# Properties

 Label 2352.3.f.j.97.3 Level $2352$ Weight $3$ Character 2352.97 Analytic conductor $64.087$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2352.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.0873581775$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4$$ x^8 - 4*x^6 + 14*x^4 - 8*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 588) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 97.3 Root $$-1.60021 + 0.923880i$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.97 Dual form 2352.3.f.j.97.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +0.480662i q^{5} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +0.480662i q^{5} -3.00000 q^{9} -5.76316 q^{11} -1.41991i q^{13} +0.832530 q^{15} +23.0671i q^{17} -10.6739i q^{19} +6.59980 q^{23} +24.7690 q^{25} +5.19615i q^{27} -6.20258 q^{29} -41.9320i q^{31} +9.98209i q^{33} -60.0929 q^{37} -2.45936 q^{39} +48.8250i q^{41} +51.5603 q^{43} -1.44198i q^{45} +19.2421i q^{47} +39.9534 q^{51} +82.1345 q^{53} -2.77013i q^{55} -18.4877 q^{57} -92.5800i q^{59} +4.99187i q^{61} +0.682497 q^{65} +2.20699 q^{67} -11.4312i q^{69} +80.5899 q^{71} -13.9088i q^{73} -42.9011i q^{75} +64.8885 q^{79} +9.00000 q^{81} -118.005i q^{83} -11.0875 q^{85} +10.7432i q^{87} -104.265i q^{89} -72.6284 q^{93} +5.13052 q^{95} -31.7875i q^{97} +17.2895 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{9}+O(q^{10})$$ 8 * q - 24 * q^9 $$8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100})$$ 8 * q - 24 * q^9 + 16 * q^23 + 72 * q^25 + 80 * q^29 + 128 * q^37 + 112 * q^43 + 96 * q^51 - 144 * q^53 - 192 * q^57 + 240 * q^65 + 64 * q^67 - 224 * q^71 + 432 * q^79 + 72 * q^81 - 96 * q^85 - 96 * q^93 + 272 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.73205i − 0.577350i
$$4$$ 0 0
$$5$$ 0.480662i 0.0961323i 0.998844 + 0.0480662i $$0.0153058\pi$$
−0.998844 + 0.0480662i $$0.984694\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −3.00000 −0.333333
$$10$$ 0 0
$$11$$ −5.76316 −0.523924 −0.261962 0.965078i $$-0.584369\pi$$
−0.261962 + 0.965078i $$0.584369\pi$$
$$12$$ 0 0
$$13$$ − 1.41991i − 0.109224i −0.998508 0.0546120i $$-0.982608\pi$$
0.998508 0.0546120i $$-0.0173922\pi$$
$$14$$ 0 0
$$15$$ 0.832530 0.0555020
$$16$$ 0 0
$$17$$ 23.0671i 1.35689i 0.734651 + 0.678445i $$0.237346\pi$$
−0.734651 + 0.678445i $$0.762654\pi$$
$$18$$ 0 0
$$19$$ − 10.6739i − 0.561783i −0.959740 0.280891i $$-0.909370\pi$$
0.959740 0.280891i $$-0.0906300\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.59980 0.286948 0.143474 0.989654i $$-0.454173\pi$$
0.143474 + 0.989654i $$0.454173\pi$$
$$24$$ 0 0
$$25$$ 24.7690 0.990759
$$26$$ 0 0
$$27$$ 5.19615i 0.192450i
$$28$$ 0 0
$$29$$ −6.20258 −0.213882 −0.106941 0.994265i $$-0.534106\pi$$
−0.106941 + 0.994265i $$0.534106\pi$$
$$30$$ 0 0
$$31$$ − 41.9320i − 1.35265i −0.736605 0.676323i $$-0.763572\pi$$
0.736605 0.676323i $$-0.236428\pi$$
$$32$$ 0 0
$$33$$ 9.98209i 0.302488i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −60.0929 −1.62413 −0.812066 0.583566i $$-0.801657\pi$$
−0.812066 + 0.583566i $$0.801657\pi$$
$$38$$ 0 0
$$39$$ −2.45936 −0.0630605
$$40$$ 0 0
$$41$$ 48.8250i 1.19085i 0.803409 + 0.595427i $$0.203017\pi$$
−0.803409 + 0.595427i $$0.796983\pi$$
$$42$$ 0 0
$$43$$ 51.5603 1.19908 0.599539 0.800346i $$-0.295351\pi$$
0.599539 + 0.800346i $$0.295351\pi$$
$$44$$ 0 0
$$45$$ − 1.44198i − 0.0320441i
$$46$$ 0 0
$$47$$ 19.2421i 0.409406i 0.978824 + 0.204703i $$0.0656229\pi$$
−0.978824 + 0.204703i $$0.934377\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 39.9534 0.783401
$$52$$ 0 0
$$53$$ 82.1345 1.54971 0.774854 0.632141i $$-0.217824\pi$$
0.774854 + 0.632141i $$0.217824\pi$$
$$54$$ 0 0
$$55$$ − 2.77013i − 0.0503660i
$$56$$ 0 0
$$57$$ −18.4877 −0.324345
$$58$$ 0 0
$$59$$ − 92.5800i − 1.56915i −0.620032 0.784576i $$-0.712881\pi$$
0.620032 0.784576i $$-0.287119\pi$$
$$60$$ 0 0
$$61$$ 4.99187i 0.0818340i 0.999163 + 0.0409170i $$0.0130279\pi$$
−0.999163 + 0.0409170i $$0.986972\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.682497 0.0105000
$$66$$ 0 0
$$67$$ 2.20699 0.0329402 0.0164701 0.999864i $$-0.494757\pi$$
0.0164701 + 0.999864i $$0.494757\pi$$
$$68$$ 0 0
$$69$$ − 11.4312i − 0.165669i
$$70$$ 0 0
$$71$$ 80.5899 1.13507 0.567535 0.823349i $$-0.307897\pi$$
0.567535 + 0.823349i $$0.307897\pi$$
