Properties

 Label 3969.2.a.t.1.2 Level $3969$ Weight $2$ Character 3969.1 Self dual yes Analytic conductor $31.693$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3969.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$31.6926245622$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.14013.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 6x + 3$$ x^4 - x^3 - 6*x^2 + 6*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 567) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.53652$$ of defining polynomial Character $$\chi$$ $$=$$ 3969.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.53652 q^{2} +0.360904 q^{4} +3.15761 q^{5} +2.51851 q^{8} +O(q^{10})$$ $$q-1.53652 q^{2} +0.360904 q^{4} +3.15761 q^{5} +2.51851 q^{8} -4.85173 q^{10} +5.74916 q^{11} +0.360904 q^{13} -4.59156 q^{16} +2.77684 q^{17} +7.23065 q^{19} +1.13959 q^{20} -8.83372 q^{22} +0.824381 q^{23} +4.97047 q^{25} -0.554537 q^{26} +4.27819 q^{29} +4.98199 q^{31} +2.01801 q^{32} -4.26668 q^{34} +7.49083 q^{37} -11.1101 q^{38} +7.95246 q^{40} -3.33138 q^{41} -7.86975 q^{43} +2.07489 q^{44} -1.26668 q^{46} -3.48149 q^{47} -7.63725 q^{50} +0.130252 q^{52} +2.91544 q^{53} +18.1536 q^{55} -6.57354 q^{58} +2.39878 q^{59} -3.20113 q^{61} -7.65494 q^{62} +6.08239 q^{64} +1.13959 q^{65} +1.89927 q^{67} +1.00217 q^{68} -1.60957 q^{71} +15.4138 q^{73} -11.5098 q^{74} +2.60957 q^{76} +5.47282 q^{79} -14.4983 q^{80} +5.11874 q^{82} -13.0348 q^{83} +8.76817 q^{85} +12.0921 q^{86} +14.4793 q^{88} -14.2677 q^{89} +0.297522 q^{92} +5.34939 q^{94} +22.8315 q^{95} -16.0053 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10})$$ 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8 $$4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100})$$ 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8 + 7 * q^10 - 5 * q^11 + 5 * q^13 - q^16 + 6 * q^17 + 8 * q^19 - 8 * q^20 - 7 * q^22 + 12 * q^23 + 8 * q^25 + q^26 + 10 * q^29 + 18 * q^31 + 10 * q^32 - 20 * q^38 + 18 * q^40 - 5 * q^41 - 7 * q^43 - 13 * q^44 + 12 * q^46 - 21 * q^47 - 38 * q^50 + 25 * q^52 + 12 * q^53 + 26 * q^55 - 7 * q^58 + 6 * q^59 + 20 * q^61 + 18 * q^62 - 23 * q^64 - 8 * q^65 - 5 * q^67 + 51 * q^68 + 9 * q^71 + 6 * q^73 - 5 * q^76 - 10 * q^79 - 2 * q^80 + 35 * q^82 + 9 * q^83 - 9 * q^85 + 22 * q^86 + 18 * q^88 - 22 * q^89 + 36 * q^92 + 15 * q^94 + 16 * q^95 + 9 * q^97

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.53652 −1.08649 −0.543243 0.839576i $$-0.682804\pi$$
−0.543243 + 0.839576i $$0.682804\pi$$
$$3$$ 0 0
$$4$$ 0.360904 0.180452
$$5$$ 3.15761 1.41212 0.706062 0.708150i $$-0.250470\pi$$
0.706062 + 0.708150i $$0.250470\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 2.51851 0.890428
$$9$$ 0 0
$$10$$ −4.85173 −1.53425
$$11$$ 5.74916 1.73344 0.866719 0.498797i $$-0.166225\pi$$
0.866719 + 0.498797i $$0.166225\pi$$
$$12$$ 0 0
$$13$$ 0.360904 0.100097 0.0500484 0.998747i $$-0.484062\pi$$
0.0500484 + 0.998747i $$0.484062\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.59156 −1.14789
$$17$$ 2.77684 0.673483 0.336741 0.941597i $$-0.390675\pi$$
0.336741 + 0.941597i $$0.390675\pi$$
$$18$$ 0 0
$$19$$ 7.23065 1.65883 0.829413 0.558636i $$-0.188675\pi$$
0.829413 + 0.558636i $$0.188675\pi$$
$$20$$ 1.13959 0.254820
$$21$$ 0 0
$$22$$ −8.83372 −1.88336
$$23$$ 0.824381 0.171895 0.0859476 0.996300i $$-0.472608\pi$$
0.0859476 + 0.996300i $$0.472608\pi$$
$$24$$ 0 0
$$25$$ 4.97047 0.994095
$$26$$ −0.554537 −0.108754
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.27819 0.794440 0.397220 0.917723i $$-0.369975\pi$$
0.397220 + 0.917723i $$0.369975\pi$$
$$30$$ 0 0
$$31$$ 4.98199 0.894791 0.447396 0.894336i $$-0.352352\pi$$
0.447396 + 0.894336i $$0.352352\pi$$
$$32$$ 2.01801 0.356738
$$33$$ 0 0
$$34$$ −4.26668 −0.731730
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.49083 1.23149 0.615743 0.787947i $$-0.288856\pi$$
0.615743 + 0.787947i $$0.288856\pi$$
$$38$$ −11.1101 −1.80229
$$39$$ 0 0
$$40$$ 7.95246 1.25739
$$41$$ −3.33138 −0.520274 −0.260137 0.965572i $$-0.583768\pi$$
−0.260137 + 0.965572i $$0.583768\pi$$
$$42$$ 0 0
$$43$$ −7.86975 −1.20013 −0.600063 0.799953i $$-0.704858\pi$$
−0.600063 + 0.799953i $$0.704858\pi$$
$$44$$ 2.07489 0.312802
$$45$$ 0 0
$$46$$ −1.26668 −0.186762
$$47$$ −3.48149 −0.507828 −0.253914 0.967227i $$-0.581718\pi$$
−0.253914 + 0.967227i $$0.581718\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −7.63725 −1.08007
$$51$$ 0 0
$$52$$ 0.130252 0.0180626
$$53$$ 2.91544 0.400467 0.200233 0.979748i $$-0.435830\pi$$
0.200233 + 0.979748i $$0.435830\pi$$
$$54$$ 0 0
$$55$$ 18.1536 2.44783
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.57354 −0.863148
$$59$$ 2.39878 0.312294 0.156147 0.987734i $$-0.450093\pi$$
