# Properties

 Label 1024.4.b.k.513.6 Level $1024$ Weight $4$ Character 1024.513 Analytic conductor $60.418$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1024.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$60.4179558459$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 36 x^{8} + 405 x^{6} + 1380 x^{4} + 420 x^{2} + 32$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 513.6 Root $$-3.94652i$$ of defining polynomial Character $$\chi$$ $$=$$ 1024.513 Dual form 1024.4.b.k.513.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.07024i q^{3} -11.6331i q^{5} +2.67171 q^{7} +25.8546 q^{9} +O(q^{10})$$ $$q+1.07024i q^{3} -11.6331i q^{5} +2.67171 q^{7} +25.8546 q^{9} +63.9525i q^{11} +50.0586i q^{13} +12.4503 q^{15} +72.4991 q^{17} -27.4961i q^{19} +2.85937i q^{21} -139.462 q^{23} -10.3299 q^{25} +56.5672i q^{27} -93.3995i q^{29} -188.682 q^{31} -68.4447 q^{33} -31.0803i q^{35} +118.886i q^{37} -53.5748 q^{39} -104.629 q^{41} +44.5275i q^{43} -300.770i q^{45} -488.151 q^{47} -335.862 q^{49} +77.5916i q^{51} +211.510i q^{53} +743.968 q^{55} +29.4275 q^{57} +402.624i q^{59} +322.538i q^{61} +69.0758 q^{63} +582.338 q^{65} +196.789i q^{67} -149.258i q^{69} +453.655 q^{71} +259.747 q^{73} -11.0555i q^{75} +170.862i q^{77} +323.190 q^{79} +637.533 q^{81} +797.471i q^{83} -843.392i q^{85} +99.9602 q^{87} +866.853 q^{89} +133.742i q^{91} -201.935i q^{93} -319.866 q^{95} -936.077 q^{97} +1653.47i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 28q^{7} - 54q^{9} + O(q^{10})$$ $$10q + 28q^{7} - 54q^{9} - 124q^{15} + 4q^{17} + 276q^{23} - 50q^{25} - 368q^{31} - 4q^{33} + 732q^{39} - 944q^{47} - 94q^{49} + 1380q^{55} - 108q^{57} - 2628q^{63} - 492q^{65} + 3468q^{71} + 296q^{73} - 4416q^{79} - 482q^{81} + 6036q^{87} - 88q^{89} - 6900q^{95} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$3$$ 1.07024i 0.205968i 0.994683 + 0.102984i $$0.0328391\pi$$
−0.994683 + 0.102984i $$0.967161\pi$$
$$4$$ 0 0
$$5$$ − 11.6331i − 1.04050i −0.854014 0.520250i $$-0.825839\pi$$
0.854014 0.520250i $$-0.174161\pi$$
$$6$$ 0 0
$$7$$ 2.67171 0.144259 0.0721293 0.997395i $$-0.477021\pi$$
0.0721293 + 0.997395i $$0.477021\pi$$
$$8$$ 0 0
$$9$$ 25.8546 0.957577
$$10$$ 0 0
$$11$$ 63.9525i 1.75295i 0.481451 + 0.876473i $$0.340110\pi$$
−0.481451 + 0.876473i $$0.659890\pi$$
$$12$$ 0 0
$$13$$ 50.0586i 1.06798i 0.845491 + 0.533990i $$0.179308\pi$$
−0.845491 + 0.533990i $$0.820692\pi$$
$$14$$ 0 0
$$15$$ 12.4503 0.214310
$$16$$ 0 0
$$17$$ 72.4991 1.03433 0.517165 0.855886i $$-0.326987\pi$$
0.517165 + 0.855886i $$0.326987\pi$$
$$18$$ 0 0
$$19$$ − 27.4961i − 0.332002i −0.986126 0.166001i $$-0.946915\pi$$
0.986126 0.166001i $$-0.0530855\pi$$
$$20$$ 0 0
$$21$$ 2.85937i 0.0297127i
$$22$$ 0 0
$$23$$ −139.462 −1.26434 −0.632170 0.774830i $$-0.717835\pi$$
−0.632170 + 0.774830i $$0.717835\pi$$
$$24$$ 0 0
$$25$$ −10.3299 −0.0826390
$$26$$ 0 0
$$27$$ 56.5672i 0.403199i
$$28$$ 0 0
$$29$$ − 93.3995i − 0.598064i −0.954243 0.299032i $$-0.903336\pi$$
0.954243 0.299032i $$-0.0966637\pi$$
$$30$$ 0 0
$$31$$ −188.682 −1.09317 −0.546584 0.837404i $$-0.684072\pi$$
−0.546584 + 0.837404i $$0.684072\pi$$
$$32$$ 0 0
$$33$$ −68.4447 −0.361051
$$34$$ 0 0
$$35$$ − 31.0803i − 0.150101i
$$36$$ 0 0
$$37$$ 118.886i 0.528238i 0.964490 + 0.264119i $$0.0850811\pi$$
−0.964490 + 0.264119i $$0.914919\pi$$
$$38$$ 0 0
$$39$$ −53.5748 −0.219970
$$40$$ 0 0
$$41$$ −104.629 −0.398545 −0.199272 0.979944i $$-0.563858\pi$$
−0.199272 + 0.979944i $$0.563858\pi$$
$$42$$ 0 0
$$43$$ 44.5275i 0.157916i 0.996878 + 0.0789579i $$0.0251593\pi$$
−0.996878 + 0.0789579i $$0.974841\pi$$
$$44$$ 0 0
$$45$$ − 300.770i − 0.996358i
$$46$$ 0 0
$$47$$ −488.151 −1.51498 −0.757491 0.652846i $$-0.773575\pi$$
−0.757491 + 0.652846i $$0.773575\pi$$
$$48$$ 0 0
$$49$$ −335.862 −0.979189
$$50$$ 0 0
$$51$$ 77.5916i 0.213039i
$$52$$ 0 0
$$53$$ 211.510i 0.548173i 0.961705 + 0.274087i $$0.0883755\pi$$
−0.961705 + 0.274087i $$0.911624\pi$$
$$54$$ 0 0
$$55$$ 743.968 1.82394
$$56$$ 0 0
$$57$$ 29.4275 0.0683819
$$58$$ 0 0
$$59$$ 402.624i 0.888426i 0.895921 + 0.444213i $$0.146517\pi$$
−0.895921 + 0.444213i $$0.853483\pi$$
$$60$$ 0 0
$$61$$ 322.538i 0.676997i 0.940967 + 0.338498i $$0.109919\pi$$
−0.940967 + 0.338498i $$0.890081\pi$$
$$62$$ 0 0
$$63$$ 69.0758 0.138139
$$64$$ 0 0
$$65$$ 582.338 1.11123
$$66$$ 0 0
$$67$$ 196.789i 0.358829i 0.983774 + 0.179415i $$0.0574203\pi$$
−0.983774 + 0.179415i $$0.942580\pi$$
