# Properties

 Label 112.4.a.f.1.1 Level $112$ Weight $4$ Character 112.1 Self dual yes Analytic conductor $6.608$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.60821392064$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 112.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} +16.0000 q^{5} +7.00000 q^{7} -23.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} +16.0000 q^{5} +7.00000 q^{7} -23.0000 q^{9} +8.00000 q^{11} +28.0000 q^{13} +32.0000 q^{15} +54.0000 q^{17} +110.000 q^{19} +14.0000 q^{21} -48.0000 q^{23} +131.000 q^{25} -100.000 q^{27} -110.000 q^{29} -12.0000 q^{31} +16.0000 q^{33} +112.000 q^{35} -246.000 q^{37} +56.0000 q^{39} +182.000 q^{41} -128.000 q^{43} -368.000 q^{45} -324.000 q^{47} +49.0000 q^{49} +108.000 q^{51} -162.000 q^{53} +128.000 q^{55} +220.000 q^{57} -810.000 q^{59} -488.000 q^{61} -161.000 q^{63} +448.000 q^{65} -244.000 q^{67} -96.0000 q^{69} +768.000 q^{71} -702.000 q^{73} +262.000 q^{75} +56.0000 q^{77} -440.000 q^{79} +421.000 q^{81} +1302.00 q^{83} +864.000 q^{85} -220.000 q^{87} +730.000 q^{89} +196.000 q^{91} -24.0000 q^{93} +1760.00 q^{95} +294.000 q^{97} -184.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 2.00000 0.384900 0.192450 0.981307i $$-0.438357\pi$$
0.192450 + 0.981307i $$0.438357\pi$$
$$4$$ 0 0
$$5$$ 16.0000 1.43108 0.715542 0.698570i $$-0.246180\pi$$
0.715542 + 0.698570i $$0.246180\pi$$
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 0 0
$$9$$ −23.0000 −0.851852
$$10$$ 0 0
$$11$$ 8.00000 0.219281 0.109640 0.993971i $$-0.465030\pi$$
0.109640 + 0.993971i $$0.465030\pi$$
$$12$$ 0 0
$$13$$ 28.0000 0.597369 0.298685 0.954352i $$-0.403452\pi$$
0.298685 + 0.954352i $$0.403452\pi$$
$$14$$ 0 0
$$15$$ 32.0000 0.550824
$$16$$ 0 0
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ 110.000 1.32820 0.664098 0.747645i $$-0.268816\pi$$
0.664098 + 0.747645i $$0.268816\pi$$
$$20$$ 0 0
$$21$$ 14.0000 0.145479
$$22$$ 0 0
$$23$$ −48.0000 −0.435161 −0.217580 0.976042i $$-0.569816\pi$$
−0.217580 + 0.976042i $$0.569816\pi$$
$$24$$ 0 0
$$25$$ 131.000 1.04800
$$26$$ 0 0
$$27$$ −100.000 −0.712778
$$28$$ 0 0
$$29$$ −110.000 −0.704362 −0.352181 0.935932i $$-0.614560\pi$$
−0.352181 + 0.935932i $$0.614560\pi$$
$$30$$ 0 0
$$31$$ −12.0000 −0.0695246 −0.0347623 0.999396i $$-0.511067\pi$$
−0.0347623 + 0.999396i $$0.511067\pi$$
$$32$$ 0 0
$$33$$ 16.0000 0.0844013
$$34$$ 0 0
$$35$$ 112.000 0.540899
$$36$$ 0 0
$$37$$ −246.000 −1.09303 −0.546516 0.837449i $$-0.684046\pi$$
−0.546516 + 0.837449i $$0.684046\pi$$
$$38$$ 0 0
$$39$$ 56.0000 0.229928
$$40$$ 0 0
$$41$$ 182.000 0.693259 0.346630 0.938002i $$-0.387326\pi$$
0.346630 + 0.938002i $$0.387326\pi$$
$$42$$ 0 0
$$43$$ −128.000 −0.453949 −0.226975 0.973901i $$-0.572883\pi$$
−0.226975 + 0.973901i $$0.572883\pi$$
$$44$$ 0 0
$$45$$ −368.000 −1.21907
$$46$$ 0 0
$$47$$ −324.000 −1.00554 −0.502769 0.864421i $$-0.667685\pi$$
−0.502769 + 0.864421i $$0.667685\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 108.000 0.296530
$$52$$ 0 0
$$53$$ −162.000 −0.419857 −0.209928 0.977717i $$-0.567323\pi$$
−0.209928 + 0.977717i $$0.567323\pi$$
$$54$$ 0 0
$$55$$ 128.000 0.313809
$$56$$ 0 0
$$57$$ 220.000 0.511223
$$58$$ 0 0
$$59$$ −810.000 −1.78734 −0.893670 0.448725i $$-0.851878\pi$$
−0.893670 + 0.448725i $$0.851878\pi$$
$$60$$ 0 0
$$61$$ −488.000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ −161.000 −0.321970
$$64$$ 0 0
$$65$$ 448.000 0.854886
$$66$$ 0 0
$$67$$ −244.000 −0.444916 −0.222458 0.974942i $$-0.571408\pi$$
−0.222458 + 0.974942i $$0.571408\pi$$
$$68$$ 0 0
$$69$$ −96.0000 −0.167493
$$70$$ 0 0
$$71$$ 768.000 1.28373 0.641865 0.766818i $$-0.278161\pi$$
0.641865 + 0.766818i $$0.278161\pi$$
$$72$$ 0 0
$$73$$ −702.000 −1.12552 −0.562759 0.826621i $$-0.690260\pi$$
−0.562759 + 0.826621i $$0.690260\pi$$
$$74$$ 0 0
$$75$$ 262.000 0.403375
$$76$$ 0 0
$$77$$ 56.0000 0.0828804
$$78$$ 0 0
$$79$$ −440.000 −0.626631 −0.313316 0.949649i $$-0.601440\pi$$
−0.313316 + 0.949649i $$0.601440\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ 0 0
$$83$$ 1302.00 1.72184 0.860922 0.508737i $$-0.169887\pi$$
0.860922 + 0.508737i $$0.169887\pi$$
$$84$$ 0 0
$$85$$ 864.000 1.10252
$$86$$ 0 0
$$87$$ −220.000 −0.271109
$$88$$ 0 0
$$89$$ 730.000 0.869436 0.434718 0.900567i $$-0.356848\pi$$
