# Properties

 Label 3525.2.a.c.1.1 Level $3525$ Weight $2$ Character 3525.1 Self dual yes Analytic conductor $28.147$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 141) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3525.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -6.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} +6.00000 q^{19} -3.00000 q^{21} -2.00000 q^{22} -3.00000 q^{23} -4.00000 q^{26} -1.00000 q^{27} +6.00000 q^{28} +3.00000 q^{29} +2.00000 q^{31} +8.00000 q^{32} -1.00000 q^{33} +4.00000 q^{34} +2.00000 q^{36} +7.00000 q^{37} -12.0000 q^{38} -2.00000 q^{39} +10.0000 q^{41} +6.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} +6.00000 q^{46} +1.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{51} +4.00000 q^{52} -4.00000 q^{53} +2.00000 q^{54} -6.00000 q^{57} -6.00000 q^{58} +8.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} -10.0000 q^{67} -4.00000 q^{68} +3.00000 q^{69} -14.0000 q^{71} +10.0000 q^{73} -14.0000 q^{74} +12.0000 q^{76} +3.00000 q^{77} +4.00000 q^{78} +17.0000 q^{79} +1.00000 q^{81} -20.0000 q^{82} -8.00000 q^{83} -6.00000 q^{84} -20.0000 q^{86} -3.00000 q^{87} +6.00000 q^{89} +6.00000 q^{91} -6.00000 q^{92} -2.00000 q^{93} -2.00000 q^{94} -8.00000 q^{96} -1.00000 q^{97} -4.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −6.00000 −1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ −2.00000 −0.426401
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 6.00000 1.13389
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 8.00000 1.41421
$$33$$ −1.00000 −0.174078
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ −12.0000 −1.94666
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 6.00000 0.925820
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 1.00000 0.145865
$$48$$ 4.00000 0.577350
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 4.00000 0.554700
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.00000 −0.794719
$$58$$ −6.00000 −0.787839
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 3.00000 0.377964
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −14.0000 −1.62747
$$75$$ 0 0
$$76$$ 12.0000 1.37649
$$77$$ 3.00000 0.341882
$$78$$ 4.00000 0.452911
$$79$$ 17.0000 1.91265 0.956325 0.292306i $$-0.0944227\pi$$
0.956325 + 0.292306i $$0.0944227\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −20.0000 −2.20863
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ −6.00000 −0.654654
$$85$$ 0 0
$$86$$ −20.0000 −2.15666
$$87$$ −3.00000 −0.321634
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ −6.00000 −0.625543
$$93$$ −2.00000 −0.207390
$$94$$ −2.00000 −0.206284
$$95$$ 0 0
$$96$$ −8.00000 −0.816497
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ −4.00000 −0.404061
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −16.0000 −1.59206 −0.796030 0.605257i $$-0.793070\pi$$
−0.796030 + 0.605257i $$0.793070\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ −11.0000 −1.08386 −0.541931 0.840423i $$-0.682307\pi$$
−0.541931 + 0.840423i $$0.682307\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ −5.00000 −0.483368 −0.241684 0.970355i $$-0.577700\pi$$
−0.241684 + 0.970355i $$0.577700\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ −12.0000 −1.13389
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 12.0000 1.12390
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000 0.184900
$$118$$ −16.0000 −1.47292
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 20.0000 1.81071
$$123$$ −10.0000 −0.901670
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ −10.0000 −0.880451
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 18.0000 1.56080
$$134$$ 20.0000 1.72774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ −2.00000 −0.169638 −0.0848189 0.996396i $$-0.527031\pi$$
−0.0848189 + 0.996396i $$0.527031\pi$$
$$140$$ 0 0
$$141$$ −1.00000 −0.0842152
$$142$$ 28.0000 2.34971
$$143$$ 2.00000 0.167248
