# Properties

 Label 6160.2.a.bf.1.3 Level $6160$ Weight $2$ Character 6160.1 Self dual yes Analytic conductor $49.188$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.1878476451$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.470683$$ of defining polynomial Character $$\chi$$ $$=$$ 6160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} +O(q^{10})$$ $$q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} -1.00000 q^{11} +0.941367 q^{13} +2.24914 q^{15} +6.49828 q^{17} +4.36641 q^{19} +2.24914 q^{21} -6.24914 q^{23} +1.00000 q^{25} -2.11727 q^{27} +8.74742 q^{29} +9.55691 q^{31} -2.24914 q^{33} +1.00000 q^{35} +4.24914 q^{37} +2.11727 q^{39} -2.13187 q^{41} -7.67418 q^{43} +2.05863 q^{45} -11.1138 q^{47} +1.00000 q^{49} +14.6155 q^{51} -4.74742 q^{53} -1.00000 q^{55} +9.82066 q^{57} +1.88273 q^{59} +9.11383 q^{61} +2.05863 q^{63} +0.941367 q^{65} -12.9966 q^{67} -14.0552 q^{69} +14.6155 q^{71} -10.4983 q^{73} +2.24914 q^{75} -1.00000 q^{77} -8.36641 q^{79} -10.9379 q^{81} +8.49828 q^{83} +6.49828 q^{85} +19.6742 q^{87} +12.3810 q^{89} +0.941367 q^{91} +21.4948 q^{93} +4.36641 q^{95} -15.3630 q^{97} -2.05863 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} + O(q^{10})$$ $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.24914 1.29854 0.649271 0.760557i $$-0.275074\pi$$
0.649271 + 0.760557i $$0.275074\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 2.05863 0.686211
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 0.941367 0.261088 0.130544 0.991443i $$-0.458328\pi$$
0.130544 + 0.991443i $$0.458328\pi$$
$$14$$ 0 0
$$15$$ 2.24914 0.580726
$$16$$ 0 0
$$17$$ 6.49828 1.57606 0.788032 0.615634i $$-0.211100\pi$$
0.788032 + 0.615634i $$0.211100\pi$$
$$18$$ 0 0
$$19$$ 4.36641 1.00172 0.500861 0.865528i $$-0.333017\pi$$
0.500861 + 0.865528i $$0.333017\pi$$
$$20$$ 0 0
$$21$$ 2.24914 0.490803
$$22$$ 0 0
$$23$$ −6.24914 −1.30304 −0.651518 0.758633i $$-0.725867\pi$$
−0.651518 + 0.758633i $$0.725867\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.11727 −0.407468
$$28$$ 0 0
$$29$$ 8.74742 1.62436 0.812178 0.583410i $$-0.198282\pi$$
0.812178 + 0.583410i $$0.198282\pi$$
$$30$$ 0 0
$$31$$ 9.55691 1.71647 0.858236 0.513255i $$-0.171560\pi$$
0.858236 + 0.513255i $$0.171560\pi$$
$$32$$ 0 0
$$33$$ −2.24914 −0.391525
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ 4.24914 0.698554 0.349277 0.937019i $$-0.386427\pi$$
0.349277 + 0.937019i $$0.386427\pi$$
$$38$$ 0 0
$$39$$ 2.11727 0.339034
$$40$$ 0 0
$$41$$ −2.13187 −0.332943 −0.166471 0.986046i $$-0.553237\pi$$
−0.166471 + 0.986046i $$0.553237\pi$$
$$42$$ 0 0
$$43$$ −7.67418 −1.17030 −0.585151 0.810925i $$-0.698965\pi$$
−0.585151 + 0.810925i $$0.698965\pi$$
$$44$$ 0 0
$$45$$ 2.05863 0.306883
$$46$$ 0 0
$$47$$ −11.1138 −1.62112 −0.810559 0.585657i $$-0.800837\pi$$
−0.810559 + 0.585657i $$0.800837\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 14.6155 2.04659
$$52$$ 0 0
$$53$$ −4.74742 −0.652109 −0.326054 0.945351i $$-0.605719\pi$$
−0.326054 + 0.945351i $$0.605719\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 9.82066 1.30078
$$58$$ 0 0
$$59$$ 1.88273 0.245111 0.122556 0.992462i $$-0.460891\pi$$
0.122556 + 0.992462i $$0.460891\pi$$
$$60$$ 0 0
$$61$$ 9.11383 1.16691 0.583453 0.812147i $$-0.301701\pi$$
0.583453 + 0.812147i $$0.301701\pi$$
$$62$$ 0 0
$$63$$ 2.05863 0.259363
$$64$$ 0 0
$$65$$ 0.941367 0.116762
$$66$$ 0 0
$$67$$ −12.9966 −1.58778 −0.793891 0.608060i $$-0.791948\pi$$
−0.793891 + 0.608060i $$0.791948\pi$$
$$68$$ 0 0
$$69$$ −14.0552 −1.69205
$$70$$ 0 0
$$71$$ 14.6155 1.73455 0.867273 0.497833i $$-0.165871\pi$$
0.867273 + 0.497833i $$0.165871\pi$$
$$72$$ 0 0
$$73$$ −10.4983 −1.22873 −0.614365 0.789022i $$-0.710588\pi$$
−0.614365 + 0.789022i $$0.710588\pi$$
$$74$$ 0 0
$$75$$ 2.24914 0.259708
$$76$$ 0 0
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ −8.36641 −0.941294 −0.470647 0.882322i $$-0.655980\pi$$
−0.470647 + 0.882322i $$0.655980\pi$$
$$80$$ 0 0
$$81$$ −10.9379 −1.21533
$$82$$ 0 0
$$83$$ 8.49828 0.932808 0.466404 0.884572i $$-0.345549\pi$$
0.466404 + 0.884572i $$0.345549\pi$$