0.434718 + 0.900567i $$0.356848\pi$$
$$90$$ 0 0
$$91$$ 196.000 0.225784
$$92$$ 0 0
$$93$$ −24.0000 −0.0267600
$$94$$ 0 0
$$95$$ 1760.00 1.90076
$$96$$ 0 0
$$97$$ 294.000 0.307744 0.153872 0.988091i $$-0.450826\pi$$
0.153872 + 0.988091i $$0.450826\pi$$
$$98$$ 0 0
$$99$$ −184.000 −0.186795
$$100$$ 0 0
$$101$$ −688.000 −0.677808 −0.338904 0.940821i $$-0.610056\pi$$
−0.338904 + 0.940821i $$0.610056\pi$$
$$102$$ 0 0
$$103$$ −1388.00 −1.32780 −0.663901 0.747820i $$-0.731101\pi$$
−0.663901 + 0.747820i $$0.731101\pi$$
$$104$$ 0 0
$$105$$ 224.000 0.208192
$$106$$ 0 0
$$107$$ −244.000 −0.220452 −0.110226 0.993907i $$-0.535157\pi$$
−0.110226 + 0.993907i $$0.535157\pi$$
$$108$$ 0 0
$$109$$ 90.0000 0.0790866 0.0395433 0.999218i $$-0.487410\pi$$
0.0395433 + 0.999218i $$0.487410\pi$$
$$110$$ 0 0
$$111$$ −492.000 −0.420708
$$112$$ 0 0
$$113$$ 1318.00 1.09723 0.548615 0.836075i $$-0.315155\pi$$
0.548615 + 0.836075i $$0.315155\pi$$
$$114$$ 0 0
$$115$$ −768.000 −0.622751
$$116$$ 0 0
$$117$$ −644.000 −0.508870
$$118$$ 0 0
$$119$$ 378.000 0.291187
$$120$$ 0 0
$$121$$ −1267.00 −0.951916
$$122$$ 0 0
$$123$$ 364.000 0.266836
$$124$$ 0 0
$$125$$ 96.0000 0.0686920
$$126$$ 0 0
$$127$$ 1776.00 1.24090 0.620451 0.784245i $$-0.286950\pi$$
0.620451 + 0.784245i $$0.286950\pi$$
$$128$$ 0 0
$$129$$ −256.000 −0.174725
$$130$$ 0 0
$$131$$ 1118.00 0.745650 0.372825 0.927902i $$-0.378389\pi$$
0.372825 + 0.927902i $$0.378389\pi$$
$$132$$ 0 0
$$133$$ 770.000 0.502011
$$134$$ 0 0
$$135$$ −1600.00 −1.02004
$$136$$ 0 0
$$137$$ 2274.00 1.41811 0.709054 0.705154i $$-0.249122\pi$$
0.709054 + 0.705154i $$0.249122\pi$$
$$138$$ 0 0
$$139$$ 210.000 0.128144 0.0640718 0.997945i $$-0.479591\pi$$
0.0640718 + 0.997945i $$0.479591\pi$$
$$140$$ 0 0
$$141$$ −648.000 −0.387032
$$142$$ 0 0
$$143$$ 224.000 0.130992
$$144$$ 0 0
$$145$$ −1760.00 −1.00800
$$146$$ 0 0
$$147$$ 98.0000 0.0549857
$$148$$ 0 0
$$149$$ −2010.00 −1.10514 −0.552569 0.833467i $$-0.686352\pi$$
−0.552569 + 0.833467i $$0.686352\pi$$
$$150$$ 0 0
$$151$$ −1112.00 −0.599293 −0.299647 0.954050i $$-0.596869\pi$$
−0.299647 + 0.954050i $$0.596869\pi$$
$$152$$ 0 0
$$153$$ −1242.00 −0.656273
$$154$$ 0 0
$$155$$ −192.000 −0.0994956
$$156$$ 0 0
$$157$$ 124.000 0.0630336 0.0315168 0.999503i $$-0.489966\pi$$
0.0315168 + 0.999503i $$0.489966\pi$$
$$158$$ 0 0
$$159$$ −324.000 −0.161603
$$160$$ 0 0
$$161$$ −336.000 −0.164475
$$162$$ 0 0
$$163$$ −2008.00 −0.964900 −0.482450 0.875924i $$-0.660253\pi$$
−0.482450 + 0.875924i $$0.660253\pi$$
$$164$$ 0 0
$$165$$ 256.000 0.120785
$$166$$ 0 0
$$167$$ −2884.00 −1.33635 −0.668176 0.744004i $$-0.732924\pi$$
−0.668176 + 0.744004i $$0.732924\pi$$
$$168$$ 0 0
$$169$$ −1413.00 −0.643150
$$170$$ 0 0
$$171$$ −2530.00 −1.13143
$$172$$ 0 0
$$173$$ 2228.00 0.979143 0.489571 0.871963i $$-0.337153\pi$$
0.489571 + 0.871963i $$0.337153\pi$$
$$174$$ 0 0
$$175$$ 917.000 0.396107
$$176$$ 0 0
$$177$$ −1620.00 −0.687947
$$178$$ 0 0
$$179$$ 820.000 0.342400 0.171200 0.985236i $$-0.445236\pi$$
0.171200 + 0.985236i $$0.445236\pi$$
$$180$$ 0 0
$$181$$ 3892.00 1.59829 0.799144 0.601140i $$-0.205287\pi$$
0.799144 + 0.601140i $$0.205287\pi$$
$$182$$ 0 0
$$183$$ −976.000 −0.394251
$$184$$ 0 0
$$185$$ −3936.00 −1.56422
$$186$$ 0 0
$$187$$ 432.000 0.168936
$$188$$ 0 0
$$189$$ −700.000 −0.269405
$$190$$ 0 0
$$191$$ 5048.00 1.91236 0.956179 0.292782i $$-0.0945810\pi$$
0.956179 + 0.292782i $$0.0945810\pi$$
$$192$$ 0 0
$$193$$ −2962.00 −1.10471 −0.552356 0.833608i $$-0.686271\pi$$
−0.552356 + 0.833608i $$0.686271\pi$$
$$194$$ 0 0
$$195$$ 896.000 0.329046
$$196$$ 0 0
$$197$$ 3334.00 1.20577 0.602887 0.797826i $$-0.294017\pi$$
0.602887 + 0.797826i $$0.294017\pi$$
$$198$$ 0 0
$$199$$ −1860.00 −0.662572 −0.331286 0.943530i $$-0.607483\pi$$
−0.331286 + 0.943530i $$0.607483\pi$$
$$200$$ 0 0
$$201$$ −488.000 −0.171248
$$202$$ 0 0
$$203$$ −770.000 −0.266224
$$204$$ 0 0
$$205$$ 2912.00 0.992112
$$206$$ 0 0
$$207$$ 1104.00 0.370692
$$208$$ 0 0
$$209$$ 880.000 0.291248
$$210$$ 0 0
$$211$$ 4268.00 1.39252 0.696259 0.717791i $$-0.254847\pi$$
0.696259 + 0.717791i $$0.254847\pi$$
$$212$$ 0 0
$$213$$ 1536.00 0.494108
$$214$$ 0 0
$$215$$ −2048.00 −0.649639
$$216$$ 0 0
$$217$$ −84.0000 −0.0262778
$$218$$ 0 0
$$219$$ −1404.00 −0.433212