$$72$$ 0 0
$$73$$ − 13.9088i − 0.190531i −0.995452 0.0952657i $$-0.969630\pi$$
0.995452 0.0952657i $$-0.0303701\pi$$
$$74$$ 0 0
$$75$$ − 42.9011i − 0.572015i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 64.8885 0.821373 0.410687 0.911777i $$-0.365289\pi$$
0.410687 + 0.911777i $$0.365289\pi$$
$$80$$ 0 0
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ − 118.005i − 1.42174i −0.703322 0.710872i $$-0.748301\pi$$
0.703322 0.710872i $$-0.251699\pi$$
$$84$$ 0 0
$$85$$ −11.0875 −0.130441
$$86$$ 0 0
$$87$$ 10.7432i 0.123485i
$$88$$ 0 0
$$89$$ − 104.265i − 1.17151i −0.810487 0.585757i $$-0.800797\pi$$
0.810487 0.585757i $$-0.199203\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −72.6284 −0.780951
$$94$$ 0 0
$$95$$ 5.13052 0.0540055
$$96$$ 0 0
$$97$$ − 31.7875i − 0.327706i −0.986485 0.163853i $$-0.947608\pi$$
0.986485 0.163853i $$-0.0523923\pi$$
$$98$$ 0 0
$$99$$ 17.2895 0.174641
$$100$$ 0 0
$$101$$ 72.5037i 0.717859i 0.933365 + 0.358929i $$0.116858\pi$$
−0.933365 + 0.358929i $$0.883142\pi$$
$$102$$ 0 0
$$103$$ − 106.963i − 1.03847i −0.854631 0.519236i $$-0.826217\pi$$
0.854631 0.519236i $$-0.173783\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −196.619 −1.83756 −0.918782 0.394765i $$-0.870826\pi$$
−0.918782 + 0.394765i $$0.870826\pi$$
$$108$$ 0 0
$$109$$ 42.6922 0.391672 0.195836 0.980637i $$-0.437258\pi$$
0.195836 + 0.980637i $$0.437258\pi$$
$$110$$ 0 0
$$111$$ 104.084i 0.937693i
$$112$$ 0 0
$$113$$ 175.501 1.55310 0.776552 0.630053i $$-0.216967\pi$$
0.776552 + 0.630053i $$0.216967\pi$$
$$114$$ 0 0
$$115$$ 3.17227i 0.0275849i
$$116$$ 0 0
$$117$$ 4.25973i 0.0364080i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −87.7860 −0.725504
$$122$$ 0 0
$$123$$ 84.5674 0.687540
$$124$$ 0 0
$$125$$ 23.9220i 0.191376i
$$126$$ 0 0
$$127$$ −31.0434 −0.244436 −0.122218 0.992503i $$-0.539001\pi$$
−0.122218 + 0.992503i $$0.539001\pi$$
$$128$$ 0 0
$$129$$ − 89.3051i − 0.692288i
$$130$$ 0 0
$$131$$ 46.5095i 0.355034i 0.984118 + 0.177517i $$0.0568065\pi$$
−0.984118 + 0.177517i $$0.943194\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.49759 −0.0185007
$$136$$ 0 0
$$137$$ 45.4654 0.331864 0.165932 0.986137i $$-0.446937\pi$$
0.165932 + 0.986137i $$0.446937\pi$$
$$138$$ 0 0
$$139$$ 138.075i 0.993343i 0.867939 + 0.496672i $$0.165445\pi$$
−0.867939 + 0.496672i $$0.834555\pi$$
$$140$$ 0 0
$$141$$ 33.3283 0.236371
$$142$$ 0 0
$$143$$ 8.18318i 0.0572250i
$$144$$ 0 0
$$145$$ − 2.98134i − 0.0205610i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 239.228 1.60556 0.802780 0.596276i $$-0.203354\pi$$
0.802780 + 0.596276i $$0.203354\pi$$
$$150$$ 0 0
$$151$$ 188.146 1.24600 0.623001 0.782221i $$-0.285913\pi$$
0.623001 + 0.782221i $$0.285913\pi$$
$$152$$ 0 0
$$153$$ − 69.2014i − 0.452297i
$$154$$ 0 0
$$155$$ 20.1551 0.130033
$$156$$ 0 0
$$157$$ − 215.239i − 1.37095i −0.728097 0.685474i $$-0.759595\pi$$
0.728097 0.685474i $$-0.240405\pi$$
$$158$$ 0 0
$$159$$ − 142.261i − 0.894724i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −149.811 −0.919083 −0.459542 0.888156i $$-0.651986\pi$$
−0.459542 + 0.888156i $$0.651986\pi$$
$$164$$ 0 0
$$165$$ −4.79801 −0.0290788
$$166$$ 0 0
$$167$$ − 137.195i − 0.821528i −0.911742 0.410764i $$-0.865262\pi$$
0.911742 0.410764i $$-0.134738\pi$$
$$168$$ 0 0
$$169$$ 166.984 0.988070
$$170$$ 0 0
$$171$$ 32.0216i 0.187261i
$$172$$ 0 0
$$173$$ − 251.181i − 1.45191i −0.687740 0.725957i $$-0.741397\pi$$
0.687740 0.725957i $$-0.258603\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −160.353 −0.905951
$$178$$ 0 0
$$179$$ 116.793 0.652477 0.326239 0.945287i $$-0.394219\pi$$
0.326239 + 0.945287i $$0.394219\pi$$
$$180$$ 0 0
$$181$$ − 117.148i − 0.647228i −0.946189 0.323614i $$-0.895102\pi$$
0.946189 0.323614i $$-0.104898\pi$$
$$182$$ 0 0
$$183$$ 8.64618 0.0472469
$$184$$ 0 0
$$185$$ − 28.8843i − 0.156131i
$$186$$ 0 0
$$187$$ − 132.940i − 0.710907i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −341.818 −1.78962 −0.894812 0.446443i $$-0.852691\pi$$
−0.894812 + 0.446443i $$0.852691\pi$$
$$192$$ 0 0
$$193$$ −224.551 −1.16348 −0.581738 0.813376i $$-0.697627\pi$$
−0.581738 + 0.813376i $$0.697627\pi$$
$$194$$ 0 0
$$195$$ − 1.18212i − 0.00606215i
$$196$$ 0 0
$$197$$ 257.109 1.30512 0.652560 0.757737i $$-0.273695\pi$$
0.652560 + 0.757737i $$0.273695\pi$$