$$84$$ 0 0
$$85$$ 6.49828 0.704838
$$86$$ 0 0
$$87$$ 19.6742 2.10929
$$88$$ 0 0
$$89$$ 12.3810 1.31238 0.656192 0.754594i $$-0.272166\pi$$
0.656192 + 0.754594i $$0.272166\pi$$
$$90$$ 0 0
$$91$$ 0.941367 0.0986821
$$92$$ 0 0
$$93$$ 21.4948 2.22891
$$94$$ 0 0
$$95$$ 4.36641 0.447984
$$96$$ 0 0
$$97$$ −15.3630 −1.55987 −0.779937 0.625859i $$-0.784749\pi$$
−0.779937 + 0.625859i $$0.784749\pi$$
$$98$$ 0 0
$$99$$ −2.05863 −0.206900
$$100$$ 0 0
$$101$$ −16.8793 −1.67955 −0.839776 0.542932i $$-0.817314\pi$$
−0.839776 + 0.542932i $$0.817314\pi$$
$$102$$ 0 0
$$103$$ 6.61555 0.651849 0.325925 0.945396i $$-0.394324\pi$$
0.325925 + 0.945396i $$0.394324\pi$$
$$104$$ 0 0
$$105$$ 2.24914 0.219494
$$106$$ 0 0
$$107$$ 14.5535 1.40694 0.703469 0.710726i $$-0.251633\pi$$
0.703469 + 0.710726i $$0.251633\pi$$
$$108$$ 0 0
$$109$$ −0.249141 −0.0238633 −0.0119317 0.999929i $$-0.503798\pi$$
−0.0119317 + 0.999929i $$0.503798\pi$$
$$110$$ 0 0
$$111$$ 9.55691 0.907102
$$112$$ 0 0
$$113$$ 10.9966 1.03447 0.517235 0.855844i $$-0.326961\pi$$
0.517235 + 0.855844i $$0.326961\pi$$
$$114$$ 0 0
$$115$$ −6.24914 −0.582735
$$116$$ 0 0
$$117$$ 1.93793 0.179162
$$118$$ 0 0
$$119$$ 6.49828 0.595696
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −4.79488 −0.432340
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 5.88273 0.522008 0.261004 0.965338i $$-0.415946\pi$$
0.261004 + 0.965338i $$0.415946\pi$$
$$128$$ 0 0
$$129$$ −17.2603 −1.51969
$$130$$ 0 0
$$131$$ 1.25258 0.109438 0.0547191 0.998502i $$-0.482574\pi$$
0.0547191 + 0.998502i $$0.482574\pi$$
$$132$$ 0 0
$$133$$ 4.36641 0.378615
$$134$$ 0 0
$$135$$ −2.11727 −0.182225
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −10.9820 −0.931477 −0.465739 0.884922i $$-0.654211\pi$$
−0.465739 + 0.884922i $$0.654211\pi$$
$$140$$ 0 0
$$141$$ −24.9966 −2.10509
$$142$$ 0 0
$$143$$ −0.941367 −0.0787210
$$144$$ 0 0
$$145$$ 8.74742 0.726434
$$146$$ 0 0
$$147$$ 2.24914 0.185506
$$148$$ 0 0
$$149$$ 0.0146079 0.00119673 0.000598363 1.00000i $$-0.499810\pi$$
0.000598363 1.00000i $$0.499810\pi$$
$$150$$ 0 0
$$151$$ 15.2457 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$152$$ 0 0
$$153$$ 13.3776 1.08151
$$154$$ 0 0
$$155$$ 9.55691 0.767630
$$156$$ 0 0
$$157$$ −5.50172 −0.439085 −0.219542 0.975603i $$-0.570456\pi$$
−0.219542 + 0.975603i $$0.570456\pi$$
$$158$$ 0 0
$$159$$ −10.6776 −0.846790
$$160$$ 0 0
$$161$$ −6.24914 −0.492501
$$162$$ 0 0
$$163$$ 18.2277 1.42770 0.713850 0.700298i $$-0.246950\pi$$
0.713850 + 0.700298i $$0.246950\pi$$
$$164$$ 0 0
$$165$$ −2.24914 −0.175095
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −12.1138 −0.931833
$$170$$ 0 0
$$171$$ 8.98883 0.687393
$$172$$ 0 0
$$173$$ 0.117266 0.00891559 0.00445780 0.999990i $$-0.498581\pi$$
0.00445780 + 0.999990i $$0.498581\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 4.23453 0.318287
$$178$$ 0 0
$$179$$ 22.5535 1.68573 0.842863 0.538128i $$-0.180868\pi$$
0.842863 + 0.538128i $$0.180868\pi$$
$$180$$ 0 0
$$181$$ 20.8793 1.55195 0.775973 0.630766i $$-0.217259\pi$$
0.775973 + 0.630766i $$0.217259\pi$$
$$182$$ 0 0
$$183$$ 20.4983 1.51528
$$184$$ 0 0
$$185$$ 4.24914 0.312403
$$186$$ 0 0
$$187$$ −6.49828 −0.475201
$$188$$ 0 0
$$189$$ −2.11727 −0.154008
$$190$$ 0 0
$$191$$ −3.11383 −0.225309 −0.112654 0.993634i $$-0.535935\pi$$
−0.112654 + 0.993634i $$0.535935\pi$$
$$192$$ 0 0
$$193$$ −6.17246 −0.444304 −0.222152 0.975012i $$-0.571308\pi$$
−0.222152 + 0.975012i $$0.571308\pi$$
$$194$$ 0 0
$$195$$ 2.11727 0.151621
$$196$$ 0 0
$$197$$ 6.73281 0.479693 0.239847 0.970811i $$-0.422903\pi$$
0.239847 + 0.970811i $$0.422903\pi$$
$$198$$ 0 0
$$199$$ 13.2311 0.937927 0.468964 0.883217i $$-0.344627\pi$$
0.468964 + 0.883217i $$0.344627\pi$$
$$200$$ 0 0
$$201$$ −29.2311 −2.06180
$$202$$ 0 0
$$203$$ 8.74742 0.613949
$$204$$ 0 0
$$205$$ −2.13187 −0.148897
$$206$$ 0 0
$$207$$ −12.8647 −0.894158
$$208$$ 0 0
$$209$$ −4.36641 −0.302031
$$210$$ 0 0
$$211$$ −23.1138 −1.59122 −0.795611 0.605808i $$-0.792850\pi$$
−0.795611 + 0.605808i $$0.792850\pi$$
$$212$$ 0 0
$$213$$ 32.8724 2.25238
$$214$$ 0 0
$$215$$ −7.67418 −0.523375
