# Properties

 Label 4900.2.e.i.2549.2 Level $4900$ Weight $2$ Character 4900.2549 Analytic conductor $39.127$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.1266969904$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2549.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4900.2549 Dual form 4900.2.e.i.2549.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} -7.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +9.00000i q^{47} +3.00000 q^{51} +3.00000i q^{53} +1.00000i q^{57} -9.00000 q^{59} -1.00000 q^{61} +7.00000i q^{67} -3.00000 q^{69} -1.00000i q^{73} +13.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{87} -15.0000 q^{89} -7.00000i q^{93} +10.0000i q^{97} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{9} + O(q^{10})$$ $$2q + 4q^{9} - 6q^{11} + 2q^{19} + 12q^{29} - 14q^{31} - 4q^{39} + 12q^{41} + 6q^{51} - 18q^{59} - 2q^{61} - 6q^{69} + 26q^{79} + 2q^{81} - 30q^{89} - 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$2451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.00000i 0.625543i 0.949828 + 0.312772i $$0.101257\pi$$
−0.949828 + 0.312772i $$0.898743\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000i 0.962250i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ − 3.00000i − 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.00000i 0.164399i 0.996616 + 0.0821995i $$0.0261945\pi$$
−0.996616 + 0.0821995i $$0.973806\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.00000i 1.31278i 0.754420 + 0.656392i $$0.227918\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ 3.00000i 0.412082i 0.978543 + 0.206041i $$0.0660580\pi$$
−0.978543 + 0.206041i $$0.933942\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000i 0.132453i
$$58$$ 0 0
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000i 0.855186i 0.903971 + 0.427593i $$0.140638\pi$$
−0.903971 + 0.427593i $$0.859362\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ − 1.00000i − 0.117041i −0.998286 0.0585206i $$-0.981362\pi$$
0.998286 0.0585206i $$-0.0186383\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.0000 1.46261 0.731307 0.682048i $$-0.238911\pi$$
0.731307 + 0.682048i $$0.238911\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 7.00000i − 0.725866i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 0 0
$$103$$ 11.0000i 1.08386i 0.840423 + 0.541931i $$0.182307\pi$$
−0.840423 + 0.541931i $$0.817693\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 15.0000i − 1.45010i −0.688694 0.725052i $$-0.741816\pi$$
0.688694 0.725052i $$-0.258184\pi$$
$$108$$ 0 0
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 4.00000i 0.369800i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.0000i 1.79415i 0.441877 + 0.897076i $$0.354313\pi$$
−0.441877 + 0.897076i $$0.645687\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 0 0
$$143$$ − 6.00000i − 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −3.00000 −0.245770 −0.122885 0.992421i $$-0.539215\pi$$
−0.122885 + 0.992421i $$0.539215\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.0000i 1.03751i 0.854922 + 0.518756i $$0.173605\pi$$
−0.854922 + 0.518756i $$0.826395\pi$$
$$158$$ 0 0
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.0000i 0.861586i 0.902451 + 0.430793i $$0.141766\pi$$
−0.902451 + 0.430793i $$0.858234\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ − 9.00000i − 0.684257i −0.939653 0.342129i $$-0.888852\pi$$
0.939653 0.342129i $$-0.111148\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 9.00000i − 0.676481i
$$178$$ 0 0
$$179$$ −21.0000 −1.56961 −0.784807 0.619740i $$-0.787238\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ − 1.00000i − 0.0739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 0 0
$$193$$ 11.0000i 0.791797i 0.918294 + 0.395899i $$0.129567\pi$$
−0.918294 + 0.395899i $$0.870433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 7.00000 0.496217 0.248108 0.968732i $$-0.420191\pi$$
0.248108 + 0.968732i $$0.420191\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000i 0.417029i
$$208$$ 0 0