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ −20.0000 −1.65521
$$147$$ −2.00000 −0.164957
$$148$$ 14.0000 1.15079
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 3.00000 0.239426 0.119713 0.992809i $$-0.461803\pi$$
0.119713 + 0.992809i $$0.461803\pi$$
$$158$$ −34.0000 −2.70489
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −9.00000 −0.709299
$$162$$ −2.00000 −0.157135
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ 20.0000 1.56174
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ −21.0000 −1.62503 −0.812514 0.582941i $$-0.801902\pi$$
−0.812514 + 0.582941i $$0.801902\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ 20.0000 1.52499
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ −8.00000 −0.601317
$$178$$ −12.0000 −0.899438
$$179$$ 5.00000 0.373718 0.186859 0.982387i $$-0.440169\pi$$
0.186859 + 0.982387i $$0.440169\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ −2.00000 −0.146254
$$188$$ 2.00000 0.145865
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 8.00000 0.577350
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 26.0000 1.84309 0.921546 0.388270i $$-0.126927\pi$$
0.921546 + 0.388270i $$0.126927\pi$$
$$200$$ 0 0
$$201$$ 10.0000 0.705346
$$202$$ 32.0000 2.25151
$$203$$ 9.00000 0.631676
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 22.0000 1.53281
$$207$$ −3.00000 −0.208514
$$208$$ −8.00000 −0.554700
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ −8.00000 −0.549442
$$213$$ 14.0000 0.959264
$$214$$ 10.0000 0.683586
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6.00000 0.407307
$$218$$ −12.0000 −0.812743
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 14.0000 0.939618
$$223$$ 12.0000 0.803579 0.401790 0.915732i $$-0.368388\pi$$
0.401790 + 0.915732i $$0.368388\pi$$
$$224$$ 24.0000 1.60357
$$225$$ 0 0
$$226$$ −20.0000 −1.33038
$$227$$ −7.00000 −0.464606 −0.232303 0.972643i $$-0.574626\pi$$
−0.232303 + 0.972643i $$0.574626\pi$$
$$228$$ −12.0000 −0.794719
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ −23.0000 −1.50678 −0.753390 0.657574i $$-0.771583\pi$$
−0.753390 + 0.657574i $$0.771583\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 16.0000 1.04151
$$237$$ −17.0000 −1.10427
$$238$$ 12.0000 0.777844
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 11.0000 0.708572 0.354286 0.935137i $$-0.384724\pi$$
0.354286 + 0.935137i $$0.384724\pi$$
$$242$$ 20.0000 1.28565
$$243$$ −1.00000 −0.0641500
$$244$$ −20.0000 −1.28037
$$245$$ 0 0
$$246$$ 20.0000 1.27515
$$247$$ 12.0000 0.763542
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 6.00000 0.377964
$$253$$ −3.00000 −0.188608
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 31.0000 1.93373 0.966863 0.255294i $$-0.0821723\pi$$
0.966863 + 0.255294i $$0.0821723\pi$$
$$258$$ 20.0000 1.24515
$$259$$ 21.0000 1.30488
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 12.0000 0.741362
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −36.0000 −2.20730
$$267$$ −6.00000 −0.367194
$$268$$ −20.0000 −1.22169
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ 8.00000 0.485071
$$273$$ −6.00000 −0.363137
$$274$$ −44.0000 −2.65814
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 27.0000 1.61068 0.805342 0.592810i $$-0.201981\pi$$
0.805342 + 0.592810i $$0.201981\pi$$
$$282$$ 2.00000 0.119098
$$283$$ 21.0000 1.24832 0.624160 0.781296i $$-0.285441\pi$$
0.624160 + 0.781296i $$0.285441\pi$$
$$284$$ −28.0000 −1.66149
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 30.0000 1.77084
$$288$$ 8.00000 0.471405
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 1.00000 0.0586210
$$292$$ 20.0000 1.17041
$$293$$ −27.0000 −1.57736 −0.788678 0.614806i $$-0.789234\pi$$
−0.788678 + 0.614806i $$0.789234\pi$$
$$294$$ 4.00000 0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1.00000 −0.0580259
$$298$$ 24.0000 1.39028
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 30.0000 1.72917
$$302$$ 28.0000 1.61122