$$68$$ 0 0
$$69$$ − 149.258i − 0.260414i
$$70$$ 0 0
$$71$$ 453.655 0.758294 0.379147 0.925336i $$-0.376217\pi$$
0.379147 + 0.925336i $$0.376217\pi$$
$$72$$ 0 0
$$73$$ 259.747 0.416454 0.208227 0.978081i $$-0.433231\pi$$
0.208227 + 0.978081i $$0.433231\pi$$
$$74$$ 0 0
$$75$$ − 11.0555i − 0.0170210i
$$76$$ 0 0
$$77$$ 170.862i 0.252877i
$$78$$ 0 0
$$79$$ 323.190 0.460275 0.230138 0.973158i $$-0.426082\pi$$
0.230138 + 0.973158i $$0.426082\pi$$
$$80$$ 0 0
$$81$$ 637.533 0.874531
$$82$$ 0 0
$$83$$ 797.471i 1.05462i 0.849672 + 0.527312i $$0.176800\pi$$
−0.849672 + 0.527312i $$0.823200\pi$$
$$84$$ 0 0
$$85$$ − 843.392i − 1.07622i
$$86$$ 0 0
$$87$$ 99.9602 0.123182
$$88$$ 0 0
$$89$$ 866.853 1.03243 0.516215 0.856459i $$-0.327341\pi$$
0.516215 + 0.856459i $$0.327341\pi$$
$$90$$ 0 0
$$91$$ 133.742i 0.154065i
$$92$$ 0 0
$$93$$ − 201.935i − 0.225158i
$$94$$ 0 0
$$95$$ −319.866 −0.345448
$$96$$ 0 0
$$97$$ −936.077 −0.979837 −0.489919 0.871768i $$-0.662974\pi$$
−0.489919 + 0.871768i $$0.662974\pi$$
$$98$$ 0 0
$$99$$ 1653.47i 1.67858i
$$100$$ 0 0
$$101$$ 2.24639i 0.00221311i 0.999999 + 0.00110656i $$0.000352228\pi$$
−0.999999 + 0.00110656i $$0.999648\pi$$
$$102$$ 0 0
$$103$$ 1388.28 1.32807 0.664036 0.747700i $$-0.268842\pi$$
0.664036 + 0.747700i $$0.268842\pi$$
$$104$$ 0 0
$$105$$ 33.2635 0.0309160
$$106$$ 0 0
$$107$$ 1161.81i 1.04969i 0.851198 + 0.524845i $$0.175877\pi$$
−0.851198 + 0.524845i $$0.824123\pi$$
$$108$$ 0 0
$$109$$ 753.489i 0.662121i 0.943610 + 0.331060i $$0.107406\pi$$
−0.943610 + 0.331060i $$0.892594\pi$$
$$110$$ 0 0
$$111$$ −127.237 −0.108800
$$112$$ 0 0
$$113$$ 67.2680 0.0560003 0.0280002 0.999608i $$-0.491086\pi$$
0.0280002 + 0.999608i $$0.491086\pi$$
$$114$$ 0 0
$$115$$ 1622.38i 1.31554i
$$116$$ 0 0
$$117$$ 1294.24i 1.02267i
$$118$$ 0 0
$$119$$ 193.696 0.149211
$$120$$ 0 0
$$121$$ −2758.92 −2.07282
$$122$$ 0 0
$$123$$ − 111.979i − 0.0820876i
$$124$$ 0 0
$$125$$ − 1333.97i − 0.954514i
$$126$$ 0 0
$$127$$ −1903.59 −1.33005 −0.665026 0.746820i $$-0.731579\pi$$
−0.665026 + 0.746820i $$0.731579\pi$$
$$128$$ 0 0
$$129$$ −47.6552 −0.0325257
$$130$$ 0 0
$$131$$ 1298.86i 0.866271i 0.901329 + 0.433136i $$0.142593\pi$$
−0.901329 + 0.433136i $$0.857407\pi$$
$$132$$ 0 0
$$133$$ − 73.4615i − 0.0478941i
$$134$$ 0 0
$$135$$ 658.054 0.419528
$$136$$ 0 0
$$137$$ −477.234 −0.297612 −0.148806 0.988866i $$-0.547543\pi$$
−0.148806 + 0.988866i $$0.547543\pi$$
$$138$$ 0 0
$$139$$ 2140.97i 1.30643i 0.757171 + 0.653217i $$0.226581\pi$$
−0.757171 + 0.653217i $$0.773419\pi$$
$$140$$ 0 0
$$141$$ − 522.440i − 0.312038i
$$142$$ 0 0
$$143$$ −3201.37 −1.87211
$$144$$ 0 0
$$145$$ −1086.53 −0.622285
$$146$$ 0 0
$$147$$ − 359.454i − 0.201682i
$$148$$ 0 0
$$149$$ 530.829i 0.291861i 0.989295 + 0.145930i $$0.0466175\pi$$
−0.989295 + 0.145930i $$0.953382\pi$$
$$150$$ 0 0
$$151$$ 2997.52 1.61546 0.807730 0.589553i $$-0.200696\pi$$
0.807730 + 0.589553i $$0.200696\pi$$
$$152$$ 0 0
$$153$$ 1874.43 0.990451
$$154$$ 0 0
$$155$$ 2194.96i 1.13744i
$$156$$ 0 0
$$157$$ 2134.06i 1.08482i 0.840115 + 0.542409i $$0.182488\pi$$
−0.840115 + 0.542409i $$0.817512\pi$$
$$158$$ 0 0
$$159$$ −226.368 −0.112906
$$160$$ 0 0
$$161$$ −372.601 −0.182392
$$162$$ 0 0
$$163$$ − 2015.52i − 0.968515i −0.874925 0.484258i $$-0.839090\pi$$
0.874925 0.484258i $$-0.160910\pi$$
$$164$$ 0 0
$$165$$ 796.227i 0.375674i
$$166$$ 0 0
$$167$$ 792.415 0.367179 0.183590 0.983003i $$-0.441228\pi$$
0.183590 + 0.983003i $$0.441228\pi$$
$$168$$ 0 0
$$169$$ −308.861 −0.140583
$$170$$ 0 0
$$171$$ − 710.900i − 0.317917i
$$172$$ 0 0
$$173$$ − 1094.03i − 0.480794i −0.970675 0.240397i $$-0.922722\pi$$
0.970675 0.240397i $$-0.0772776\pi$$
$$174$$ 0 0
$$175$$ −27.5984 −0.0119214
$$176$$ 0 0
$$177$$ −430.905 −0.182988
$$178$$ 0 0
$$179$$ − 602.526i − 0.251592i −0.992056 0.125796i $$-0.959852\pi$$
0.992056 0.125796i $$-0.0401484\pi$$
$$180$$ 0 0
$$181$$ − 3702.50i − 1.52047i −0.649649 0.760234i $$-0.725084\pi$$
0.649649 0.760234i $$-0.274916\pi$$
$$182$$ 0 0
$$183$$ −345.194 −0.139440
$$184$$ 0 0
$$185$$ 1383.02 0.549631
$$186$$ 0 0
$$187$$ 4636.50i 1.81312i
$$188$$ 0 0
$$189$$ 151.131i 0.0581649i
$$190$$ 0 0
$$191$$ −3216.39 −1.21848 −0.609240 0.792986i $$-0.708525\pi$$
−0.609240 + 0.792986i $$0.708525\pi$$
$$192$$ 0 0
$$193$$ 2852.57 1.06390 0.531950 0.846776i $$-0.321459\pi$$
0.531950 + 0.846776i $$0.321459\pi$$
$$194$$ 0 0
$$195$$ 623.243i 0.228879i
$$196$$ 0 0