$$220$$ 0 0
$$221$$ 1512.00 0.460218
$$222$$ 0 0
$$223$$ 5432.00 1.63118 0.815591 0.578629i $$-0.196412\pi$$
0.815591 + 0.578629i $$0.196412\pi$$
$$224$$ 0 0
$$225$$ −3013.00 −0.892741
$$226$$ 0 0
$$227$$ 2046.00 0.598228 0.299114 0.954217i $$-0.403309\pi$$
0.299114 + 0.954217i $$0.403309\pi$$
$$228$$ 0 0
$$229$$ −2980.00 −0.859930 −0.429965 0.902846i $$-0.641474\pi$$
−0.429965 + 0.902846i $$0.641474\pi$$
$$230$$ 0 0
$$231$$ 112.000 0.0319007
$$232$$ 0 0
$$233$$ 4458.00 1.25345 0.626724 0.779241i $$-0.284395\pi$$
0.626724 + 0.779241i $$0.284395\pi$$
$$234$$ 0 0
$$235$$ −5184.00 −1.43901
$$236$$ 0 0
$$237$$ −880.000 −0.241190
$$238$$ 0 0
$$239$$ −4440.00 −1.20167 −0.600836 0.799372i $$-0.705166\pi$$
−0.600836 + 0.799372i $$0.705166\pi$$
$$240$$ 0 0
$$241$$ 3302.00 0.882575 0.441287 0.897366i $$-0.354522\pi$$
0.441287 + 0.897366i $$0.354522\pi$$
$$242$$ 0 0
$$243$$ 3542.00 0.935059
$$244$$ 0 0
$$245$$ 784.000 0.204441
$$246$$ 0 0
$$247$$ 3080.00 0.793424
$$248$$ 0 0
$$249$$ 2604.00 0.662738
$$250$$ 0 0
$$251$$ −1582.00 −0.397829 −0.198914 0.980017i $$-0.563742\pi$$
−0.198914 + 0.980017i $$0.563742\pi$$
$$252$$ 0 0
$$253$$ −384.000 −0.0954224
$$254$$ 0 0
$$255$$ 1728.00 0.424359
$$256$$ 0 0
$$257$$ 2354.00 0.571356 0.285678 0.958326i $$-0.407781\pi$$
0.285678 + 0.958326i $$0.407781\pi$$
$$258$$ 0 0
$$259$$ −1722.00 −0.413127
$$260$$ 0 0
$$261$$ 2530.00 0.600012
$$262$$ 0 0
$$263$$ 3872.00 0.907824 0.453912 0.891046i $$-0.350028\pi$$
0.453912 + 0.891046i $$0.350028\pi$$
$$264$$ 0 0
$$265$$ −2592.00 −0.600850
$$266$$ 0 0
$$267$$ 1460.00 0.334646
$$268$$ 0 0
$$269$$ 180.000 0.0407985 0.0203992 0.999792i $$-0.493506\pi$$
0.0203992 + 0.999792i $$0.493506\pi$$
$$270$$ 0 0
$$271$$ −2032.00 −0.455480 −0.227740 0.973722i $$-0.573134\pi$$
−0.227740 + 0.973722i $$0.573134\pi$$
$$272$$ 0 0
$$273$$ 392.000 0.0869045
$$274$$ 0 0
$$275$$ 1048.00 0.229806
$$276$$ 0 0
$$277$$ −5426.00 −1.17696 −0.588478 0.808513i $$-0.700273\pi$$
−0.588478 + 0.808513i $$0.700273\pi$$
$$278$$ 0 0
$$279$$ 276.000 0.0592247
$$280$$ 0 0
$$281$$ 842.000 0.178753 0.0893764 0.995998i $$-0.471513\pi$$
0.0893764 + 0.995998i $$0.471513\pi$$
$$282$$ 0 0
$$283$$ 3782.00 0.794405 0.397202 0.917731i $$-0.369981\pi$$
0.397202 + 0.917731i $$0.369981\pi$$
$$284$$ 0 0
$$285$$ 3520.00 0.731603
$$286$$ 0 0
$$287$$ 1274.00 0.262027
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ 588.000 0.118451
$$292$$ 0 0
$$293$$ −4312.00 −0.859760 −0.429880 0.902886i $$-0.641444\pi$$
−0.429880 + 0.902886i $$0.641444\pi$$
$$294$$ 0 0
$$295$$ −12960.0 −2.55783
$$296$$ 0 0
$$297$$ −800.000 −0.156299
$$298$$ 0 0
$$299$$ −1344.00 −0.259952
$$300$$ 0 0
$$301$$ −896.000 −0.171577
$$302$$ 0 0
$$303$$ −1376.00 −0.260888
$$304$$ 0 0
$$305$$ −7808.00 −1.46585
$$306$$ 0 0
$$307$$ −2674.00 −0.497112 −0.248556 0.968618i $$-0.579956\pi$$
−0.248556 + 0.968618i $$0.579956\pi$$
$$308$$ 0 0
$$309$$ −2776.00 −0.511072
$$310$$ 0 0
$$311$$ 3768.00 0.687021 0.343511 0.939149i $$-0.388384\pi$$
0.343511 + 0.939149i $$0.388384\pi$$
$$312$$ 0 0
$$313$$ 2438.00 0.440268 0.220134 0.975470i $$-0.429351\pi$$
0.220134 + 0.975470i $$0.429351\pi$$
$$314$$ 0 0
$$315$$ −2576.00 −0.460766
$$316$$ 0 0
$$317$$ −3186.00 −0.564491 −0.282245 0.959342i $$-0.591079\pi$$
−0.282245 + 0.959342i $$0.591079\pi$$
$$318$$ 0 0
$$319$$ −880.000 −0.154453
$$320$$ 0 0
$$321$$ −488.000 −0.0848520
$$322$$ 0 0
$$323$$ 5940.00 1.02325
$$324$$ 0 0
$$325$$ 3668.00 0.626043
$$326$$ 0 0
$$327$$ 180.000 0.0304404
$$328$$ 0 0
$$329$$ −2268.00 −0.380057
$$330$$ 0 0
$$331$$ −8672.00 −1.44005 −0.720025 0.693949i $$-0.755869\pi$$
−0.720025 + 0.693949i $$0.755869\pi$$
$$332$$ 0 0
$$333$$ 5658.00 0.931101
$$334$$ 0 0
$$335$$ −3904.00 −0.636711
$$336$$ 0 0
$$337$$ 814.000 0.131577 0.0657884 0.997834i $$-0.479044\pi$$
0.0657884 + 0.997834i $$0.479044\pi$$
$$338$$ 0 0
$$339$$ 2636.00 0.422324
$$340$$ 0 0
$$341$$ −96.0000 −0.0152454
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 0 0
$$345$$ −1536.00 −0.239697
$$346$$ 0 0
$$347$$ −9344.00 −1.44557 −0.722784 0.691074i $$-0.757138\pi$$
−0.722784 + 0.691074i $$0.757138\pi$$
$$348$$ 0 0
$$349$$ −5180.00 −0.794496 −0.397248 0.917711i $$-0.630035\pi$$
−0.397248 + 0.917711i $$0.630035\pi$$
$$350$$ 0 0
$$351$$ −2800.00 −0.425792
