −0.401440
$$323$$ 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
$$329$$ 0.526989 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
$$333$$ 4.24860 0.232822
$$334$$ −35.6412 −1.95020
$$335$$ 21.0242 1.14867
$$336$$ −4.25901 −0.232348
$$337$$ −11.1648 −0.608187 −0.304094 0.952642i $$-0.598354\pi$$
−0.304094 + 0.952642i $$0.598354\pi$$
$$338$$ −18.2125 −0.990632
$$339$$ 18.7368 1.01765
$$340$$ 2.12588 0.115292
$$341$$ 0 0
$$342$$ −7.78600 −0.421019
$$343$$ 1.00000 0.0539949
$$344$$ 28.6260 1.54341
$$345$$ 11.8116 0.635918
$$346$$ 18.0596 0.970889
$$347$$ 22.5872 1.21255 0.606273 0.795256i $$-0.292664\pi$$
0.606273 + 0.795256i $$0.292664\pi$$
$$348$$ −0.702237 −0.0376438
$$349$$ −27.9315 −1.49514 −0.747568 0.664185i $$-0.768779\pi$$
−0.747568 + 0.664185i $$0.768779\pi$$
$$350$$ −1.09899 −0.0587437
$$351$$ −5.04502 −0.269283
$$352$$ 0 0
$$353$$ −16.5478 −0.880751 −0.440376 0.897814i $$-0.645155\pi$$
−0.440376 + 0.897814i $$0.645155\pi$$
$$354$$ −11.6572 −0.619574
$$355$$ −27.4432 −1.45654
$$356$$ 1.64951 0.0874236
$$357$$ 6.36842 0.337053
$$358$$ 8.18006 0.432330
$$359$$ −22.0305 −1.16272 −0.581362 0.813645i $$-0.697480\pi$$
−0.581362 + 0.813645i $$0.697480\pi$$
$$360$$ 6.52699 0.344003
$$361$$ 9.33863 0.491507
$$362$$ −19.8504 −1.04331
$$363$$ 0 0
$$364$$ −0.702237 −0.0368072
$$365$$ 31.2847 1.63751
$$366$$ −2.92520 −0.152902
$$367$$ −19.3836 −1.01182 −0.505909 0.862587i $$-0.668843\pi$$
−0.505909 + 0.862587i $$0.668843\pi$$
$$368$$ −20.9765 −1.09347
$$369$$ 0.646809 0.0336715
$$370$$ −14.9025 −0.774742
$$371$$ 3.72161 0.193216
$$372$$ −1.05398 −0.0546463
$$373$$ −29.2549 −1.51476 −0.757380 0.652975i $$-0.773521\pi$$
−0.757380 + 0.652975i $$0.773521\pi$$
$$374$$ 0 0
$$375$$ −10.1890 −0.526159
$$376$$ 1.43426 0.0739663
$$377$$ 25.4522 1.31085
$$378$$ −1.46260 −0.0752279
$$379$$ 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 $$-0.648389\pi$$
−0.449476 + 0.893293i $$0.648389\pi$$
$$384$$ −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
$$392$$ 2.72161 0.137462
$$393$$ −4.00000 −0.201773
$$394$$ −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
$$399$$ 5.32340 0.266504
$$400$$ −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
$$406$$ 7.37883 0.366205
$$407$$ 0 0
$$408$$ 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
$$413$$ 7.97021 0.392189
$$414$$ −7.20359 −0.354037
$$415$$ −31.5720 −1.54981
$$416$$ −3.96540 −0.194420
$$417$$ 15.4432 0.756258
$$418$$ 0 0
$$419$$ 7.17380 0.350463 0.175231 0.984527i $$-0.443933\pi$$
0.175231 + 0.984527i $$0.443933\pi$$
$$420$$ 0.333816 0.0162886
$$421$$ 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
$$427$$ 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 $$0.542776\pi$$
$$432$$ −4.25901 −0.204912
$$433$$ −25.6412 −1.23224 −0.616119 0.787653i $$-0.711296\pi$$
