# Properties

 Label 4650.2.a.bg.1.1 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4650.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -2.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +12.0000 q^{37} -8.00000 q^{38} -4.00000 q^{39} +10.0000 q^{41} -2.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} +4.00000 q^{58} +2.00000 q^{59} +10.0000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +6.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +12.0000 q^{74} -8.00000 q^{76} -8.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} -4.00000 q^{87} -4.00000 q^{88} +8.00000 q^{91} +8.00000 q^{92} +1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +18.0000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ −4.00000 −0.852803
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000 0.377964
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 12.0000 1.97279 0.986394 0.164399i $$-0.0525685\pi$$
0.986394 + 0.164399i $$0.0525685\pi$$
$$38$$ −8.00000 −1.29777
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 4.00000 0.554700
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 8.00000 1.05963
$$58$$ 4.00000 0.525226
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −1.00000 −0.127000
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 6.00000 0.733017 0.366508 0.930415i $$-0.380553\pi$$
0.366508 + 0.930415i $$0.380553\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 12.0000 1.39497
$$75$$ 0 0
$$76$$ −8.00000 −0.917663
$$77$$ −8.00000 −0.911685
$$78$$ −4.00000 −0.452911
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ −4.00000 −0.428845
$$88$$ −4.00000 −0.426401
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 8.00000 0.834058
$$93$$ 1.00000 0.103695
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ 2.00000 0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 4.00000 0.369800
$$118$$ 2.00000 0.184115
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ −10.0000 −0.901670
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −16.0000 −1.38738
$$134$$ 6.00000 0.518321
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 6.00000 0.503509
$$143$$ −16.0000 −1.33799
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 3.00000 0.247436
$$148$$ 12.0000 0.986394
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −8.00000 −0.648886
$$153$$ −2.00000 −0.161690
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 1.00000 0.0785674
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ −8.00000 −0.609994
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ −2.00000 −0.150329
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 8.00000 0.592999
$$183$$ −10.0000 −0.739221
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 8.00000 0.585018
$$188$$ 4.00000 0.291730
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −6.00000 −0.423207
$$202$$ −18.0000 −1.26648
$$203$$ 8.00000 0.561490
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 8.00000 0.556038
$$208$$ 4.00000 0.277350
$$209$$ 32.0000 2.21349
$$210$$ 0 0
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −6.00000 −0.411113
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ −2.00000 −0.135769
$$218$$ 18.0000 1.21911
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ −12.0000 −0.805387
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 8.00000 0.529813
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 4.00000 0.262613
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 2.00000 0.130189
$$237$$ 8.00000 0.519656
$$238$$ −4.00000 −0.259281
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ −32.0000 −2.03611
$$248$$ −1.00000 −0.0635001
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −32.0000 −2.01182
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 10.0000 0.617802
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ −16.0000 −0.981023
$$267$$ 0 0
$$268$$ 6.00000 0.366508
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −8.00000 −0.484182
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ −16.0000 −0.946100
$$287$$ 20.0000 1.18056
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −18.0000 −1.05518
$$292$$ 4.00000 0.234082
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 12.0000 0.697486
$$297$$ 4.00000 0.232104
$$298$$ −18.0000 −1.04271
$$299$$ 32.0000 1.85061
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ −16.0000 −0.920697
$$303$$ 18.0000 1.03407
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ −14.0000 −0.796432
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −20.0000 −1.13047 −0.565233 0.824931i $$-0.691214\pi$$
−0.565233 + 0.824931i $$0.691214\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −16.0000 −0.895828
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 16.0000 0.891645
$$323$$ 16.0000 0.890264
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 6.00000 0.332309
$$327$$ −18.0000 −0.995402
$$328$$ 10.0000 0.552158