$$197$$ − 2275.50i − 0.822956i −0.911420 0.411478i $$-0.865013\pi$$
0.911420 0.411478i $$-0.134987\pi$$
$$198$$ 0 0
$$199$$ −747.136 −0.266146 −0.133073 0.991106i $$-0.542484\pi$$
−0.133073 + 0.991106i $$0.542484\pi$$
$$200$$ 0 0
$$201$$ −210.612 −0.0739074
$$202$$ 0 0
$$203$$ − 249.536i − 0.0862758i
$$204$$ 0 0
$$205$$ 1217.17i 0.414685i
$$206$$ 0 0
$$207$$ −3605.73 −1.21070
$$208$$ 0 0
$$209$$ 1758.44 0.581981
$$210$$ 0 0
$$211$$ 3149.64i 1.02763i 0.857901 + 0.513816i $$0.171769\pi$$
−0.857901 + 0.513816i $$0.828231\pi$$
$$212$$ 0 0
$$213$$ 485.520i 0.156185i
$$214$$ 0 0
$$215$$ 517.995 0.164311
$$216$$ 0 0
$$217$$ −504.102 −0.157699
$$218$$ 0 0
$$219$$ 277.993i 0.0857763i
$$220$$ 0 0
$$221$$ 3629.20i 1.10464i
$$222$$ 0 0
$$223$$ 358.053 0.107520 0.0537601 0.998554i $$-0.482879\pi$$
0.0537601 + 0.998554i $$0.482879\pi$$
$$224$$ 0 0
$$225$$ −267.075 −0.0791332
$$226$$ 0 0
$$227$$ 4886.67i 1.42881i 0.699733 + 0.714404i $$0.253302\pi$$
−0.699733 + 0.714404i $$0.746698\pi$$
$$228$$ 0 0
$$229$$ 2022.37i 0.583589i 0.956481 + 0.291794i $$0.0942523\pi$$
−0.956481 + 0.291794i $$0.905748\pi$$
$$230$$ 0 0
$$231$$ −182.864 −0.0520847
$$232$$ 0 0
$$233$$ 926.479 0.260496 0.130248 0.991481i $$-0.458423\pi$$
0.130248 + 0.991481i $$0.458423\pi$$
$$234$$ 0 0
$$235$$ 5678.73i 1.57634i
$$236$$ 0 0
$$237$$ 345.892i 0.0948021i
$$238$$ 0 0
$$239$$ −792.472 −0.214480 −0.107240 0.994233i $$-0.534201\pi$$
−0.107240 + 0.994233i $$0.534201\pi$$
$$240$$ 0 0
$$241$$ −1449.01 −0.387299 −0.193650 0.981071i $$-0.562033\pi$$
−0.193650 + 0.981071i $$0.562033\pi$$
$$242$$ 0 0
$$243$$ 2209.63i 0.583325i
$$244$$ 0 0
$$245$$ 3907.13i 1.01885i
$$246$$ 0 0
$$247$$ 1376.42 0.354572
$$248$$ 0 0
$$249$$ −853.487 −0.217219
$$250$$ 0 0
$$251$$ − 5062.94i − 1.27319i −0.771199 0.636594i $$-0.780343\pi$$
0.771199 0.636594i $$-0.219657\pi$$
$$252$$ 0 0
$$253$$ − 8918.93i − 2.21632i
$$254$$ 0 0
$$255$$ 902.634 0.221667
$$256$$ 0 0
$$257$$ −4708.87 −1.14292 −0.571461 0.820629i $$-0.693623\pi$$
−0.571461 + 0.820629i $$0.693623\pi$$
$$258$$ 0 0
$$259$$ 317.629i 0.0762028i
$$260$$ 0 0
$$261$$ − 2414.81i − 0.572692i
$$262$$ 0 0
$$263$$ 2967.82 0.695830 0.347915 0.937526i $$-0.386890\pi$$
0.347915 + 0.937526i $$0.386890\pi$$
$$264$$ 0 0
$$265$$ 2460.53 0.570374
$$266$$ 0 0
$$267$$ 927.743i 0.212648i
$$268$$ 0 0
$$269$$ − 938.519i − 0.212723i −0.994328 0.106362i $$-0.966080\pi$$
0.994328 0.106362i $$-0.0339201\pi$$
$$270$$ 0 0
$$271$$ 8058.74 1.80640 0.903199 0.429223i $$-0.141212\pi$$
0.903199 + 0.429223i $$0.141212\pi$$
$$272$$ 0 0
$$273$$ −143.136 −0.0317326
$$274$$ 0 0
$$275$$ − 660.622i − 0.144862i
$$276$$ 0 0
$$277$$ 682.325i 0.148003i 0.997258 + 0.0740017i $$0.0235770\pi$$
−0.997258 + 0.0740017i $$0.976423\pi$$
$$278$$ 0 0
$$279$$ −4878.29 −1.04679
$$280$$ 0 0
$$281$$ −5899.10 −1.25235 −0.626175 0.779682i $$-0.715381\pi$$
−0.626175 + 0.779682i $$0.715381\pi$$
$$282$$ 0 0
$$283$$ − 961.519i − 0.201966i −0.994888 0.100983i $$-0.967801\pi$$
0.994888 0.100983i $$-0.0321988\pi$$
$$284$$ 0 0
$$285$$ − 342.334i − 0.0711513i
$$286$$ 0 0
$$287$$ −279.538 −0.0574935
$$288$$ 0 0
$$289$$ 343.118 0.0698388
$$290$$ 0 0
$$291$$ − 1001.83i − 0.201815i
$$292$$ 0 0
$$293$$ 5024.51i 1.00183i 0.865498 + 0.500913i $$0.167002\pi$$
−0.865498 + 0.500913i $$0.832998\pi$$
$$294$$ 0 0
$$295$$ 4683.78 0.924407
$$296$$ 0 0
$$297$$ −3617.62 −0.706786
$$298$$ 0 0
$$299$$ − 6981.26i − 1.35029i
$$300$$ 0 0
$$301$$ 118.964i 0.0227807i
$$302$$ 0 0
$$303$$ −2.40419 −0.000455831 0
$$304$$ 0 0
$$305$$ 3752.13 0.704415
$$306$$ 0 0
$$307$$ − 3868.67i − 0.719207i −0.933105 0.359603i $$-0.882912\pi$$
0.933105 0.359603i $$-0.117088\pi$$
$$308$$ 0 0
$$309$$ 1485.80i 0.273541i
$$310$$ 0 0
$$311$$ 5796.70 1.05692 0.528458 0.848960i $$-0.322771\pi$$
0.528458 + 0.848960i $$0.322771\pi$$
$$312$$ 0 0
$$313$$ 8362.62 1.51017 0.755085 0.655627i $$-0.227596\pi$$
0.755085 + 0.655627i $$0.227596\pi$$
$$314$$ 0 0
$$315$$ − 803.569i − 0.143733i
$$316$$ 0 0
$$317$$ 487.064i 0.0862973i 0.999069 + 0.0431487i $$0.0137389\pi$$
−0.999069 + 0.0431487i $$0.986261\pi$$
$$318$$ 0 0
$$319$$ 5973.13 1.04837
$$320$$ 0 0
$$321$$ −1243.42 −0.216203
$$322$$ 0 0
$$323$$ − 1993.44i − 0.343400i
$$324$$ 0 0
$$325$$ − 517.099i − 0.0882569i
$$326$$ 0 0
$$327$$ −806.416 −0.136376
$$328$$ 0 0
$$329$$ −1304.20 −0.218549
$$330$$ 0 0
$$331$$ − 3801.21i − 0.631218i −0.948889 0.315609i $$-0.897791\pi$$
0.948889 0.315609i $$-0.102209\pi$$
$$332$$ 0 0
$$333$$ 3073.76i 0.505828i