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1.00000 0.0675737
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.00000i 0.199117i 0.995032 + 0.0995585i $$0.0317430\pi$$
−0.995032 + 0.0995585i $$0.968257\pi$$
$$228$$ 0 0
$$229$$ −11.0000 −0.726900 −0.363450 0.931614i $$-0.618401\pi$$
−0.363450 + 0.931614i $$0.618401\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 21.0000i − 1.37576i −0.725826 0.687878i $$-0.758542\pi$$
0.725826 0.687878i $$-0.241458\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 13.0000i 0.844441i
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −0.0644157 −0.0322078 0.999481i $$-0.510254\pi$$
−0.0322078 + 0.999481i $$0.510254\pi$$
$$242$$ 0 0
$$243$$ 16.0000i 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000i 0.127257i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ − 9.00000i − 0.565825i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 3.00000i − 0.187135i −0.995613 0.0935674i $$-0.970173\pi$$
0.995613 0.0935674i $$-0.0298271\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ − 3.00000i − 0.184988i −0.995713 0.0924940i $$-0.970516\pi$$
0.995713 0.0924940i $$-0.0294839\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 15.0000i − 0.917985i
$$268$$ 0 0
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.0000i 0.781094i 0.920583 + 0.390547i $$0.127714\pi$$
−0.920583 + 0.390547i $$0.872286\pi$$
$$278$$ 0 0
$$279$$ −14.0000 −0.838158
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ 29.0000i 1.72387i 0.507018 + 0.861936i $$0.330748\pi$$
−0.507018 + 0.861936i $$0.669252\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 15.0000i − 0.870388i
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 15.0000i 0.861727i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 0 0
$$309$$ −11.0000 −0.625768
$$310$$ 0 0
$$311$$ −27.0000 −1.53103 −0.765515 0.643418i $$-0.777516\pi$$
−0.765515 + 0.643418i $$0.777516\pi$$
$$312$$ 0 0
$$313$$ 23.0000i 1.30004i 0.759918 + 0.650018i $$0.225239\pi$$
−0.759918 + 0.650018i $$0.774761\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.00000i 0.505490i 0.967533 + 0.252745i $$0.0813334\pi$$
−0.967533 + 0.252745i $$0.918667\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 15.0000 0.837218
$$322$$ 0 0
$$323$$ − 3.00000i − 0.166924i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1.00000i 0.0553001i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −13.0000 −0.714545 −0.357272 0.934000i $$-0.616293\pi$$
−0.357272 + 0.934000i $$0.616293\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 34.0000i 1.85210i 0.377403 + 0.926049i $$0.376817\pi$$
−0.377403 + 0.926049i $$0.623183\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 21.0000 1.13721
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 9.00000i − 0.483145i −0.970383 0.241573i $$-0.922337\pi$$
0.970383 0.241573i $$-0.0776632\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ −10.0000 −0.533761
$$352$$ 0 0
$$353$$ − 21.0000i − 1.11772i −0.829263 0.558859i $$-0.811239\pi$$
0.829263 0.558859i $$-0.188761\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ − 2.00000i − 0.104973i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 5.00000i − 0.260998i −0.991448 0.130499i $$-0.958342\pi$$
0.991448 0.130499i $$-0.0416579\pi$$
$$368$$ 0 0
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 25.0000i − 1.29445i −0.762299 0.647225i $$-0.775929\pi$$
0.762299 0.647225i $$-0.224071\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ − 33.0000i − 1.68622i −0.537740 0.843111i $$-0.680722\pi$$
0.537740 0.843111i $$-0.319278\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 8.00000i − 0.406663i
$$388$$ 0 0
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 0 0
$$393$$ 3.00000i 0.151330i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 37.0000i 1.85698i 0.371361 + 0.928488i $$0.378891\pi$$
−0.371361 + 0.928488i $$0.621109\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 0 0
$$403$$ − 14.0000i − 0.697390i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 3.00000i − 0.148704i
$$408$$ 0 0