$$216$$ 0 0
$$217$$ 9.55691 0.648766
$$218$$ 0 0
$$219$$ −23.6121 −1.59556
$$220$$ 0 0
$$221$$ 6.11727 0.411492
$$222$$ 0 0
$$223$$ −10.3810 −0.695164 −0.347582 0.937650i $$-0.612997\pi$$
−0.347582 + 0.937650i $$0.612997\pi$$
$$224$$ 0 0
$$225$$ 2.05863 0.137242
$$226$$ 0 0
$$227$$ −16.1104 −1.06928 −0.534642 0.845079i $$-0.679554\pi$$
−0.534642 + 0.845079i $$0.679554\pi$$
$$228$$ 0 0
$$229$$ 24.2897 1.60511 0.802555 0.596578i $$-0.203473\pi$$
0.802555 + 0.596578i $$0.203473\pi$$
$$230$$ 0 0
$$231$$ −2.24914 −0.147983
$$232$$ 0 0
$$233$$ 6.88617 0.451128 0.225564 0.974228i $$-0.427578\pi$$
0.225564 + 0.974228i $$0.427578\pi$$
$$234$$ 0 0
$$235$$ −11.1138 −0.724986
$$236$$ 0 0
$$237$$ −18.8172 −1.22231
$$238$$ 0 0
$$239$$ 11.1353 0.720283 0.360142 0.932898i $$-0.382728\pi$$
0.360142 + 0.932898i $$0.382728\pi$$
$$240$$ 0 0
$$241$$ 2.36641 0.152434 0.0762168 0.997091i $$-0.475716\pi$$
0.0762168 + 0.997091i $$0.475716\pi$$
$$242$$ 0 0
$$243$$ −18.2491 −1.17068
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 4.11039 0.261538
$$248$$ 0 0
$$249$$ 19.1138 1.21129
$$250$$ 0 0
$$251$$ −25.1070 −1.58474 −0.792368 0.610043i $$-0.791152\pi$$
−0.792368 + 0.610043i $$0.791152\pi$$
$$252$$ 0 0
$$253$$ 6.24914 0.392880
$$254$$ 0 0
$$255$$ 14.6155 0.915261
$$256$$ 0 0
$$257$$ −2.86469 −0.178694 −0.0893472 0.996001i $$-0.528478\pi$$
−0.0893472 + 0.996001i $$0.528478\pi$$
$$258$$ 0 0
$$259$$ 4.24914 0.264029
$$260$$ 0 0
$$261$$ 18.0077 1.11465
$$262$$ 0 0
$$263$$ −3.76547 −0.232189 −0.116094 0.993238i $$-0.537037\pi$$
−0.116094 + 0.993238i $$0.537037\pi$$
$$264$$ 0 0
$$265$$ −4.74742 −0.291632
$$266$$ 0 0
$$267$$ 27.8466 1.70419
$$268$$ 0 0
$$269$$ 8.94137 0.545165 0.272582 0.962132i $$-0.412122\pi$$
0.272582 + 0.962132i $$0.412122\pi$$
$$270$$ 0 0
$$271$$ −21.4948 −1.30572 −0.652859 0.757479i $$-0.726431\pi$$
−0.652859 + 0.757479i $$0.726431\pi$$
$$272$$ 0 0
$$273$$ 2.11727 0.128143
$$274$$ 0 0
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 7.64820 0.459536 0.229768 0.973245i $$-0.426203\pi$$
0.229768 + 0.973245i $$0.426203\pi$$
$$278$$ 0 0
$$279$$ 19.6742 1.17786
$$280$$ 0 0
$$281$$ −28.6155 −1.70706 −0.853530 0.521043i $$-0.825543\pi$$
−0.853530 + 0.521043i $$0.825543\pi$$
$$282$$ 0 0
$$283$$ 2.87930 0.171156 0.0855782 0.996331i $$-0.472726\pi$$
0.0855782 + 0.996331i $$0.472726\pi$$
$$284$$ 0 0
$$285$$ 9.82066 0.581726
$$286$$ 0 0
$$287$$ −2.13187 −0.125841
$$288$$ 0 0
$$289$$ 25.2277 1.48398
$$290$$ 0 0
$$291$$ −34.5535 −2.02556
$$292$$ 0 0
$$293$$ −12.9414 −0.756043 −0.378021 0.925797i $$-0.623395\pi$$
−0.378021 + 0.925797i $$0.623395\pi$$
$$294$$ 0 0
$$295$$ 1.88273 0.109617
$$296$$ 0 0
$$297$$ 2.11727 0.122856
$$298$$ 0 0
$$299$$ −5.88273 −0.340207
$$300$$ 0 0
$$301$$ −7.67418 −0.442332
$$302$$ 0 0
$$303$$ −37.9639 −2.18097
$$304$$ 0 0
$$305$$ 9.11383 0.521856
$$306$$ 0 0
$$307$$ −0.498281 −0.0284384 −0.0142192 0.999899i $$-0.504526\pi$$
−0.0142192 + 0.999899i $$0.504526\pi$$
$$308$$ 0 0
$$309$$ 14.8793 0.846454
$$310$$ 0 0
$$311$$ 23.4396 1.32914 0.664570 0.747226i $$-0.268615\pi$$
0.664570 + 0.747226i $$0.268615\pi$$
$$312$$ 0 0
$$313$$ 9.36984 0.529615 0.264807 0.964301i $$-0.414692\pi$$
0.264807 + 0.964301i $$0.414692\pi$$
$$314$$ 0 0
$$315$$ 2.05863 0.115991
$$316$$ 0 0
$$317$$ −9.97852 −0.560449 −0.280225 0.959934i $$-0.590409\pi$$
−0.280225 + 0.959934i $$0.590409\pi$$
$$318$$ 0 0
$$319$$ −8.74742 −0.489762
$$320$$ 0 0
$$321$$ 32.7328 1.82697
$$322$$ 0 0
$$323$$ 28.3741 1.57878
$$324$$ 0 0
$$325$$ 0.941367 0.0522176
$$326$$ 0 0
$$327$$ −0.560352 −0.0309875
$$328$$ 0 0
$$329$$ −11.1138 −0.612725
$$330$$ 0 0
$$331$$ 20.4362 1.12328 0.561638 0.827383i $$-0.310171\pi$$
0.561638 + 0.827383i $$0.310171\pi$$
$$332$$ 0 0
$$333$$ 8.74742 0.479356
$$334$$ 0 0
$$335$$ −12.9966 −0.710078
$$336$$ 0 0
$$337$$ 7.88273 0.429400 0.214700 0.976680i $$-0.431123\pi$$
0.214700 + 0.976680i $$0.431123\pi$$
$$338$$ 0 0
$$339$$ 24.7328 1.34330
$$340$$ 0 0
$$341$$ −9.55691 −0.517536
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −14.0552 −0.756706
$$346$$ 0 0