$$334$$ 0 0
$$335$$ 2289.27 0.373361
$$336$$ 0 0
$$337$$ 1795.31 0.290199 0.145099 0.989417i $$-0.453650\pi$$
0.145099 + 0.989417i $$0.453650\pi$$
$$338$$ 0 0
$$339$$ 71.9931i 0.0115343i
$$340$$ 0 0
$$341$$ − 12066.7i − 1.91627i
$$342$$ 0 0
$$343$$ −1813.72 −0.285515
$$344$$ 0 0
$$345$$ −1736.34 −0.270960
$$346$$ 0 0
$$347$$ − 2782.23i − 0.430426i −0.976567 0.215213i $$-0.930955\pi$$
0.976567 0.215213i $$-0.0690445\pi$$
$$348$$ 0 0
$$349$$ 10413.4i 1.59718i 0.601876 + 0.798589i $$0.294420\pi$$
−0.601876 + 0.798589i $$0.705580\pi$$
$$350$$ 0 0
$$351$$ −2831.68 −0.430609
$$352$$ 0 0
$$353$$ 10644.3 1.60493 0.802466 0.596698i $$-0.203521\pi$$
0.802466 + 0.596698i $$0.203521\pi$$
$$354$$ 0 0
$$355$$ − 5277.43i − 0.789005i
$$356$$ 0 0
$$357$$ 207.302i 0.0307327i
$$358$$ 0 0
$$359$$ −7459.42 −1.09664 −0.548319 0.836269i $$-0.684732\pi$$
−0.548319 + 0.836269i $$0.684732\pi$$
$$360$$ 0 0
$$361$$ 6102.96 0.889775
$$362$$ 0 0
$$363$$ − 2952.72i − 0.426935i
$$364$$ 0 0
$$365$$ − 3021.68i − 0.433320i
$$366$$ 0 0
$$367$$ −6251.35 −0.889149 −0.444574 0.895742i $$-0.646645\pi$$
−0.444574 + 0.895742i $$0.646645\pi$$
$$368$$ 0 0
$$369$$ −2705.14 −0.381637
$$370$$ 0 0
$$371$$ 565.094i 0.0790787i
$$372$$ 0 0
$$373$$ 12603.3i 1.74953i 0.484552 + 0.874763i $$0.338983\pi$$
−0.484552 + 0.874763i $$0.661017\pi$$
$$374$$ 0 0
$$375$$ 1427.68 0.196600
$$376$$ 0 0
$$377$$ 4675.45 0.638721
$$378$$ 0 0
$$379$$ 1674.47i 0.226943i 0.993541 + 0.113472i $$0.0361971\pi$$
−0.993541 + 0.113472i $$0.963803\pi$$
$$380$$ 0 0
$$381$$ − 2037.31i − 0.273949i
$$382$$ 0 0
$$383$$ 2880.38 0.384283 0.192142 0.981367i $$-0.438457\pi$$
0.192142 + 0.981367i $$0.438457\pi$$
$$384$$ 0 0
$$385$$ 1987.66 0.263119
$$386$$ 0 0
$$387$$ 1151.24i 0.151217i
$$388$$ 0 0
$$389$$ − 13073.3i − 1.70397i −0.523567 0.851984i $$-0.675399\pi$$
0.523567 0.851984i $$-0.324601\pi$$
$$390$$ 0 0
$$391$$ −10110.9 −1.30774
$$392$$ 0 0
$$393$$ −1390.09 −0.178424
$$394$$ 0 0
$$395$$ − 3759.72i − 0.478916i
$$396$$ 0 0
$$397$$ − 6021.44i − 0.761228i −0.924734 0.380614i $$-0.875713\pi$$
0.924734 0.380614i $$-0.124287\pi$$
$$398$$ 0 0
$$399$$ 78.6216 0.00986467
$$400$$ 0 0
$$401$$ −12722.6 −1.58437 −0.792187 0.610278i $$-0.791058\pi$$
−0.792187 + 0.610278i $$0.791058\pi$$
$$402$$ 0 0
$$403$$ − 9445.14i − 1.16748i
$$404$$ 0 0
$$405$$ − 7416.51i − 0.909949i
$$406$$ 0 0
$$407$$ −7603.08 −0.925972
$$408$$ 0 0
$$409$$ 232.991 0.0281678 0.0140839 0.999901i $$-0.495517\pi$$
0.0140839 + 0.999901i $$0.495517\pi$$
$$410$$ 0 0
$$411$$ − 510.756i − 0.0612987i
$$412$$ 0 0
$$413$$ 1075.69i 0.128163i
$$414$$ 0 0
$$415$$ 9277.08 1.09734
$$416$$ 0 0
$$417$$ −2291.35 −0.269084
$$418$$ 0 0
$$419$$ − 8663.03i − 1.01006i −0.863101 0.505032i $$-0.831481\pi$$
0.863101 0.505032i $$-0.168519\pi$$
$$420$$ 0 0
$$421$$ 11749.9i 1.36023i 0.733107 + 0.680113i $$0.238069\pi$$
−0.733107 + 0.680113i $$0.761931\pi$$
$$422$$ 0 0
$$423$$ −12620.9 −1.45071
$$424$$ 0 0
$$425$$ −748.907 −0.0854760
$$426$$ 0 0
$$427$$ 861.727i 0.0976626i
$$428$$ 0 0
$$429$$ − 3426.24i − 0.385596i
$$430$$ 0 0
$$431$$ 8737.57 0.976506 0.488253 0.872702i $$-0.337634\pi$$
0.488253 + 0.872702i $$0.337634\pi$$
$$432$$ 0 0
$$433$$ 11627.5 1.29049 0.645247 0.763974i $$-0.276755\pi$$
0.645247 + 0.763974i $$0.276755\pi$$
$$434$$ 0 0
$$435$$ − 1162.85i − 0.128171i
$$436$$ 0 0
$$437$$ 3834.66i 0.419763i
$$438$$ 0 0
$$439$$ −17631.8 −1.91690 −0.958450 0.285261i $$-0.907920\pi$$
−0.958450 + 0.285261i $$0.907920\pi$$
$$440$$ 0 0
$$441$$ −8683.57 −0.937649
$$442$$ 0 0
$$443$$ − 6434.41i − 0.690085i −0.938587 0.345043i $$-0.887864\pi$$
0.938587 0.345043i $$-0.112136\pi$$
$$444$$ 0 0
$$445$$ − 10084.2i − 1.07424i
$$446$$ 0 0
$$447$$ −568.116 −0.0601140
$$448$$ 0 0
$$449$$ −12926.5 −1.35867 −0.679334 0.733830i $$-0.737731\pi$$
−0.679334 + 0.733830i $$0.737731\pi$$
$$450$$ 0 0
$$451$$ − 6691.30i − 0.698627i
$$452$$ 0 0
$$453$$ 3208.07i 0.332734i
$$454$$ 0 0
$$455$$ 1555.84 0.160305
$$456$$ 0 0
$$457$$ −9320.32 −0.954018 −0.477009 0.878898i $$-0.658279\pi$$
−0.477009 + 0.878898i $$0.658279\pi$$
$$458$$ 0 0
$$459$$ 4101.07i 0.417041i
$$460$$ 0 0
$$461$$ 18222.1i 1.84098i 0.390771 + 0.920488i $$0.372208\pi$$
−0.390771 + 0.920488i $$0.627792\pi$$
$$462$$ 0 0
$$463$$ 7038.37 0.706482 0.353241 0.935532i $$-0.385080\pi$$
0.353241 + 0.935532i $$0.385080\pi$$
$$464$$ 0 0
$$465$$ −2349.14 −0.234277
$$466$$ 0 0
$$467$$ − 8487.77i − 0.841043i −0.907283 0.420522i $$-0.861847\pi$$