$$352$$ 0 0
$$353$$ 12178.0 1.83617 0.918087 0.396379i $$-0.129733\pi$$
0.918087 + 0.396379i $$0.129733\pi$$
$$354$$ 0 0
$$355$$ 12288.0 1.83712
$$356$$ 0 0
$$357$$ 756.000 0.112078
$$358$$ 0 0
$$359$$ −440.000 −0.0646861 −0.0323431 0.999477i $$-0.510297\pi$$
−0.0323431 + 0.999477i $$0.510297\pi$$
$$360$$ 0 0
$$361$$ 5241.00 0.764106
$$362$$ 0 0
$$363$$ −2534.00 −0.366393
$$364$$ 0 0
$$365$$ −11232.0 −1.61071
$$366$$ 0 0
$$367$$ 9816.00 1.39616 0.698080 0.716019i $$-0.254038\pi$$
0.698080 + 0.716019i $$0.254038\pi$$
$$368$$ 0 0
$$369$$ −4186.00 −0.590554
$$370$$ 0 0
$$371$$ −1134.00 −0.158691
$$372$$ 0 0
$$373$$ −442.000 −0.0613563 −0.0306781 0.999529i $$-0.509767\pi$$
−0.0306781 + 0.999529i $$0.509767\pi$$
$$374$$ 0 0
$$375$$ 192.000 0.0264396
$$376$$ 0 0
$$377$$ −3080.00 −0.420764
$$378$$ 0 0
$$379$$ 3960.00 0.536706 0.268353 0.963321i $$-0.413521\pi$$
0.268353 + 0.963321i $$0.413521\pi$$
$$380$$ 0 0
$$381$$ 3552.00 0.477623
$$382$$ 0 0
$$383$$ −6708.00 −0.894942 −0.447471 0.894298i $$-0.647675\pi$$
−0.447471 + 0.894298i $$0.647675\pi$$
$$384$$ 0 0
$$385$$ 896.000 0.118609
$$386$$ 0 0
$$387$$ 2944.00 0.386697
$$388$$ 0 0
$$389$$ −13350.0 −1.74003 −0.870015 0.493025i $$-0.835891\pi$$
−0.870015 + 0.493025i $$0.835891\pi$$
$$390$$ 0 0
$$391$$ −2592.00 −0.335251
$$392$$ 0 0
$$393$$ 2236.00 0.287001
$$394$$ 0 0
$$395$$ −7040.00 −0.896762
$$396$$ 0 0
$$397$$ −1356.00 −0.171425 −0.0857125 0.996320i $$-0.527317\pi$$
−0.0857125 + 0.996320i $$0.527317\pi$$
$$398$$ 0 0
$$399$$ 1540.00 0.193224
$$400$$ 0 0
$$401$$ 6222.00 0.774843 0.387421 0.921903i $$-0.373366\pi$$
0.387421 + 0.921903i $$0.373366\pi$$
$$402$$ 0 0
$$403$$ −336.000 −0.0415319
$$404$$ 0 0
$$405$$ 6736.00 0.826456
$$406$$ 0 0
$$407$$ −1968.00 −0.239681
$$408$$ 0 0
$$409$$ 5150.00 0.622619 0.311309 0.950309i $$-0.399232\pi$$
0.311309 + 0.950309i $$0.399232\pi$$
$$410$$ 0 0
$$411$$ 4548.00 0.545830
$$412$$ 0 0
$$413$$ −5670.00 −0.675551
$$414$$ 0 0
$$415$$ 20832.0 2.46410
$$416$$ 0 0
$$417$$ 420.000 0.0493225
$$418$$ 0 0
$$419$$ −2310.00 −0.269334 −0.134667 0.990891i $$-0.542996\pi$$
−0.134667 + 0.990891i $$0.542996\pi$$
$$420$$ 0 0
$$421$$ 1262.00 0.146095 0.0730476 0.997328i $$-0.476727\pi$$
0.0730476 + 0.997328i $$0.476727\pi$$
$$422$$ 0 0
$$423$$ 7452.00 0.856569
$$424$$ 0 0
$$425$$ 7074.00 0.807387
$$426$$ 0 0
$$427$$ −3416.00 −0.387147
$$428$$ 0 0
$$429$$ 448.000 0.0504188
$$430$$ 0 0
$$431$$ 4488.00 0.501576 0.250788 0.968042i $$-0.419310\pi$$
0.250788 + 0.968042i $$0.419310\pi$$
$$432$$ 0 0
$$433$$ 17038.0 1.89098 0.945490 0.325652i $$-0.105584\pi$$
0.945490 + 0.325652i $$0.105584\pi$$
$$434$$ 0 0
$$435$$ −3520.00 −0.387979
$$436$$ 0 0
$$437$$ −5280.00 −0.577979
$$438$$ 0 0
$$439$$ −16200.0 −1.76124 −0.880619 0.473824i $$-0.842873\pi$$
−0.880619 + 0.473824i $$0.842873\pi$$
$$440$$ 0 0
$$441$$ −1127.00 −0.121693
$$442$$ 0 0
$$443$$ 8772.00 0.940791 0.470395 0.882456i $$-0.344111\pi$$
0.470395 + 0.882456i $$0.344111\pi$$
$$444$$ 0 0
$$445$$ 11680.0 1.24424
$$446$$ 0 0
$$447$$ −4020.00 −0.425368
$$448$$ 0 0
$$449$$ 2130.00 0.223877 0.111939 0.993715i $$-0.464294\pi$$
0.111939 + 0.993715i $$0.464294\pi$$
$$450$$ 0 0
$$451$$ 1456.00 0.152019
$$452$$ 0 0
$$453$$ −2224.00 −0.230668
$$454$$ 0 0
$$455$$ 3136.00 0.323116
$$456$$ 0 0
$$457$$ 10534.0 1.07825 0.539124 0.842226i $$-0.318755\pi$$
0.539124 + 0.842226i $$0.318755\pi$$
$$458$$ 0 0
$$459$$ −5400.00 −0.549129
$$460$$ 0 0
$$461$$ −9268.00 −0.936342 −0.468171 0.883638i $$-0.655087\pi$$
−0.468171 + 0.883638i $$0.655087\pi$$
$$462$$ 0 0
$$463$$ 9392.00 0.942728 0.471364 0.881939i $$-0.343762\pi$$
0.471364 + 0.881939i $$0.343762\pi$$
$$464$$ 0 0
$$465$$ −384.000 −0.0382959
$$466$$ 0 0
$$467$$ 10806.0 1.07075 0.535377 0.844613i $$-0.320170\pi$$
0.535377 + 0.844613i $$0.320170\pi$$
$$468$$ 0 0
$$469$$ −1708.00 −0.168162
$$470$$ 0 0
$$471$$ 248.000 0.0242616
$$472$$ 0 0
$$473$$ −1024.00 −0.0995424
$$474$$ 0 0
$$475$$ 14410.0 1.39195
$$476$$ 0 0
$$477$$ 3726.00 0.357656
$$478$$ 0 0
$$479$$ −4940.00 −0.471220 −0.235610 0.971848i $$-0.575709\pi$$
−0.235610 + 0.971848i $$0.575709\pi$$
$$480$$ 0 0
$$481$$ −6888.00 −0.652943
$$482$$ 0 0
$$483$$ −672.000 −0.0633065
$$484$$ 0 0
$$485$$ 4704.00 0.440407
$$486$$ 0 0