0.156147 + 0.987734i $$0.450093\pi$$
$$60$$ 0 0
$$61$$ −3.20113 −0.409862 −0.204931 0.978776i $$-0.565697\pi$$
−0.204931 + 0.978776i $$0.565697\pi$$
$$62$$ −7.65494 −0.972178
$$63$$ 0 0
$$64$$ 6.08239 0.760298
$$65$$ 1.13959 0.141349
$$66$$ 0 0
$$67$$ 1.89927 0.232033 0.116017 0.993247i $$-0.462987\pi$$
0.116017 + 0.993247i $$0.462987\pi$$
$$68$$ 1.00217 0.121531
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.60957 −0.191021 −0.0955104 0.995428i $$-0.530448\pi$$
−0.0955104 + 0.995428i $$0.530448\pi$$
$$72$$ 0 0
$$73$$ 15.4138 1.80404 0.902022 0.431689i $$-0.142082\pi$$
0.902022 + 0.431689i $$0.142082\pi$$
$$74$$ −11.5098 −1.33799
$$75$$ 0 0
$$76$$ 2.60957 0.299338
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.47282 0.615740 0.307870 0.951428i $$-0.400384\pi$$
0.307870 + 0.951428i $$0.400384\pi$$
$$80$$ −14.4983 −1.62096
$$81$$ 0 0
$$82$$ 5.11874 0.565270
$$83$$ −13.0348 −1.43076 −0.715380 0.698735i $$-0.753746\pi$$
−0.715380 + 0.698735i $$0.753746\pi$$
$$84$$ 0 0
$$85$$ 8.76817 0.951041
$$86$$ 12.0921 1.30392
$$87$$ 0 0
$$88$$ 14.4793 1.54350
$$89$$ −14.2677 −1.51237 −0.756185 0.654358i $$-0.772939\pi$$
−0.756185 + 0.654358i $$0.772939\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0.297522 0.0310188
$$93$$ 0 0
$$94$$ 5.34939 0.551748
$$95$$ 22.8315 2.34247
$$96$$ 0 0
$$97$$ −16.0053 −1.62509 −0.812547 0.582896i $$-0.801920\pi$$
−0.812547 + 0.582896i $$0.801920\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.79386 0.179386
$$101$$ 3.00749 0.299257 0.149628 0.988742i $$-0.452192\pi$$
0.149628 + 0.988742i $$0.452192\pi$$
$$102$$ 0 0
$$103$$ −9.72063 −0.957802 −0.478901 0.877869i $$-0.658965\pi$$
−0.478901 + 0.877869i $$0.658965\pi$$
$$104$$ 0.908940 0.0891289
$$105$$ 0 0
$$106$$ −4.47964 −0.435101
$$107$$ −10.4243 −1.00775 −0.503877 0.863775i $$-0.668093\pi$$
−0.503877 + 0.863775i $$0.668093\pi$$
$$108$$ 0 0
$$109$$ −4.67427 −0.447714 −0.223857 0.974622i $$-0.571865\pi$$
−0.223857 + 0.974622i $$0.571865\pi$$
$$110$$ −27.8934 −2.65953
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −4.68664 −0.440882 −0.220441 0.975400i $$-0.570750\pi$$
−0.220441 + 0.975400i $$0.570750\pi$$
$$114$$ 0 0
$$115$$ 2.60307 0.242737
$$116$$ 1.54402 0.143358
$$117$$ 0 0
$$118$$ −3.68578 −0.339304
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 22.0529 2.00481
$$122$$ 4.91860 0.445310
$$123$$ 0 0
$$124$$ 1.79802 0.161467
$$125$$ −0.0932326 −0.00833898
$$126$$ 0 0
$$127$$ −9.15945 −0.812770 −0.406385 0.913702i $$-0.633211\pi$$
−0.406385 + 0.913702i $$0.633211\pi$$
$$128$$ −13.3818 −1.18279
$$129$$ 0 0
$$130$$ −1.75101 −0.153574
$$131$$ 9.21165 0.804825 0.402413 0.915458i $$-0.368172\pi$$
0.402413 + 0.915458i $$0.368172\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −2.91828 −0.252101
$$135$$ 0 0
$$136$$ 6.99350 0.599688
$$137$$ −6.82438 −0.583046 −0.291523 0.956564i $$-0.594162\pi$$
−0.291523 + 0.956564i $$0.594162\pi$$
$$138$$ 0 0
$$139$$ 5.68664 0.482334 0.241167 0.970484i $$-0.422470\pi$$
0.241167 + 0.970484i $$0.422470\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2.47314 0.207541
$$143$$ 2.07489 0.173511
$$144$$ 0 0
$$145$$ 13.5088 1.12185
$$146$$ −23.6836 −1.96007
$$147$$ 0 0
$$148$$ 2.70347 0.222224
$$149$$ −21.5885 −1.76860 −0.884301 0.466918i $$-0.845364\pi$$
−0.884301 + 0.466918i $$0.845364\pi$$
$$150$$ 0 0
$$151$$ −5.55553 −0.452102 −0.226051 0.974115i $$-0.572582\pi$$
−0.226051 + 0.974115i $$0.572582\pi$$
$$152$$ 18.2105 1.47706
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 15.7311 1.26356
$$156$$ 0 0
$$157$$ −6.07120 −0.484534 −0.242267 0.970210i $$-0.577891\pi$$
−0.242267 + 0.970210i $$0.577891\pi$$
$$158$$ −8.40911 −0.668993
$$159$$ 0 0
$$160$$ 6.37209 0.503758
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3.85259 −0.301758 −0.150879 0.988552i $$-0.548210\pi$$
−0.150879 + 0.988552i $$0.548210\pi$$
$$164$$ −1.20231 −0.0938844
$$165$$ 0 0
$$166$$ 20.0283 1.55450
$$167$$ −3.53837 −0.273807 −0.136904 0.990584i $$-0.543715\pi$$
−0.136904 + 0.990584i $$0.543715\pi$$
$$168$$ 0 0
$$169$$ −12.8697 −0.989981
$$170$$ −13.4725 −1.03329
$$171$$ 0 0
$$172$$ −2.84022 −0.216565
$$173$$ −9.85358 −0.749154 −0.374577 0.927196i $$-0.622212\pi$$
−0.374577 + 0.927196i $$0.622212\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −26.3976 −1.98979
$$177$$ 0 0
$$178$$ 21.9226 1.64317
$$179$$ 19.8971 1.48718 0.743590 0.668636i $$-0.233122\pi$$
0.743590 + 0.668636i $$0.233122\pi$$
$$180$$ 0 0
$$181$$ −12.0930 −0.898869 −0.449434 0.893313i $$-0.648374\pi$$
−0.449434 + 0.893313i $$0.648374\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2.07621 0.153060
$$185$$ 23.6531 1.73901
$$186$$ 0 0
$$187$$ 15.9645 1.16744
$$188$$ −1.25648 −0.0916385
$$189$$ 0 0