$$198$$ 0 0
$$199$$ 245.479i 1.23356i 0.787134 + 0.616782i $$0.211564\pi$$
−0.787134 + 0.616782i $$0.788436\pi$$
$$200$$ 0 0
$$201$$ − 3.82262i − 0.0190180i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −23.4683 −0.114480
$$206$$ 0 0
$$207$$ −19.7994 −0.0956492
$$208$$ 0 0
$$209$$ 61.5152i 0.294331i
$$210$$ 0 0
$$211$$ 95.8210 0.454128 0.227064 0.973880i $$-0.427087\pi$$
0.227064 + 0.973880i $$0.427087\pi$$
$$212$$ 0 0
$$213$$ − 139.586i − 0.655333i
$$214$$ 0 0
$$215$$ 24.7831i 0.115270i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −24.0907 −0.110003
$$220$$ 0 0
$$221$$ 32.7533 0.148205
$$222$$ 0 0
$$223$$ 94.2091i 0.422462i 0.977436 + 0.211231i $$0.0677473\pi$$
−0.977436 + 0.211231i $$0.932253\pi$$
$$224$$ 0 0
$$225$$ −74.3069 −0.330253
$$226$$ 0 0
$$227$$ 224.253i 0.987899i 0.869491 + 0.493949i $$0.164447\pi$$
−0.869491 + 0.493949i $$0.835553\pi$$
$$228$$ 0 0
$$229$$ − 366.724i − 1.60142i −0.599055 0.800708i $$-0.704457\pi$$
0.599055 0.800708i $$-0.295543\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 69.6741 0.299030 0.149515 0.988759i $$-0.452229\pi$$
0.149515 + 0.988759i $$0.452229\pi$$
$$234$$ 0 0
$$235$$ −9.24893 −0.0393572
$$236$$ 0 0
$$237$$ − 112.390i − 0.474220i
$$238$$ 0 0
$$239$$ 214.544 0.897674 0.448837 0.893614i $$-0.351838\pi$$
0.448837 + 0.893614i $$0.351838\pi$$
$$240$$ 0 0
$$241$$ − 163.395i − 0.677989i −0.940788 0.338994i $$-0.889913\pi$$
0.940788 0.338994i $$-0.110087\pi$$
$$242$$ 0 0
$$243$$ − 15.5885i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −15.1560 −0.0613601
$$248$$ 0 0
$$249$$ −204.390 −0.820844
$$250$$ 0 0
$$251$$ − 330.546i − 1.31692i −0.752617 0.658458i $$-0.771209\pi$$
0.752617 0.658458i $$-0.228791\pi$$
$$252$$ 0 0
$$253$$ −38.0357 −0.150339
$$254$$ 0 0
$$255$$ 19.2041i 0.0753101i
$$256$$ 0 0
$$257$$ − 160.529i − 0.624626i −0.949979 0.312313i $$-0.898896\pi$$
0.949979 0.312313i $$-0.101104\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 18.6077 0.0712940
$$262$$ 0 0
$$263$$ 106.515 0.405002 0.202501 0.979282i $$-0.435093\pi$$
0.202501 + 0.979282i $$0.435093\pi$$
$$264$$ 0 0
$$265$$ 39.4789i 0.148977i
$$266$$ 0 0
$$267$$ −180.592 −0.676374
$$268$$ 0 0
$$269$$ 155.454i 0.577895i 0.957345 + 0.288947i $$0.0933052\pi$$
−0.957345 + 0.288947i $$0.906695\pi$$
$$270$$ 0 0
$$271$$ − 516.080i − 1.90435i −0.305549 0.952176i $$-0.598840\pi$$
0.305549 0.952176i $$-0.401160\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −142.748 −0.519082
$$276$$ 0 0
$$277$$ 200.202 0.722750 0.361375 0.932421i $$-0.382308\pi$$
0.361375 + 0.932421i $$0.382308\pi$$
$$278$$ 0 0
$$279$$ 125.796i 0.450882i
$$280$$ 0 0
$$281$$ 228.093 0.811720 0.405860 0.913935i $$-0.366972\pi$$
0.405860 + 0.913935i $$0.366972\pi$$
$$282$$ 0 0
$$283$$ − 260.710i − 0.921236i −0.887598 0.460618i $$-0.847628\pi$$
0.887598 0.460618i $$-0.152372\pi$$
$$284$$ 0 0
$$285$$ − 8.88632i − 0.0311801i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −243.092 −0.841150
$$290$$ 0 0
$$291$$ −55.0576 −0.189201
$$292$$ 0 0
$$293$$ − 349.885i − 1.19415i −0.802186 0.597074i $$-0.796330\pi$$
0.802186 0.597074i $$-0.203670\pi$$
$$294$$ 0 0
$$295$$ 44.4996 0.150846
$$296$$ 0 0
$$297$$ − 29.9463i − 0.100829i
$$298$$ 0 0
$$299$$ − 9.37113i − 0.0313416i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 125.580 0.414456
$$304$$ 0 0
$$305$$ −2.39940 −0.00786689
$$306$$ 0 0
$$307$$ − 146.898i − 0.478495i −0.970959 0.239247i $$-0.923099\pi$$
0.970959 0.239247i $$-0.0769007\pi$$
$$308$$ 0 0
$$309$$ −185.265 −0.599562
$$310$$ 0 0
$$311$$ − 80.7340i − 0.259595i −0.991541 0.129797i $$-0.958567\pi$$
0.991541 0.129797i $$-0.0414327\pi$$
$$312$$ 0 0
$$313$$ − 154.339i − 0.493096i −0.969131 0.246548i $$-0.920704\pi$$
0.969131 0.246548i $$-0.0792963\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −188.545 −0.594780 −0.297390 0.954756i $$-0.596116\pi$$
−0.297390 + 0.954756i $$0.596116\pi$$
$$318$$ 0 0
$$319$$ 35.7465 0.112058
$$320$$ 0 0
$$321$$ 340.555i 1.06092i
$$322$$ 0 0
$$323$$ 246.216 0.762277
$$324$$ 0 0
$$325$$ − 35.1697i − 0.108215i
$$326$$ 0 0
$$327$$ − 73.9451i − 0.226132i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −146.787 −0.443466 −0.221733 0.975107i $$-0.571171\pi$$
−0.221733 + 0.975107i $$0.571171\pi$$
$$332$$ 0 0
$$333$$ 180.279 0.541377