$$487$$ 5216.00 0.485338 0.242669 0.970109i $$-0.421977\pi$$
0.242669 + 0.970109i $$0.421977\pi$$
$$488$$ 0 0
$$489$$ −4016.00 −0.371390
$$490$$ 0 0
$$491$$ −4412.00 −0.405521 −0.202760 0.979228i $$-0.564991\pi$$
−0.202760 + 0.979228i $$0.564991\pi$$
$$492$$ 0 0
$$493$$ −5940.00 −0.542645
$$494$$ 0 0
$$495$$ −2944.00 −0.267319
$$496$$ 0 0
$$497$$ 5376.00 0.485204
$$498$$ 0 0
$$499$$ −19060.0 −1.70991 −0.854953 0.518706i $$-0.826414\pi$$
−0.854953 + 0.518706i $$0.826414\pi$$
$$500$$ 0 0
$$501$$ −5768.00 −0.514362
$$502$$ 0 0
$$503$$ −12768.0 −1.13180 −0.565902 0.824473i $$-0.691472\pi$$
−0.565902 + 0.824473i $$0.691472\pi$$
$$504$$ 0 0
$$505$$ −11008.0 −0.969999
$$506$$ 0 0
$$507$$ −2826.00 −0.247548
$$508$$ 0 0
$$509$$ −5500.00 −0.478945 −0.239473 0.970903i $$-0.576975\pi$$
−0.239473 + 0.970903i $$0.576975\pi$$
$$510$$ 0 0
$$511$$ −4914.00 −0.425406
$$512$$ 0 0
$$513$$ −11000.0 −0.946709
$$514$$ 0 0
$$515$$ −22208.0 −1.90020
$$516$$ 0 0
$$517$$ −2592.00 −0.220495
$$518$$ 0 0
$$519$$ 4456.00 0.376872
$$520$$ 0 0
$$521$$ −7338.00 −0.617051 −0.308526 0.951216i $$-0.599836\pi$$
−0.308526 + 0.951216i $$0.599836\pi$$
$$522$$ 0 0
$$523$$ 17582.0 1.46999 0.734997 0.678070i $$-0.237183\pi$$
0.734997 + 0.678070i $$0.237183\pi$$
$$524$$ 0 0
$$525$$ 1834.00 0.152462
$$526$$ 0 0
$$527$$ −648.000 −0.0535623
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ 0 0
$$531$$ 18630.0 1.52255
$$532$$ 0 0
$$533$$ 5096.00 0.414132
$$534$$ 0 0
$$535$$ −3904.00 −0.315485
$$536$$ 0 0
$$537$$ 1640.00 0.131790
$$538$$ 0 0
$$539$$ 392.000 0.0313259
$$540$$ 0 0
$$541$$ −1618.00 −0.128583 −0.0642914 0.997931i $$-0.520479\pi$$
−0.0642914 + 0.997931i $$0.520479\pi$$
$$542$$ 0 0
$$543$$ 7784.00 0.615181
$$544$$ 0 0
$$545$$ 1440.00 0.113179
$$546$$ 0 0
$$547$$ −16144.0 −1.26192 −0.630958 0.775817i $$-0.717338\pi$$
−0.630958 + 0.775817i $$0.717338\pi$$
$$548$$ 0 0
$$549$$ 11224.0 0.872548
$$550$$ 0 0
$$551$$ −12100.0 −0.935531
$$552$$ 0 0
$$553$$ −3080.00 −0.236844
$$554$$ 0 0
$$555$$ −7872.00 −0.602068
$$556$$ 0 0
$$557$$ 4654.00 0.354033 0.177016 0.984208i $$-0.443355\pi$$
0.177016 + 0.984208i $$0.443355\pi$$
$$558$$ 0 0
$$559$$ −3584.00 −0.271175
$$560$$ 0 0
$$561$$ 864.000 0.0650234
$$562$$ 0 0
$$563$$ −10078.0 −0.754418 −0.377209 0.926128i $$-0.623116\pi$$
−0.377209 + 0.926128i $$0.623116\pi$$
$$564$$ 0 0
$$565$$ 21088.0 1.57023
$$566$$ 0 0
$$567$$ 2947.00 0.218276
$$568$$ 0 0
$$569$$ −5930.00 −0.436904 −0.218452 0.975848i $$-0.570101\pi$$
−0.218452 + 0.975848i $$0.570101\pi$$
$$570$$ 0 0
$$571$$ 19048.0 1.39603 0.698016 0.716082i $$-0.254067\pi$$
0.698016 + 0.716082i $$0.254067\pi$$
$$572$$ 0 0
$$573$$ 10096.0 0.736067
$$574$$ 0 0
$$575$$ −6288.00 −0.456048
$$576$$ 0 0
$$577$$ −14366.0 −1.03651 −0.518253 0.855227i $$-0.673418\pi$$
−0.518253 + 0.855227i $$0.673418\pi$$
$$578$$ 0 0
$$579$$ −5924.00 −0.425204
$$580$$ 0 0
$$581$$ 9114.00 0.650796
$$582$$ 0 0
$$583$$ −1296.00 −0.0920666
$$584$$ 0 0
$$585$$ −10304.0 −0.728236
$$586$$ 0 0
$$587$$ 3626.00 0.254959 0.127480 0.991841i $$-0.459311\pi$$
0.127480 + 0.991841i $$0.459311\pi$$
$$588$$ 0 0
$$589$$ −1320.00 −0.0923424
$$590$$ 0 0
$$591$$ 6668.00 0.464103
$$592$$ 0 0
$$593$$ −1062.00 −0.0735432 −0.0367716 0.999324i $$-0.511707\pi$$
−0.0367716 + 0.999324i $$0.511707\pi$$
$$594$$ 0 0
$$595$$ 6048.00 0.416712
$$596$$ 0 0
$$597$$ −3720.00 −0.255024
$$598$$ 0 0
$$599$$ 10200.0 0.695761 0.347880 0.937539i $$-0.386902\pi$$
0.347880 + 0.937539i $$0.386902\pi$$
$$600$$ 0 0
$$601$$ −25158.0 −1.70751 −0.853757 0.520671i $$-0.825682\pi$$
−0.853757 + 0.520671i $$0.825682\pi$$
$$602$$ 0 0
$$603$$ 5612.00 0.379002
$$604$$ 0 0
$$605$$ −20272.0 −1.36227
$$606$$ 0 0
$$607$$ −25664.0 −1.71609 −0.858047 0.513570i $$-0.828323\pi$$
−0.858047 + 0.513570i $$0.828323\pi$$
$$608$$ 0 0
$$609$$ −1540.00 −0.102470
$$610$$ 0 0
$$611$$ −9072.00 −0.600677
$$612$$ 0 0
$$613$$ 19018.0 1.25307 0.626533 0.779395i $$-0.284473\pi$$
0.626533 + 0.779395i $$0.284473\pi$$
$$614$$ 0 0
$$615$$ 5824.00 0.381864
$$616$$ 0 0
$$617$$ 17334.0 1.13102 0.565511 0.824741i $$-0.308679\pi$$
0.565511 + 0.824741i $$0.308679\pi$$
$$618$$ 0 0
$$619$$ −18730.0 −1.21619 −0.608096 0.793864i $$-0.708066\pi$$
−0.608096 + 0.793864i $$0.708066\pi$$
$$620$$ 0 0
$$621$$ 4800.00 0.310173
$$622$$ 0 0