−0.616119 + 0.787653i $$0.711296\pi$$
$$434$$ 11.0748 0.531608
$$435$$ −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 $$0.691077\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 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
$$448$$ 7.36842 0.348125
$$449$$ −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 0
$$452$$ 2.60806 0.122673
$$453$$ 4.12878 0.193987
$$454$$ −4.68556 −0.219904
$$455$$ −12.0990 −0.567210
$$456$$ 14.4882 0.678474
$$457$$ −4.53595 −0.212183 −0.106091 0.994356i $$-0.533834\pi$$
−0.106091 + 0.994356i $$0.533834\pi$$
$$458$$ 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 $$-0.657031\pi$$
−0.473560 + 0.880762i $$0.657031\pi$$
$$468$$ −0.702237 −0.0324609
$$469$$ 8.76663 0.404805
$$470$$ −1.84848 −0.0852638
$$471$$ −0.946021 −0.0435904
$$472$$ 21.6918 0.998447
$$473$$ 0 0
$$474$$ 16.7368 0.768749
$$475$$ 4.00000 0.183533
$$476$$ 0.886447 0.0406302
$$477$$ 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
$$483$$ 4.92520 0.224104
$$484$$ 0 0
$$485$$ −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
$$490$$ −3.50761 −0.158458
$$491$$ −26.6766 −1.20390 −0.601949 0.798535i $$-0.705609\pi$$
−0.601949 + 0.798535i $$0.705609\pi$$
$$492$$ 0.0900320 0.00405895
$$493$$ −32.1288 −1.44701
$$494$$ 39.2805 1.76731
$$495$$ 0 0
$$496$$ 32.2493 1.44804
$$497$$ −11.4432 −0.513299
$$498$$ 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 $$0.738179\pi$$
$$504$$ 2.72161 0.121230
$$505$$ −10.8352 −0.482159
$$506$$ 0 0
$$507$$ 12.4522 0.553021
$$508$$ 0.317138 0.0140707
$$509$$ −18.9944 −0.841912 −0.420956 0.907081i $$-0.638305\pi$$
−0.420956 + 0.907081i $$0.638305\pi$$
$$510$$ −22.3380 −0.989142
$$511$$ 13.0450 0.577078
$$512$$ 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 0
$$518$$ −6.21400 −0.273027
$$519$$ −12.3476 −0.541999
$$520$$ −32.9288 −1.44402
$$521$$ 25.2430 1.10592 0.552958 0.833209i $$-0.313499\pi$$
0.552958 + 0.833209i $$0.313499\pi$$
$$522$$ 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
$$525$$ 0.751399 0.0327937
$$526$$ 7.43551 0.324204
$$527$$ −48.2217 −2.10057
$$528$$ 0 0
$$529$$ 1.25756 0.0546767
$$530$$ −13.0540 −0.567029
$$531$$ 7.97021 0.345878
$$532$$ 0.740987 0.0321258
$$533$$ −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 $$-0.561307\pi$$
−0.191414 + 0.981509i $$0.561307\pi$$
$$542$$ −37.0380 −1.59092
$$543$$ 13.5720 0.582430
$$544$$ 5.00560 0.214613
$$545$$ −30.6385 −1.31241
$$546$$ 7.37883 0.315785
$$547$$ 29.4737 1.26020 0.630102 0.776513i $$-0.283013\pi$$
0.630102 + 0.776513i $$0.283013\pi$$
$$548$$ −0.664734 −0.0283960
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −26.8567 −1.14413
$$552$$ 13.4045 0.570532
$$553$$ −11.4432 −0.486615
$$554$$ −36.3117 −1.54274
$$555$$ 10.1890 0.432500