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 12.0000 0.657596
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 3.00000 0.163178
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ −8.00000 −0.432590
$$343$$ −20.0000 −1.07990
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ −4.00000 −0.213201
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ −2.00000 −0.106299
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.00000 0.211702
$$358$$ −20.0000 −1.05703
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 18.0000 0.946059
$$363$$ −5.00000 −0.262432
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 1.00000 0.0518476
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ 16.0000 0.824042
$$378$$ −2.00000 −0.102869
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 18.0000 0.920960
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ −8.00000 −0.406663
$$388$$ 18.0000 0.913812
$$389$$ −28.0000 −1.41966 −0.709828 0.704375i $$-0.751227\pi$$
−0.709828 + 0.704375i $$0.751227\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ −3.00000 −0.151523
$$393$$ −10.0000 −0.504433
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ 4.00000 0.199750 0.0998752 0.995000i $$-0.468156\pi$$
0.0998752 + 0.995000i $$0.468156\pi$$
$$402$$ −6.00000 −0.299253
$$403$$ −4.00000 −0.199254
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ −48.0000 −2.37927
$$408$$ 2.00000 0.0990148
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 14.0000 0.689730
$$413$$ 4.00000 0.196827
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ −4.00000 −0.195881
$$418$$ 32.0000 1.56517
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 24.0000 1.16830
$$423$$ 4.00000 0.194487
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 20.0000 0.967868
$$428$$ −8.00000 −0.386695
$$429$$ 16.0000 0.772487
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ −64.0000 −3.06154
$$438$$ −4.00000 −0.191127
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −8.00000 −0.380521
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −12.0000 −0.569495
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 18.0000 0.851371
$$448$$ 2.00000 0.0944911
$$449$$ 28.0000 1.32140 0.660701 0.750649i $$-0.270259\pi$$
0.660701 + 0.750649i $$0.270259\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 6.00000 0.282216
$$453$$ 16.0000 0.751746
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 8.00000 0.374634
$$457$$ −36.0000 −1.68401 −0.842004 0.539471i $$-0.818624\pi$$
−0.842004 + 0.539471i $$0.818624\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −40.0000 −1.86299 −0.931493 0.363760i $$-0.881493\pi$$
−0.931493 + 0.363760i $$0.881493\pi$$
$$462$$ 8.00000 0.372194
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 2.00000 0.0920575
$$473$$ 32.0000 1.47136
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ −6.00000 −0.274721
$$478$$ −4.00000 −0.182956
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ 10.0000 0.455488
$$483$$ −16.0000 −0.728025
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −6.00000 −0.271329
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ −8.00000 −0.360302
$$494$$ −32.0000 −1.43975
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ 12.0000 0.538274
$$498$$ 4.00000 0.179244
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 20.0000 0.892644
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ −3.00000 −0.133235
$$508$$ −4.00000 −0.177471
$$509$$ 16.0000 0.709188 0.354594 0.935020i $$-0.384619\pi$$
0.354594 + 0.935020i $$0.384619\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000 0.0441942
$$513$$ 8.00000 0.353209
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ −16.0000 −0.703679
$$518$$ 24.0000 1.05450
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 4.00000 0.175075
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 10.0000 0.436852
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.00000 0.0871214
$$528$$ 4.00000 0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 2.00000 0.0867926
$$532$$ −16.0000 −0.693688
$$533$$ 40.0000 1.73259
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.00000 0.259161
$$537$$ 20.0000 0.863064
$$538$$ −12.0000 −0.517357
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 8.00000 0.343629
$$543$$ −18.0000 −0.772454
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ −2.00000 −0.0855138 −0.0427569 0.999086i $$-0.513614\pi$$
−0.0427569 + 0.999086i $$0.513614\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ −8.00000 −0.340503
$$553$$ −16.0000 −0.680389
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ −6.00000 −0.254228 −0.127114 0.991888i $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ −1.00000 −0.0423334
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 6.00000 0.253095
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −2.00000 −0.0840663
$$567$$ 2.00000 0.0839921
$$568$$ 6.00000 0.251754