$$409$$ −11.0000 −0.543915 −0.271957 0.962309i $$-0.587671\pi$$
−0.271957 + 0.962309i $$0.587671\pi$$
$$410$$ 0 0
$$411$$ −21.0000 −1.03585
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 20.0000i − 0.979404i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ 18.0000i 0.875190i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −15.0000 −0.722525 −0.361262 0.932464i $$-0.617654\pi$$
−0.361262 + 0.932464i $$0.617654\pi$$
$$432$$ 0 0
$$433$$ − 10.0000i − 0.480569i −0.970702 0.240285i $$-0.922759\pi$$
0.970702 0.240285i $$-0.0772408\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.00000i 0.143509i
$$438$$ 0 0
$$439$$ 1.00000 0.0477274 0.0238637 0.999715i $$-0.492403\pi$$
0.0238637 + 0.999715i $$0.492403\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 9.00000i − 0.427603i −0.976877 0.213801i $$-0.931415\pi$$
0.976877 0.213801i $$-0.0685846\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 3.00000i − 0.141895i
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −18.0000 −0.847587
$$452$$ 0 0
$$453$$ 17.0000i 0.798730i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 23.0000i − 1.07589i −0.842978 0.537947i $$-0.819200\pi$$
0.842978 0.537947i $$-0.180800\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 21.0000i 0.971764i 0.874024 + 0.485882i $$0.161502\pi$$
−0.874024 + 0.485882i $$0.838498\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ 12.0000i 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ 3.00000 0.137073 0.0685367 0.997649i $$-0.478167\pi$$
0.0685367 + 0.997649i $$0.478167\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 19.0000i 0.860972i 0.902597 + 0.430486i $$0.141658\pi$$
−0.902597 + 0.430486i $$0.858342\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ − 18.0000i − 0.810679i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −11.0000 −0.492428 −0.246214 0.969216i $$-0.579187\pi$$
−0.246214 + 0.969216i $$0.579187\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ −3.00000 −0.132973 −0.0664863 0.997787i $$-0.521179\pi$$
−0.0664863 + 0.997787i $$0.521179\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 5.00000i 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 27.0000i − 1.18746i
$$518$$ 0 0
$$519$$ 9.00000 0.395056
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ 0 0
$$523$$ − 1.00000i − 0.0437269i −0.999761 0.0218635i $$-0.993040\pi$$
0.999761 0.0218635i $$-0.00695991\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 21.0000i 0.914774i
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ −18.0000 −0.781133
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 21.0000i − 0.906217i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 35.0000 1.50477 0.752384 0.658725i $$-0.228904\pi$$
0.752384 + 0.658725i $$0.228904\pi$$
$$542$$ 0 0
$$543$$ − 10.0000i − 0.429141i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 33.0000i 1.39825i 0.714997 + 0.699127i $$0.246428\pi$$
−0.714997 + 0.699127i $$0.753572\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ 9.00000i 0.379305i 0.981851 + 0.189652i $$0.0607361\pi$$
−0.981851 + 0.189652i $$0.939264\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 9.00000 0.377300 0.188650 0.982044i $$-0.439589\pi$$
0.188650 + 0.982044i $$0.439589\pi$$
$$570$$ 0 0
$$571$$ 29.0000 1.21361 0.606806 0.794850i $$-0.292450\pi$$
0.606806 + 0.794850i $$0.292450\pi$$
$$572$$ 0 0
$$573$$ − 9.00000i − 0.375980i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.00000i 0.0416305i 0.999783 + 0.0208153i $$0.00662619\pi$$
−0.999783 + 0.0208153i $$0.993374\pi$$
$$578$$ 0 0
$$579$$ −11.0000 −0.457144
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 9.00000i − 0.372742i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 0 0
$$589$$ −7.00000 −0.288430
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ − 21.0000i − 0.862367i −0.902264 0.431183i $$-0.858096\pi$$
0.902264 0.431183i $$-0.141904\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 7.00000i 0.286491i
$$598$$ 0 0