$$556$$ 2.14961 0.0911636
$$557$$ −14.8954 −0.631139 −0.315569 0.948903i $$-0.602196\pi$$
−0.315569 + 0.948903i $$0.602196\pi$$
$$558$$ 11.0748 0.468834
$$559$$ −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 $$0.447363\pi$$
$$564$$ 0.0733538 0.00308875
$$565$$ 44.9348 1.89042
$$566$$ −32.6420 −1.37205
$$567$$ 1.00000 0.0419961
$$568$$ −31.1440 −1.30677
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ −18.6724 −0.782103
$$571$$ −25.5512 −1.06928 −0.534642 0.845079i $$-0.679553\pi$$
−0.534642 + 0.845079i $$0.679553\pi$$
$$572$$ 0 0
$$573$$ −9.42240 −0.393626
$$574$$ −0.946021 −0.0394862
$$575$$ 3.70079 0.154334
$$576$$ 7.36842 0.307018
$$577$$ −29.5124 −1.22862 −0.614309 0.789065i $$-0.710565\pi$$
−0.614309 + 0.789065i $$0.710565\pi$$
$$578$$ −34.4541 −1.43310
$$579$$ 10.1288 0.420938
$$580$$ −1.68411 −0.0699288
$$581$$ −13.1648 −0.546169
$$582$$ 2.73684 0.113446
$$583$$ 0 0
$$584$$ 35.5035 1.46914
$$585$$ −12.0990 −0.500232
$$586$$ 17.6829 0.730472
$$587$$ 31.3955 1.29583 0.647916 0.761712i $$-0.275641\pi$$
0.647916 + 0.761712i $$0.275641\pi$$
$$588$$ 0.139194 0.00574027
$$589$$ −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 $$0.448073\pi$$
$$594$$ 0 0
$$595$$ 15.2728 0.626123
$$596$$ 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 $$0.726268\pi$$
$$602$$ −15.3836 −0.626991
$$603$$ 8.76663 0.357005
$$604$$ 0.574702 0.0233843
$$605$$ 0 0
$$606$$ 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
$$609$$ −5.04502 −0.204434
$$610$$ −7.01523 −0.284038
$$611$$ −2.65867 −0.107558
$$612$$ 0.886447 0.0358325
$$613$$ 33.9917 1.37291 0.686456 0.727171i $$-0.259165\pi$$
0.686456 + 0.727171i $$0.259165\pi$$
$$614$$ −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 $$-0.202668\pi$$
0.804061 + 0.594546i $$0.202668\pi$$
$$620$$ −2.52766 −0.101513
$$621$$ 4.92520 0.197641
$$622$$ −11.7008 −0.469159
$$623$$ 11.8504 0.474776
$$624$$ 21.4868 0.860160
$$625$$ −28.1924 −1.12770
$$626$$ −21.8296 −0.872485
$$627$$ 0 0
$$628$$ −0.131681 −0.00525463
$$629$$ 27.0569 1.07883
$$630$$ −3.50761 −0.139747
$$631$$ 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
$$637$$ −5.04502 −0.199891
$$638$$ 0 0
$$639$$ −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 $$-0.534475\pi$$
−0.108094 + 0.994141i $$0.534475\pi$$
$$644$$ 0.685559 0.0270148
$$645$$ 25.2244 0.993210
$$646$$ −49.5845 −1.95088
$$647$$ −9.26383 −0.364199 −0.182099 0.983280i $$-0.558289\pi$$
−0.182099 + 0.983280i $$0.558289\pi$$
$$648$$ 2.72161 0.106915
$$649$$ 0 0
$$650$$ 5.54445 0.217471
$$651$$ −7.57201 −0.296770
$$652$$ 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
$$658$$ −0.770774 −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
$$663$$ −32.1288 −1.24778
$$664$$ −35.8296 −1.39046
$$665$$ 12.7666 0.495069
$$666$$ −6.21400 −0.240788