0.907283 0.420522i $$-0.138153\pi$$
$$468$$ 0 0
$$469$$ 525.761i 0.0517642i
$$470$$ 0 0
$$471$$ −2283.96 −0.223438
$$472$$ 0 0
$$473$$ −2847.65 −0.276818
$$474$$ 0 0
$$475$$ 284.031i 0.0274363i
$$476$$ 0 0
$$477$$ 5468.51i 0.524918i
$$478$$ 0 0
$$479$$ 587.317 0.0560234 0.0280117 0.999608i $$-0.491082\pi$$
0.0280117 + 0.999608i $$0.491082\pi$$
$$480$$ 0 0
$$481$$ −5951.28 −0.564148
$$482$$ 0 0
$$483$$ − 398.773i − 0.0375669i
$$484$$ 0 0
$$485$$ 10889.5i 1.01952i
$$486$$ 0 0
$$487$$ 8366.45 0.778481 0.389240 0.921136i $$-0.372738\pi$$
0.389240 + 0.921136i $$0.372738\pi$$
$$488$$ 0 0
$$489$$ 2157.10 0.199483
$$490$$ 0 0
$$491$$ − 2162.76i − 0.198786i −0.995048 0.0993929i $$-0.968310\pi$$
0.995048 0.0993929i $$-0.0316901\pi$$
$$492$$ 0 0
$$493$$ − 6771.38i − 0.618596i
$$494$$ 0 0
$$495$$ 19235.0 1.74656
$$496$$ 0 0
$$497$$ 1212.03 0.109390
$$498$$ 0 0
$$499$$ − 16071.8i − 1.44183i −0.693026 0.720913i $$-0.743723\pi$$
0.693026 0.720913i $$-0.256277\pi$$
$$500$$ 0 0
$$501$$ 848.077i 0.0756273i
$$502$$ 0 0
$$503$$ −12570.2 −1.11427 −0.557137 0.830421i $$-0.688100\pi$$
−0.557137 + 0.830421i $$0.688100\pi$$
$$504$$ 0 0
$$505$$ 26.1326 0.00230274
$$506$$ 0 0
$$507$$ − 330.556i − 0.0289556i
$$508$$ 0 0
$$509$$ − 16801.4i − 1.46308i −0.681797 0.731542i $$-0.738801\pi$$
0.681797 0.731542i $$-0.261199\pi$$
$$510$$ 0 0
$$511$$ 693.968 0.0600770
$$512$$ 0 0
$$513$$ 1555.38 0.133863
$$514$$ 0 0
$$515$$ − 16150.1i − 1.38186i
$$516$$ 0 0
$$517$$ − 31218.5i − 2.65568i
$$518$$ 0 0
$$519$$ 1170.87 0.0990283
$$520$$ 0 0
$$521$$ 6612.98 0.556085 0.278042 0.960569i $$-0.410314\pi$$
0.278042 + 0.960569i $$0.410314\pi$$
$$522$$ 0 0
$$523$$ − 7253.92i − 0.606485i −0.952913 0.303243i $$-0.901931\pi$$
0.952913 0.303243i $$-0.0980693\pi$$
$$524$$ 0 0
$$525$$ − 29.5370i − 0.00245543i
$$526$$ 0 0
$$527$$ −13679.3 −1.13070
$$528$$ 0 0
$$529$$ 7282.60 0.598553
$$530$$ 0 0
$$531$$ 10409.7i 0.850737i
$$532$$ 0 0
$$533$$ − 5237.59i − 0.425638i
$$534$$ 0 0
$$535$$ 13515.5 1.09220
$$536$$ 0 0
$$537$$ 644.849 0.0518199
$$538$$ 0 0
$$539$$ − 21479.2i − 1.71647i
$$540$$ 0 0
$$541$$ 15511.9i 1.23273i 0.787461 + 0.616365i $$0.211395\pi$$
−0.787461 + 0.616365i $$0.788605\pi$$
$$542$$ 0 0
$$543$$ 3962.57 0.313168
$$544$$ 0 0
$$545$$ 8765.44 0.688936
$$546$$ 0 0
$$547$$ 18510.4i 1.44689i 0.690382 + 0.723445i $$0.257442\pi$$
−0.690382 + 0.723445i $$0.742558\pi$$
$$548$$ 0 0
$$549$$ 8339.09i 0.648277i
$$550$$ 0 0
$$551$$ −2568.12 −0.198558
$$552$$ 0 0
$$553$$ 863.469 0.0663986
$$554$$ 0 0
$$555$$ 1480.17i 0.113207i
$$556$$ 0 0
$$557$$ 7141.59i 0.543266i 0.962401 + 0.271633i $$0.0875637\pi$$
−0.962401 + 0.271633i $$0.912436\pi$$
$$558$$ 0 0
$$559$$ −2228.98 −0.168651
$$560$$ 0 0
$$561$$ −4962.18 −0.373446
$$562$$ 0 0
$$563$$ 4594.86i 0.343961i 0.985100 + 0.171981i $$0.0550167\pi$$
−0.985100 + 0.171981i $$0.944983\pi$$
$$564$$ 0 0
$$565$$ − 782.538i − 0.0582683i
$$566$$ 0 0
$$567$$ 1703.30 0.126159
$$568$$ 0 0
$$569$$ 2806.05 0.206741 0.103371 0.994643i $$-0.467037\pi$$
0.103371 + 0.994643i $$0.467037\pi$$
$$570$$ 0 0
$$571$$ − 17025.4i − 1.24779i −0.781506 0.623897i $$-0.785548\pi$$
0.781506 0.623897i $$-0.214452\pi$$
$$572$$ 0 0
$$573$$ − 3442.31i − 0.250968i
$$574$$ 0 0
$$575$$ 1440.62 0.104484
$$576$$ 0 0
$$577$$ 7206.84 0.519973 0.259987 0.965612i $$-0.416282\pi$$
0.259987 + 0.965612i $$0.416282\pi$$
$$578$$ 0 0
$$579$$ 3052.95i 0.219130i
$$580$$ 0 0
$$581$$ 2130.61i 0.152138i
$$582$$ 0 0
$$583$$ −13526.6 −0.960918
$$584$$ 0 0
$$585$$ 15056.1 1.06409
$$586$$ 0 0
$$587$$ − 14675.2i − 1.03188i −0.856626 0.515938i $$-0.827443\pi$$
0.856626 0.515938i $$-0.172557\pi$$
$$588$$ 0 0
$$589$$ 5188.01i 0.362934i
$$590$$ 0 0
$$591$$ 2435.33 0.169503
$$592$$ 0 0
$$593$$ 4758.60 0.329531 0.164766 0.986333i $$-0.447313\pi$$
0.164766 + 0.986333i $$0.447313\pi$$
$$594$$ 0 0
$$595$$ − 2253.29i − 0.155254i
$$596$$ 0 0
$$597$$ − 799.617i − 0.0548176i
$$598$$ 0 0
$$599$$ 14256.4 0.972455 0.486227 0.873832i $$-0.338373\pi$$
0.486227 + 0.873832i $$0.338373\pi$$
$$600$$ 0 0
$$601$$ 10385.2 0.704862 0.352431 0.935838i $$-0.385355\pi$$
0.352431 + 0.935838i $$0.385355\pi$$
$$602$$ 0 0
$$603$$ 5087.89i 0.343606i
$$604$$ 0 0
$$605$$ 32094.9i 2.15677i
$$606$$ 0 0
$$607$$ 16243.6 1.08618 0.543088 0.839676i $$-0.317255\pi$$
0.543088 + 0.839676i $$0.317255\pi$$
$$608$$ 0 0
$$609$$ 267.064 0.0177701
$$610$$ 0 0
$$611$$ − 24436.1i − 1.61797i
$$612$$ 0 0