$$303$$ 16.0000 0.919176
$$304$$ −24.0000 −1.37649
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ −23.0000 −1.31268 −0.656340 0.754466i $$-0.727896\pi$$
−0.656340 + 0.754466i $$0.727896\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 11.0000 0.625768
$$310$$ 0 0
$$311$$ −23.0000 −1.30421 −0.652105 0.758129i $$-0.726114\pi$$
−0.652105 + 0.758129i $$0.726114\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ 34.0000 1.91265
$$317$$ 3.00000 0.168497 0.0842484 0.996445i $$-0.473151\pi$$
0.0842484 + 0.996445i $$0.473151\pi$$
$$318$$ −8.00000 −0.448618
$$319$$ 3.00000 0.167968
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ 18.0000 1.00310
$$323$$ −12.0000 −0.667698
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ −6.00000 −0.331801
$$328$$ 0 0
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 7.00000 0.383598
$$334$$ 42.0000 2.29814
$$335$$ 0 0
$$336$$ 12.0000 0.654654
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ 18.0000 0.979071
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 2.00000 0.108306
$$342$$ −12.0000 −0.648886
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 16.0000 0.856460 0.428230 0.903670i $$-0.359137\pi$$
0.428230 + 0.903670i $$0.359137\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 8.00000 0.426401
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ 6.00000 0.317554
$$358$$ −10.0000 −0.528516
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −24.0000 −1.26141
$$363$$ 10.0000 0.524864
$$364$$ 12.0000 0.628971
$$365$$ 0 0
$$366$$ −20.0000 −1.04542
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ −4.00000 −0.207390
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000 0.309016
$$378$$ 6.00000 0.308607
$$379$$ 35.0000 1.79783 0.898915 0.438124i $$-0.144357\pi$$
0.898915 + 0.438124i $$0.144357\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 36.0000 1.84192
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ 10.0000 0.508329
$$388$$ −2.00000 −0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 16.0000 0.806068
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −52.0000 −2.60652
$$399$$ −18.0000 −0.901127
$$400$$ 0 0
$$401$$ 4.00000 0.199750 0.0998752 0.995000i $$-0.468156\pi$$
0.0998752 + 0.995000i $$0.468156\pi$$
$$402$$ −20.0000 −0.997509
$$403$$ 4.00000 0.199254
$$404$$ −32.0000 −1.59206
$$405$$ 0 0
$$406$$ −18.0000 −0.893325
$$407$$ 7.00000 0.346977
$$408$$ 0 0
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ −22.0000 −1.08386
$$413$$ 24.0000 1.18096
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 16.0000 0.784465
$$417$$ 2.00000 0.0979404
$$418$$ −12.0000 −0.586939
$$419$$ 25.0000 1.22133 0.610665 0.791889i $$-0.290902\pi$$
0.610665 + 0.791889i $$0.290902\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ −32.0000 −1.55774
$$423$$ 1.00000 0.0486217
$$424$$ 0 0
$$425$$ 0 0
$$426$$ −28.0000 −1.35660
$$427$$ −30.0000 −1.45180
$$428$$ −10.0000 −0.483368
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 10.0000 0.481683 0.240842 0.970564i $$-0.422577\pi$$
0.240842 + 0.970564i $$0.422577\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ −12.0000 −0.576018
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ −18.0000 −0.861057
$$438$$ 20.0000 0.955637
$$439$$ −15.0000 −0.715911 −0.357955 0.933739i $$-0.616526\pi$$
−0.357955 + 0.933739i $$0.616526\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 8.00000 0.380521
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −14.0000 −0.664411
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ 12.0000 0.567581
$$448$$ −24.0000 −1.13389
$$449$$ 7.00000 0.330350 0.165175 0.986264i $$-0.447181\pi$$
0.165175 + 0.986264i $$0.447181\pi$$
$$450$$ 0 0
$$451$$ 10.0000 0.470882
$$452$$ 20.0000 0.940721
$$453$$ 14.0000 0.657777
$$454$$ 14.0000 0.657053
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.0000 0.608114 0.304057 0.952654i $$-0.401659\pi$$