$$667$$ −24.8477 −0.962107
$$668$$ 3.39194 0.131238
$$669$$ −1.90997 −0.0738436
$$670$$ −30.7499 −1.18797
$$671$$ 0 0
$$672$$ 0.786003 0.0303207
$$673$$ −21.8712 −0.843073 −0.421537 0.906811i $$-0.638509\pi$$
−0.421537 + 0.906811i $$0.638509\pi$$
$$674$$ 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
$$679$$ −1.87122 −0.0718108
$$680$$ 41.5666 1.59401
$$681$$ 3.20359 0.122762
$$682$$ 0 0
$$683$$ 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
$$686$$ −1.46260 −0.0558423
$$687$$ −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 $$-0.411754\pi$$
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
$$696$$ −13.7306 −0.520456
$$697$$ 4.11915 0.156024
$$698$$ 40.8525 1.54629
$$699$$ 16.5872 0.627387
$$700$$ 0.104590 0.00395314
$$701$$ 39.2936 1.48410 0.742050 0.670345i $$-0.233854\pi$$
0.742050 + 0.670345i $$0.233854\pi$$
$$702$$ 7.37883 0.278496
$$703$$ 22.6170 0.853017
$$704$$ 0 0
$$705$$ 1.26383 0.0475986
$$706$$ 24.2028 0.910885
$$707$$ −4.51803 −0.169918
$$708$$ 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
$$714$$ −9.31444 −0.348584
$$715$$ 0 0
$$716$$ −0.778489 −0.0290935
$$717$$ −2.91623 −0.108909
$$718$$ 32.2217 1.20250
$$719$$ −7.41344 −0.276475 −0.138237 0.990399i $$-0.544144\pi$$
−0.138237 + 0.990399i $$0.544144\pi$$
$$720$$ −10.2140 −0.380653
$$721$$ −10.6468 −0.396508
$$722$$ −13.6587 −0.508323
$$723$$ 6.09899 0.226824
$$724$$ 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
$$729$$ 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
$$735$$ 2.39821 0.0884592
$$736$$ 3.87122 0.142695
$$737$$ 0 0
$$738$$ −0.946021 −0.0348235
$$739$$ 26.7756 0.984956 0.492478 0.870325i $$-0.336091\pi$$
0.492478 + 0.870325i $$0.336091\pi$$
$$740$$ 1.41825 0.0521360
$$741$$ −26.8567 −0.986604
$$742$$ −5.44322 −0.199827
$$743$$ 33.8027 1.24010 0.620050 0.784562i $$-0.287112\pi$$
0.620050 + 0.784562i $$0.287112\pi$$
$$744$$ −20.6081 −0.755528
$$745$$ 23.6018 0.864703
$$746$$ 42.7881 1.56658
$$747$$ −13.1648 −0.481676
$$748$$ 0 0
$$749$$ −15.9702 −0.583539
$$750$$ 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
$$756$$ 0.139194 0.00506244
$$757$$ −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 $$0.575146\pi$$
$$762$$ −3.33237 −0.120719
$$763$$ −12.7756 −0.462507
$$764$$ −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 $$-0.320493\pi$$
0.534518 + 0.845157i $$0.320493\pi$$
$$774$$ −15.3836 −0.552954
$$775$$ −5.68960 −0.204376
$$776$$ −5.09273 −0.182818
$$777$$ 4.24860 0.152418
$$778$$ −29.3836 −1.05345
$$779$$ 3.44322 0.123366
$$780$$ −1.68411 −0.0603008
$$781$$ 0 0
$$782$$ −45.8755 −1.64050
$$783$$ −5.04502 −0.180294
$$784$$ −4.25901 −0.152108
$$785$$ −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
$$791$$ 18.7368 0.666205
$$792$$ 0 0