$$190$$ −35.0812 −2.54506
$$191$$ −9.71964 −0.703288 −0.351644 0.936134i $$-0.614377\pi$$
−0.351644 + 0.936134i $$0.614377\pi$$
$$192$$ 0 0
$$193$$ −2.30185 −0.165691 −0.0828454 0.996562i $$-0.526401\pi$$
−0.0828454 + 0.996562i $$0.526401\pi$$
$$194$$ 24.5925 1.76564
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −5.06470 −0.360845 −0.180422 0.983589i $$-0.557746\pi$$
−0.180422 + 0.983589i $$0.557746\pi$$
$$198$$ 0 0
$$199$$ 9.95332 0.705572 0.352786 0.935704i $$-0.385234\pi$$
0.352786 + 0.935704i $$0.385234\pi$$
$$200$$ 12.5182 0.885169
$$201$$ 0 0
$$202$$ −4.62108 −0.325138
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.5192 −0.734691
$$206$$ 14.9360 1.04064
$$207$$ 0 0
$$208$$ −1.65711 −0.114900
$$209$$ 41.5702 2.87547
$$210$$ 0 0
$$211$$ −23.3007 −1.60408 −0.802042 0.597267i $$-0.796253\pi$$
−0.802042 + 0.597267i $$0.796253\pi$$
$$212$$ 1.05219 0.0722650
$$213$$ 0 0
$$214$$ 16.0172 1.09491
$$215$$ −24.8496 −1.69473
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 7.18212 0.486435
$$219$$ 0 0
$$220$$ 6.55170 0.441715
$$221$$ 1.00217 0.0674134
$$222$$ 0 0
$$223$$ 11.8754 0.795235 0.397618 0.917551i $$-0.369837\pi$$
0.397618 + 0.917551i $$0.369837\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 7.20113 0.479012
$$227$$ 5.96081 0.395633 0.197816 0.980239i $$-0.436615\pi$$
0.197816 + 0.980239i $$0.436615\pi$$
$$228$$ 0 0
$$229$$ 23.7206 1.56750 0.783752 0.621074i $$-0.213304\pi$$
0.783752 + 0.621074i $$0.213304\pi$$
$$230$$ −3.99968 −0.263731
$$231$$ 0 0
$$232$$ 10.7747 0.707392
$$233$$ 13.8592 0.907948 0.453974 0.891015i $$-0.350006\pi$$
0.453974 + 0.891015i $$0.350006\pi$$
$$234$$ 0 0
$$235$$ −10.9932 −0.717116
$$236$$ 0.865728 0.0563541
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −22.3426 −1.44522 −0.722610 0.691256i $$-0.757058\pi$$
−0.722610 + 0.691256i $$0.757058\pi$$
$$240$$ 0 0
$$241$$ −9.72063 −0.626161 −0.313080 0.949727i $$-0.601361\pi$$
−0.313080 + 0.949727i $$0.601361\pi$$
$$242$$ −33.8847 −2.17819
$$243$$ 0 0
$$244$$ −1.15530 −0.0739604
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.60957 0.166043
$$248$$ 12.5472 0.796747
$$249$$ 0 0
$$250$$ 0.143254 0.00906018
$$251$$ −6.51950 −0.411507 −0.205754 0.978604i $$-0.565965\pi$$
−0.205754 + 0.978604i $$0.565965\pi$$
$$252$$ 0 0
$$253$$ 4.73950 0.297970
$$254$$ 14.0737 0.883063
$$255$$ 0 0
$$256$$ 8.39661 0.524788
$$257$$ 5.53002 0.344953 0.172477 0.985014i $$-0.444823\pi$$
0.172477 + 0.985014i $$0.444823\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0.411283 0.0255067
$$261$$ 0 0
$$262$$ −14.1539 −0.874431
$$263$$ −8.69499 −0.536156 −0.268078 0.963397i $$-0.586388\pi$$
−0.268078 + 0.963397i $$0.586388\pi$$
$$264$$ 0 0
$$265$$ 9.20581 0.565509
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0.685455 0.0418709
$$269$$ −3.77901 −0.230410 −0.115205 0.993342i $$-0.536753\pi$$
−0.115205 + 0.993342i $$0.536753\pi$$
$$270$$ 0 0
$$271$$ −6.61990 −0.402130 −0.201065 0.979578i $$-0.564440\pi$$
−0.201065 + 0.979578i $$0.564440\pi$$
$$272$$ −12.7500 −0.773083
$$273$$ 0 0
$$274$$ 10.4858 0.633472
$$275$$ 28.5761 1.72320
$$276$$ 0 0
$$277$$ −25.2658 −1.51808 −0.759038 0.651046i $$-0.774330\pi$$
−0.759038 + 0.651046i $$0.774330\pi$$
$$278$$ −8.73765 −0.524049
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −7.42442 −0.442904 −0.221452 0.975171i $$-0.571080\pi$$
−0.221452 + 0.975171i $$0.571080\pi$$
$$282$$ 0 0
$$283$$ −15.4203 −0.916640 −0.458320 0.888787i $$-0.651549\pi$$
−0.458320 + 0.888787i $$0.651549\pi$$
$$284$$ −0.580900 −0.0344701
$$285$$ 0 0
$$286$$ −3.18812 −0.188518
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −9.28916 −0.546421
$$290$$ −20.7567 −1.21887
$$291$$ 0 0
$$292$$ 5.56289 0.325543
$$293$$ 30.5797 1.78649 0.893243 0.449574i $$-0.148424\pi$$
0.893243 + 0.449574i $$0.148424\pi$$
$$294$$ 0 0
$$295$$ 7.57440 0.440999
$$296$$ 18.8657 1.09655
$$297$$ 0 0
$$298$$ 33.1713 1.92156
$$299$$ 0.297522 0.0172061
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.53620 0.491203
$$303$$ 0 0
$$304$$ −33.1999 −1.90415
$$305$$ −10.1079 −0.578776
$$306$$ 0 0
$$307$$ 28.7794 1.64252 0.821262 0.570551i $$-0.193270\pi$$
0.821262 + 0.570551i $$0.193270\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −24.1713 −1.37284
$$311$$ −2.57255 −0.145876 −0.0729380 0.997336i $$-0.523238\pi$$
−0.0729380 + 0.997336i $$0.523238\pi$$
$$312$$ 0 0
$$313$$ 18.5145 1.04650 0.523250 0.852179i $$-0.324719\pi$$
0.523250 + 0.852179i $$0.324719\pi$$
$$314$$ 9.32854 0.526440
$$315$$ 0 0
$$316$$ 1.97516 0.111111
$$317$$ −15.5182 −0.871588 −0.435794 0.900046i $$-0.643532\pi$$
−0.435794 + 0.900046i $$0.643532\pi$$
$$318$$ 0 0
$$319$$ 24.5960 1.37711
$$320$$ 19.2058 1.07364
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 20.0784 1.11719