$$569$$ −28.0000 −1.17382 −0.586911 0.809652i $$-0.699656\pi$$
−0.586911 + 0.809652i $$0.699656\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −16.0000 −0.668994
$$573$$ −18.0000 −0.751961
$$574$$ 20.0000 0.834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 6.00000 0.249783 0.124892 0.992170i $$-0.460142\pi$$
0.124892 + 0.992170i $$0.460142\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ −18.0000 −0.746124
$$583$$ 24.0000 0.993978
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 12.0000 0.493197
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ −16.0000 −0.654836
$$598$$ 32.0000 1.30858
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ −16.0000 −0.652111
$$603$$ 6.00000 0.244339
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ −2.00000 −0.0808452
$$613$$ −8.00000 −0.323117 −0.161558 0.986863i $$-0.551652\pi$$
−0.161558 + 0.986863i $$0.551652\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ −30.0000 −1.20289
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ −32.0000 −1.27796
$$628$$ 22.0000 0.877896
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −24.0000 −0.953914
$$634$$ 22.0000 0.873732
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −12.0000 −0.475457
$$638$$ −16.0000 −0.633446
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −4.00000 −0.157991 −0.0789953 0.996875i $$-0.525171\pi$$
−0.0789953 + 0.996875i $$0.525171\pi$$
$$642$$ 8.00000 0.315735
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ 16.0000 0.629512
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ 2.00000 0.0783862
$$652$$ 6.00000 0.234978
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 4.00000 0.156055
$$658$$ 8.00000 0.311872
$$659$$ −2.00000 −0.0779089 −0.0389545 0.999241i $$-0.512403\pi$$
−0.0389545 + 0.999241i $$0.512403\pi$$
$$660$$ 0 0
$$661$$ 46.0000 1.78919 0.894596 0.446875i $$-0.147463\pi$$
0.894596 + 0.446875i $$0.147463\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 8.00000 0.310694
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 12.0000 0.464991
$$667$$ 32.0000 1.23904
$$668$$ 8.00000 0.309529
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ −2.00000 −0.0771517
$$673$$ −8.00000 −0.308377 −0.154189 0.988041i $$-0.549276\pi$$
−0.154189 + 0.988041i $$0.549276\pi$$
$$674$$ 32.0000 1.23259
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 36.0000 1.38155
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 4.00000 0.153168
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ −8.00000 −0.305888
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ −22.0000 −0.839352
$$688$$ −8.00000 −0.304997
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ −8.00000 −0.303895
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ −4.00000 −0.151620
$$697$$ −20.0000 −0.757554
$$698$$ −30.0000 −1.13552
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ −96.0000 −3.62071
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −34.0000 −1.27961
$$707$$ −36.0000 −1.35392
$$708$$ −2.00000 −0.0751646
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ −8.00000 −0.299602
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 4.00000 0.149383
$$718$$ 10.0000 0.373197
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 45.0000 1.67473
$$723$$ −10.0000 −0.371904
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 2.00000 0.0741759 0.0370879 0.999312i $$-0.488192\pi$$
0.0370879 + 0.999312i $$0.488192\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ −10.0000 −0.369611
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ −24.0000 −0.884051
$$738$$ 10.0000 0.368105
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 32.0000 1.17555
$$742$$ −12.0000 −0.440534
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ −4.00000 −0.146352
$$748$$ 8.00000 0.292509
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 4.00000 0.145865
$$753$$ −20.0000 −0.728841
$$754$$ 16.0000 0.582686
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ −36.0000 −1.30844 −0.654221 0.756303i $$-0.727003\pi$$
−0.654221 + 0.756303i $$0.727003\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ −20.0000 −0.724999 −0.362500 0.931984i $$-0.618077\pi$$
−0.362500 + 0.931984i $$0.618077\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 36.0000 1.30329
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 8.00000 0.288863
$$768$$ −1.00000 −0.0360844
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 2.00000 0.0719816
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ −24.0000 −0.860995
$$778$$ −28.0000 −1.00385
$$779$$ −80.0000 −2.86630
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ −16.0000 −0.572159
$$783$$ −4.00000 −0.142948
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −10.0000 −0.356688
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ −10.0000 −0.356235
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ −4.00000 −0.142134
$$793$$ 40.0000 1.42044