$$793$$ −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 $$-0.723220\pi$$
−0.645185 + 0.764027i $$0.723220\pi$$
$$798$$ −7.78600 −0.275622
$$799$$ 3.35609 0.118730
$$800$$ 0.590602 0.0208809
$$801$$ 11.8504 0.418713
$$802$$ −14.0000 −0.494357
$$803$$ 0 0
$$804$$ 1.22026 0.0430354
$$805$$ 11.8116 0.416306
$$806$$ −55.8726 −1.96803
$$807$$ −0.886447 −0.0312044
$$808$$ −12.2963 −0.432583
$$809$$ −44.4882 −1.56412 −0.782062 0.623201i $$-0.785832\pi$$
−0.782062 + 0.623201i $$0.785832\pi$$
$$810$$ −3.50761 −0.123245
$$811$$ −7.65307 −0.268736 −0.134368 0.990932i $$-0.542900\pi$$
−0.134368 + 0.990932i $$0.542900\pi$$
$$812$$ −0.702237 −0.0246437
$$813$$ 25.3234 0.888131
$$814$$ 0 0
$$815$$ 21.0242 0.736445
$$816$$ −27.1232 −0.949501
$$817$$ 55.9917 1.95890
$$818$$ 55.8421 1.95247
$$819$$ −5.04502 −0.176287
$$820$$ 0.215915 0.00754009
$$821$$ 44.3691 1.54849 0.774246 0.632885i $$-0.218129\pi$$
0.774246 + 0.632885i $$0.218129\pi$$
$$822$$ 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 0
$$826$$ −11.6572 −0.405607
$$827$$ −39.7126 −1.38094 −0.690472 0.723359i $$-0.742597\pi$$
−0.690472 + 0.723359i $$0.742597\pi$$
$$828$$ 0.685559 0.0238248
$$829$$ −3.90997 −0.135799 −0.0678994 0.997692i $$-0.521630\pi$$
−0.0678994 + 0.997692i $$0.521630\pi$$
$$830$$ 46.1772 1.60283
$$831$$ 24.8269 0.861235
$$832$$ −37.1738 −1.28877
$$833$$ 6.36842 0.220653
$$834$$ −22.5872 −0.782132
$$835$$ 58.4405 2.02242
$$836$$ 0 0
$$837$$ −7.57201 −0.261727
$$838$$ −10.4924 −0.362453
$$839$$ −9.58097 −0.330772 −0.165386 0.986229i $$-0.552887\pi$$
−0.165386 + 0.986229i $$0.552887\pi$$
$$840$$ 6.52699 0.225203
$$841$$ −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 $$0.419830\pi$$
$$854$$ −2.92520 −0.100098
$$855$$ 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 $$0.557281\pi$$
$$858$$ 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
$$861$$ 0.646809 0.0220432
$$862$$ 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
$$868$$ −1.05398 −0.0357744
$$869$$ 0 0
$$870$$ 17.6960 0.599950
$$871$$ −44.2278 −1.49860
$$872$$ −34.7702 −1.17747
$$873$$ −1.87122 −0.0633311
$$874$$ −38.3476 −1.29713
$$875$$ −10.1890 −0.344452
$$876$$ 1.81579 0.0613499
$$877$$ −59.1149 −1.99617 −0.998084 0.0618724i $$-0.980293\pi$$
−0.998084 + 0.0618724i $$0.980293\pi$$
$$878$$ 34.6212 1.16841
$$879$$ −12.0900 −0.407787
$$880$$ 0 0
$$881$$ 30.3982 1.02414 0.512071 0.858943i $$-0.328879\pi$$
0.512071 + 0.858943i $$0.328879\pi$$
$$882$$ −1.46260 −0.0492483
$$883$$ −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
$$889$$ 2.27839 0.0764147
$$890$$ −41.5666 −1.39332
$$891$$ 0 0
$$892$$ −0.265856 −0.00890153
$$893$$ 2.80538 0.0938784
$$894$$ −14.3941 −0.481409
$$895$$ −13.4128 −0.448339
$$896$$ −12.3490 −0.412553
$$897$$ −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 0