$$347$$ −13.5569 −0.727773 −0.363887 0.931443i $$-0.618550\pi$$
−0.363887 + 0.931443i $$0.618550\pi$$
$$348$$ 0 0
$$349$$ −10.7328 −0.574514 −0.287257 0.957853i $$-0.592743\pi$$
−0.287257 + 0.957853i $$0.592743\pi$$
$$350$$ 0 0
$$351$$ −1.99312 −0.106385
$$352$$ 0 0
$$353$$ −20.8578 −1.11015 −0.555075 0.831801i $$-0.687310\pi$$
−0.555075 + 0.831801i $$0.687310\pi$$
$$354$$ 0 0
$$355$$ 14.6155 0.775713
$$356$$ 0 0
$$357$$ 14.6155 0.773537
$$358$$ 0 0
$$359$$ −0.366407 −0.0193382 −0.00966911 0.999953i $$-0.503078\pi$$
−0.00966911 + 0.999953i $$0.503078\pi$$
$$360$$ 0 0
$$361$$ 0.0655089 0.00344783
$$362$$ 0 0
$$363$$ 2.24914 0.118049
$$364$$ 0 0
$$365$$ −10.4983 −0.549505
$$366$$ 0 0
$$367$$ −20.8432 −1.08801 −0.544003 0.839083i $$-0.683092\pi$$
−0.544003 + 0.839083i $$0.683092\pi$$
$$368$$ 0 0
$$369$$ −4.38875 −0.228469
$$370$$ 0 0
$$371$$ −4.74742 −0.246474
$$372$$ 0 0
$$373$$ −5.37758 −0.278440 −0.139220 0.990261i $$-0.544460\pi$$
−0.139220 + 0.990261i $$0.544460\pi$$
$$374$$ 0 0
$$375$$ 2.24914 0.116145
$$376$$ 0 0
$$377$$ 8.23453 0.424100
$$378$$ 0 0
$$379$$ −3.53093 −0.181372 −0.0906860 0.995880i $$-0.528906\pi$$
−0.0906860 + 0.995880i $$0.528906\pi$$
$$380$$ 0 0
$$381$$ 13.2311 0.677850
$$382$$ 0 0
$$383$$ −1.38445 −0.0707422 −0.0353711 0.999374i $$-0.511261\pi$$
−0.0353711 + 0.999374i $$0.511261\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ 0 0
$$387$$ −15.7983 −0.803074
$$388$$ 0 0
$$389$$ 15.7294 0.797511 0.398756 0.917057i $$-0.369442\pi$$
0.398756 + 0.917057i $$0.369442\pi$$
$$390$$ 0 0
$$391$$ −40.6087 −2.05367
$$392$$ 0 0
$$393$$ 2.81722 0.142110
$$394$$ 0 0
$$395$$ −8.36641 −0.420960
$$396$$ 0 0
$$397$$ −17.6121 −0.883926 −0.441963 0.897033i $$-0.645718\pi$$
−0.441963 + 0.897033i $$0.645718\pi$$
$$398$$ 0 0
$$399$$ 9.82066 0.491648
$$400$$ 0 0
$$401$$ −32.8172 −1.63881 −0.819407 0.573212i $$-0.805697\pi$$
−0.819407 + 0.573212i $$0.805697\pi$$
$$402$$ 0 0
$$403$$ 8.99656 0.448151
$$404$$ 0 0
$$405$$ −10.9379 −0.543510
$$406$$ 0 0
$$407$$ −4.24914 −0.210622
$$408$$ 0 0
$$409$$ −33.3561 −1.64935 −0.824676 0.565605i $$-0.808643\pi$$
−0.824676 + 0.565605i $$0.808643\pi$$
$$410$$ 0 0
$$411$$ 22.4914 1.10942
$$412$$ 0 0
$$413$$ 1.88273 0.0926433
$$414$$ 0 0
$$415$$ 8.49828 0.417164
$$416$$ 0 0
$$417$$ −24.7000 −1.20956
$$418$$ 0 0
$$419$$ −12.3449 −0.603089 −0.301544 0.953452i $$-0.597502\pi$$
−0.301544 + 0.953452i $$0.597502\pi$$
$$420$$ 0 0
$$421$$ −8.87930 −0.432750 −0.216375 0.976310i $$-0.569423\pi$$
−0.216375 + 0.976310i $$0.569423\pi$$
$$422$$ 0 0
$$423$$ −22.8793 −1.11243
$$424$$ 0 0
$$425$$ 6.49828 0.315213
$$426$$ 0 0
$$427$$ 9.11383 0.441049
$$428$$ 0 0
$$429$$ −2.11727 −0.102223
$$430$$ 0 0
$$431$$ −18.2784 −0.880437 −0.440219 0.897891i $$-0.645099\pi$$
−0.440219 + 0.897891i $$0.645099\pi$$
$$432$$ 0 0
$$433$$ 6.86469 0.329896 0.164948 0.986302i $$-0.447254\pi$$
0.164948 + 0.986302i $$0.447254\pi$$
$$434$$ 0 0
$$435$$ 19.6742 0.943305
$$436$$ 0 0
$$437$$ −27.2863 −1.30528
$$438$$ 0 0
$$439$$ 11.8466 0.565409 0.282705 0.959207i $$-0.408768\pi$$
0.282705 + 0.959207i $$0.408768\pi$$
$$440$$ 0 0
$$441$$ 2.05863 0.0980302
$$442$$ 0 0
$$443$$ −17.8827 −0.849634 −0.424817 0.905279i $$-0.639662\pi$$
−0.424817 + 0.905279i $$0.639662\pi$$
$$444$$ 0 0
$$445$$ 12.3810 0.586916
$$446$$ 0 0
$$447$$ 0.0328552 0.00155400
$$448$$ 0 0
$$449$$ −16.8172 −0.793654 −0.396827 0.917893i $$-0.629889\pi$$
−0.396827 + 0.917893i $$0.629889\pi$$
$$450$$ 0 0
$$451$$ 2.13187 0.100386
$$452$$ 0 0
$$453$$ 34.2897 1.61107
$$454$$ 0 0
$$455$$ 0.941367 0.0441320
$$456$$ 0 0
$$457$$ −12.2277 −0.571986 −0.285993 0.958232i $$-0.592323\pi$$
−0.285993 + 0.958232i $$0.592323\pi$$
$$458$$ 0 0
$$459$$ −13.7586 −0.642196
$$460$$ 0 0
$$461$$ −16.6155 −0.773863 −0.386932 0.922108i $$-0.626465\pi$$
−0.386932 + 0.922108i $$0.626465\pi$$
$$462$$ 0 0
$$463$$ −18.4837 −0.859009 −0.429505 0.903065i $$-0.641312\pi$$
−0.429505 + 0.903065i $$0.641312\pi$$
$$464$$ 0 0
$$465$$ 21.4948 0.996799
$$466$$ 0 0
$$467$$ −24.3956 −1.12889 −0.564447 0.825469i $$-0.690911\pi$$
−0.564447 + 0.825469i $$0.690911\pi$$
$$468$$ 0 0