$$334$$ 0 0
$$335$$ 1.06082i 0.00316662i
$$336$$ 0 0
$$337$$ 101.231 0.300388 0.150194 0.988657i $$-0.452010\pi$$
0.150194 + 0.988657i $$0.452010\pi$$
$$338$$ 0 0
$$339$$ − 303.976i − 0.896685i
$$340$$ 0 0
$$341$$ 241.661i 0.708684i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 5.49453 0.0159262
$$346$$ 0 0
$$347$$ −351.689 −1.01351 −0.506756 0.862089i $$-0.669156\pi$$
−0.506756 + 0.862089i $$0.669156\pi$$
$$348$$ 0 0
$$349$$ − 88.3780i − 0.253232i −0.991952 0.126616i $$-0.959588\pi$$
0.991952 0.126616i $$-0.0404116\pi$$
$$350$$ 0 0
$$351$$ 7.37808 0.0210202
$$352$$ 0 0
$$353$$ 520.542i 1.47462i 0.675552 + 0.737312i $$0.263905\pi$$
−0.675552 + 0.737312i $$0.736095\pi$$
$$354$$ 0 0
$$355$$ 38.7365i 0.109117i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 514.265 1.43249 0.716246 0.697848i $$-0.245859\pi$$
0.716246 + 0.697848i $$0.245859\pi$$
$$360$$ 0 0
$$361$$ 247.069 0.684400
$$362$$ 0 0
$$363$$ 152.050i 0.418870i
$$364$$ 0 0
$$365$$ 6.68542 0.0183162
$$366$$ 0 0
$$367$$ − 417.686i − 1.13811i −0.822300 0.569054i $$-0.807310\pi$$
0.822300 0.569054i $$-0.192690\pi$$
$$368$$ 0 0
$$369$$ − 146.475i − 0.396951i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −140.314 −0.376176 −0.188088 0.982152i $$-0.560229\pi$$
−0.188088 + 0.982152i $$0.560229\pi$$
$$374$$ 0 0
$$375$$ 41.4342 0.110491
$$376$$ 0 0
$$377$$ 8.80712i 0.0233611i
$$378$$ 0 0
$$379$$ −153.298 −0.404480 −0.202240 0.979336i $$-0.564822\pi$$
−0.202240 + 0.979336i $$0.564822\pi$$
$$380$$ 0 0
$$381$$ 53.7687i 0.141125i
$$382$$ 0 0
$$383$$ − 121.717i − 0.317799i −0.987295 0.158900i $$-0.949205\pi$$
0.987295 0.158900i $$-0.0507946\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −154.681 −0.399692
$$388$$ 0 0
$$389$$ 401.271 1.03155 0.515773 0.856725i $$-0.327505\pi$$
0.515773 + 0.856725i $$0.327505\pi$$
$$390$$ 0 0
$$391$$ 152.238i 0.389356i
$$392$$ 0 0
$$393$$ 80.5568 0.204979
$$394$$ 0 0
$$395$$ 31.1894i 0.0789605i
$$396$$ 0 0
$$397$$ 60.2251i 0.151700i 0.997119 + 0.0758502i $$0.0241671\pi$$
−0.997119 + 0.0758502i $$0.975833\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −374.308 −0.933436 −0.466718 0.884406i $$-0.654564\pi$$
−0.466718 + 0.884406i $$0.654564\pi$$
$$402$$ 0 0
$$403$$ −59.5398 −0.147741
$$404$$ 0 0
$$405$$ 4.32595i 0.0106814i
$$406$$ 0 0
$$407$$ 346.325 0.850921
$$408$$ 0 0
$$409$$ 614.609i 1.50271i 0.659897 + 0.751356i $$0.270600\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$410$$ 0 0
$$411$$ − 78.7484i − 0.191602i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 56.7203 0.136675
$$416$$ 0 0
$$417$$ 239.152 0.573507
$$418$$ 0 0
$$419$$ − 129.067i − 0.308035i −0.988068 0.154017i $$-0.950779\pi$$
0.988068 0.154017i $$-0.0492212\pi$$
$$420$$ 0 0
$$421$$ −697.880 −1.65767 −0.828836 0.559492i $$-0.810996\pi$$
−0.828836 + 0.559492i $$0.810996\pi$$
$$422$$ 0 0
$$423$$ − 57.7263i − 0.136469i
$$424$$ 0 0
$$425$$ 571.349i 1.34435i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 14.1737 0.0330389
$$430$$ 0 0
$$431$$ 277.092 0.642906 0.321453 0.946926i $$-0.395829\pi$$
0.321453 + 0.946926i $$0.395829\pi$$
$$432$$ 0 0
$$433$$ 822.794i 1.90022i 0.311919 + 0.950109i $$0.399028\pi$$
−0.311919 + 0.950109i $$0.600972\pi$$
$$434$$ 0 0
$$435$$ −5.16384 −0.0118709
$$436$$ 0 0
$$437$$ − 70.4454i − 0.161202i
$$438$$ 0 0
$$439$$ 365.827i 0.833320i 0.909062 + 0.416660i $$0.136799\pi$$
−0.909062 + 0.416660i $$0.863201\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 573.392 1.29434 0.647169 0.762346i $$-0.275953\pi$$
0.647169 + 0.762346i $$0.275953\pi$$
$$444$$ 0 0
$$445$$ 50.1161 0.112620
$$446$$ 0 0
$$447$$ − 414.356i − 0.926970i
$$448$$ 0 0
$$449$$ 227.961 0.507708 0.253854 0.967243i $$-0.418302\pi$$
0.253854 + 0.967243i $$0.418302\pi$$
$$450$$ 0 0
$$451$$ − 281.386i − 0.623917i
$$452$$ 0 0
$$453$$ − 325.879i − 0.719380i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −343.124 −0.750818 −0.375409 0.926859i $$-0.622498\pi$$
−0.375409 + 0.926859i $$0.622498\pi$$
$$458$$ 0 0
$$459$$ −119.860 −0.261134
$$460$$ 0 0
$$461$$ − 611.314i − 1.32606i −0.748593 0.663030i $$-0.769270\pi$$
0.748593 0.663030i $$-0.230730\pi$$
$$462$$ 0 0
$$463$$ 67.2682 0.145288 0.0726439 0.997358i $$-0.476856\pi$$
0.0726439 + 0.997358i $$0.476856\pi$$
$$464$$ 0 0
$$465$$ − 34.9097i − 0.0750746i
$$466$$ 0 0