$$613$$ 707.817i 0.0466369i 0.999728 + 0.0233185i $$0.00742317\pi$$
−0.999728 + 0.0233185i $$0.992577\pi$$
$$614$$ 0 0
$$615$$ −1302.66 −0.0854121
$$616$$ 0 0
$$617$$ −11575.9 −0.755316 −0.377658 0.925945i $$-0.623271\pi$$
−0.377658 + 0.925945i $$0.623271\pi$$
$$618$$ 0 0
$$619$$ − 25959.4i − 1.68562i −0.538214 0.842808i $$-0.680901\pi$$
0.538214 0.842808i $$-0.319099\pi$$
$$620$$ 0 0
$$621$$ − 7888.97i − 0.509780i
$$622$$ 0 0
$$623$$ 2315.98 0.148937
$$624$$ 0 0
$$625$$ −16809.5 −1.07581
$$626$$ 0 0
$$627$$ 1881.96i 0.119870i
$$628$$ 0 0
$$629$$ 8619.15i 0.546372i
$$630$$ 0 0
$$631$$ −10224.8 −0.645079 −0.322539 0.946556i $$-0.604537\pi$$
−0.322539 + 0.946556i $$0.604537\pi$$
$$632$$ 0 0
$$633$$ −3370.88 −0.211660
$$634$$ 0 0
$$635$$ 22144.8i 1.38392i
$$636$$ 0 0
$$637$$ − 16812.8i − 1.04576i
$$638$$ 0 0
$$639$$ 11729.0 0.726125
$$640$$ 0 0
$$641$$ 19804.4 1.22032 0.610162 0.792277i $$-0.291104\pi$$
0.610162 + 0.792277i $$0.291104\pi$$
$$642$$ 0 0
$$643$$ 22175.9i 1.36008i 0.733174 + 0.680041i $$0.238038\pi$$
−0.733174 + 0.680041i $$0.761962\pi$$
$$644$$ 0 0
$$645$$ 554.380i 0.0338429i
$$646$$ 0 0
$$647$$ 9232.26 0.560985 0.280493 0.959856i $$-0.409502\pi$$
0.280493 + 0.959856i $$0.409502\pi$$
$$648$$ 0 0
$$649$$ −25748.8 −1.55736
$$650$$ 0 0
$$651$$ − 539.511i − 0.0324810i
$$652$$ 0 0
$$653$$ − 27857.1i − 1.66942i −0.550690 0.834710i $$-0.685635\pi$$
0.550690 0.834710i $$-0.314365\pi$$
$$654$$ 0 0
$$655$$ 15109.8 0.901355
$$656$$ 0 0
$$657$$ 6715.66 0.398786
$$658$$ 0 0
$$659$$ − 5498.55i − 0.325028i −0.986706 0.162514i $$-0.948040\pi$$
0.986706 0.162514i $$-0.0519602\pi$$
$$660$$ 0 0
$$661$$ − 11469.6i − 0.674908i −0.941342 0.337454i $$-0.890434\pi$$
0.941342 0.337454i $$-0.109566\pi$$
$$662$$ 0 0
$$663$$ −3884.13 −0.227522
$$664$$ 0 0
$$665$$ −854.587 −0.0498338
$$666$$ 0 0
$$667$$ 13025.7i 0.756156i
$$668$$ 0 0
$$669$$ 383.204i 0.0221458i
$$670$$ 0 0
$$671$$ −20627.1 −1.18674
$$672$$ 0 0
$$673$$ −28428.2 −1.62827 −0.814135 0.580676i $$-0.802788\pi$$
−0.814135 + 0.580676i $$0.802788\pi$$
$$674$$ 0 0
$$675$$ − 584.333i − 0.0333200i
$$676$$ 0 0
$$677$$ 23995.5i 1.36222i 0.732181 + 0.681110i $$0.238502\pi$$
−0.732181 + 0.681110i $$0.761498\pi$$
$$678$$ 0 0
$$679$$ −2500.92 −0.141350
$$680$$ 0 0
$$681$$ −5229.92 −0.294289
$$682$$ 0 0
$$683$$ 13506.0i 0.756650i 0.925673 + 0.378325i $$0.123500\pi$$
−0.925673 + 0.378325i $$0.876500\pi$$
$$684$$ 0 0
$$685$$ 5551.73i 0.309665i
$$686$$ 0 0
$$687$$ −2164.42 −0.120201
$$688$$ 0 0
$$689$$ −10587.9 −0.585439
$$690$$ 0 0
$$691$$ 29500.1i 1.62408i 0.583605 + 0.812038i $$0.301642\pi$$
−0.583605 + 0.812038i $$0.698358\pi$$
$$692$$ 0 0
$$693$$ 4417.57i 0.242150i
$$694$$ 0 0
$$695$$ 24906.1 1.35934
$$696$$ 0 0
$$697$$ −7585.52 −0.412227
$$698$$ 0 0
$$699$$ 991.557i 0.0536540i
$$700$$ 0 0
$$701$$ − 33227.5i − 1.79028i −0.445787 0.895139i $$-0.647076\pi$$
0.445787 0.895139i $$-0.352924\pi$$
$$702$$ 0 0
$$703$$ 3268.91 0.175376
$$704$$ 0 0
$$705$$ −6077.62 −0.324676
$$706$$ 0 0
$$707$$ 6.00170i 0 0.000319261i
$$708$$ 0 0
$$709$$ − 6448.04i − 0.341553i −0.985310 0.170777i $$-0.945372\pi$$
0.985310 0.170777i $$-0.0546277\pi$$
$$710$$ 0 0
$$711$$ 8355.95 0.440749
$$712$$ 0 0
$$713$$ 26313.9 1.38214
$$714$$ 0 0
$$715$$ 37242.0i 1.94793i
$$716$$ 0 0
$$717$$ − 848.138i − 0.0441761i
$$718$$ 0 0
$$719$$ −6494.67 −0.336871 −0.168436 0.985713i $$-0.553872\pi$$
−0.168436 + 0.985713i $$0.553872\pi$$
$$720$$ 0 0
$$721$$ 3709.08 0.191586
$$722$$ 0 0
$$723$$ − 1550.80i − 0.0797714i
$$724$$ 0 0
$$725$$ 964.806i 0.0494234i
$$726$$ 0 0
$$727$$ −24866.4 −1.26856 −0.634280 0.773103i $$-0.718704\pi$$
−0.634280 + 0.773103i $$0.718704\pi$$
$$728$$ 0 0
$$729$$ 14848.5 0.754384
$$730$$ 0 0
$$731$$ 3228.20i 0.163337i
$$732$$ 0 0
$$733$$ 21092.1i 1.06283i 0.847112 + 0.531414i $$0.178339\pi$$
−0.847112 + 0.531414i $$0.821661\pi$$
$$734$$ 0 0
$$735$$ −4181.58 −0.209850
$$736$$ 0 0
$$737$$ −12585.1 −0.629008
$$738$$ 0 0
$$739$$ 11952.7i 0.594977i 0.954725 + 0.297489i $$0.0961491\pi$$
−0.954725 + 0.297489i $$0.903851\pi$$
$$740$$ 0 0
$$741$$ 1473.10i 0.0730305i
$$742$$ 0 0
$$743$$ 5622.43 0.277614 0.138807 0.990319i $$-0.455673\pi$$
0.138807 + 0.990319i $$0.455673\pi$$
$$744$$ 0 0
$$745$$ 6175.21 0.303681
$$746$$ 0 0
$$747$$ 20618.3i 1.00988i
$$748$$ 0 0
$$749$$ 3104.02i 0.151427i
$$750$$ 0 0
$$751$$ −32314.9 −1.57016 −0.785079 0.619396i $$-0.787378\pi$$
−0.785079 + 0.619396i $$0.787378\pi$$
$$752$$ 0 0