0.304057 + 0.952654i $$0.401659\pi$$
$$458$$ −8.00000 −0.373815
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 31.0000 1.44381 0.721907 0.691990i $$-0.243266\pi$$
0.721907 + 0.691990i $$0.243266\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 28.0000 1.30127 0.650635 0.759390i $$-0.274503\pi$$
0.650635 + 0.759390i $$0.274503\pi$$
$$464$$ −12.0000 −0.557086
$$465$$ 0 0
$$466$$ 46.0000 2.13091
$$467$$ 21.0000 0.971764 0.485882 0.874024i $$-0.338498\pi$$
0.485882 + 0.874024i $$0.338498\pi$$
$$468$$ 4.00000 0.184900
$$469$$ −30.0000 −1.38527
$$470$$ 0 0
$$471$$ −3.00000 −0.138233
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 34.0000 1.56167
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ −4.00000 −0.183147
$$478$$ −48.0000 −2.19547
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ 14.0000 0.638345
$$482$$ −22.0000 −1.00207
$$483$$ 9.00000 0.409514
$$484$$ −20.0000 −0.909091
$$485$$ 0 0
$$486$$ 2.00000 0.0907218
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ −6.00000 −0.271329
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ −20.0000 −0.901670
$$493$$ −6.00000 −0.270226
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ −42.0000 −1.88396
$$498$$ −16.0000 −0.716977
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 0 0
$$501$$ 21.0000 0.938211
$$502$$ −36.0000 −1.60676
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ 9.00000 0.399704
$$508$$ 16.0000 0.709885
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 30.0000 1.32712
$$512$$ −32.0000 −1.41421
$$513$$ −6.00000 −0.264906
$$514$$ −62.0000 −2.73470
$$515$$ 0 0
$$516$$ −20.0000 −0.880451
$$517$$ 1.00000 0.0439799
$$518$$ −42.0000 −1.84537
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 13.0000 0.568450 0.284225 0.958758i $$-0.408264\pi$$
0.284225 + 0.958758i $$0.408264\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ −4.00000 −0.174243
$$528$$ 4.00000 0.174078
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 36.0000 1.56080
$$533$$ 20.0000 0.866296
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −5.00000 −0.215766
$$538$$ 24.0000 1.03471
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 34.0000 1.46177 0.730887 0.682498i $$-0.239107\pi$$
0.730887 + 0.682498i $$0.239107\pi$$
$$542$$ 30.0000 1.28861
$$543$$ −12.0000 −0.514969
$$544$$ −16.0000 −0.685994
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 44.0000 1.87959
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 18.0000 0.766826
$$552$$ 0 0
$$553$$ 51.0000 2.16874
$$554$$ 52.0000 2.20927
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 20.0000 0.845910
$$560$$ 0 0
$$561$$ 2.00000 0.0844401
$$562$$ −54.0000 −2.27785
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ −2.00000 −0.0842152
$$565$$ 0 0
$$566$$ −42.0000 −1.76539
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ 15.0000 0.628833 0.314416 0.949285i $$-0.398191\pi$$
0.314416 + 0.949285i $$0.398191\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 18.0000 0.751961
$$574$$ −60.0000 −2.50435
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 26.0000 1.08146
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ −2.00000 −0.0829027
$$583$$ −4.00000 −0.165663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 54.0000 2.23072
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ −4.00000 −0.164957
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ 8.00000 0.329076
$$592$$ −28.0000 −1.15079
$$593$$ 26.0000 1.06769 0.533846 0.845582i $$-0.320746\pi$$
0.533846 + 0.845582i $$0.320746\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −24.0000 −0.983078
$$597$$ −26.0000 −1.06411
$$598$$ 12.0000 0.490716
$$599$$ −39.0000 −1.59350 −0.796748 0.604311i $$-0.793448\pi$$
−0.796748 + 0.604311i $$0.793448\pi$$
$$600$$ 0 0
$$601$$ 7.00000 0.285536 0.142768 0.989756i $$-0.454400\pi$$
0.142768 + 0.989756i $$0.454400\pi$$
$$602$$ −60.0000 −2.44542
$$603$$ −10.0000 −0.407231
$$604$$ −28.0000 −1.13930