$$467$$ 93.7498i 0.200749i 0.994950 + 0.100375i $$0.0320041\pi$$
−0.994950 + 0.100375i $$0.967996\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −372.804 −0.791517
$$472$$ 0 0
$$473$$ −297.150 −0.628225
$$474$$ 0 0
$$475$$ − 264.381i − 0.556591i
$$476$$ 0 0
$$477$$ −246.403 −0.516569
$$478$$ 0 0
$$479$$ 353.796i 0.738613i 0.929308 + 0.369306i $$0.120405\pi$$
−0.929308 + 0.369306i $$0.879595\pi$$
$$480$$ 0 0
$$481$$ 85.3265i 0.177394i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 15.2790 0.0315032
$$486$$ 0 0
$$487$$ −952.677 −1.95621 −0.978107 0.208101i $$-0.933272\pi$$
−0.978107 + 0.208101i $$0.933272\pi$$
$$488$$ 0 0
$$489$$ 259.479i 0.530633i
$$490$$ 0 0
$$491$$ 818.649 1.66731 0.833655 0.552286i $$-0.186244\pi$$
0.833655 + 0.552286i $$0.186244\pi$$
$$492$$ 0 0
$$493$$ − 143.076i − 0.290214i
$$494$$ 0 0
$$495$$ 8.31039i 0.0167887i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 24.0821 0.0482607 0.0241304 0.999709i $$-0.492318\pi$$
0.0241304 + 0.999709i $$0.492318\pi$$
$$500$$ 0 0
$$501$$ −237.629 −0.474310
$$502$$ 0 0
$$503$$ 420.445i 0.835874i 0.908476 + 0.417937i $$0.137247\pi$$
−0.908476 + 0.417937i $$0.862753\pi$$
$$504$$ 0 0
$$505$$ −34.8498 −0.0690094
$$506$$ 0 0
$$507$$ − 289.225i − 0.570463i
$$508$$ 0 0
$$509$$ 239.869i 0.471255i 0.971843 + 0.235627i $$0.0757145\pi$$
−0.971843 + 0.235627i $$0.924285\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 55.4631 0.108115
$$514$$ 0 0
$$515$$ 51.4128 0.0998307
$$516$$ 0 0
$$517$$ − 110.895i − 0.214498i
$$518$$ 0 0
$$519$$ −435.059 −0.838263
$$520$$ 0 0
$$521$$ − 26.8565i − 0.0515479i −0.999668 0.0257739i $$-0.991795\pi$$
0.999668 0.0257739i $$-0.00820501\pi$$
$$522$$ 0 0
$$523$$ 223.737i 0.427796i 0.976856 + 0.213898i $$0.0686160\pi$$
−0.976856 + 0.213898i $$0.931384\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 967.252 1.83539
$$528$$ 0 0
$$529$$ −485.443 −0.917661
$$530$$ 0 0
$$531$$ 277.740i 0.523051i
$$532$$ 0 0
$$533$$ 69.3272 0.130070
$$534$$ 0 0
$$535$$ − 94.5074i − 0.176649i
$$536$$ 0 0
$$537$$ − 202.292i − 0.376708i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 986.190 1.82290 0.911451 0.411410i $$-0.134963\pi$$
0.911451 + 0.411410i $$0.134963\pi$$
$$542$$ 0 0
$$543$$ −202.907 −0.373678
$$544$$ 0 0
$$545$$ 20.5205i 0.0376523i
$$546$$ 0 0
$$547$$ −735.369 −1.34437 −0.672183 0.740385i $$-0.734643\pi$$
−0.672183 + 0.740385i $$0.734643\pi$$
$$548$$ 0 0
$$549$$ − 14.9756i − 0.0272780i
$$550$$ 0 0
$$551$$ 66.2056i 0.120155i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −50.0291 −0.0901426
$$556$$ 0 0
$$557$$ 418.568 0.751469 0.375735 0.926727i $$-0.377390\pi$$
0.375735 + 0.926727i $$0.377390\pi$$
$$558$$ 0 0
$$559$$ − 73.2111i − 0.130968i
$$560$$ 0 0
$$561$$ −230.258 −0.410442
$$562$$ 0 0
$$563$$ 746.738i 1.32635i 0.748462 + 0.663177i $$0.230792\pi$$
−0.748462 + 0.663177i $$0.769208\pi$$
$$564$$ 0 0
$$565$$ 84.3565i 0.149304i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −288.710 −0.507398 −0.253699 0.967283i $$-0.581647\pi$$
−0.253699 + 0.967283i $$0.581647\pi$$
$$570$$ 0 0
$$571$$ 916.207 1.60457 0.802283 0.596944i $$-0.203618\pi$$
0.802283 + 0.596944i $$0.203618\pi$$
$$572$$ 0 0
$$573$$ 592.047i 1.03324i
$$574$$ 0 0
$$575$$ 163.470 0.284296
$$576$$ 0 0
$$577$$ − 992.068i − 1.71936i −0.510836 0.859678i $$-0.670664\pi$$
0.510836 0.859678i $$-0.329336\pi$$
$$578$$ 0 0
$$579$$ 388.934i 0.671733i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −473.354 −0.811929
$$584$$ 0 0
$$585$$ −2.04749 −0.00349998
$$586$$ 0 0
$$587$$ 154.965i 0.263996i 0.991250 + 0.131998i $$0.0421392\pi$$
−0.991250 + 0.131998i $$0.957861\pi$$
$$588$$ 0 0
$$589$$ −447.577 −0.759893
$$590$$ 0 0
$$591$$ − 445.325i − 0.753512i
$$592$$ 0 0
$$593$$ 1004.17i 1.69337i 0.532094 + 0.846685i $$0.321405\pi$$
−0.532094 + 0.846685i $$0.678595\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 425.182 0.712198
$$598$$ 0 0
$$599$$ 796.917 1.33041 0.665206 0.746660i $$-0.268344\pi$$
0.665206 + 0.746660i $$0.268344\pi$$
$$600$$ 0 0
$$601$$ 467.002i 0.777041i 0.921440 + 0.388521i $$0.127014\pi$$
−0.921440 + 0.388521i $$0.872986\pi$$
$$602$$ 0 0
$$603$$ −6.62098 −0.0109801
$$604$$ 0 0
$$605$$ − 42.1953i − 0.0697444i
$$606$$ 0 0
$$607$$ 910.216i 1.49953i 0.661703 + 0.749766i $$0.269834\pi$$