$$599$$ 27.0000 1.10319 0.551595 0.834112i $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ 14.0000i 0.570124i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 47.0000i − 1.90767i −0.300329 0.953836i $$-0.597097\pi$$
0.300329 0.953836i $$-0.402903\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ 0 0
$$613$$ − 25.0000i − 1.00974i −0.863195 0.504870i $$-0.831540\pi$$
0.863195 0.504870i $$-0.168460\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 0 0
$$619$$ 31.0000 1.24600 0.622998 0.782224i $$-0.285915\pi$$
0.622998 + 0.782224i $$0.285915\pi$$
$$620$$ 0 0
$$621$$ −15.0000 −0.601929
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 3.00000i − 0.119808i
$$628$$ 0 0
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ − 4.00000i − 0.158986i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 21.0000i − 0.825595i −0.910823 0.412798i $$-0.864552\pi$$
0.910823 0.412798i $$-0.135448\pi$$
$$648$$ 0 0
$$649$$ 27.0000 1.05984
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 39.0000i 1.52619i 0.646288 + 0.763094i $$0.276321\pi$$
−0.646288 + 0.763094i $$0.723679\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 11.0000 0.427850 0.213925 0.976850i $$-0.431375\pi$$
0.213925 + 0.976850i $$0.431375\pi$$
$$662$$ 0 0
$$663$$ 6.00000i 0.233021i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 18.0000i 0.696963i
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ 3.00000 0.115814
$$672$$ 0 0
$$673$$ 14.0000i 0.539660i 0.962908 + 0.269830i $$0.0869676\pi$$
−0.962908 + 0.269830i $$0.913032\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 27.0000i − 1.03769i −0.854867 0.518847i $$-0.826361\pi$$
0.854867 0.518847i $$-0.173639\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −3.00000 −0.114960
$$682$$ 0 0
$$683$$ 21.0000i 0.803543i 0.915740 + 0.401771i $$0.131605\pi$$
−0.915740 + 0.401771i $$0.868395\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 11.0000i − 0.419676i
$$688$$ 0 0
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ −13.0000 −0.494543 −0.247272 0.968946i $$-0.579534\pi$$
−0.247272 + 0.968946i $$0.579534\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 18.0000i − 0.681799i
$$698$$ 0 0
$$699$$ 21.0000 0.794293
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 1.00000i 0.0377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.00000 0.0375558 0.0187779 0.999824i $$-0.494022\pi$$
0.0187779 + 0.999824i $$0.494022\pi$$
$$710$$ 0 0
$$711$$ 26.0000 0.975076
$$712$$ 0 0
$$713$$ − 21.0000i − 0.786456i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0000i 0.448148i
$$718$$ 0 0
$$719$$ 21.0000 0.783168 0.391584 0.920142i $$-0.371927\pi$$
0.391584 + 0.920142i $$0.371927\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ − 1.00000i − 0.0371904i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 32.0000i − 1.18681i −0.804902 0.593407i $$-0.797782\pi$$
0.804902 0.593407i $$-0.202218\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ − 25.0000i − 0.923396i −0.887037 0.461698i $$-0.847240\pi$$
0.887037 0.461698i $$-0.152760\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 21.0000i − 0.773545i
$$738$$ 0 0
$$739$$ 19.0000 0.698926 0.349463 0.936950i $$-0.386364\pi$$
0.349463 + 0.936950i $$0.386364\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ − 48.0000i − 1.76095i −0.474093 0.880475i $$-0.657224\pi$$
0.474093 0.880475i $$-0.342776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.0000i 0.878114i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −25.0000 −0.912263 −0.456131 0.889912i $$-0.650765\pi$$
−0.456131 + 0.889912i $$0.650765\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 0 0
$$759$$ 9.00000 0.326679
$$760$$ 0 0
$$761$$ 3.00000 0.108750 0.0543750 0.998521i $$-0.482683\pi$$
0.0543750 + 0.998521i $$0.482683\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 18.0000i − 0.649942i
$$768$$ 0 0
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ 3.00000 0.108042
$$772$$ 0 0
$$773$$ − 33.0000i − 1.18693i −0.804861 0.593464i $$-0.797760\pi$$
0.804861 0.593464i $$-0.202240\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 30.0000i 1.07211i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 31.0000i 1.10503i 0.833503 + 0.552515i $$0.186332\pi$$