$$623$$ 5110.00 0.328616
$$624$$ 0 0
$$625$$ −14839.0 −0.949696
$$626$$ 0 0
$$627$$ 1760.00 0.112101
$$628$$ 0 0
$$629$$ −13284.0 −0.842079
$$630$$ 0 0
$$631$$ 6928.00 0.437083 0.218541 0.975828i $$-0.429870\pi$$
0.218541 + 0.975828i $$0.429870\pi$$
$$632$$ 0 0
$$633$$ 8536.00 0.535980
$$634$$ 0 0
$$635$$ 28416.0 1.77583
$$636$$ 0 0
$$637$$ 1372.00 0.0853385
$$638$$ 0 0
$$639$$ −17664.0 −1.09355
$$640$$ 0 0
$$641$$ 16302.0 1.00451 0.502255 0.864720i $$-0.332504\pi$$
0.502255 + 0.864720i $$0.332504\pi$$
$$642$$ 0 0
$$643$$ −4718.00 −0.289362 −0.144681 0.989478i $$-0.546216\pi$$
−0.144681 + 0.989478i $$0.546216\pi$$
$$644$$ 0 0
$$645$$ −4096.00 −0.250046
$$646$$ 0 0
$$647$$ 21436.0 1.30253 0.651264 0.758851i $$-0.274239\pi$$
0.651264 + 0.758851i $$0.274239\pi$$
$$648$$ 0 0
$$649$$ −6480.00 −0.391930
$$650$$ 0 0
$$651$$ −168.000 −0.0101143
$$652$$ 0 0
$$653$$ 4458.00 0.267159 0.133580 0.991038i $$-0.457353\pi$$
0.133580 + 0.991038i $$0.457353\pi$$
$$654$$ 0 0
$$655$$ 17888.0 1.06709
$$656$$ 0 0
$$657$$ 16146.0 0.958775
$$658$$ 0 0
$$659$$ 26640.0 1.57473 0.787365 0.616487i $$-0.211445\pi$$
0.787365 + 0.616487i $$0.211445\pi$$
$$660$$ 0 0
$$661$$ 7432.00 0.437324 0.218662 0.975801i $$-0.429831\pi$$
0.218662 + 0.975801i $$0.429831\pi$$
$$662$$ 0 0
$$663$$ 3024.00 0.177138
$$664$$ 0 0
$$665$$ 12320.0 0.718420
$$666$$ 0 0
$$667$$ 5280.00 0.306510
$$668$$ 0 0
$$669$$ 10864.0 0.627842
$$670$$ 0 0
$$671$$ −3904.00 −0.224608
$$672$$ 0 0
$$673$$ 58.0000 0.00332204 0.00166102 0.999999i $$-0.499471\pi$$
0.00166102 + 0.999999i $$0.499471\pi$$
$$674$$ 0 0
$$675$$ −13100.0 −0.746991
$$676$$ 0 0
$$677$$ −21516.0 −1.22146 −0.610729 0.791840i $$-0.709124\pi$$
−0.610729 + 0.791840i $$0.709124\pi$$
$$678$$ 0 0
$$679$$ 2058.00 0.116316
$$680$$ 0 0
$$681$$ 4092.00 0.230258
$$682$$ 0 0
$$683$$ −18108.0 −1.01447 −0.507235 0.861808i $$-0.669332\pi$$
−0.507235 + 0.861808i $$0.669332\pi$$
$$684$$ 0 0
$$685$$ 36384.0 2.02943
$$686$$ 0 0
$$687$$ −5960.00 −0.330987
$$688$$ 0 0
$$689$$ −4536.00 −0.250810
$$690$$ 0 0
$$691$$ 10078.0 0.554827 0.277413 0.960751i $$-0.410523\pi$$
0.277413 + 0.960751i $$0.410523\pi$$
$$692$$ 0 0
$$693$$ −1288.00 −0.0706018
$$694$$ 0 0
$$695$$ 3360.00 0.183384
$$696$$ 0 0
$$697$$ 9828.00 0.534092
$$698$$ 0 0
$$699$$ 8916.00 0.482452
$$700$$ 0 0
$$701$$ 18762.0 1.01089 0.505443 0.862860i $$-0.331329\pi$$
0.505443 + 0.862860i $$0.331329\pi$$
$$702$$ 0 0
$$703$$ −27060.0 −1.45176
$$704$$ 0 0
$$705$$ −10368.0 −0.553874
$$706$$ 0 0
$$707$$ −4816.00 −0.256187
$$708$$ 0 0
$$709$$ 6810.00 0.360726 0.180363 0.983600i $$-0.442273\pi$$
0.180363 + 0.983600i $$0.442273\pi$$
$$710$$ 0 0
$$711$$ 10120.0 0.533797
$$712$$ 0 0
$$713$$ 576.000 0.0302544
$$714$$ 0 0
$$715$$ 3584.00 0.187460
$$716$$ 0 0
$$717$$ −8880.00 −0.462524
$$718$$ 0 0
$$719$$ −4860.00 −0.252083 −0.126041 0.992025i $$-0.540227\pi$$
−0.126041 + 0.992025i $$0.540227\pi$$
$$720$$ 0 0
$$721$$ −9716.00 −0.501862
$$722$$ 0 0
$$723$$ 6604.00 0.339703
$$724$$ 0 0
$$725$$ −14410.0 −0.738171
$$726$$ 0 0
$$727$$ 13636.0 0.695641 0.347821 0.937561i $$-0.386922\pi$$
0.347821 + 0.937561i $$0.386922\pi$$
$$728$$ 0 0
$$729$$ −4283.00 −0.217599
$$730$$ 0 0
$$731$$ −6912.00 −0.349726
$$732$$ 0 0
$$733$$ 2088.00 0.105214 0.0526071 0.998615i $$-0.483247\pi$$
0.0526071 + 0.998615i $$0.483247\pi$$
$$734$$ 0 0
$$735$$ 1568.00 0.0786892
$$736$$ 0 0
$$737$$ −1952.00 −0.0975615
$$738$$ 0 0
$$739$$ 5160.00 0.256852 0.128426 0.991719i $$-0.459008\pi$$
0.128426 + 0.991719i $$0.459008\pi$$
$$740$$ 0 0
$$741$$ 6160.00 0.305389
$$742$$ 0 0
$$743$$ 28152.0 1.39004 0.695018 0.718992i $$-0.255396\pi$$
0.695018 + 0.718992i $$0.255396\pi$$
$$744$$ 0 0
$$745$$ −32160.0 −1.58155
$$746$$ 0 0
$$747$$ −29946.0 −1.46676
$$748$$ 0 0
$$749$$ −1708.00 −0.0833230
$$750$$ 0 0
$$751$$ 16808.0 0.816688 0.408344 0.912828i $$-0.366106\pi$$
0.408344 + 0.912828i $$0.366106\pi$$
$$752$$ 0 0
$$753$$ −3164.00 −0.153124
$$754$$ 0 0
$$755$$ −17792.0 −0.857639
$$756$$ 0 0
$$757$$ 21674.0 1.04063 0.520314 0.853975i $$-0.325815\pi$$
0.520314 + 0.853975i $$0.325815\pi$$
$$758$$ 0 0
$$759$$ −768.000 −0.0367281
$$760$$ 0 0
$$761$$ 7422.00 0.353544 0.176772 0.984252i $$-0.443434\pi$$
0.176772 + 0.984252i $$0.443434\pi$$
$$762$$ 0 0
$$763$$ 630.000 0.0298919