−0.661703 + 0.749766i $$0.730166\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 27.3221 0.0447170
$$612$$ 0 0
$$613$$ 775.800 1.26558 0.632790 0.774324i $$-0.281910\pi$$
0.632790 + 0.774324i $$0.281910\pi$$
$$614$$ 0 0
$$615$$ 40.6483i 0.0660948i
$$616$$ 0 0
$$617$$ 974.803 1.57991 0.789954 0.613166i $$-0.210104\pi$$
0.789954 + 0.613166i $$0.210104\pi$$
$$618$$ 0 0
$$619$$ − 199.260i − 0.321906i −0.986962 0.160953i $$-0.948543\pi$$
0.986962 0.160953i $$-0.0514568\pi$$
$$620$$ 0 0
$$621$$ 34.2935i 0.0552231i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 607.726 0.972361
$$626$$ 0 0
$$627$$ 106.548 0.169932
$$628$$ 0 0
$$629$$ − 1386.17i − 2.20377i
$$630$$ 0 0
$$631$$ −751.062 −1.19027 −0.595136 0.803625i $$-0.702902\pi$$
−0.595136 + 0.803625i $$0.702902\pi$$
$$632$$ 0 0
$$633$$ − 165.967i − 0.262191i
$$634$$ 0 0
$$635$$ − 14.9214i − 0.0234982i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −241.770 −0.378356
$$640$$ 0 0
$$641$$ −1138.28 −1.77578 −0.887891 0.460053i $$-0.847830\pi$$
−0.887891 + 0.460053i $$0.847830\pi$$
$$642$$ 0 0
$$643$$ 647.823i 1.00750i 0.863849 + 0.503751i $$0.168047\pi$$
−0.863849 + 0.503751i $$0.831953\pi$$
$$644$$ 0 0
$$645$$ 42.9255 0.0665512
$$646$$ 0 0
$$647$$ 882.943i 1.36467i 0.731039 + 0.682336i $$0.239036\pi$$
−0.731039 + 0.682336i $$0.760964\pi$$
$$648$$ 0 0
$$649$$ 533.554i 0.822116i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 545.117 0.834788 0.417394 0.908726i $$-0.362944\pi$$
0.417394 + 0.908726i $$0.362944\pi$$
$$654$$ 0 0
$$655$$ −22.3553 −0.0341302
$$656$$ 0 0
$$657$$ 41.7264i 0.0635104i
$$658$$ 0 0
$$659$$ −698.290 −1.05962 −0.529810 0.848116i $$-0.677737\pi$$
−0.529810 + 0.848116i $$0.677737\pi$$
$$660$$ 0 0
$$661$$ 571.725i 0.864940i 0.901648 + 0.432470i $$0.142358\pi$$
−0.901648 + 0.432470i $$0.857642\pi$$
$$662$$ 0 0
$$663$$ − 56.7303i − 0.0855661i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −40.9358 −0.0613730
$$668$$ 0 0
$$669$$ 163.175 0.243909
$$670$$ 0 0
$$671$$ − 28.7690i − 0.0428748i
$$672$$ 0 0
$$673$$ 221.015 0.328403 0.164202 0.986427i $$-0.447495\pi$$
0.164202 + 0.986427i $$0.447495\pi$$
$$674$$ 0 0
$$675$$ 128.703i 0.190672i
$$676$$ 0 0
$$677$$ 470.666i 0.695222i 0.937639 + 0.347611i $$0.113007\pi$$
−0.937639 + 0.347611i $$0.886993\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 388.418 0.570364
$$682$$ 0 0
$$683$$ −215.960 −0.316193 −0.158097 0.987424i $$-0.550536\pi$$
−0.158097 + 0.987424i $$0.550536\pi$$
$$684$$ 0 0
$$685$$ 21.8535i 0.0319029i
$$686$$ 0 0
$$687$$ −635.185 −0.924578
$$688$$ 0 0
$$689$$ − 116.624i − 0.169265i
$$690$$ 0 0
$$691$$ − 80.1141i − 0.115939i −0.998318 0.0579697i $$-0.981537\pi$$
0.998318 0.0579697i $$-0.0184627\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −66.3672 −0.0954924
$$696$$ 0 0
$$697$$ −1126.25 −1.61586
$$698$$ 0 0
$$699$$ − 120.679i − 0.172645i
$$700$$ 0 0
$$701$$ −821.973 −1.17257 −0.586286 0.810104i $$-0.699411\pi$$
−0.586286 + 0.810104i $$0.699411\pi$$
$$702$$ 0 0
$$703$$ 641.423i 0.912409i
$$704$$ 0 0
$$705$$ 16.0196i 0.0227229i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 575.456 0.811644 0.405822 0.913952i $$-0.366985\pi$$
0.405822 + 0.913952i $$0.366985\pi$$
$$710$$ 0 0
$$711$$ −194.665 −0.273791
$$712$$ 0 0
$$713$$ − 276.743i − 0.388139i
$$714$$ 0 0
$$715$$ −3.93334 −0.00550117
$$716$$ 0 0
$$717$$ − 371.601i − 0.518272i
$$718$$ 0 0
$$719$$ 1213.93i 1.68836i 0.536063 + 0.844178i $$0.319911\pi$$
−0.536063 + 0.844178i $$0.680089\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −283.009 −0.391437
$$724$$ 0 0
$$725$$ −153.632 −0.211906
$$726$$ 0 0
$$727$$ 379.498i 0.522005i 0.965338 + 0.261003i $$0.0840532\pi$$
−0.965338 + 0.261003i $$0.915947\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −0.0370370
$$730$$ 0 0
$$731$$ 1189.35i 1.62702i
$$732$$ 0 0
$$733$$ − 1250.46i − 1.70595i −0.521954 0.852973i $$-0.674797\pi$$
0.521954 0.852973i $$-0.325203\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.7193 −0.0172582
$$738$$ 0 0
$$739$$ −501.234 −0.678260 −0.339130 0.940740i $$-0.610133\pi$$
−0.339130 + 0.940740i $$0.610133\pi$$
$$740$$ 0 0
$$741$$ 26.2509i 0.0354263i
$$742$$ 0 0
$$743$$ −444.584 −0.598363 −0.299181 0.954196i $$-0.596714\pi$$
−0.299181 + 0.954196i $$0.596714\pi$$
$$744$$ 0 0
$$745$$ 114.988i 0.154346i
$$746$$ 0 0