$$469$$ −12.9966 −0.600125
$$470$$ 0 0
$$471$$ −12.3741 −0.570170
$$472$$ 0 0
$$473$$ 7.67418 0.352859
$$474$$ 0 0
$$475$$ 4.36641 0.200344
$$476$$ 0 0
$$477$$ −9.77320 −0.447484
$$478$$ 0 0
$$479$$ 27.7655 1.26864 0.634318 0.773072i $$-0.281281\pi$$
0.634318 + 0.773072i $$0.281281\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ −14.0552 −0.639534
$$484$$ 0 0
$$485$$ −15.3630 −0.697596
$$486$$ 0 0
$$487$$ −10.0958 −0.457484 −0.228742 0.973487i $$-0.573461\pi$$
−0.228742 + 0.973487i $$0.573461\pi$$
$$488$$ 0 0
$$489$$ 40.9966 1.85393
$$490$$ 0 0
$$491$$ 32.1104 1.44912 0.724561 0.689211i $$-0.242043\pi$$
0.724561 + 0.689211i $$0.242043\pi$$
$$492$$ 0 0
$$493$$ 56.8432 2.56009
$$494$$ 0 0
$$495$$ −2.05863 −0.0925287
$$496$$ 0 0
$$497$$ 14.6155 0.655597
$$498$$ 0 0
$$499$$ 8.79488 0.393713 0.196857 0.980432i $$-0.436927\pi$$
0.196857 + 0.980432i $$0.436927\pi$$
$$500$$ 0 0
$$501$$ 17.9931 0.803874
$$502$$ 0 0
$$503$$ 15.0034 0.668970 0.334485 0.942401i $$-0.391438\pi$$
0.334485 + 0.942401i $$0.391438\pi$$
$$504$$ 0 0
$$505$$ −16.8793 −0.751119
$$506$$ 0 0
$$507$$ −27.2457 −1.21002
$$508$$ 0 0
$$509$$ 21.8759 0.969630 0.484815 0.874617i $$-0.338887\pi$$
0.484815 + 0.874617i $$0.338887\pi$$
$$510$$ 0 0
$$511$$ −10.4983 −0.464417
$$512$$ 0 0
$$513$$ −9.24485 −0.408170
$$514$$ 0 0
$$515$$ 6.61555 0.291516
$$516$$ 0 0
$$517$$ 11.1138 0.488786
$$518$$ 0 0
$$519$$ 0.263748 0.0115773
$$520$$ 0 0
$$521$$ 14.0292 0.614631 0.307316 0.951608i $$-0.400569\pi$$
0.307316 + 0.951608i $$0.400569\pi$$
$$522$$ 0 0
$$523$$ −1.14992 −0.0502825 −0.0251412 0.999684i $$-0.508004\pi$$
−0.0251412 + 0.999684i $$0.508004\pi$$
$$524$$ 0 0
$$525$$ 2.24914 0.0981605
$$526$$ 0 0
$$527$$ 62.1035 2.70527
$$528$$ 0 0
$$529$$ 16.0518 0.697902
$$530$$ 0 0
$$531$$ 3.87586 0.168198
$$532$$ 0 0
$$533$$ −2.00688 −0.0869274
$$534$$ 0 0
$$535$$ 14.5535 0.629202
$$536$$ 0 0
$$537$$ 50.7259 2.18899
$$538$$ 0 0
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −6.47680 −0.278459 −0.139230 0.990260i $$-0.544463\pi$$
−0.139230 + 0.990260i $$0.544463\pi$$
$$542$$ 0 0
$$543$$ 46.9605 2.01527
$$544$$ 0 0
$$545$$ −0.249141 −0.0106720
$$546$$ 0 0
$$547$$ −12.9966 −0.555693 −0.277846 0.960626i $$-0.589621\pi$$
−0.277846 + 0.960626i $$0.589621\pi$$
$$548$$ 0 0
$$549$$ 18.7620 0.800744
$$550$$ 0 0
$$551$$ 38.1948 1.62715
$$552$$ 0 0
$$553$$ −8.36641 −0.355776
$$554$$ 0 0
$$555$$ 9.55691 0.405668
$$556$$ 0 0
$$557$$ −16.4914 −0.698763 −0.349382 0.936981i $$-0.613608\pi$$
−0.349382 + 0.936981i $$0.613608\pi$$
$$558$$ 0 0
$$559$$ −7.22422 −0.305552
$$560$$ 0 0
$$561$$ −14.6155 −0.617069
$$562$$ 0 0
$$563$$ 16.7620 0.706435 0.353218 0.935541i $$-0.385088\pi$$
0.353218 + 0.935541i $$0.385088\pi$$
$$564$$ 0 0
$$565$$ 10.9966 0.462629
$$566$$ 0 0
$$567$$ −10.9379 −0.459350
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ −12.7328 −0.532852 −0.266426 0.963855i $$-0.585843\pi$$
−0.266426 + 0.963855i $$0.585843\pi$$
$$572$$ 0 0
$$573$$ −7.00344 −0.292573
$$574$$ 0 0
$$575$$ −6.24914 −0.260607
$$576$$ 0 0
$$577$$ −11.8613 −0.493790 −0.246895 0.969042i $$-0.579410\pi$$
−0.246895 + 0.969042i $$0.579410\pi$$
$$578$$ 0 0
$$579$$ −13.8827 −0.576947
$$580$$ 0 0
$$581$$ 8.49828 0.352568
$$582$$ 0 0
$$583$$ 4.74742 0.196618
$$584$$ 0 0
$$585$$ 1.93793 0.0801235
$$586$$ 0 0
$$587$$ 8.60094 0.354999 0.177499 0.984121i $$-0.443199\pi$$
0.177499 + 0.984121i $$0.443199\pi$$
$$588$$ 0 0
$$589$$ 41.7294 1.71943
$$590$$ 0 0
$$591$$ 15.1430 0.622902
$$592$$ 0 0
$$593$$ 10.7328 0.440744 0.220372 0.975416i $$-0.429273\pi$$
0.220372 + 0.975416i $$0.429273\pi$$
$$594$$ 0 0
$$595$$ 6.49828 0.266404
$$596$$ 0 0
$$597$$ 29.7586 1.21794
$$598$$ 0 0
$$599$$ −16.0812 −0.657059 −0.328529 0.944494i $$-0.606553\pi$$
−0.328529 + 0.944494i $$0.606553\pi$$
$$600$$ 0 0
$$601$$ −34.0889 −1.39052 −0.695258 0.718760i $$-0.744710\pi$$
−0.695258 + 0.718760i $$0.744710\pi$$
$$602$$ 0 0
$$603$$ −26.7552 −1.08955
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −11.6742 −0.473840 −0.236920 0.971529i $$-0.576138\pi$$
−0.236920 + 0.971529i $$0.576138\pi$$
$$608$$ 0 0
$$609$$ 19.6742 0.797238