$$605$$ 0 0
$$606$$ −32.0000 −1.29991
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 48.0000 1.94666
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ 2.00000 0.0809113
$$612$$ −4.00000 −0.161690
$$613$$ −27.0000 −1.09052 −0.545260 0.838267i $$-0.683569\pi$$
−0.545260 + 0.838267i $$0.683569\pi$$
$$614$$ 46.0000 1.85641
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −36.0000 −1.44931 −0.724653 0.689114i $$-0.758000\pi$$
−0.724653 + 0.689114i $$0.758000\pi$$
$$618$$ −22.0000 −0.884970
$$619$$ −19.0000 −0.763674 −0.381837 0.924230i $$-0.624709\pi$$
−0.381837 + 0.924230i $$0.624709\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 46.0000 1.84443
$$623$$ 18.0000 0.721155
$$624$$ 8.00000 0.320256
$$625$$ 0 0
$$626$$ 12.0000 0.479616
$$627$$ −6.00000 −0.239617
$$628$$ 6.00000 0.239426
$$629$$ −14.0000 −0.558217
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ 0 0
$$633$$ −16.0000 −0.635943
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 8.00000 0.317221
$$637$$ 4.00000 0.158486
$$638$$ −6.00000 −0.237542
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ 17.0000 0.671460 0.335730 0.941958i $$-0.391017\pi$$
0.335730 + 0.941958i $$0.391017\pi$$
$$642$$ −10.0000 −0.394669
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ −18.0000 −0.709299
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ −6.00000 −0.235159
$$652$$ 12.0000 0.469956
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 12.0000 0.469237
$$655$$ 0 0
$$656$$ −40.0000 −1.56174
$$657$$ 10.0000 0.390137
$$658$$ −6.00000 −0.233904
$$659$$ −10.0000 −0.389545 −0.194772 0.980848i $$-0.562397\pi$$
−0.194772 + 0.980848i $$0.562397\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 4.00000 0.155347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −14.0000 −0.542489
$$667$$ −9.00000 −0.348481
$$668$$ −42.0000 −1.62503
$$669$$ −12.0000 −0.463947
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ −24.0000 −0.925820
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ −18.0000 −0.692308
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ 20.0000 0.768095
$$679$$ −3.00000 −0.115129
$$680$$ 0 0
$$681$$ 7.00000 0.268241
$$682$$ −4.00000 −0.153168
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 0 0
$$686$$ 30.0000 1.14541
$$687$$ −4.00000 −0.152610
$$688$$ −40.0000 −1.52499
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ 34.0000 1.29342 0.646710 0.762736i $$-0.276144\pi$$
0.646710 + 0.762736i $$0.276144\pi$$
$$692$$ 0 0
$$693$$ 3.00000 0.113961
$$694$$ −24.0000 −0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ −32.0000 −1.21122
$$699$$ 23.0000 0.869940
$$700$$ 0 0
$$701$$ −1.00000 −0.0377695 −0.0188847 0.999822i $$-0.506012\pi$$
−0.0188847 + 0.999822i $$0.506012\pi$$
$$702$$ 4.00000 0.150970
$$703$$ 42.0000 1.58406
$$704$$ −8.00000 −0.301511
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −48.0000 −1.80523
$$708$$ −16.0000 −0.601317
$$709$$ −35.0000 −1.31445 −0.657226 0.753693i $$-0.728270\pi$$
−0.657226 + 0.753693i $$0.728270\pi$$
$$710$$ 0 0
$$711$$ 17.0000 0.637550
$$712$$ 0 0
$$713$$ −6.00000 −0.224702
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ −24.0000 −0.896296
$$718$$ 30.0000 1.11959
$$719$$ 2.00000 0.0745874 0.0372937 0.999304i $$-0.488126\pi$$
0.0372937 + 0.999304i $$0.488126\pi$$
$$720$$ 0 0
$$721$$ −33.0000 −1.22898
$$722$$ −34.0000 −1.26535
$$723$$ −11.0000 −0.409094
$$724$$ 24.0000 0.891953
$$725$$ 0 0
$$726$$ −20.0000 −0.742270
$$727$$ −10.0000 −0.370879 −0.185440 0.982656i $$-0.559371\pi$$
−0.185440 + 0.982656i $$0.559371\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 20.0000 0.739221
$$733$$ −41.0000 −1.51437 −0.757185 0.653201i $$-0.773426\pi$$
−0.757185 + 0.653201i $$0.773426\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ −24.0000 −0.884652
$$737$$ −10.0000 −0.368355
$$738$$ −20.0000 −0.736210
$$739$$ 17.0000 0.625355 0.312678 0.949859i $$-0.398774\pi$$