$$764$$ 0 0
$$765$$ −19872.0 −0.939181
$$766$$ 0 0
$$767$$ −22680.0 −1.06770
$$768$$ 0 0
$$769$$ 13790.0 0.646658 0.323329 0.946287i $$-0.395198\pi$$
0.323329 + 0.946287i $$0.395198\pi$$
$$770$$ 0 0
$$771$$ 4708.00 0.219915
$$772$$ 0 0
$$773$$ −6232.00 −0.289973 −0.144987 0.989434i $$-0.546314\pi$$
−0.144987 + 0.989434i $$0.546314\pi$$
$$774$$ 0 0
$$775$$ −1572.00 −0.0728618
$$776$$ 0 0
$$777$$ −3444.00 −0.159013
$$778$$ 0 0
$$779$$ 20020.0 0.920784
$$780$$ 0 0
$$781$$ 6144.00 0.281498
$$782$$ 0 0
$$783$$ 11000.0 0.502054
$$784$$ 0 0
$$785$$ 1984.00 0.0902064
$$786$$ 0 0
$$787$$ 1766.00 0.0799887 0.0399943 0.999200i $$-0.487266\pi$$
0.0399943 + 0.999200i $$0.487266\pi$$
$$788$$ 0 0
$$789$$ 7744.00 0.349422
$$790$$ 0 0
$$791$$ 9226.00 0.414714
$$792$$ 0 0
$$793$$ −13664.0 −0.611883
$$794$$ 0 0
$$795$$ −5184.00 −0.231267
$$796$$ 0 0
$$797$$ 1204.00 0.0535105 0.0267552 0.999642i $$-0.491483\pi$$
0.0267552 + 0.999642i $$0.491483\pi$$
$$798$$ 0 0
$$799$$ −17496.0 −0.774673
$$800$$ 0 0
$$801$$ −16790.0 −0.740631
$$802$$ 0 0
$$803$$ −5616.00 −0.246805
$$804$$ 0 0
$$805$$ −5376.00 −0.235378
$$806$$ 0 0
$$807$$ 360.000 0.0157033
$$808$$ 0 0
$$809$$ −7050.00 −0.306384 −0.153192 0.988196i $$-0.548955\pi$$
−0.153192 + 0.988196i $$0.548955\pi$$
$$810$$ 0 0
$$811$$ −23282.0 −1.00807 −0.504033 0.863684i $$-0.668151\pi$$
−0.504033 + 0.863684i $$0.668151\pi$$
$$812$$ 0 0
$$813$$ −4064.00 −0.175315
$$814$$ 0 0
$$815$$ −32128.0 −1.38085
$$816$$ 0 0
$$817$$ −14080.0 −0.602934
$$818$$ 0 0
$$819$$ −4508.00 −0.192335
$$820$$ 0 0
$$821$$ 10142.0 0.431131 0.215565 0.976489i $$-0.430841\pi$$
0.215565 + 0.976489i $$0.430841\pi$$
$$822$$ 0 0
$$823$$ 9192.00 0.389323 0.194662 0.980870i $$-0.437639\pi$$
0.194662 + 0.980870i $$0.437639\pi$$
$$824$$ 0 0
$$825$$ 2096.00 0.0884525
$$826$$ 0 0
$$827$$ 46716.0 1.96430 0.982149 0.188104i $$-0.0602344\pi$$
0.982149 + 0.188104i $$0.0602344\pi$$
$$828$$ 0 0
$$829$$ 11240.0 0.470906 0.235453 0.971886i $$-0.424343\pi$$
0.235453 + 0.971886i $$0.424343\pi$$
$$830$$ 0 0
$$831$$ −10852.0 −0.453010
$$832$$ 0 0
$$833$$ 2646.00 0.110058
$$834$$ 0 0
$$835$$ −46144.0 −1.91243
$$836$$ 0 0
$$837$$ 1200.00 0.0495556
$$838$$ 0 0
$$839$$ −700.000 −0.0288042 −0.0144021 0.999896i $$-0.504584\pi$$
−0.0144021 + 0.999896i $$0.504584\pi$$
$$840$$ 0 0
$$841$$ −12289.0 −0.503875
$$842$$ 0 0
$$843$$ 1684.00 0.0688019
$$844$$ 0 0
$$845$$ −22608.0 −0.920401
$$846$$ 0 0
$$847$$ −8869.00 −0.359790
$$848$$ 0 0
$$849$$ 7564.00 0.305767
$$850$$ 0 0
$$851$$ 11808.0 0.475644
$$852$$ 0 0
$$853$$ −37492.0 −1.50493 −0.752463 0.658635i $$-0.771134\pi$$
−0.752463 + 0.658635i $$0.771134\pi$$
$$854$$ 0 0
$$855$$ −40480.0 −1.61917
$$856$$ 0 0
$$857$$ 28894.0 1.15169 0.575846 0.817558i $$-0.304673\pi$$
0.575846 + 0.817558i $$0.304673\pi$$
$$858$$ 0 0
$$859$$ 2770.00 0.110025 0.0550123 0.998486i $$-0.482480\pi$$
0.0550123 + 0.998486i $$0.482480\pi$$
$$860$$ 0 0
$$861$$ 2548.00 0.100854
$$862$$ 0 0
$$863$$ −17688.0 −0.697690 −0.348845 0.937180i $$-0.613426\pi$$
−0.348845 + 0.937180i $$0.613426\pi$$
$$864$$ 0 0
$$865$$ 35648.0 1.40124
$$866$$ 0 0
$$867$$ −3994.00 −0.156451
$$868$$ 0 0
$$869$$ −3520.00 −0.137408
$$870$$ 0 0
$$871$$ −6832.00 −0.265779
$$872$$ 0 0
$$873$$ −6762.00 −0.262152
$$874$$ 0 0
$$875$$ 672.000 0.0259631
$$876$$ 0 0
$$877$$ −33566.0 −1.29241 −0.646205 0.763164i $$-0.723645\pi$$
−0.646205 + 0.763164i $$0.723645\pi$$
$$878$$ 0 0
$$879$$ −8624.00 −0.330922
$$880$$ 0 0
$$881$$ −16758.0 −0.640853 −0.320426 0.947273i $$-0.603826\pi$$
−0.320426 + 0.947273i $$0.603826\pi$$
$$882$$ 0 0
$$883$$ −11468.0 −0.437066 −0.218533 0.975830i $$-0.570127\pi$$
−0.218533 + 0.975830i $$0.570127\pi$$
$$884$$ 0 0
$$885$$ −25920.0 −0.984510
$$886$$ 0 0
$$887$$ 50356.0 1.90619 0.953094 0.302674i $$-0.0978793\pi$$
0.953094 + 0.302674i $$0.0978793\pi$$
$$888$$ 0 0
$$889$$ 12432.0 0.469017
$$890$$ 0 0
$$891$$ 3368.00 0.126636
$$892$$ 0 0
$$893$$ −35640.0 −1.33555
$$894$$ 0 0
$$895$$ 13120.0 0.490004
$$896$$ 0 0
$$897$$ −2688.00 −0.100055
$$898$$ 0 0
$$899$$ 1320.00 0.0489705
$$900$$ 0 0
$$901$$ −8748.00 −0.323461
$$902$$ 0 0
$$903$$ −1792.00 −0.0660399
$$904$$ 0 0
$$905$$ 62272.0 2.28728
$$906$$ 0 0
$$907$$ 8716.00 0.319085 0.159542 0.987191i $$-0.448998\pi$$
0.159542 + 0.987191i $$0.448998\pi$$