−0.833503 + 0.552515i $$0.813668\pi$$
$$788$$ 0 0
$$789$$ 3.00000 0.106803
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 2.00000i − 0.0710221i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ 27.0000 0.955191
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 0 0
$$803$$ 3.00000i 0.105868i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 3.00000i − 0.105605i
$$808$$ 0 0
$$809$$ 33.0000 1.16022 0.580109 0.814539i $$-0.303010\pi$$
0.580109 + 0.814539i $$0.303010\pi$$
$$810$$ 0 0
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 0 0
$$813$$ 11.0000i 0.385787i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 4.00000i − 0.139942i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 27.0000 0.942306 0.471153 0.882051i $$-0.343838\pi$$
0.471153 + 0.882051i $$0.343838\pi$$
$$822$$ 0 0
$$823$$ 5.00000i 0.174289i 0.996196 + 0.0871445i $$0.0277742\pi$$
−0.996196 + 0.0871445i $$0.972226\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ 1.00000 0.0347314 0.0173657 0.999849i $$-0.494472\pi$$
0.0173657 + 0.999849i $$0.494472\pi$$
$$830$$ 0 0
$$831$$ −13.0000 −0.450965
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 35.0000i − 1.20978i
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 30.0000i 1.03325i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −29.0000 −0.995277
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ − 22.0000i − 0.753266i −0.926363 0.376633i $$-0.877082\pi$$
0.926363 0.376633i $$-0.122918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 15.0000i − 0.512390i −0.966625 0.256195i $$-0.917531\pi$$
0.966625 0.256195i $$-0.0824690\pi$$
$$858$$ 0 0
$$859$$ −23.0000 −0.784750 −0.392375 0.919805i $$-0.628346\pi$$
−0.392375 + 0.919805i $$0.628346\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 51.0000i 1.73606i 0.496512 + 0.868030i $$0.334614\pi$$
−0.496512 + 0.868030i $$0.665386\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 8.00000i 0.271694i
$$868$$ 0 0
$$869$$ −39.0000 −1.32298
$$870$$ 0 0
$$871$$ −14.0000 −0.474372
$$872$$ 0 0
$$873$$ 20.0000i 0.676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13.0000i 0.438979i 0.975615 + 0.219489i $$0.0704391\pi$$
−0.975615 + 0.219489i $$0.929561\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 39.0000i − 1.30949i −0.755849 0.654746i $$-0.772776\pi$$
0.755849 0.654746i $$-0.227224\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 0 0
$$893$$ 9.00000i 0.301174i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ 0 0
$$899$$ −42.0000 −1.40078
$$900$$ 0 0
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 41.0000i − 1.36138i −0.732570 0.680691i $$-0.761680\pi$$
0.732570 0.680691i $$-0.238320\pi$$
$$908$$ 0 0
$$909$$ 30.0000 0.995037
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ − 36.0000i − 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.00000 0.0329870 0.0164935 0.999864i $$-0.494750\pi$$
0.0164935 + 0.999864i $$0.494750\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 22.0000i 0.722575i
$$928$$ 0 0
$$929$$ 9.00000 0.295280 0.147640 0.989041i $$-0.452832\pi$$
0.147640 + 0.989041i $$0.452832\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ − 27.0000i − 0.883940i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 26.0000i − 0.849383i −0.905338 0.424691i $$-0.860383\pi$$
0.905338 0.424691i $$-0.139617\pi$$
$$938$$ 0 0
$$939$$ −23.0000 −0.750577
$$940$$ 0 0
$$941$$ 27.0000 0.880175 0.440087 0.897955i $$-0.354947\pi$$
0.440087 + 0.897955i $$0.354947\pi$$
$$942$$ 0 0
$$943$$ 18.0000i 0.586161i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 45.0000i 1.46230i 0.682215 + 0.731152i $$0.261017\pi$$
−0.682215 + 0.731152i $$0.738983\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ −9.00000 −0.291845
$$952$$ 0 0
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 18.0000i − 0.581857i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ − 30.0000i − 0.966736i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ 0 0