$$747$$ 354.014i 0.473914i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −239.335 −0.318688 −0.159344 0.987223i $$-0.550938\pi$$
−0.159344 + 0.987223i $$0.550938\pi$$
$$752$$ 0 0
$$753$$ −572.522 −0.760322
$$754$$ 0 0
$$755$$ 90.4347i 0.119781i
$$756$$ 0 0
$$757$$ 249.486 0.329572 0.164786 0.986329i $$-0.447307\pi$$
0.164786 + 0.986329i $$0.447307\pi$$
$$758$$ 0 0
$$759$$ 65.8797i 0.0867981i
$$760$$ 0 0
$$761$$ 1375.68i 1.80773i 0.427821 + 0.903864i $$0.359281\pi$$
−0.427821 + 0.903864i $$0.640719\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 33.2624 0.0434803
$$766$$ 0 0
$$767$$ −131.455 −0.171389
$$768$$ 0 0
$$769$$ 528.594i 0.687379i 0.939083 + 0.343689i $$0.111677\pi$$
−0.939083 + 0.343689i $$0.888323\pi$$
$$770$$ 0 0
$$771$$ −278.044 −0.360628
$$772$$ 0 0
$$773$$ 61.9934i 0.0801984i 0.999196 + 0.0400992i $$0.0127674\pi$$
−0.999196 + 0.0400992i $$0.987233\pi$$
$$774$$ 0 0
$$775$$ − 1038.61i − 1.34015i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 521.152 0.669001
$$780$$ 0 0
$$781$$ −464.453 −0.594690
$$782$$ 0 0
$$783$$ − 32.2296i − 0.0411616i
$$784$$ 0 0
$$785$$ 103.457 0.131792
$$786$$ 0 0
$$787$$ 163.568i 0.207837i 0.994586 + 0.103919i $$0.0331381\pi$$
−0.994586 + 0.103919i $$0.966862\pi$$
$$788$$ 0 0
$$789$$ − 184.490i − 0.233828i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 7.08802 0.00893823
$$794$$ 0 0
$$795$$ 68.3794 0.0860119
$$796$$ 0 0
$$797$$ − 533.394i − 0.669253i −0.942351 0.334626i $$-0.891390\pi$$
0.942351 0.334626i $$-0.108610\pi$$
$$798$$ 0 0
$$799$$ −443.860 −0.555519
$$800$$ 0 0
$$801$$ 312.794i 0.390505i
$$802$$ 0 0
$$803$$ 80.1586i 0.0998239i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 269.254 0.333648
$$808$$ 0 0
$$809$$ −550.274 −0.680191 −0.340095 0.940391i $$-0.610459\pi$$
−0.340095 + 0.940391i $$0.610459\pi$$
$$810$$ 0 0
$$811$$ − 415.532i − 0.512370i −0.966628 0.256185i $$-0.917534\pi$$
0.966628 0.256185i $$-0.0824656\pi$$
$$812$$ 0 0
$$813$$ −893.876 −1.09948
$$814$$ 0 0
$$815$$ − 72.0082i − 0.0883536i
$$816$$ 0 0
$$817$$ − 550.348i − 0.673621i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 869.191 1.05870 0.529349 0.848404i $$-0.322436\pi$$
0.529349 + 0.848404i $$0.322436\pi$$
$$822$$ 0 0
$$823$$ 421.783 0.512494 0.256247 0.966611i $$-0.417514\pi$$
0.256247 + 0.966611i $$0.417514\pi$$
$$824$$ 0 0
$$825$$ 247.246i 0.299692i
$$826$$ 0 0
$$827$$ −70.7290 −0.0855248 −0.0427624 0.999085i $$-0.513616\pi$$
−0.0427624 + 0.999085i $$0.513616\pi$$
$$828$$ 0 0
$$829$$ − 1268.85i − 1.53058i −0.643685 0.765290i $$-0.722595\pi$$
0.643685 0.765290i $$-0.277405\pi$$
$$830$$ 0 0
$$831$$ − 346.759i − 0.417280i
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 65.9445 0.0789754
$$836$$ 0 0
$$837$$ 217.885 0.260317
$$838$$ 0 0
$$839$$ − 613.254i − 0.730935i −0.930824 0.365467i $$-0.880909\pi$$
0.930824 0.365467i $$-0.119091\pi$$
$$840$$ 0 0
$$841$$ −802.528 −0.954254
$$842$$ 0 0
$$843$$ − 395.069i − 0.468647i
$$844$$ 0 0
$$845$$ 80.2627i 0.0949855i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −451.563 −0.531876
$$850$$ 0 0
$$851$$ −396.601 −0.466041
$$852$$ 0 0
$$853$$ − 668.244i − 0.783404i −0.920092 0.391702i $$-0.871886\pi$$
0.920092 0.391702i $$-0.128114\pi$$
$$854$$ 0 0
$$855$$ −15.3916 −0.0180018
$$856$$ 0 0
$$857$$ 519.712i 0.606431i 0.952922 + 0.303216i $$0.0980602\pi$$
−0.952922 + 0.303216i $$0.901940\pi$$
$$858$$ 0 0
$$859$$ 715.553i 0.833007i 0.909134 + 0.416504i $$0.136745\pi$$
−0.909134 + 0.416504i $$0.863255\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 342.716 0.397122 0.198561 0.980089i $$-0.436373\pi$$
0.198561 + 0.980089i $$0.436373\pi$$
$$864$$ 0 0
$$865$$ 120.733 0.139576
$$866$$ 0 0
$$867$$ 421.048i 0.485638i
$$868$$ 0 0
$$869$$ −373.963 −0.430337
$$870$$ 0 0
$$871$$ − 3.13374i − 0.00359786i
$$872$$ 0 0
$$873$$ 95.3625i 0.109235i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −158.550 −0.180787 −0.0903934 0.995906i $$-0.528812\pi$$
−0.0903934 + 0.995906i $$0.528812\pi$$
$$878$$ 0 0
$$879$$ −606.019 −0.689442
$$880$$ 0 0
$$881$$ 1100.63i 1.24930i 0.780906 + 0.624648i $$0.214758\pi$$
−0.780906 + 0.624648i $$0.785242\pi$$
$$882$$ 0 0
$$883$$ −255.888 −0.289794 −0.144897 0.989447i $$-0.546285\pi$$
−0.144897 + 0.989447i $$0.546285\pi$$
$$884$$ 0 0
$$885$$ − 77.0756i − 0.0870911i