$$610$$ 0 0
$$611$$ −10.4622 −0.423255
$$612$$ 0 0
$$613$$ −34.2637 −1.38390 −0.691950 0.721946i $$-0.743248\pi$$
−0.691950 + 0.721946i $$0.743248\pi$$
$$614$$ 0 0
$$615$$ −4.79488 −0.193348
$$616$$ 0 0
$$617$$ 39.3707 1.58500 0.792502 0.609869i $$-0.208778\pi$$
0.792502 + 0.609869i $$0.208778\pi$$
$$618$$ 0 0
$$619$$ −16.1104 −0.647531 −0.323766 0.946137i $$-0.604949\pi$$
−0.323766 + 0.946137i $$0.604949\pi$$
$$620$$ 0 0
$$621$$ 13.2311 0.530946
$$622$$ 0 0
$$623$$ 12.3810 0.496035
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −9.82066 −0.392199
$$628$$ 0 0
$$629$$ 27.6121 1.10097
$$630$$ 0 0
$$631$$ −13.8827 −0.552663 −0.276331 0.961062i $$-0.589119\pi$$
−0.276331 + 0.961062i $$0.589119\pi$$
$$632$$ 0 0
$$633$$ −51.9862 −2.06627
$$634$$ 0 0
$$635$$ 5.88273 0.233449
$$636$$ 0 0
$$637$$ 0.941367 0.0372983
$$638$$ 0 0
$$639$$ 30.0881 1.19026
$$640$$ 0 0
$$641$$ −39.3415 −1.55390 −0.776948 0.629565i $$-0.783233\pi$$
−0.776948 + 0.629565i $$0.783233\pi$$
$$642$$ 0 0
$$643$$ −38.3595 −1.51275 −0.756376 0.654137i $$-0.773032\pi$$
−0.756376 + 0.654137i $$0.773032\pi$$
$$644$$ 0 0
$$645$$ −17.2603 −0.679624
$$646$$ 0 0
$$647$$ −25.7294 −1.01153 −0.505763 0.862672i $$-0.668789\pi$$
−0.505763 + 0.862672i $$0.668789\pi$$
$$648$$ 0 0
$$649$$ −1.88273 −0.0739038
$$650$$ 0 0
$$651$$ 21.4948 0.842449
$$652$$ 0 0
$$653$$ 18.4768 0.723053 0.361526 0.932362i $$-0.382256\pi$$
0.361526 + 0.932362i $$0.382256\pi$$
$$654$$ 0 0
$$655$$ 1.25258 0.0489423
$$656$$ 0 0
$$657$$ −21.6121 −0.843169
$$658$$ 0 0
$$659$$ 39.3776 1.53393 0.766966 0.641687i $$-0.221765\pi$$
0.766966 + 0.641687i $$0.221765\pi$$
$$660$$ 0 0
$$661$$ −34.2208 −1.33103 −0.665517 0.746383i $$-0.731789\pi$$
−0.665517 + 0.746383i $$0.731789\pi$$
$$662$$ 0 0
$$663$$ 13.7586 0.534339
$$664$$ 0 0
$$665$$ 4.36641 0.169322
$$666$$ 0 0
$$667$$ −54.6639 −2.11659
$$668$$ 0 0
$$669$$ −23.3484 −0.902700
$$670$$ 0 0
$$671$$ −9.11383 −0.351835
$$672$$ 0 0
$$673$$ −24.4622 −0.942948 −0.471474 0.881880i $$-0.656278\pi$$
−0.471474 + 0.881880i $$0.656278\pi$$
$$674$$ 0 0
$$675$$ −2.11727 −0.0814936
$$676$$ 0 0
$$677$$ −39.1070 −1.50300 −0.751501 0.659732i $$-0.770670\pi$$
−0.751501 + 0.659732i $$0.770670\pi$$
$$678$$ 0 0
$$679$$ −15.3630 −0.589577
$$680$$ 0 0
$$681$$ −36.2345 −1.38851
$$682$$ 0 0
$$683$$ −23.3776 −0.894518 −0.447259 0.894404i $$-0.647600\pi$$
−0.447259 + 0.894404i $$0.647600\pi$$
$$684$$ 0 0
$$685$$ 10.0000 0.382080
$$686$$ 0 0
$$687$$ 54.6310 2.08430
$$688$$ 0 0
$$689$$ −4.46907 −0.170258
$$690$$ 0 0
$$691$$ −8.49828 −0.323290 −0.161645 0.986849i $$-0.551680\pi$$
−0.161645 + 0.986849i $$0.551680\pi$$
$$692$$ 0 0
$$693$$ −2.05863 −0.0782010
$$694$$ 0 0
$$695$$ −10.9820 −0.416569
$$696$$ 0 0
$$697$$ −13.8535 −0.524739
$$698$$ 0 0
$$699$$ 15.4880 0.585809
$$700$$ 0 0
$$701$$ 14.9751 0.565601 0.282800 0.959179i $$-0.408737\pi$$
0.282800 + 0.959179i $$0.408737\pi$$
$$702$$ 0 0
$$703$$ 18.5535 0.699758
$$704$$ 0 0
$$705$$ −24.9966 −0.941425
$$706$$ 0 0
$$707$$ −16.8793 −0.634811
$$708$$ 0 0
$$709$$ 40.7259 1.52949 0.764747 0.644330i $$-0.222864\pi$$
0.764747 + 0.644330i $$0.222864\pi$$
$$710$$ 0 0
$$711$$ −17.2234 −0.645927
$$712$$ 0 0
$$713$$ −59.7225 −2.23663
$$714$$ 0 0
$$715$$ −0.941367 −0.0352051
$$716$$ 0 0
$$717$$ 25.0449 0.935318
$$718$$ 0 0
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 6.61555 0.246376
$$722$$ 0 0
$$723$$ 5.32238 0.197942
$$724$$ 0 0
$$725$$ 8.74742 0.324871
$$726$$ 0 0
$$727$$ 51.6413 1.91527 0.957635 0.287984i $$-0.0929848\pi$$
0.957635 + 0.287984i $$0.0929848\pi$$
$$728$$ 0 0
$$729$$ −8.23109 −0.304855
$$730$$ 0 0
$$731$$ −49.8690 −1.84447
$$732$$ 0 0
$$733$$ −42.0000 −1.55131 −0.775653 0.631160i $$-0.782579\pi$$
−0.775653 + 0.631160i $$0.782579\pi$$
$$734$$ 0 0
$$735$$ 2.24914 0.0829608
$$736$$ 0 0
$$737$$ 12.9966 0.478735
$$738$$ 0 0
$$739$$ 49.1070 1.80643 0.903214 0.429190i $$-0.141201\pi$$
0.903214 + 0.429190i $$0.141201\pi$$
$$740$$ 0 0
$$741$$ 9.24485 0.339618
$$742$$ 0 0
$$743$$ 53.2311 1.95286 0.976430 0.215836i $$-0.0692475\pi$$
0.976430 + 0.215836i $$0.0692475\pi$$
$$744$$ 0 0