$$908$$ 0 0
$$909$$ 15824.0 0.577392
$$910$$ 0 0
$$911$$ −7632.00 −0.277563 −0.138781 0.990323i $$-0.544318\pi$$
−0.138781 + 0.990323i $$0.544318\pi$$
$$912$$ 0 0
$$913$$ 10416.0 0.377568
$$914$$ 0 0
$$915$$ −15616.0 −0.564207
$$916$$ 0 0
$$917$$ 7826.00 0.281829
$$918$$ 0 0
$$919$$ 23080.0 0.828443 0.414221 0.910176i $$-0.364054\pi$$
0.414221 + 0.910176i $$0.364054\pi$$
$$920$$ 0 0
$$921$$ −5348.00 −0.191338
$$922$$ 0 0
$$923$$ 21504.0 0.766861
$$924$$ 0 0
$$925$$ −32226.0 −1.14550
$$926$$ 0 0
$$927$$ 31924.0 1.13109
$$928$$ 0 0
$$929$$ 45110.0 1.59312 0.796561 0.604558i $$-0.206650\pi$$
0.796561 + 0.604558i $$0.206650\pi$$
$$930$$ 0 0
$$931$$ 5390.00 0.189742
$$932$$ 0 0
$$933$$ 7536.00 0.264435
$$934$$ 0 0
$$935$$ 6912.00 0.241761
$$936$$ 0 0
$$937$$ 16674.0 0.581340 0.290670 0.956823i $$-0.406122\pi$$
0.290670 + 0.956823i $$0.406122\pi$$
$$938$$ 0 0
$$939$$ 4876.00 0.169459
$$940$$ 0 0
$$941$$ 43832.0 1.51847 0.759236 0.650815i $$-0.225573\pi$$
0.759236 + 0.650815i $$0.225573\pi$$
$$942$$ 0 0
$$943$$ −8736.00 −0.301679
$$944$$ 0 0
$$945$$ −11200.0 −0.385541
$$946$$ 0 0
$$947$$ 736.000 0.0252553 0.0126277 0.999920i $$-0.495980\pi$$
0.0126277 + 0.999920i $$0.495980\pi$$
$$948$$ 0 0
$$949$$ −19656.0 −0.672351
$$950$$ 0 0
$$951$$ −6372.00 −0.217273
$$952$$ 0 0
$$953$$ 38138.0 1.29634 0.648169 0.761496i $$-0.275535\pi$$
0.648169 + 0.761496i $$0.275535\pi$$
$$954$$ 0 0
$$955$$ 80768.0 2.73674
$$956$$ 0 0
$$957$$ −1760.00 −0.0594490
$$958$$ 0 0
$$959$$ 15918.0 0.535995
$$960$$ 0 0
$$961$$ −29647.0 −0.995166
$$962$$ 0 0
$$963$$ 5612.00 0.187792
$$964$$ 0 0
$$965$$ −47392.0 −1.58094
$$966$$ 0 0
$$967$$ −26224.0 −0.872086 −0.436043 0.899926i $$-0.643620\pi$$
−0.436043 + 0.899926i $$0.643620\pi$$
$$968$$ 0 0
$$969$$ 11880.0 0.393850
$$970$$ 0 0
$$971$$ −18762.0 −0.620084 −0.310042 0.950723i $$-0.600343\pi$$
−0.310042 + 0.950723i $$0.600343\pi$$
$$972$$ 0 0
$$973$$ 1470.00 0.0484337
$$974$$ 0 0
$$975$$ 7336.00 0.240964
$$976$$ 0 0
$$977$$ 38394.0 1.25725 0.628625 0.777709i $$-0.283618\pi$$
0.628625 + 0.777709i $$0.283618\pi$$
$$978$$ 0 0
$$979$$ 5840.00 0.190651
$$980$$ 0 0
$$981$$ −2070.00 −0.0673700
$$982$$ 0 0
$$983$$ −5388.00 −0.174822 −0.0874112 0.996172i $$-0.527859\pi$$
−0.0874112 + 0.996172i $$0.527859\pi$$
$$984$$ 0 0
$$985$$ 53344.0 1.72556
$$986$$ 0 0
$$987$$ −4536.00 −0.146284
$$988$$ 0 0
$$989$$ 6144.00 0.197541
$$990$$ 0 0
$$991$$ −25472.0 −0.816493 −0.408247 0.912872i $$-0.633860\pi$$
−0.408247 + 0.912872i $$0.633860\pi$$
$$992$$ 0 0
$$993$$ −17344.0 −0.554275
$$994$$ 0 0
$$995$$ −29760.0 −0.948196
$$996$$ 0 0
$$997$$ −17096.0 −0.543065 −0.271532 0.962429i $$-0.587530\pi$$
−0.271532 + 0.962429i $$0.587530\pi$$
$$998$$ 0 0
$$999$$ 24600.0 0.779089
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 112.4.a.f.1.1 1
3.2 odd 2 1008.4.a.c.1.1 1
4.3 odd 2 7.4.a.a.1.1 1
7.6 odd 2 784.4.a.g.1.1 1
8.3 odd 2 448.4.a.i.1.1 1
8.5 even 2 448.4.a.e.1.1 1
12.11 even 2 63.4.a.b.1.1 1
20.3 even 4 175.4.b.b.99.2 2
20.7 even 4 175.4.b.b.99.1 2
20.19 odd 2 175.4.a.b.1.1 1
28.3 even 6 49.4.c.b.30.1 2
28.11 odd 6 49.4.c.c.30.1 2
28.19 even 6 49.4.c.b.18.1 2
28.23 odd 6 49.4.c.c.18.1 2
28.27 even 2 49.4.a.b.1.1 1
44.43 even 2 847.4.a.b.1.1 1
52.51 odd 2 1183.4.a.b.1.1 1
60.59 even 2 1575.4.a.e.1.1 1
68.67 odd 2 2023.4.a.a.1.1 1
84.11 even 6 441.4.e.h.226.1 2
84.23 even 6 441.4.e.h.361.1 2
84.47 odd 6 441.4.e.e.361.1 2
84.59 odd 6 441.4.e.e.226.1 2
84.83 odd 2 441.4.a.i.1.1 1
140.139 even 2 1225.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 4.3 odd 2
49.4.a.b.1.1 1 28.27 even 2
49.4.c.b.18.1 2 28.19 even 6
49.4.c.b.30.1 2 28.3 even 6
49.4.c.c.18.1 2 28.23 odd 6
49.4.c.c.30.1 2 28.11 odd 6
63.4.a.b.1.1 1 12.11 even 2
112.4.a.f.1.1 1 1.1 even 1 trivial
175.4.a.b.1.1 1 20.19 odd 2
175.4.b.b.99.1 2 20.7 even 4
175.4.b.b.99.2 2 20.3 even 4
441.4.a.i.1.1 1 84.83 odd 2
441.4.e.e.226.1 2 84.59 odd 6
441.4.e.e.361.1 2 84.47 odd 6
441.4.e.h.226.1 2 84.11 even 6
441.4.e.h.361.1 2 84.23 even 6
448.4.a.e.1.1 1 8.5 even 2
448.4.a.i.1.1 1 8.3 odd 2
784.4.a.g.1.1 1 7.6 odd 2
847.4.a.b.1.1 1 44.43 even 2
1008.4.a.c.1.1 1 3.2 odd 2
1183.4.a.b.1.1 1 52.51 odd 2
1225.4.a.j.1.1 1 140.139 even 2
1575.4.a.e.1.1 1 60.59 even 2
2023.4.a.a.1.1 1 68.67 odd 2