$$794$$ 34.0000 1.20661
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 16.0000 0.566394
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 4.00000 0.141245
$$803$$ −16.0000 −0.564628
$$804$$ −6.00000 −0.211604
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 12.0000 0.422420
$$808$$ −18.0000 −0.633238
$$809$$ 28.0000 0.984428 0.492214 0.870474i $$-0.336188\pi$$
0.492214 + 0.870474i $$0.336188\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 8.00000 0.280745
$$813$$ −8.00000 −0.280572
$$814$$ −48.0000 −1.68240
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 64.0000 2.23908
$$818$$ 6.00000 0.209785
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ −4.00000 −0.139601 −0.0698005 0.997561i $$-0.522236\pi$$
−0.0698005 + 0.997561i $$0.522236\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 4.00000 0.138675
$$833$$ 6.00000 0.207888
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 32.0000 1.10674
$$837$$ 1.00000 0.0345651
$$838$$ −10.0000 −0.345444
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 2.00000 0.0689246
$$843$$ −6.00000 −0.206651
$$844$$ 24.0000 0.826114
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 10.0000 0.343604
$$848$$ −6.00000 −0.206041
$$849$$ 2.00000 0.0686398
$$850$$ 0 0
$$851$$ 96.0000 3.29084
$$852$$ −6.00000 −0.205557
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 16.0000 0.546231
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ −20.0000 −0.681598
$$862$$ −30.0000 −1.02180
$$863$$ −16.0000 −0.544646 −0.272323 0.962206i $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −4.00000 −0.135926
$$867$$ 13.0000 0.441503
$$868$$ −2.00000 −0.0678844
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 18.0000 0.609557
$$873$$ 18.0000 0.609208
$$874$$ −64.0000 −2.16483
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −40.0000 −1.34993
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 56.0000 1.88030 0.940148 0.340766i $$-0.110687\pi$$
0.940148 + 0.340766i $$0.110687\pi$$
$$888$$ −12.0000 −0.402694
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 8.00000 0.267860
$$893$$ −32.0000 −1.07084
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ −32.0000 −1.06845
$$898$$ 28.0000 0.934372
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ −40.0000 −1.33185
$$903$$ 16.0000 0.532447
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 8.00000 0.264906
$$913$$ 16.0000 0.529523
$$914$$ −36.0000 −1.19077
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ 20.0000 0.660458
$$918$$ 2.00000 0.0660098
$$919$$ 4.00000 0.131948 0.0659739 0.997821i $$-0.478985\pi$$
0.0659739 + 0.997821i $$0.478985\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ −40.0000 −1.31733
$$923$$ 24.0000 0.789970
$$924$$ 8.00000 0.263181
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ 14.0000 0.459820
$$928$$ 4.00000 0.131306
$$929$$ 32.0000 1.04989 0.524943 0.851137i $$-0.324087\pi$$
0.524943 + 0.851137i $$0.324087\pi$$
$$930$$ 0 0
$$931$$ 24.0000 0.786568
$$932$$ 10.0000 0.327561
$$933$$ 30.0000 0.982156
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 12.0000 0.391814
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ −22.0000 −0.716799
$$943$$ 80.0000 2.60516
$$944$$ 2.00000 0.0650945
$$945$$ 0 0
$$946$$ 32.0000 1.04041
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ −4.00000 −0.129641
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −4.00000 −0.129369
$$957$$ 16.0000 0.517207
$$958$$ −10.0000 −0.323085
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 48.0000 1.54758
$$963$$ −8.00000 −0.257796
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 5.00000 0.160706
$$969$$ −16.0000 −0.513994
$$970$$ 0 0
$$971$$ −2.00000 −0.0641831 −0.0320915 0.999485i $$-0.510217\pi$$
−0.0320915 + 0.999485i $$0.510217\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 8.00000 0.256468
$$974$$ 12.0000 0.384505
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 46.0000 1.47167 0.735835 0.677161i $$-0.236790\pi$$
0.735835 + 0.677161i $$0.236790\pi$$
$$978$$ −6.00000 −0.191859
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ −8.00000 −0.255290
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ −8.00000 −0.254643
$$988$$ −32.0000 −1.01806
$$989$$ −64.0000 −2.03508
$$990$$ 0 0
$$991$$ −56.0000 −1.77890 −0.889449 0.457034i $$-0.848912\pi$$
−0.889449 + 0.457034i $$0.848912\pi$$
$$992$$ −1.00000 −0.0317500
$$993$$ 12.0000 0.380808
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ −12.0000 −0.379663
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 4650.2.a.bg.1.1 1
5.2 odd 4 4650.2.d.b.3349.2 2
5.3 odd 4 4650.2.d.b.3349.1 2
5.4 even 2 930.2.a.h.1.1 1
15.14 odd 2 2790.2.a.p.1.1 1
20.19 odd 2 7440.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.h.1.1 1 5.4 even 2
2790.2.a.p.1.1 1 15.14 odd 2
4650.2.a.bg.1.1 1 1.1 even 1 trivial
4650.2.d.b.3349.1 2 5.3 odd 4
4650.2.d.b.3349.2 2 5.2 odd 4
7440.2.a.m.1.1 1 20.19 odd 2