$$745$$ 0.0146079 0.000535192 0
$$746$$ 0 0
$$747$$ 17.4948 0.640103
$$748$$ 0 0
$$749$$ 14.5535 0.531772
$$750$$ 0 0
$$751$$ 31.6121 1.15354 0.576771 0.816906i $$-0.304312\pi$$
0.576771 + 0.816906i $$0.304312\pi$$
$$752$$ 0 0
$$753$$ −56.4691 −2.05785
$$754$$ 0 0
$$755$$ 15.2457 0.554848
$$756$$ 0 0
$$757$$ −1.24570 −0.0452758 −0.0226379 0.999744i $$-0.507206\pi$$
−0.0226379 + 0.999744i $$0.507206\pi$$
$$758$$ 0 0
$$759$$ 14.0552 0.510171
$$760$$ 0 0
$$761$$ −10.6009 −0.384284 −0.192142 0.981367i $$-0.561543\pi$$
−0.192142 + 0.981367i $$0.561543\pi$$
$$762$$ 0 0
$$763$$ −0.249141 −0.00901949
$$764$$ 0 0
$$765$$ 13.3776 0.483667
$$766$$ 0 0
$$767$$ 1.77234 0.0639956
$$768$$ 0 0
$$769$$ −19.1284 −0.689789 −0.344895 0.938641i $$-0.612085\pi$$
−0.344895 + 0.938641i $$0.612085\pi$$
$$770$$ 0 0
$$771$$ −6.44309 −0.232042
$$772$$ 0 0
$$773$$ 39.9931 1.43845 0.719226 0.694776i $$-0.244496\pi$$
0.719226 + 0.694776i $$0.244496\pi$$
$$774$$ 0 0
$$775$$ 9.55691 0.343294
$$776$$ 0 0
$$777$$ 9.55691 0.342852
$$778$$ 0 0
$$779$$ −9.30863 −0.333516
$$780$$ 0 0
$$781$$ −14.6155 −0.522985
$$782$$ 0 0
$$783$$ −18.5206 −0.661873
$$784$$ 0 0
$$785$$ −5.50172 −0.196365
$$786$$ 0 0
$$787$$ −37.9931 −1.35431 −0.677154 0.735841i $$-0.736787\pi$$
−0.677154 + 0.735841i $$0.736787\pi$$
$$788$$ 0 0
$$789$$ −8.46907 −0.301507
$$790$$ 0 0
$$791$$ 10.9966 0.390993
$$792$$ 0 0
$$793$$ 8.57946 0.304665
$$794$$ 0 0
$$795$$ −10.6776 −0.378696
$$796$$ 0 0
$$797$$ −20.3810 −0.721933 −0.360966 0.932579i $$-0.617553\pi$$
−0.360966 + 0.932579i $$0.617553\pi$$
$$798$$ 0 0
$$799$$ −72.2208 −2.55499
$$800$$ 0 0
$$801$$ 25.4880 0.900573
$$802$$ 0 0
$$803$$ 10.4983 0.370476
$$804$$ 0 0
$$805$$ −6.24914 −0.220253
$$806$$ 0 0
$$807$$ 20.1104 0.707919
$$808$$ 0 0
$$809$$ −31.7294 −1.11555 −0.557773 0.829994i $$-0.688344\pi$$
−0.557773 + 0.829994i $$0.688344\pi$$
$$810$$ 0 0
$$811$$ −43.5095 −1.52782 −0.763912 0.645321i $$-0.776724\pi$$
−0.763912 + 0.645321i $$0.776724\pi$$
$$812$$ 0 0
$$813$$ −48.3449 −1.69553
$$814$$ 0 0
$$815$$ 18.2277 0.638487
$$816$$ 0 0
$$817$$ −33.5086 −1.17232
$$818$$ 0 0
$$819$$ 1.93793 0.0677167
$$820$$ 0 0
$$821$$ −24.7766 −0.864711 −0.432355 0.901703i $$-0.642317\pi$$
−0.432355 + 0.901703i $$0.642317\pi$$
$$822$$ 0 0
$$823$$ −28.8647 −1.00616 −0.503080 0.864240i $$-0.667800\pi$$
−0.503080 + 0.864240i $$0.667800\pi$$
$$824$$ 0 0
$$825$$ −2.24914 −0.0783050
$$826$$ 0 0
$$827$$ −34.2277 −1.19021 −0.595106 0.803647i $$-0.702890\pi$$
−0.595106 + 0.803647i $$0.702890\pi$$
$$828$$ 0 0
$$829$$ 23.6381 0.820985 0.410492 0.911864i $$-0.365357\pi$$
0.410492 + 0.911864i $$0.365357\pi$$
$$830$$ 0 0
$$831$$ 17.2019 0.596727
$$832$$ 0 0
$$833$$ 6.49828 0.225152
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ −20.2345 −0.699408
$$838$$ 0 0
$$839$$ 24.9053 0.859826 0.429913 0.902870i $$-0.358544\pi$$
0.429913 + 0.902870i $$0.358544\pi$$
$$840$$ 0 0
$$841$$ 47.5174 1.63853
$$842$$ 0 0
$$843$$ −64.3604 −2.21669
$$844$$ 0 0
$$845$$ −12.1138 −0.416728
$$846$$ 0 0
$$847$$ 1.00000 0.0343604
$$848$$ 0 0
$$849$$ 6.47594 0.222254
$$850$$ 0 0
$$851$$ −26.5535 −0.910241
$$852$$ 0 0
$$853$$ 22.9966 0.787387 0.393694 0.919242i $$-0.371197\pi$$
0.393694 + 0.919242i $$0.371197\pi$$
$$854$$ 0 0
$$855$$ 8.98883 0.307411
$$856$$ 0 0
$$857$$ −31.2603 −1.06783 −0.533916 0.845538i $$-0.679280\pi$$
−0.533916 + 0.845538i $$0.679280\pi$$
$$858$$ 0 0
$$859$$ −20.3449 −0.694160 −0.347080 0.937836i $$-0.612827\pi$$
−0.347080 + 0.937836i $$0.612827\pi$$
$$860$$ 0 0
$$861$$ −4.79488 −0.163409
$$862$$ 0 0
$$863$$ 58.3595 1.98658 0.993291 0.115644i $$-0.0368930\pi$$
0.993291 + 0.115644i $$0.0368930\pi$$
$$864$$ 0 0
$$865$$ 0.117266 0.00398717
$$866$$ 0 0
$$867$$ 56.7405 1.92701
$$868$$ 0 0
$$869$$ 8.36641 0.283811
$$870$$ 0 0
$$871$$ −12.2345 −0.414551
$$872$$ 0 0
$$873$$ −31.6267 −1.07040
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 11.9639 0.403992 0.201996 0.979386i $$-0.435257\pi$$
0.201996 + 0.979386i $$0.435257\pi$$
$$878$$ 0 0
$$879$$ −29.1070 −0.981753
$$880$$ 0 0
$$881$$ 5.88961 0.198426 0.0992130 0.995066i $$-0.468367\pi$$