$$324$$ 0 0
$$325$$ 1.79386 0.0995056
$$326$$ 5.91960 0.327856
$$327$$ 0 0
$$328$$ −8.39011 −0.463266
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 31.5904 1.73636 0.868182 0.496246i $$-0.165289\pi$$
0.868182 + 0.496246i $$0.165289\pi$$
$$332$$ −4.70433 −0.258183
$$333$$ 0 0
$$334$$ 5.43679 0.297488
$$335$$ 5.99716 0.327660
$$336$$ 0 0
$$337$$ −11.4081 −0.621440 −0.310720 0.950502i $$-0.600570\pi$$
−0.310720 + 0.950502i $$0.600570\pi$$
$$338$$ 19.7747 1.07560
$$339$$ 0 0
$$340$$ 3.16446 0.171617
$$341$$ 28.6422 1.55106
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −19.8200 −1.06862
$$345$$ 0 0
$$346$$ 15.1403 0.813945
$$347$$ 4.08140 0.219101 0.109550 0.993981i $$-0.465059\pi$$
0.109550 + 0.993981i $$0.465059\pi$$
$$348$$ 0 0
$$349$$ 24.2779 1.29956 0.649782 0.760120i $$-0.274860\pi$$
0.649782 + 0.760120i $$0.274860\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 11.6019 0.618383
$$353$$ −34.0790 −1.81384 −0.906922 0.421299i $$-0.861574\pi$$
−0.906922 + 0.421299i $$0.861574\pi$$
$$354$$ 0 0
$$355$$ −5.08239 −0.269745
$$356$$ −5.14926 −0.272910
$$357$$ 0 0
$$358$$ −30.5724 −1.61580
$$359$$ −5.48931 −0.289715 −0.144857 0.989453i $$-0.546272\pi$$
−0.144857 + 0.989453i $$0.546272\pi$$
$$360$$ 0 0
$$361$$ 33.2823 1.75170
$$362$$ 18.5812 0.976608
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 48.6706 2.54754
$$366$$ 0 0
$$367$$ 13.6391 0.711955 0.355978 0.934495i $$-0.384148\pi$$
0.355978 + 0.934495i $$0.384148\pi$$
$$368$$ −3.78519 −0.197317
$$369$$ 0 0
$$370$$ −36.3435 −1.88941
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 8.92379 0.462056 0.231028 0.972947i $$-0.425791\pi$$
0.231028 + 0.972947i $$0.425791\pi$$
$$374$$ −24.5298 −1.26841
$$375$$ 0 0
$$376$$ −8.76817 −0.452184
$$377$$ 1.54402 0.0795209
$$378$$ 0 0
$$379$$ 29.7035 1.52576 0.762882 0.646537i $$-0.223784\pi$$
0.762882 + 0.646537i $$0.223784\pi$$
$$380$$ 8.23999 0.422703
$$381$$ 0 0
$$382$$ 14.9344 0.764113
$$383$$ −17.7101 −0.904944 −0.452472 0.891779i $$-0.649458\pi$$
−0.452472 + 0.891779i $$0.649458\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.53685 0.180021
$$387$$ 0 0
$$388$$ −5.77638 −0.293251
$$389$$ −21.3255 −1.08125 −0.540624 0.841264i $$-0.681812\pi$$
−0.540624 + 0.841264i $$0.681812\pi$$
$$390$$ 0 0
$$391$$ 2.28917 0.115768
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 7.78203 0.392053
$$395$$ 17.2810 0.869501
$$396$$ 0 0
$$397$$ 2.23566 0.112205 0.0561024 0.998425i $$-0.482133\pi$$
0.0561024 + 0.998425i $$0.482133\pi$$
$$398$$ −15.2935 −0.766594
$$399$$ 0 0
$$400$$ −22.8222 −1.14111
$$401$$ −8.73702 −0.436306 −0.218153 0.975915i $$-0.570003\pi$$
−0.218153 + 0.975915i $$0.570003\pi$$
$$402$$ 0 0
$$403$$ 1.79802 0.0895656
$$404$$ 1.08542 0.0540014
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 43.0660 2.13470
$$408$$ 0 0
$$409$$ 28.6920 1.41873 0.709363 0.704843i $$-0.248983\pi$$
0.709363 + 0.704843i $$0.248983\pi$$
$$410$$ 16.1630 0.798232
$$411$$ 0 0
$$412$$ −3.50821 −0.172837
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −41.1589 −2.02041
$$416$$ 0.728309 0.0357083
$$417$$ 0 0
$$418$$ −63.8736 −3.12416
$$419$$ 8.64442 0.422307 0.211154 0.977453i $$-0.432278\pi$$
0.211154 + 0.977453i $$0.432278\pi$$
$$420$$ 0 0
$$421$$ 18.4669 0.900024 0.450012 0.893022i $$-0.351420\pi$$
0.450012 + 0.893022i $$0.351420\pi$$
$$422$$ 35.8020 1.74282
$$423$$ 0 0
$$424$$ 7.34257 0.356587
$$425$$ 13.8022 0.669506
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −3.76216 −0.181851
$$429$$ 0 0
$$430$$ 38.1819 1.84130
$$431$$ 5.80736 0.279731 0.139865 0.990171i $$-0.455333\pi$$
0.139865 + 0.990171i $$0.455333\pi$$
$$432$$ 0 0
$$433$$ −3.63877 −0.174868 −0.0874341 0.996170i $$-0.527867\pi$$
−0.0874341 + 0.996170i $$0.527867\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1.68696 −0.0807908
$$437$$ 5.96081 0.285144
$$438$$ 0 0
$$439$$ 18.6619 0.890684 0.445342 0.895361i $$-0.353082\pi$$
0.445342 + 0.895361i $$0.353082\pi$$
$$440$$ 45.7200 2.17961
$$441$$ 0 0
$$442$$ −1.53986 −0.0732437
$$443$$ 26.0911 1.23962 0.619812 0.784750i $$-0.287209\pi$$
0.619812 + 0.784750i $$0.287209\pi$$
$$444$$ 0 0
$$445$$ −45.0517 −2.13565
$$446$$ −18.2468 −0.864012
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.63824 0.266085 0.133042 0.991110i $$-0.457525\pi$$
0.133042 + 0.991110i $$0.457525\pi$$
$$450$$ 0 0
$$451$$ −19.1526 −0.901862
$$452$$ −1.69142 −0.0795579
$$453$$ 0 0
$$454$$ −9.15892 −0.429849
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −19.9923 −0.935201 −0.467601 0.883940i $$-0.654881\pi$$
−0.467601 + 0.883940i $$0.654881\pi$$
$$458$$ −36.4473 −1.70307
$$459$$ 0 0
$$460$$ 0.939457 0.0438024
$$461$$ 37.4495 1.74420 0.872098 0.489332i $$-0.162759\pi$$