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ 51.0000 1.63667 0.818334 0.574743i $$-0.194898\pi$$
0.818334 + 0.574743i $$0.194898\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 27.0000i − 0.863807i −0.901920 0.431903i $$-0.857842\pi$$
0.901920 0.431903i $$-0.142158\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 33.0000i 1.05254i 0.850319 + 0.526268i $$0.176409\pi$$
−0.850319 + 0.526268i $$0.823591\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ −19.0000 −0.603555 −0.301777 0.953378i $$-0.597580\pi$$
−0.301777 + 0.953378i $$0.597580\pi$$
$$992$$ 0 0
$$993$$ − 13.0000i − 0.412543i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 59.0000i − 1.86855i −0.356555 0.934274i $$-0.616049\pi$$
0.356555 0.934274i $$-0.383951\pi$$
$$998$$ 0 0
$$999$$ −5.00000 −0.158193
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 4900.2.e.i.2549.2 2
5.2 odd 4 196.2.a.b.1.1 1
5.3 odd 4 4900.2.a.g.1.1 1
5.4 even 2 inner 4900.2.e.i.2549.1 2
7.2 even 3 700.2.r.b.249.1 4
7.4 even 3 700.2.r.b.149.2 4
7.6 odd 2 4900.2.e.h.2549.1 2
15.2 even 4 1764.2.a.a.1.1 1
20.7 even 4 784.2.a.d.1.1 1
35.2 odd 12 28.2.e.a.25.1 yes 2
35.4 even 6 700.2.r.b.149.1 4
35.9 even 6 700.2.r.b.249.2 4
35.12 even 12 196.2.e.a.165.1 2
35.13 even 4 4900.2.a.n.1.1 1
35.17 even 12 196.2.e.a.177.1 2
35.18 odd 12 700.2.i.c.401.1 2
35.23 odd 12 700.2.i.c.501.1 2
35.27 even 4 196.2.a.a.1.1 1
35.32 odd 12 28.2.e.a.9.1 2
35.34 odd 2 4900.2.e.h.2549.2 2
40.27 even 4 3136.2.a.s.1.1 1
40.37 odd 4 3136.2.a.h.1.1 1
60.47 odd 4 7056.2.a.f.1.1 1
105.2 even 12 252.2.k.c.109.1 2
105.17 odd 12 1764.2.k.b.1549.1 2
105.32 even 12 252.2.k.c.37.1 2
105.47 odd 12 1764.2.k.b.361.1 2
105.62 odd 4 1764.2.a.j.1.1 1
140.27 odd 4 784.2.a.g.1.1 1
140.47 odd 12 784.2.i.d.753.1 2
140.67 even 12 112.2.i.b.65.1 2
140.87 odd 12 784.2.i.d.177.1 2
140.107 even 12 112.2.i.b.81.1 2
280.27 odd 4 3136.2.a.k.1.1 1
280.37 odd 12 448.2.i.e.193.1 2
280.67 even 12 448.2.i.c.65.1 2
280.107 even 12 448.2.i.c.193.1 2
280.237 even 4 3136.2.a.v.1.1 1
280.277 odd 12 448.2.i.e.65.1 2
315.2 even 12 2268.2.l.a.109.1 2
315.32 even 12 2268.2.l.a.541.1 2
315.67 odd 12 2268.2.l.h.541.1 2
315.137 even 12 2268.2.i.h.2053.1 2
315.142 odd 12 2268.2.l.h.109.1 2
315.212 even 12 2268.2.i.h.865.1 2
315.247 odd 12 2268.2.i.a.865.1 2
315.277 odd 12 2268.2.i.a.2053.1 2
420.107 odd 12 1008.2.s.p.865.1 2
420.167 even 4 7056.2.a.bw.1.1 1
420.347 odd 12 1008.2.s.p.289.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.e.a.9.1 2 35.32 odd 12
28.2.e.a.25.1 yes 2 35.2 odd 12
112.2.i.b.65.1 2 140.67 even 12
112.2.i.b.81.1 2 140.107 even 12
196.2.a.a.1.1 1 35.27 even 4
196.2.a.b.1.1 1 5.2 odd 4
196.2.e.a.165.1 2 35.12 even 12
196.2.e.a.177.1 2 35.17 even 12
252.2.k.c.37.1 2 105.32 even 12
252.2.k.c.109.1 2 105.2 even 12
448.2.i.c.65.1 2 280.67 even 12
448.2.i.c.193.1 2 280.107 even 12
448.2.i.e.65.1 2 280.277 odd 12
448.2.i.e.193.1 2 280.37 odd 12
700.2.i.c.401.1 2 35.18 odd 12
700.2.i.c.501.1 2 35.23 odd 12
700.2.r.b.149.1 4 35.4 even 6
700.2.r.b.149.2 4 7.4 even 3
700.2.r.b.249.1 4 7.2 even 3
700.2.r.b.249.2 4 35.9 even 6
784.2.a.d.1.1 1 20.7 even 4
784.2.a.g.1.1 1 140.27 odd 4
784.2.i.d.177.1 2 140.87 odd 12
784.2.i.d.753.1 2 140.47 odd 12
1008.2.s.p.289.1 2 420.347 odd 12
1008.2.s.p.865.1 2 420.107 odd 12
1764.2.a.a.1.1 1 15.2 even 4
1764.2.a.j.1.1 1 105.62 odd 4
1764.2.k.b.361.1 2 105.47 odd 12
1764.2.k.b.1549.1 2 105.17 odd 12
2268.2.i.a.865.1 2 315.247 odd 12
2268.2.i.a.2053.1 2 315.277 odd 12
2268.2.i.h.865.1 2 315.212 even 12
2268.2.i.h.2053.1 2 315.137 even 12
2268.2.l.a.109.1 2 315.2 even 12
2268.2.l.a.541.1 2 315.32 even 12
2268.2.l.h.109.1 2 315.142 odd 12
2268.2.l.h.541.1 2 315.67 odd 12
3136.2.a.h.1.1 1 40.37 odd 4
3136.2.a.k.1.1 1 280.27 odd 4
3136.2.a.s.1.1 1 40.27 even 4
3136.2.a.v.1.1 1 280.237 even 4
4900.2.a.g.1.1 1 5.3 odd 4
4900.2.a.n.1.1 1 35.13 even 4
4900.2.e.h.2549.1 2 7.6 odd 2
4900.2.e.h.2549.2 2 35.34 odd 2
4900.2.e.i.2549.1 2 5.4 even 2 inner
4900.2.e.i.2549.2 2 1.1 even 1 trivial
7056.2.a.f.1.1 1 60.47 odd 4
7056.2.a.bw.1.1 1 420.167 even 4