0.0992130 + 0.995066i $$0.468367\pi$$
$$882$$ 0 0
$$883$$ 19.4880 0.655822 0.327911 0.944709i $$-0.393655\pi$$
0.327911 + 0.944709i $$0.393655\pi$$
$$884$$ 0 0
$$885$$ 4.23453 0.142342
$$886$$ 0 0
$$887$$ −50.4622 −1.69435 −0.847177 0.531310i $$-0.821700\pi$$
−0.847177 + 0.531310i $$0.821700\pi$$
$$888$$ 0 0
$$889$$ 5.88273 0.197301
$$890$$ 0 0
$$891$$ 10.9379 0.366434
$$892$$ 0 0
$$893$$ −48.5275 −1.62391
$$894$$ 0 0
$$895$$ 22.5535 0.753880
$$896$$ 0 0
$$897$$ −13.2311 −0.441773
$$898$$ 0 0
$$899$$ 83.5984 2.78816
$$900$$ 0 0
$$901$$ −30.8501 −1.02777
$$902$$ 0 0
$$903$$ −17.2603 −0.574387
$$904$$ 0 0
$$905$$ 20.8793 0.694051
$$906$$ 0 0
$$907$$ 16.8432 0.559269 0.279635 0.960106i $$-0.409787\pi$$
0.279635 + 0.960106i $$0.409787\pi$$
$$908$$ 0 0
$$909$$ −34.7483 −1.15253
$$910$$ 0 0
$$911$$ 38.6155 1.27939 0.639695 0.768629i $$-0.279061\pi$$
0.639695 + 0.768629i $$0.279061\pi$$
$$912$$ 0 0
$$913$$ −8.49828 −0.281252
$$914$$ 0 0
$$915$$ 20.4983 0.677652
$$916$$ 0 0
$$917$$ 1.25258 0.0413638
$$918$$ 0 0
$$919$$ −17.2818 −0.570074 −0.285037 0.958517i $$-0.592006\pi$$
−0.285037 + 0.958517i $$0.592006\pi$$
$$920$$ 0 0
$$921$$ −1.12070 −0.0369285
$$922$$ 0 0
$$923$$ 13.7586 0.452870
$$924$$ 0 0
$$925$$ 4.24914 0.139711
$$926$$ 0 0
$$927$$ 13.6190 0.447306
$$928$$ 0 0
$$929$$ 2.91539 0.0956508 0.0478254 0.998856i $$-0.484771\pi$$
0.0478254 + 0.998856i $$0.484771\pi$$
$$930$$ 0 0
$$931$$ 4.36641 0.143103
$$932$$ 0 0
$$933$$ 52.7191 1.72594
$$934$$ 0 0
$$935$$ −6.49828 −0.212517
$$936$$ 0 0
$$937$$ 22.5795 0.737639 0.368819 0.929501i $$-0.379762\pi$$
0.368819 + 0.929501i $$0.379762\pi$$
$$938$$ 0 0
$$939$$ 21.0741 0.687727
$$940$$ 0 0
$$941$$ −21.7655 −0.709534 −0.354767 0.934955i $$-0.615440\pi$$
−0.354767 + 0.934955i $$0.615440\pi$$
$$942$$ 0 0
$$943$$ 13.3224 0.433836
$$944$$ 0 0
$$945$$ −2.11727 −0.0688747
$$946$$ 0 0
$$947$$ 1.49484 0.0485759 0.0242879 0.999705i $$-0.492268\pi$$
0.0242879 + 0.999705i $$0.492268\pi$$
$$948$$ 0 0
$$949$$ −9.88273 −0.320807
$$950$$ 0 0
$$951$$ −22.4431 −0.727767
$$952$$ 0 0
$$953$$ −4.83098 −0.156491 −0.0782453 0.996934i $$-0.524932\pi$$
−0.0782453 + 0.996934i $$0.524932\pi$$
$$954$$ 0 0
$$955$$ −3.11383 −0.100761
$$956$$ 0 0
$$957$$ −19.6742 −0.635976
$$958$$ 0 0
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 60.3346 1.94628
$$962$$ 0 0
$$963$$ 29.9603 0.965456
$$964$$ 0 0
$$965$$ −6.17246 −0.198699
$$966$$ 0 0
$$967$$ 51.9278 1.66989 0.834943 0.550336i $$-0.185501\pi$$
0.834943 + 0.550336i $$0.185501\pi$$
$$968$$ 0 0
$$969$$ 63.8174 2.05011
$$970$$ 0 0
$$971$$ −59.6933 −1.91565 −0.957824 0.287354i $$-0.907224\pi$$
−0.957824 + 0.287354i $$0.907224\pi$$
$$972$$ 0 0
$$973$$ −10.9820 −0.352065
$$974$$ 0 0
$$975$$ 2.11727 0.0678068
$$976$$ 0 0
$$977$$ 31.2311 0.999171 0.499586 0.866265i $$-0.333486\pi$$
0.499586 + 0.866265i $$0.333486\pi$$
$$978$$ 0 0
$$979$$ −12.3810 −0.395699
$$980$$ 0 0
$$981$$ −0.512889 −0.0163753
$$982$$ 0 0
$$983$$ 22.6155 0.721324 0.360662 0.932697i $$-0.382551\pi$$
0.360662 + 0.932697i $$0.382551\pi$$
$$984$$ 0 0
$$985$$ 6.73281 0.214525
$$986$$ 0 0
$$987$$ −24.9966 −0.795649
$$988$$ 0 0
$$989$$ 47.9570 1.52494
$$990$$ 0 0
$$991$$ −22.4102 −0.711884 −0.355942 0.934508i $$-0.615840\pi$$
−0.355942 + 0.934508i $$0.615840\pi$$
$$992$$ 0 0
$$993$$ 45.9639 1.45862
$$994$$ 0 0
$$995$$ 13.2311 0.419454
$$996$$ 0 0
$$997$$ −27.3415 −0.865914 −0.432957 0.901415i $$-0.642530\pi$$
−0.432957 + 0.901415i $$0.642530\pi$$
$$998$$ 0 0
$$999$$ −8.99656 −0.284639
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 6160.2.a.bf.1.3 3
4.3 odd 2 770.2.a.m.1.1 3
12.11 even 2 6930.2.a.ce.1.2 3
20.3 even 4 3850.2.c.ba.1849.1 6
20.7 even 4 3850.2.c.ba.1849.6 6
20.19 odd 2 3850.2.a.bt.1.3 3
28.27 even 2 5390.2.a.ca.1.3 3
44.43 even 2 8470.2.a.ci.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.1 3 4.3 odd 2
3850.2.a.bt.1.3 3 20.19 odd 2
3850.2.c.ba.1849.1 6 20.3 even 4
3850.2.c.ba.1849.6 6 20.7 even 4
5390.2.a.ca.1.3 3 28.27 even 2
6160.2.a.bf.1.3 3 1.1 even 1 trivial
6930.2.a.ce.1.2 3 12.11 even 2
8470.2.a.ci.1.1 3 44.43 even 2