0.872098 + 0.489332i $$0.162759\pi$$
$$462$$ 0 0
$$463$$ 2.45513 0.114099 0.0570497 0.998371i $$-0.481831\pi$$
0.0570497 + 0.998371i $$0.481831\pi$$
$$464$$ −19.6436 −0.911929
$$465$$ 0 0
$$466$$ −21.2950 −0.986473
$$467$$ 33.4107 1.54606 0.773032 0.634367i $$-0.218739\pi$$
0.773032 + 0.634367i $$0.218739\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 16.8913 0.779136
$$471$$ 0 0
$$472$$ 6.04135 0.278076
$$473$$ −45.2445 −2.08034
$$474$$ 0 0
$$475$$ 35.9398 1.64903
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 34.3299 1.57021
$$479$$ 34.6936 1.58519 0.792595 0.609748i $$-0.208729\pi$$
0.792595 + 0.609748i $$0.208729\pi$$
$$480$$ 0 0
$$481$$ 2.70347 0.123268
$$482$$ 14.9360 0.680315
$$483$$ 0 0
$$484$$ 7.95896 0.361771
$$485$$ −50.5385 −2.29483
$$486$$ 0 0
$$487$$ 0.959817 0.0434935 0.0217467 0.999764i $$-0.493077\pi$$
0.0217467 + 0.999764i $$0.493077\pi$$
$$488$$ −8.06207 −0.364953
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15.7738 −0.711862 −0.355931 0.934512i $$-0.615836\pi$$
−0.355931 + 0.934512i $$0.615836\pi$$
$$492$$ 0 0
$$493$$ 11.8799 0.535042
$$494$$ −4.00966 −0.180403
$$495$$ 0 0
$$496$$ −22.8751 −1.02712
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −19.1287 −0.856320 −0.428160 0.903703i $$-0.640838\pi$$
−0.428160 + 0.903703i $$0.640838\pi$$
$$500$$ −0.0336480 −0.00150478
$$501$$ 0 0
$$502$$ 10.0174 0.447097
$$503$$ −33.3898 −1.48878 −0.744388 0.667747i $$-0.767259\pi$$
−0.744388 + 0.667747i $$0.767259\pi$$
$$504$$ 0 0
$$505$$ 9.49648 0.422588
$$506$$ −7.28235 −0.323740
$$507$$ 0 0
$$508$$ −3.30568 −0.146666
$$509$$ 33.5176 1.48564 0.742822 0.669489i $$-0.233487\pi$$
0.742822 + 0.669489i $$0.233487\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 13.8619 0.612617
$$513$$ 0 0
$$514$$ −8.49701 −0.374787
$$515$$ −30.6939 −1.35254
$$516$$ 0 0
$$517$$ −20.0157 −0.880287
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 2.87007 0.125861
$$521$$ −26.7243 −1.17081 −0.585407 0.810740i $$-0.699065\pi$$
−0.585407 + 0.810740i $$0.699065\pi$$
$$522$$ 0 0
$$523$$ −17.0644 −0.746173 −0.373086 0.927797i $$-0.621701\pi$$
−0.373086 + 0.927797i $$0.621701\pi$$
$$524$$ 3.32452 0.145232
$$525$$ 0 0
$$526$$ 13.3600 0.582526
$$527$$ 13.8342 0.602626
$$528$$ 0 0
$$529$$ −22.3204 −0.970452
$$530$$ −14.1449 −0.614417
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.20231 −0.0520777
$$534$$ 0 0
$$535$$ −32.9158 −1.42307
$$536$$ 4.78334 0.206609
$$537$$ 0 0
$$538$$ 5.80654 0.250338
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 12.3426 0.530648 0.265324 0.964159i $$-0.414521\pi$$
0.265324 + 0.964159i $$0.414521\pi$$
$$542$$ 10.1716 0.436909
$$543$$ 0 0
$$544$$ 5.60370 0.240257
$$545$$ −14.7595 −0.632227
$$546$$ 0 0
$$547$$ 23.4424 1.00233 0.501163 0.865353i $$-0.332906\pi$$
0.501163 + 0.865353i $$0.332906\pi$$
$$548$$ −2.46294 −0.105212
$$549$$ 0 0
$$550$$ −43.9078 −1.87223
$$551$$ 30.9341 1.31784
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 38.8215 1.64937
$$555$$ 0 0
$$556$$ 2.05233 0.0870381
$$557$$ 24.5115 1.03858 0.519292 0.854597i $$-0.326196\pi$$
0.519292 + 0.854597i $$0.326196\pi$$
$$558$$ 0 0
$$559$$ −2.84022 −0.120129
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 11.4078 0.481209
$$563$$ −13.3714 −0.563538 −0.281769 0.959482i $$-0.590921\pi$$
−0.281769 + 0.959482i $$0.590921\pi$$
$$564$$ 0 0
$$565$$ −14.7985 −0.622580
$$566$$ 23.6936 0.995916
$$567$$ 0 0
$$568$$ −4.05372 −0.170090
$$569$$ 14.2488 0.597341 0.298670 0.954356i $$-0.403457\pi$$
0.298670 + 0.954356i $$0.403457\pi$$
$$570$$ 0 0
$$571$$ −29.3237 −1.22716 −0.613579 0.789633i $$-0.710271\pi$$
−0.613579 + 0.789633i $$0.710271\pi$$
$$572$$ 0.748837 0.0313105
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.09756 0.170880
$$576$$ 0 0
$$577$$ 18.1001 0.753516 0.376758 0.926312i $$-0.377039\pi$$
0.376758 + 0.926312i $$0.377039\pi$$
$$578$$ 14.2730 0.593679
$$579$$ 0 0
$$580$$ 4.87539 0.202440
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 16.7613 0.694184
$$584$$ 38.8197 1.60637
$$585$$ 0 0
$$586$$ −46.9865 −1.94099
$$587$$ 6.52804 0.269441 0.134721 0.990884i $$-0.456986\pi$$
0.134721 + 0.990884i $$0.456986\pi$$
$$588$$ 0 0
$$589$$ 36.0230 1.48430
$$590$$ −11.6382 −0.479139
$$591$$ 0 0
$$592$$ −34.3946 −1.41361
$$593$$ −25.4405 −1.04471 −0.522357 0.852727i $$-0.674947\pi$$
−0.522357 + 0.852727i $$0.674947\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −7.79138 −0.319147
$$597$$ 0 0
$$598$$ −0.457150 −0.0186942
$$599$$ −6.17931 −0.252480 −0.126240 0.992000i $$-0.540291\pi$$
−0.126240 + 0.992000i $$0.540291\pi$$
$$600$$ 0 0
$$601$$ 12.9344 0.527607 0.263804 0.964576i $$-0.415023\pi$$
0.263804 + 0.964576i $$0.415023\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −2.00501 −0.0815828