$$753$$ 5418.58 0.262236
$$754$$ 0 0
$$755$$ − 34870.5i − 1.68089i
$$756$$ 0 0
$$757$$ 17950.4i 0.861847i 0.902389 + 0.430923i $$0.141812\pi$$
−0.902389 + 0.430923i $$0.858188\pi$$
$$758$$ 0 0
$$759$$ 9545.42 0.456491
$$760$$ 0 0
$$761$$ −13108.2 −0.624404 −0.312202 0.950016i $$-0.601067\pi$$
−0.312202 + 0.950016i $$0.601067\pi$$
$$762$$ 0 0
$$763$$ 2013.10i 0.0955166i
$$764$$ 0 0
$$765$$ − 21805.5i − 1.03056i
$$766$$ 0 0
$$767$$ −20154.8 −0.948822
$$768$$ 0 0
$$769$$ 23661.2 1.10955 0.554776 0.832000i $$-0.312804\pi$$
0.554776 + 0.832000i $$0.312804\pi$$
$$770$$ 0 0
$$771$$ − 5039.63i − 0.235406i
$$772$$ 0 0
$$773$$ − 30222.4i − 1.40624i −0.711071 0.703120i $$-0.751790\pi$$
0.711071 0.703120i $$-0.248210\pi$$
$$774$$ 0 0
$$775$$ 1949.06 0.0903384
$$776$$ 0 0
$$777$$ −339.940 −0.0156954
$$778$$ 0 0
$$779$$ 2876.89i 0.132318i
$$780$$ 0 0
$$781$$ 29012.3i 1.32925i
$$782$$ 0 0
$$783$$ 5283.35 0.241139
$$784$$ 0 0
$$785$$ 24825.8 1.12875
$$786$$ 0 0
$$787$$ 29544.2i 1.33817i 0.743188 + 0.669083i $$0.233313\pi$$
−0.743188 + 0.669083i $$0.766687\pi$$
$$788$$ 0 0
$$789$$ 3176.28i 0.143319i
$$790$$ 0 0
$$791$$ 179.720 0.00807853
$$792$$ 0 0
$$793$$ −16145.8 −0.723020
$$794$$ 0 0
$$795$$ 2633.36i 0.117479i
$$796$$ 0 0
$$797$$ 9665.91i 0.429591i 0.976659 + 0.214795i $$0.0689085\pi$$
−0.976659 + 0.214795i $$0.931092\pi$$
$$798$$ 0 0
$$799$$ −35390.5 −1.56699
$$800$$ 0 0
$$801$$ 22412.1 0.988631
$$802$$ 0 0
$$803$$ 16611.5i 0.730021i
$$804$$ 0 0
$$805$$ 4334.52i 0.189778i
$$806$$ 0 0
$$807$$ 1004.44 0.0438142
$$808$$ 0 0
$$809$$ 15807.0 0.686952 0.343476 0.939162i $$-0.388396\pi$$
0.343476 + 0.939162i $$0.388396\pi$$
$$810$$ 0 0
$$811$$ 16295.5i 0.705565i 0.935705 + 0.352783i $$0.114764\pi$$
−0.935705 + 0.352783i $$0.885236\pi$$
$$812$$ 0 0
$$813$$ 8624.81i 0.372061i
$$814$$ 0 0
$$815$$ −23446.9 −1.00774
$$816$$ 0 0
$$817$$ 1224.33 0.0524284
$$818$$ 0 0
$$819$$ 3457.84i 0.147529i
$$820$$ 0 0
$$821$$ 436.743i 0.0185657i 0.999957 + 0.00928286i $$0.00295487\pi$$
−0.999957 + 0.00928286i $$0.997045\pi$$
$$822$$ 0 0
$$823$$ −5633.49 −0.238604 −0.119302 0.992858i $$-0.538066\pi$$
−0.119302 + 0.992858i $$0.538066\pi$$
$$824$$ 0 0
$$825$$ 707.026 0.0298369
$$826$$ 0 0
$$827$$ − 22394.7i − 0.941645i −0.882228 0.470822i $$-0.843957\pi$$
0.882228 0.470822i $$-0.156043\pi$$
$$828$$ 0 0
$$829$$ 8768.39i 0.367357i 0.982986 + 0.183678i $$0.0588005\pi$$
−0.982986 + 0.183678i $$0.941199\pi$$
$$830$$ 0 0
$$831$$ −730.253 −0.0304840
$$832$$ 0 0
$$833$$ −24349.7 −1.01281
$$834$$ 0 0
$$835$$ − 9218.27i − 0.382050i
$$836$$ 0 0
$$837$$ − 10673.2i − 0.440764i
$$838$$ 0 0
$$839$$ 30644.3 1.26098 0.630488 0.776199i $$-0.282855\pi$$
0.630488 + 0.776199i $$0.282855\pi$$
$$840$$ 0 0
$$841$$ 15665.5 0.642319
$$842$$ 0 0
$$843$$ − 6313.47i − 0.257945i
$$844$$ 0 0
$$845$$ 3593.02i 0.146277i
$$846$$ 0 0
$$847$$ −7371.03 −0.299022
$$848$$ 0 0
$$849$$ 1029.06 0.0415986
$$850$$ 0 0
$$851$$ − 16580.1i − 0.667871i
$$852$$ 0 0
$$853$$ − 43559.2i − 1.74846i −0.485508 0.874232i $$-0.661365\pi$$
0.485508 0.874232i $$-0.338635\pi$$
$$854$$ 0 0
$$855$$ −8270.00 −0.330793
$$856$$ 0 0
$$857$$ 41788.5 1.66566 0.832828 0.553533i $$-0.186721\pi$$
0.832828 + 0.553533i $$0.186721\pi$$
$$858$$ 0 0
$$859$$ 16849.6i 0.669267i 0.942348 + 0.334633i $$0.108612\pi$$
−0.942348 + 0.334633i $$0.891388\pi$$
$$860$$ 0 0
$$861$$ − 299.174i − 0.0118418i
$$862$$ 0 0
$$863$$ −27636.7 −1.09011 −0.545054 0.838401i $$-0.683491\pi$$
−0.545054 + 0.838401i $$0.683491\pi$$
$$864$$ 0 0
$$865$$ −12727.0 −0.500266
$$866$$ 0 0
$$867$$ 367.220i 0.0143846i
$$868$$ 0 0
$$869$$ 20668.8i 0.806837i
$$870$$ 0 0
$$871$$ −9850.95 −0.383223
$$872$$ 0 0
$$873$$ −24201.9 −0.938270
$$874$$ 0 0
$$875$$ − 3563.98i − 0.137697i
$$876$$ 0 0
$$877$$ − 16855.0i − 0.648977i −0.945890 0.324488i $$-0.894808\pi$$
0.945890 0.324488i $$-0.105192\pi$$
$$878$$ 0 0
$$879$$ −5377.45 −0.206344
$$880$$ 0 0
$$881$$ 13330.0 0.509759 0.254880 0.966973i $$-0.417964\pi$$
0.254880 + 0.966973i $$0.417964\pi$$
$$882$$ 0 0
$$883$$ 35598.7i 1.35673i 0.734725 + 0.678365i $$0.237311\pi$$
−0.734725 + 0.678365i $$0.762689\pi$$
$$884$$ 0 0
$$885$$ 5012.78i 0.190399i
$$886$$ 0 0
$$887$$ 48821.3 1.84810 0.924048 0.382278i $$-0.124860\pi$$
0.924048 + 0.382278i $$0.124860\pi$$
$$888$$ 0 0
$$889$$ −5085.84 −0.191871
$$890$$ 0 0
$$891$$ 40771.8i 1.53301i
$$892$$ 0 0
$$893$$ 13422.2i 0.502977i
$$894$$ 0 0
$$895$$ −7009.27 −0.261781
$$896$$ 0 0