$$886$$ 0 0
$$887$$ 1569.23i 1.76915i 0.466400 + 0.884574i $$0.345551\pi$$
−0.466400 + 0.884574i $$0.654449\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −51.8685 −0.0582138
$$892$$ 0 0
$$893$$ 205.388 0.229997
$$894$$ 0 0
$$895$$ 56.1381i 0.0627241i
$$896$$ 0 0
$$897$$ −16.2313 −0.0180951
$$898$$ 0 0
$$899$$ 260.087i 0.289307i
$$900$$ 0 0
$$901$$ 1894.61i 2.10278i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 56.3087 0.0622196
$$906$$ 0 0
$$907$$ −876.460 −0.966328 −0.483164 0.875530i $$-0.660513\pi$$
−0.483164 + 0.875530i $$0.660513\pi$$
$$908$$ 0 0
$$909$$ − 217.511i − 0.239286i
$$910$$ 0 0
$$911$$ 1778.49 1.95224 0.976118 0.217241i $$-0.0697057\pi$$
0.976118 + 0.217241i $$0.0697057\pi$$
$$912$$ 0 0
$$913$$ 680.080i 0.744885i
$$914$$ 0 0
$$915$$ 4.15589i 0.00454195i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 264.863 0.288208 0.144104 0.989563i $$-0.453970\pi$$
0.144104 + 0.989563i $$0.453970\pi$$
$$920$$ 0 0
$$921$$ −254.435 −0.276259
$$922$$ 0 0
$$923$$ − 114.431i − 0.123977i
$$924$$ 0 0
$$925$$ −1488.44 −1.60912
$$926$$ 0 0
$$927$$ 320.888i 0.346157i
$$928$$ 0 0
$$929$$ − 789.390i − 0.849720i −0.905259 0.424860i $$-0.860323\pi$$
0.905259 0.424860i $$-0.139677\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −139.835 −0.149877
$$934$$ 0 0
$$935$$ 63.8989 0.0683411
$$936$$ 0 0
$$937$$ 1446.06i 1.54329i 0.636053 + 0.771645i $$0.280566\pi$$
−0.636053 + 0.771645i $$0.719434\pi$$
$$938$$ 0 0
$$939$$ −267.323 −0.284689
$$940$$ 0 0
$$941$$ 1201.10i 1.27640i 0.769869 + 0.638202i $$0.220322\pi$$
−0.769869 + 0.638202i $$0.779678\pi$$
$$942$$ 0 0
$$943$$ 322.235i 0.341713i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 838.720 0.885660 0.442830 0.896606i $$-0.353974\pi$$
0.442830 + 0.896606i $$0.353974\pi$$
$$948$$ 0 0
$$949$$ −19.7492 −0.0208106
$$950$$ 0 0
$$951$$ 326.570i 0.343396i
$$952$$ 0 0
$$953$$ 1377.68 1.44562 0.722812 0.691045i $$-0.242849\pi$$
0.722812 + 0.691045i $$0.242849\pi$$
$$954$$ 0 0
$$955$$ − 164.299i − 0.172041i
$$956$$ 0 0
$$957$$ − 61.9147i − 0.0646967i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −797.295 −0.829652
$$962$$ 0 0
$$963$$ 589.858 0.612521
$$964$$ 0 0
$$965$$ − 107.933i − 0.111848i
$$966$$ 0 0
$$967$$ −848.834 −0.877802 −0.438901 0.898536i $$-0.644632\pi$$
−0.438901 + 0.898536i $$0.644632\pi$$
$$968$$ 0 0
$$969$$ − 426.458i − 0.440101i
$$970$$ 0 0
$$971$$ − 1262.08i − 1.29977i −0.760031 0.649887i $$-0.774816\pi$$
0.760031 0.649887i $$-0.225184\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −60.9158 −0.0624777
$$976$$ 0 0
$$977$$ 644.168 0.659332 0.329666 0.944098i $$-0.393064\pi$$
0.329666 + 0.944098i $$0.393064\pi$$
$$978$$ 0 0
$$979$$ 600.895i 0.613784i
$$980$$ 0 0
$$981$$ −128.077 −0.130557
$$982$$ 0 0
$$983$$ − 1213.77i − 1.23476i −0.786667 0.617378i $$-0.788195\pi$$
0.786667 0.617378i $$-0.211805\pi$$
$$984$$ 0 0
$$985$$ 123.582i 0.125464i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 340.288 0.344072
$$990$$ 0 0
$$991$$ −1691.12 −1.70648 −0.853241 0.521517i $$-0.825366\pi$$
−0.853241 + 0.521517i $$0.825366\pi$$
$$992$$ 0 0
$$993$$ 254.243i 0.256035i
$$994$$ 0 0
$$995$$ −117.992 −0.118585
$$996$$ 0 0
$$997$$ 1129.58i 1.13297i 0.824071 + 0.566487i $$0.191698\pi$$
−0.824071 + 0.566487i $$0.808302\pi$$
$$998$$ 0 0
$$999$$ − 312.252i − 0.312564i
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 2352.3.f.j.97.3 8
4.3 odd 2 588.3.d.c.97.7 yes 8
7.6 odd 2 inner 2352.3.f.j.97.6 8
12.11 even 2 1764.3.d.h.685.4 8
28.3 even 6 588.3.m.e.313.3 8
28.11 odd 6 588.3.m.f.313.2 8
28.19 even 6 588.3.m.f.325.2 8
28.23 odd 6 588.3.m.e.325.3 8
28.27 even 2 588.3.d.c.97.2 8
84.11 even 6 1764.3.z.l.901.3 8
84.23 even 6 1764.3.z.m.325.2 8
84.47 odd 6 1764.3.z.l.325.3 8
84.59 odd 6 1764.3.z.m.901.2 8
84.83 odd 2 1764.3.d.h.685.5 8

By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.d.c.97.2 8 28.27 even 2
588.3.d.c.97.7 yes 8 4.3 odd 2
588.3.m.e.313.3 8 28.3 even 6
588.3.m.e.325.3 8 28.23 odd 6
588.3.m.f.313.2 8 28.11 odd 6
588.3.m.f.325.2 8 28.19 even 6
1764.3.d.h.685.4 8 12.11 even 2
1764.3.d.h.685.5 8 84.83 odd 2
1764.3.z.l.325.3 8 84.47 odd 6
1764.3.z.l.901.3 8 84.11 even 6
1764.3.z.m.325.2 8 84.23 even 6
1764.3.z.m.901.2 8 84.59 odd 6
2352.3.f.j.97.3 8 1.1 even 1 trivial
2352.3.f.j.97.6 8 7.6 odd 2 inner