$$605$$ 69.6342 2.83103
$$606$$ 0 0
$$607$$ 23.0756 0.936608 0.468304 0.883567i $$-0.344865\pi$$
0.468304 + 0.883567i $$0.344865\pi$$
$$608$$ 14.5916 0.591766
$$609$$ 0 0
$$610$$ 15.5310 0.628832
$$611$$ −1.25648 −0.0508319
$$612$$ 0 0
$$613$$ −17.6273 −0.711958 −0.355979 0.934494i $$-0.615853\pi$$
−0.355979 + 0.934494i $$0.615853\pi$$
$$614$$ −44.2201 −1.78458
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −21.6441 −0.871358 −0.435679 0.900102i $$-0.643492\pi$$
−0.435679 + 0.900102i $$0.643492\pi$$
$$618$$ 0 0
$$619$$ 9.57941 0.385029 0.192514 0.981294i $$-0.438336\pi$$
0.192514 + 0.981294i $$0.438336\pi$$
$$620$$ 5.67743 0.228011
$$621$$ 0 0
$$622$$ 3.95278 0.158492
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −25.1468 −1.00587
$$626$$ −28.4479 −1.13701
$$627$$ 0 0
$$628$$ −2.19112 −0.0874352
$$629$$ 20.8008 0.829384
$$630$$ 0 0
$$631$$ −31.1742 −1.24103 −0.620514 0.784196i $$-0.713076\pi$$
−0.620514 + 0.784196i $$0.713076\pi$$
$$632$$ 13.7833 0.548272
$$633$$ 0 0
$$634$$ 23.8441 0.946968
$$635$$ −28.9219 −1.14773
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −37.7924 −1.49621
$$639$$ 0 0
$$640$$ −42.2543 −1.67025
$$641$$ 8.25214 0.325940 0.162970 0.986631i $$-0.447893\pi$$
0.162970 + 0.986631i $$0.447893\pi$$
$$642$$ 0 0
$$643$$ −24.4318 −0.963495 −0.481748 0.876310i $$-0.659998\pi$$
−0.481748 + 0.876310i $$0.659998\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −30.8509 −1.21381
$$647$$ 38.6864 1.52092 0.760461 0.649384i $$-0.224973\pi$$
0.760461 + 0.649384i $$0.224973\pi$$
$$648$$ 0 0
$$649$$ 13.7910 0.541343
$$650$$ −2.75631 −0.108111
$$651$$ 0 0
$$652$$ −1.39041 −0.0544528
$$653$$ −7.65430 −0.299536 −0.149768 0.988721i $$-0.547853\pi$$
−0.149768 + 0.988721i $$0.547853\pi$$
$$654$$ 0 0
$$655$$ 29.0867 1.13651
$$656$$ 15.2962 0.597217
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −38.5145 −1.50031 −0.750156 0.661261i $$-0.770022\pi$$
−0.750156 + 0.661261i $$0.770022\pi$$
$$660$$ 0 0
$$661$$ −32.2132 −1.25295 −0.626474 0.779443i $$-0.715502\pi$$
−0.626474 + 0.779443i $$0.715502\pi$$
$$662$$ −48.5393 −1.88654
$$663$$ 0 0
$$664$$ −32.8284 −1.27399
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.52686 0.136561
$$668$$ −1.27701 −0.0494091
$$669$$ 0 0
$$670$$ −9.21478 −0.355998
$$671$$ −18.4038 −0.710470
$$672$$ 0 0
$$673$$ 1.26136 0.0486218 0.0243109 0.999704i $$-0.492261\pi$$
0.0243109 + 0.999704i $$0.492261\pi$$
$$674$$ 17.5288 0.675186
$$675$$ 0 0
$$676$$ −4.64474 −0.178644
$$677$$ 14.8318 0.570031 0.285016 0.958523i $$-0.408001\pi$$
0.285016 + 0.958523i $$0.408001\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 22.0827 0.846833
$$681$$ 0 0
$$682$$ −44.0095 −1.68521
$$683$$ 5.53120 0.211646 0.105823 0.994385i $$-0.466252\pi$$
0.105823 + 0.994385i $$0.466252\pi$$
$$684$$ 0 0
$$685$$ −21.5487 −0.823334
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 36.1344 1.37761
$$689$$ 1.05219 0.0400854
$$690$$ 0 0
$$691$$ −8.63792 −0.328602 −0.164301 0.986410i $$-0.552537\pi$$
−0.164301 + 0.986410i $$0.552537\pi$$
$$692$$ −3.55620 −0.135186
$$693$$ 0 0
$$694$$ −6.27116 −0.238050
$$695$$ 17.9562 0.681116
$$696$$ 0 0
$$697$$ −9.25070 −0.350395
$$698$$ −37.3035 −1.41196
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −26.5897 −1.00428 −0.502140 0.864786i $$-0.667454\pi$$
−0.502140 + 0.864786i $$0.667454\pi$$
$$702$$ 0 0
$$703$$ 54.1636 2.04282
$$704$$ 34.9686 1.31793
$$705$$ 0 0
$$706$$ 52.3632 1.97072
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.1261 0.380294 0.190147 0.981756i $$-0.439104\pi$$
0.190147 + 0.981756i $$0.439104\pi$$
$$710$$ 7.80921 0.293074
$$711$$ 0 0
$$712$$ −35.9333 −1.34666
$$713$$ 4.10705 0.153810
$$714$$ 0 0
$$715$$ 6.55170 0.245020
$$716$$ 7.18094 0.268364
$$717$$ 0 0
$$718$$ 8.43445 0.314771
$$719$$ 32.3876 1.20785 0.603927 0.797040i $$-0.293602\pi$$
0.603927 + 0.797040i $$0.293602\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −51.1391 −1.90320
$$723$$ 0 0
$$724$$ −4.36443 −0.162203
$$725$$ 21.2646 0.789749
$$726$$ 0 0
$$727$$ −25.2348 −0.935907 −0.467953 0.883753i $$-0.655008\pi$$
−0.467953 + 0.883753i $$0.655008\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −74.7835 −2.76786
$$731$$ −21.8530 −0.808264
$$732$$ 0 0
$$733$$ −2.84672 −0.105146 −0.0525731 0.998617i $$-0.516742\pi$$
−0.0525731 + 0.998617i $$0.516742\pi$$
$$734$$ −20.9568 −0.773529
$$735$$ 0 0
$$736$$ 1.66361 0.0613215
$$737$$ 10.9192 0.402215
$$738$$ 0 0
$$739$$ 24.5417 0.902779 0.451390 0.892327i $$-0.350928\pi$$
0.451390 + 0.892327i $$0.350928\pi$$
$$740$$ 8.53649 0.313808
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −4.59070 −0.168416 −0.0842082 0.996448i $$-0.526836\pi$$