$$903$$ 10.5180 0.350018
$$904$$ 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
$$910$$ 17.6960 0.586616
$$911$$ 6.82687 0.226184 0.113092 0.993584i $$-0.463924\pi$$
0.113092 + 0.993584i $$0.463924\pi$$
$$912$$ −22.6724 −0.750760
$$913$$ 0 0
$$914$$ 6.63428 0.219442
$$915$$ 4.79641 0.158565
$$916$$ −2.54848 −0.0842043
$$917$$ −4.00000 −0.132092
$$918$$ −9.31444 −0.307422
$$919$$ −12.0692 −0.398126 −0.199063 0.979987i $$-0.563790\pi$$
−0.199063 + 0.979987i $$0.563790\pi$$
$$920$$ 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 $$-0.354551\pi$$
0.441205 + 0.897406i $$0.354551\pi$$
$$930$$ 26.5597 0.870926
$$931$$ 5.32340 0.174468
$$932$$ 2.30885 0.0756288
$$933$$ 8.00000 0.261908
$$934$$ 29.9356 0.979523
$$935$$ 0 0
$$936$$ −13.7306 −0.448798
$$937$$ 14.9944 0.489846 0.244923 0.969543i $$-0.421237\pi$$
0.244923 + 0.969543i $$0.421237\pi$$
$$938$$ −12.8221 −0.418655
$$939$$ 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
$$945$$ 2.39821 0.0780137
$$946$$ 0 0
$$947$$ −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
$$952$$ 17.3324 0.561745
$$953$$ 2.14064 0.0693422 0.0346711 0.999399i $$-0.488962\pi$$
0.0346711 + 0.999399i $$0.488962\pi$$
$$954$$ −5.44322 −0.176231
$$955$$ −22.5969 −0.731217
$$956$$ −0.405923 −0.0131285
$$957$$ 0 0
$$958$$ −16.9861 −0.548796
$$959$$ −4.77559 −0.154212
$$960$$ 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
$$966$$ −7.20359 −0.231772
$$967$$ 1.53326 0.0493062 0.0246531 0.999696i $$-0.492152\pi$$
0.0246531 + 0.999696i $$0.492152\pi$$
$$968$$ 0 0
$$969$$ 33.9017 1.08908
$$970$$ 6.56351 0.210742
$$971$$ −26.5574 −0.852269 −0.426135 0.904660i $$-0.640125\pi$$
−0.426135 + 0.904660i $$0.640125\pi$$
$$972$$ 0.139194 0.00446465
$$973$$ 15.4432 0.495087
$$974$$ −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
$$979$$ 0 0
$$980$$ 0.333816 0.0106634
$$981$$ −12.7756 −0.407893
$$982$$ 39.0171 1.24509
$$983$$ −53.0361 −1.69159 −0.845794 0.533510i $$-0.820873\pi$$
−0.845794 + 0.533510i $$0.820873\pi$$
$$984$$ 1.76036 0.0561183
$$985$$ 5.41411 0.172508
$$986$$ 46.9915 1.49651
$$987$$ 0.526989 0.0167743
$$988$$ −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
$$993$$ 23.4432 0.743948
$$994$$ 16.7368 0.530860
$$995$$ 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 2541.2.a.bg.1.2 3
3.2 odd 2 7623.2.a.cd.1.2 3
11.10 odd 2 231.2.a.e.1.2 3
33.32 even 2 693.2.a.l.1.2 3
44.43 even 2 3696.2.a.bo.1.2 3
55.54 odd 2 5775.2.a.bp.1.2 3
77.76 even 2 1617.2.a.t.1.2 3
231.230 odd 2 4851.2.a.bi.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 11.10 odd 2
693.2.a.l.1.2 3 33.32 even 2
1617.2.a.t.1.2 3 77.76 even 2
2541.2.a.bg.1.2 3 1.1 even 1 trivial
3696.2.a.bo.1.2 3 44.43 even 2
4851.2.a.bi.1.2 3 231.230 odd 2
5775.2.a.bp.1.2 3 55.54 odd 2
7623.2.a.cd.1.2 3 3.2 odd 2