$$897$$ 7471.64 0.278117
$$898$$ 0 0
$$899$$ 17622.8i 0.653785i
$$900$$ 0 0
$$901$$ 15334.3i 0.566992i
$$902$$ 0 0
$$903$$ −127.321 −0.00469210
$$904$$ 0 0
$$905$$ −43071.7 −1.58205
$$906$$ 0 0
$$907$$ − 7523.90i − 0.275443i −0.990471 0.137722i $$-0.956022\pi$$
0.990471 0.137722i $$-0.0439779\pi$$
$$908$$ 0 0
$$909$$ 58.0796i 0.00211923i
$$910$$ 0 0
$$911$$ −26016.0 −0.946155 −0.473077 0.881021i $$-0.656857\pi$$
−0.473077 + 0.881021i $$0.656857\pi$$
$$912$$ 0 0
$$913$$ −51000.2 −1.84870
$$914$$ 0 0
$$915$$ 4015.69i 0.145087i
$$916$$ 0 0
$$917$$ 3470.16i 0.124967i
$$918$$ 0 0
$$919$$ 24082.1 0.864413 0.432206 0.901775i $$-0.357735\pi$$
0.432206 + 0.901775i $$0.357735\pi$$
$$920$$ 0 0
$$921$$ 4140.41 0.148134
$$922$$ 0 0
$$923$$ 22709.3i 0.809844i
$$924$$ 0 0
$$925$$ − 1228.08i − 0.0436530i
$$926$$ 0 0
$$927$$ 35893.5 1.27173
$$928$$ 0 0
$$929$$ 9324.93 0.329323 0.164661 0.986350i $$-0.447347\pi$$
0.164661 + 0.986350i $$0.447347\pi$$
$$930$$ 0 0
$$931$$ 9234.89i 0.325093i
$$932$$ 0 0
$$933$$ 6203.87i 0.217691i
$$934$$ 0 0
$$935$$ 53937.0 1.88656
$$936$$ 0 0
$$937$$ 15535.2 0.541636 0.270818 0.962631i $$-0.412706\pi$$
0.270818 + 0.962631i $$0.412706\pi$$
$$938$$ 0 0
$$939$$ 8950.03i 0.311047i
$$940$$ 0 0
$$941$$ − 48118.3i − 1.66696i −0.552547 0.833482i $$-0.686344\pi$$
0.552547 0.833482i $$-0.313656\pi$$
$$942$$ 0 0
$$943$$ 14591.8 0.503896
$$944$$ 0 0
$$945$$ 1758.13 0.0605205
$$946$$ 0 0
$$947$$ 5396.67i 0.185183i 0.995704 + 0.0925915i $$0.0295151\pi$$
−0.995704 + 0.0925915i $$0.970485\pi$$
$$948$$ 0 0
$$949$$ 13002.6i 0.444765i
$$950$$ 0 0
$$951$$ −521.277 −0.0177745
$$952$$ 0 0
$$953$$ −21759.7 −0.739629 −0.369815 0.929106i $$-0.620579\pi$$
−0.369815 + 0.929106i $$0.620579\pi$$
$$954$$ 0 0
$$955$$ 37416.7i 1.26783i
$$956$$ 0 0
$$957$$ 6392.70i 0.215932i
$$958$$ 0 0
$$959$$ −1275.03 −0.0429331
$$960$$ 0 0
$$961$$ 5809.78 0.195018
$$962$$ 0 0
$$963$$ 30038.2i 1.00516i
$$964$$ 0 0
$$965$$ − 33184.4i − 1.10699i
$$966$$ 0 0
$$967$$ −19338.5 −0.643108 −0.321554 0.946891i $$-0.604205\pi$$
−0.321554 + 0.946891i $$0.604205\pi$$
$$968$$ 0 0
$$969$$ 2133.47 0.0707294
$$970$$ 0 0
$$971$$ − 5918.37i − 0.195602i −0.995206 0.0978009i $$-0.968819\pi$$
0.995206 0.0978009i $$-0.0311809\pi$$
$$972$$ 0 0
$$973$$ 5720.03i 0.188464i
$$974$$ 0 0
$$975$$ 553.421 0.0181781
$$976$$ 0 0
$$977$$ −17841.9 −0.584249 −0.292125 0.956380i $$-0.594362\pi$$
−0.292125 + 0.956380i $$0.594362\pi$$
$$978$$ 0 0
$$979$$ 55437.4i 1.80979i
$$980$$ 0 0
$$981$$ 19481.1i 0.634032i
$$982$$ 0 0
$$983$$ 61512.0 1.99586 0.997929 0.0643304i $$-0.0204912\pi$$
0.997929 + 0.0643304i $$0.0204912\pi$$
$$984$$ 0 0
$$985$$ −26471.2 −0.856286
$$986$$ 0 0
$$987$$ − 1395.81i − 0.0450142i
$$988$$ 0 0
$$989$$ − 6209.89i − 0.199659i
$$990$$ 0 0
$$991$$ −1827.73 −0.0585870 −0.0292935 0.999571i $$-0.509326\pi$$
−0.0292935 + 0.999571i $$0.509326\pi$$
$$992$$ 0 0
$$993$$ 4068.21 0.130011
$$994$$ 0 0
$$995$$ 8691.53i 0.276925i
$$996$$ 0 0
$$997$$ 35379.4i 1.12385i 0.827188 + 0.561925i $$0.189939\pi$$
−0.827188 + 0.561925i $$0.810061\pi$$
$$998$$ 0 0
$$999$$ −6725.07 −0.212985
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 1024.4.b.k.513.6 10
4.3 odd 2 1024.4.b.j.513.5 10
8.3 odd 2 1024.4.b.j.513.6 10
8.5 even 2 inner 1024.4.b.k.513.5 10
16.3 odd 4 1024.4.a.n.1.6 10
16.5 even 4 1024.4.a.m.1.6 10
16.11 odd 4 1024.4.a.n.1.5 10
16.13 even 4 1024.4.a.m.1.5 10
32.3 odd 8 16.4.e.a.5.3 10
32.5 even 8 64.4.e.a.17.3 10
32.11 odd 8 128.4.e.b.33.3 10
32.13 even 8 128.4.e.a.97.3 10
32.19 odd 8 128.4.e.b.97.3 10
32.21 even 8 128.4.e.a.33.3 10
32.27 odd 8 16.4.e.a.13.3 yes 10
32.29 even 8 64.4.e.a.49.3 10
96.5 odd 8 576.4.k.a.145.2 10
96.29 odd 8 576.4.k.a.433.2 10
96.35 even 8 144.4.k.a.37.3 10
96.59 even 8 144.4.k.a.109.3 10

By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.3 10 32.3 odd 8
16.4.e.a.13.3 yes 10 32.27 odd 8
64.4.e.a.17.3 10 32.5 even 8
64.4.e.a.49.3 10 32.29 even 8
128.4.e.a.33.3 10 32.21 even 8
128.4.e.a.97.3 10 32.13 even 8
128.4.e.b.33.3 10 32.11 odd 8
128.4.e.b.97.3 10 32.19 odd 8
144.4.k.a.37.3 10 96.35 even 8
144.4.k.a.109.3 10 96.59 even 8
576.4.k.a.145.2 10 96.5 odd 8
576.4.k.a.433.2 10 96.29 odd 8
1024.4.a.m.1.5 10 16.13 even 4
1024.4.a.m.1.6 10 16.5 even 4
1024.4.a.n.1.5 10 16.11 odd 4
1024.4.a.n.1.6 10 16.3 odd 4
1024.4.b.j.513.5 10 4.3 odd 2
1024.4.b.j.513.6 10 8.3 odd 2
1024.4.b.k.513.5 10 8.5 even 2 inner
1024.4.b.k.513.6 10 1.1 even 1 trivial