−0.0842082 + 0.996448i $$0.526836\pi$$
$$744$$ 0 0
$$745$$ −68.1681 −2.49748
$$746$$ −13.7116 −0.502018
$$747$$ 0 0
$$748$$ 5.76165 0.210667
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 31.4356 1.14710 0.573551 0.819170i $$-0.305566\pi$$
0.573551 + 0.819170i $$0.305566\pi$$
$$752$$ 15.9855 0.582930
$$753$$ 0 0
$$754$$ −2.37242 −0.0863983
$$755$$ −17.5422 −0.638425
$$756$$ 0 0
$$757$$ 29.5432 1.07376 0.536882 0.843657i $$-0.319602\pi$$
0.536882 + 0.843657i $$0.319602\pi$$
$$758$$ −45.6401 −1.65772
$$759$$ 0 0
$$760$$ 57.5015 2.08580
$$761$$ −46.4873 −1.68516 −0.842582 0.538567i $$-0.818966\pi$$
−0.842582 + 0.538567i $$0.818966\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −3.50785 −0.126910
$$765$$ 0 0
$$766$$ 27.2120 0.983209
$$767$$ 0.865728 0.0312596
$$768$$ 0 0
$$769$$ −14.1706 −0.511006 −0.255503 0.966808i $$-0.582241\pi$$
−0.255503 + 0.966808i $$0.582241\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −0.830747 −0.0298992
$$773$$ −34.0085 −1.22320 −0.611600 0.791167i $$-0.709474\pi$$
−0.611600 + 0.791167i $$0.709474\pi$$
$$774$$ 0 0
$$775$$ 24.7628 0.889507
$$776$$ −40.3096 −1.44703
$$777$$ 0 0
$$778$$ 32.7672 1.17476
$$779$$ −24.0880 −0.863043
$$780$$ 0 0
$$781$$ −9.25368 −0.331123
$$782$$ −3.51737 −0.125781
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −19.1705 −0.684223
$$786$$ 0 0
$$787$$ 29.1059 1.03751 0.518757 0.854921i $$-0.326395\pi$$
0.518757 + 0.854921i $$0.326395\pi$$
$$788$$ −1.82787 −0.0651151
$$789$$ 0 0
$$790$$ −26.5527 −0.944701
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −1.15530 −0.0410259
$$794$$ −3.43515 −0.121909
$$795$$ 0 0
$$796$$ 3.59219 0.127322
$$797$$ −18.5038 −0.655439 −0.327720 0.944775i $$-0.606280\pi$$
−0.327720 + 0.944775i $$0.606280\pi$$
$$798$$ 0 0
$$799$$ −9.66754 −0.342013
$$800$$ 10.0305 0.354631
$$801$$ 0 0
$$802$$ 13.4246 0.474040
$$803$$ 88.6162 3.12720
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −2.76270 −0.0973118
$$807$$ 0 0
$$808$$ 7.57440 0.266466
$$809$$ −41.0813 −1.44434 −0.722172 0.691714i $$-0.756856\pi$$
−0.722172 + 0.691714i $$0.756856\pi$$
$$810$$ 0 0
$$811$$ 43.1361 1.51471 0.757357 0.653001i $$-0.226490\pi$$
0.757357 + 0.653001i $$0.226490\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −66.1719 −2.31932
$$815$$ −12.1650 −0.426120
$$816$$ 0 0
$$817$$ −56.9034 −1.99080
$$818$$ −44.0859 −1.54143
$$819$$ 0 0
$$820$$ −3.79641 −0.132576
$$821$$ −13.1748 −0.459802 −0.229901 0.973214i $$-0.573840\pi$$
−0.229901 + 0.973214i $$0.573840\pi$$
$$822$$ 0 0
$$823$$ −11.9817 −0.417654 −0.208827 0.977953i $$-0.566965\pi$$
−0.208827 + 0.977953i $$0.566965\pi$$
$$824$$ −24.4815 −0.852853
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −29.9879 −1.04278 −0.521391 0.853318i $$-0.674587\pi$$
−0.521391 + 0.853318i $$0.674587\pi$$
$$828$$ 0 0
$$829$$ −26.9238 −0.935102 −0.467551 0.883966i $$-0.654864\pi$$
−0.467551 + 0.883966i $$0.654864\pi$$
$$830$$ 63.2416 2.19515
$$831$$ 0 0
$$832$$ 2.19516 0.0761034
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −11.1728 −0.386650
$$836$$ 15.0028 0.518884
$$837$$ 0 0
$$838$$ −13.2823 −0.458831
$$839$$ −11.2238 −0.387490 −0.193745 0.981052i $$-0.562063\pi$$
−0.193745 + 0.981052i $$0.562063\pi$$
$$840$$ 0 0
$$841$$ −10.6971 −0.368864
$$842$$ −28.3749 −0.977864
$$843$$ 0 0
$$844$$ −8.40930 −0.289460
$$845$$ −40.6376 −1.39798
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −13.3864 −0.459691
$$849$$ 0 0
$$850$$ −21.2074 −0.727408
$$851$$ 6.17529 0.211686
$$852$$ 0 0
$$853$$ 12.1934 0.417496 0.208748 0.977970i $$-0.433061\pi$$
0.208748 + 0.977970i $$0.433061\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −26.2537 −0.897332
$$857$$ −37.4382 −1.27887 −0.639433 0.768847i $$-0.720831\pi$$
−0.639433 + 0.768847i $$0.720831\pi$$
$$858$$ 0 0
$$859$$ −0.757734 −0.0258535 −0.0129268 0.999916i $$-0.504115\pi$$
−0.0129268 + 0.999916i $$0.504115\pi$$
$$860$$ −8.96830 −0.305817
$$861$$ 0 0
$$862$$ −8.92314 −0.303923
$$863$$ −23.5592 −0.801965 −0.400983 0.916086i $$-0.631331\pi$$
−0.400983 + 0.916086i $$0.631331\pi$$
$$864$$ 0 0
$$865$$ −31.1137 −1.05790
$$866$$ 5.59106 0.189992
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 31.4641 1.06735
$$870$$ 0 0
$$871$$ 0.685455 0.0232258
$$872$$ −11.7722 −0.398657
$$873$$ 0 0
$$874$$ −9.15892 −0.309805
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 36.6739 1.23839 0.619196 0.785237i $$-0.287459\pi$$
0.619196 + 0.785237i $$0.287459\pi$$
$$878$$ −28.6744 −0.967716
$$879$$ 0 0
$$880$$ −83.3532 −2.80984
$$881$$ −39.4357 −1.32862 −0.664311 0.747456i $$-0.731275\pi$$
−0.664311 + 0.747456i $$0.731275\pi$$
$$882$$ 0 0
$$883$$ −8.76912 −0.295105 −0.147552 0.989054i $$-0.547139\pi$$
−0.147552 + 0.989054i $$0.547139\pi$$