0.312678 + 0.949859i $$0.398774\pi$$
$$740$$ 0 0
$$741$$ −12.0000 −0.440831
$$742$$ 24.0000 0.881068
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ −8.00000 −0.292705
$$748$$ −4.00000 −0.146254
$$749$$ −15.0000 −0.548088
$$750$$ 0 0
$$751$$ 2.00000 0.0729810 0.0364905 0.999334i $$-0.488382\pi$$
0.0364905 + 0.999334i $$0.488382\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ −18.0000 −0.655956
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ −6.00000 −0.218218
$$757$$ 18.0000 0.654221 0.327111 0.944986i $$-0.393925\pi$$
0.327111 + 0.944986i $$0.393925\pi$$
$$758$$ −70.0000 −2.54251
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 18.0000 0.651644
$$764$$ −36.0000 −1.30243
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 16.0000 0.577727
$$768$$ −16.0000 −0.577350
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −31.0000 −1.11644
$$772$$ −8.00000 −0.287926
$$773$$ 36.0000 1.29483 0.647415 0.762138i $$-0.275850\pi$$
0.647415 + 0.762138i $$0.275850\pi$$
$$774$$ −20.0000 −0.718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −21.0000 −0.753371
$$778$$ −12.0000 −0.430221
$$779$$ 60.0000 2.14972
$$780$$ 0 0
$$781$$ −14.0000 −0.500959
$$782$$ −12.0000 −0.429119
$$783$$ −3.00000 −0.107211
$$784$$ −8.00000 −0.285714
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ −16.0000 −0.569976
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 30.0000 1.06668
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ −4.00000 −0.141955
$$795$$ 0 0
$$796$$ 52.0000 1.84309
$$797$$ −29.0000 −1.02723 −0.513616 0.858020i $$-0.671695\pi$$
−0.513616 + 0.858020i $$0.671695\pi$$
$$798$$ 36.0000 1.27439
$$799$$ −2.00000 −0.0707549
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ −8.00000 −0.282490
$$803$$ 10.0000 0.352892
$$804$$ 20.0000 0.705346
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 12.0000 0.422420
$$808$$ 0 0
$$809$$ 53.0000 1.86338 0.931690 0.363253i $$-0.118334\pi$$
0.931690 + 0.363253i $$0.118334\pi$$
$$810$$ 0 0
$$811$$ −29.0000 −1.01833 −0.509164 0.860670i $$-0.670045\pi$$
−0.509164 + 0.860670i $$0.670045\pi$$
$$812$$ 18.0000 0.631676
$$813$$ 15.0000 0.526073
$$814$$ −14.0000 −0.490700
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ 60.0000 2.09913
$$818$$ 64.0000 2.23771
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 44.0000 1.53468
$$823$$ 39.0000 1.35945 0.679727 0.733465i $$-0.262098\pi$$
0.679727 + 0.733465i $$0.262098\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ 24.0000 0.834562 0.417281 0.908778i $$-0.362983\pi$$
0.417281 + 0.908778i $$0.362983\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ −16.0000 −0.554700
$$833$$ −4.00000 −0.138592
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ −2.00000 −0.0691301
$$838$$ −50.0000 −1.72722
$$839$$ −31.0000 −1.07024 −0.535119 0.844776i $$-0.679733\pi$$
−0.535119 + 0.844776i $$0.679733\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 40.0000 1.37849
$$843$$ −27.0000 −0.929929
$$844$$ 32.0000 1.10149
$$845$$ 0 0
$$846$$ −2.00000 −0.0687614
$$847$$ −30.0000 −1.03081
$$848$$ 16.0000 0.549442
$$849$$ −21.0000 −0.720718
$$850$$ 0 0
$$851$$ −21.0000 −0.719871
$$852$$ 28.0000 0.959264
$$853$$ 34.0000 1.16414 0.582069 0.813139i $$-0.302243\pi$$
0.582069 + 0.813139i $$0.302243\pi$$
$$854$$ 60.0000 2.05316
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 4.00000 0.136558
$$859$$ −18.0000 −0.614152 −0.307076 0.951685i $$-0.599351\pi$$
−0.307076 + 0.951685i $$0.599351\pi$$
$$860$$ 0 0
$$861$$ −30.0000 −1.02240
$$862$$ −20.0000 −0.681203
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 12.0000 0.407307
$$869$$ 17.0000 0.576686
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 0 0
$$873$$ −1.00000 −0.0338449
$$874$$ 36.0000 1.21772
$$875$$ 0 0
$$876$$ −20.0000 −0.675737
$$877$$ −50.0000 −1.68838 −0.844190 0.536044i $$-0.819918\pi$$
−0.844190 + 0.536044i $$0.819918\pi$$
$$878$$ 30.0000 1.01245
$$879$$ 27.0000 0.910687
$$880$$ 0 0
$$881$$ −53.0000 −1.78562 −0.892808 0.450438i $$-0.851268\pi$$
−0.892808 + 0.450438i $$0.851268\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 24.0000 0.803579
$$893$$ 6.00000 0.200782
$$894$$ −24.0000 −0.802680
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ −14.0000 −0.467186
$$899$$ 6.00000 0.200111
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ −20.0000 −0.665927
$$903$$ −30.0000 −0.998337
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −28.0000 −0.930238
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ −14.0000 −0.464606
$$909$$ −16.0000 −0.530687
$$910$$ 0 0
$$911$$ 50.0000 1.65657 0.828287 0.560304i $$-0.189316\pi$$
0.828287 + 0.560304i $$0.189316\pi$$
$$912$$ 24.0000 0.794719
$$913$$ −8.00000 −0.264761
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ 8.00000 0.264327
$$917$$ −18.0000 −0.594412
$$918$$ −4.00000 −0.132020
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 23.0000 0.757876
$$922$$ −62.0000 −2.04186
$$923$$ −28.0000 −0.921631
$$924$$ −6.00000 −0.197386
$$925$$ 0 0
$$926$$ −56.0000 −1.84027
$$927$$ −11.0000 −0.361287
$$928$$ 24.0000 0.787839
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ −46.0000 −1.50678
$$933$$ 23.0000 0.752986
$$934$$ −42.0000 −1.37428
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 60.0000 1.95907
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 6.00000 0.195491
$$943$$ −30.0000 −0.976934
$$944$$ −32.0000 −1.04151
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ 38.0000 1.23483 0.617417 0.786636i $$-0.288179\pi$$
0.617417 + 0.786636i $$0.288179\pi$$
$$948$$ −34.0000 −1.10427
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ −3.00000 −0.0972817
$$952$$ 0 0
$$953$$ −31.0000 −1.00419 −0.502094 0.864813i $$-0.667437\pi$$
−0.502094 + 0.864813i $$0.667437\pi$$
$$954$$ 8.00000 0.259010
$$955$$ 0 0
$$956$$ 48.0000 1.55243
$$957$$ −3.00000 −0.0969762
$$958$$ −24.0000 −0.775405
$$959$$ 66.0000 2.13125
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −28.0000 −0.902756
$$963$$ −5.00000 −0.161123
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ −18.0000 −0.579141
$$967$$ 13.0000 0.418052 0.209026 0.977910i $$-0.432971\pi$$
0.209026 + 0.977910i $$0.432971\pi$$
$$968$$ 0 0
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ 49.0000 1.57248 0.786242 0.617918i $$-0.212024\pi$$
0.786242 + 0.617918i $$0.212024\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ −6.00000 −0.192351
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 40.0000 1.28037
$$977$$ −28.0000 −0.895799 −0.447900 0.894084i $$-0.647828\pi$$
−0.447900 + 0.894084i $$0.647828\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ −72.0000 −2.29761
$$983$$ −60.0000 −1.91370 −0.956851 0.290578i $$-0.906153\pi$$
−0.956851 + 0.290578i $$0.906153\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ −3.00000 −0.0954911
$$988$$ 24.0000 0.763542
$$989$$ −30.0000 −0.953945
$$990$$ 0 0
$$991$$ −57.0000 −1.81066 −0.905332 0.424704i $$-0.860378\pi$$
−0.905332 + 0.424704i $$0.860378\pi$$
$$992$$ 16.0000 0.508001
$$993$$ −4.00000 −0.126936
$$994$$ 84.0000 2.66432
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ −40.0000 −1.26681 −0.633406 0.773819i $$-0.718344\pi$$
−0.633406 + 0.773819i $$0.718344\pi$$
$$998$$ 44.0000 1.39280
$$999$$ −7.00000 −0.221470
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 3525.2.a.c.1.1 1
5.4 even 2 141.2.a.e.1.1 1
15.14 odd 2 423.2.a.b.1.1 1
20.19 odd 2 2256.2.a.e.1.1 1
35.34 odd 2 6909.2.a.k.1.1 1
40.19 odd 2 9024.2.a.bq.1.1 1
40.29 even 2 9024.2.a.n.1.1 1
60.59 even 2 6768.2.a.n.1.1 1
235.234 odd 2 6627.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
141.2.a.e.1.1 1 5.4 even 2
423.2.a.b.1.1 1 15.14 odd 2
2256.2.a.e.1.1 1 20.19 odd 2
3525.2.a.c.1.1 1 1.1 even 1 trivial
6627.2.a.i.1.1 1 235.234 odd 2
6768.2.a.n.1.1 1 60.59 even 2
6909.2.a.k.1.1 1 35.34 odd 2
9024.2.a.n.1.1 1 40.29 even 2
9024.2.a.bq.1.1 1 40.19 odd 2