$$884$$ 0.361688 0.0121649
$$885$$ 0 0
$$886$$ −40.0895 −1.34683
$$887$$ 7.82601 0.262772 0.131386 0.991331i $$-0.458057\pi$$
0.131386 + 0.991331i $$0.458057\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 69.2230 2.32036
$$891$$ 0 0
$$892$$ 4.28587 0.143502
$$893$$ −25.1734 −0.842397
$$894$$ 0 0
$$895$$ 62.8272 2.10008
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −8.66329 −0.289098
$$899$$ 21.3139 0.710858
$$900$$ 0 0
$$901$$ 8.09571 0.269707
$$902$$ 29.4285 0.979860
$$903$$ 0 0
$$904$$ −11.8033 −0.392573
$$905$$ −38.1851 −1.26931
$$906$$ 0 0
$$907$$ 51.9332 1.72442 0.862208 0.506555i $$-0.169081\pi$$
0.862208 + 0.506555i $$0.169081\pi$$
$$908$$ 2.15128 0.0713927
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −36.1524 −1.19778 −0.598891 0.800830i $$-0.704392\pi$$
−0.598891 + 0.800830i $$0.704392\pi$$
$$912$$ 0 0
$$913$$ −74.9394 −2.48013
$$914$$ 30.7187 1.01608
$$915$$ 0 0
$$916$$ 8.56086 0.282859
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −11.1768 −0.368690 −0.184345 0.982862i $$-0.559016\pi$$
−0.184345 + 0.982862i $$0.559016\pi$$
$$920$$ 6.55585 0.216140
$$921$$ 0 0
$$922$$ −57.5420 −1.89504
$$923$$ −0.580900 −0.0191206
$$924$$ 0 0
$$925$$ 37.2330 1.22421
$$926$$ −3.77236 −0.123967
$$927$$ 0 0
$$928$$ 8.63345 0.283407
$$929$$ 32.8384 1.07739 0.538696 0.842500i $$-0.318917\pi$$
0.538696 + 0.842500i $$0.318917\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 5.00185 0.163841
$$933$$ 0 0
$$934$$ −51.3364 −1.67978
$$935$$ 50.4096 1.64857
$$936$$ 0 0
$$937$$ 25.9566 0.847965 0.423983 0.905670i $$-0.360632\pi$$
0.423983 + 0.905670i $$0.360632\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −3.96748 −0.129405
$$941$$ 29.4679 0.960627 0.480314 0.877097i $$-0.340523\pi$$
0.480314 + 0.877097i $$0.340523\pi$$
$$942$$ 0 0
$$943$$ −2.74632 −0.0894326
$$944$$ −11.0141 −0.358479
$$945$$ 0 0
$$946$$ 69.5192 2.26026
$$947$$ 11.5227 0.374439 0.187219 0.982318i $$-0.440052\pi$$
0.187219 + 0.982318i $$0.440052\pi$$
$$948$$ 0 0
$$949$$ 5.56289 0.180579
$$950$$ −55.2223 −1.79165
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −29.2912 −0.948835 −0.474417 0.880300i $$-0.657341\pi$$
−0.474417 + 0.880300i $$0.657341\pi$$
$$954$$ 0 0
$$955$$ −30.6908 −0.993130
$$956$$ −8.06352 −0.260793
$$957$$ 0 0
$$958$$ −53.3075 −1.72229
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.17981 −0.199349
$$962$$ −4.15394 −0.133929
$$963$$ 0 0
$$964$$ −3.50821 −0.112992
$$965$$ −7.26834 −0.233976
$$966$$ 0 0
$$967$$ 42.0803 1.35321 0.676606 0.736345i $$-0.263450\pi$$
0.676606 + 0.736345i $$0.263450\pi$$
$$968$$ 55.5403 1.78513
$$969$$ 0 0
$$970$$ 77.6536 2.49331
$$971$$ −35.1549 −1.12817 −0.564087 0.825715i $$-0.690772\pi$$
−0.564087 + 0.825715i $$0.690772\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −1.47478 −0.0472551
$$975$$ 0 0
$$976$$ 14.6981 0.470476
$$977$$ 4.43025 0.141736 0.0708682 0.997486i $$-0.477423\pi$$
0.0708682 + 0.997486i $$0.477423\pi$$
$$978$$ 0 0
$$979$$ −82.0271 −2.62160
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 24.2368 0.773428
$$983$$ 22.1601 0.706798 0.353399 0.935473i $$-0.385026\pi$$
0.353399 + 0.935473i $$0.385026\pi$$
$$984$$ 0 0
$$985$$ −15.9923 −0.509558
$$986$$ −18.2537 −0.581316
$$987$$ 0 0
$$988$$ 0.941804 0.0299628
$$989$$ −6.48767 −0.206296
$$990$$ 0 0
$$991$$ −36.7203 −1.16646 −0.583229 0.812307i $$-0.698211\pi$$
−0.583229 + 0.812307i $$0.698211\pi$$
$$992$$ 10.0537 0.319206
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 31.4286 0.996355
$$996$$ 0 0
$$997$$ 39.9031 1.26374 0.631872 0.775073i $$-0.282287\pi$$
0.631872 + 0.775073i $$0.282287\pi$$
$$998$$ 29.3918 0.930380
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3969.2.a.t.1.2 4
3.2 odd 2 3969.2.a.w.1.3 4
7.3 odd 6 567.2.e.d.163.3 yes 8
7.5 odd 6 567.2.e.d.487.3 yes 8
7.6 odd 2 3969.2.a.s.1.2 4
21.5 even 6 567.2.e.c.487.2 yes 8
21.17 even 6 567.2.e.c.163.2 8
21.20 even 2 3969.2.a.x.1.3 4
63.5 even 6 567.2.h.k.298.3 8
63.31 odd 6 567.2.g.k.541.3 8
63.38 even 6 567.2.h.k.352.3 8
63.40 odd 6 567.2.h.j.298.2 8
63.47 even 6 567.2.g.j.109.2 8
63.52 odd 6 567.2.h.j.352.2 8
63.59 even 6 567.2.g.j.541.2 8
63.61 odd 6 567.2.g.k.109.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.e.c.163.2 8 21.17 even 6
567.2.e.c.487.2 yes 8 21.5 even 6
567.2.e.d.163.3 yes 8 7.3 odd 6
567.2.e.d.487.3 yes 8 7.5 odd 6
567.2.g.j.109.2 8 63.47 even 6
567.2.g.j.541.2 8 63.59 even 6
567.2.g.k.109.3 8 63.61 odd 6
567.2.g.k.541.3 8 63.31 odd 6
567.2.h.j.298.2 8 63.40 odd 6
567.2.h.j.352.2 8 63.52 odd 6
567.2.h.k.298.3 8 63.5 even 6
567.2.h.k.352.3 8 63.38 even 6
3969.2.a.s.1.2 4 7.6 odd 2
3969.2.a.t.1.2 4 1.1 even 1 trivial
3969.2.a.w.1.3 4 3.2 odd 2
3969.2.a.x.1.3 4 21.20 even 2