Properties

 Label 196.2.a.a.1.1 Level $196$ Weight $2$ Character 196.1 Self dual yes Analytic conductor $1.565$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 196.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$1.56506787962$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 196.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} +3.00000 q^{15} -3.00000 q^{17} +1.00000 q^{19} +3.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} -6.00000 q^{29} +7.00000 q^{31} +3.00000 q^{33} -1.00000 q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} +6.00000 q^{45} +9.00000 q^{47} +3.00000 q^{51} +3.00000 q^{53} +9.00000 q^{55} -1.00000 q^{57} -9.00000 q^{59} +1.00000 q^{61} +6.00000 q^{65} -7.00000 q^{67} -3.00000 q^{69} +1.00000 q^{73} -4.00000 q^{75} -13.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +9.00000 q^{85} +6.00000 q^{87} -15.0000 q^{89} -7.00000 q^{93} -3.00000 q^{95} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 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
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 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
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 0 0
$$33$$ 3.00000 0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 6.00000 0.894427
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 0 0
$$55$$ 9.00000 1.21356
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 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 $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ 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
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.0000 −1.46261 −0.731307 0.682048i $$-0.761089\pi$$
−0.731307 + 0.682048i $$0.761089\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 9.00000 0.976187
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 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
$$93$$ −7.00000 −0.725866
$$94$$ 0 0
$$95$$ −3.00000 −0.307794
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −15.0000 −1.49256 −0.746278 0.665635i $$-0.768161\pi$$
−0.746278 + 0.665635i $$0.768161\pi$$
$$102$$ 0 0
$$103$$ −11.0000 −1.08386 −0.541931 0.840423i $$-0.682307\pi$$
−0.541931 + 0.840423i $$0.682307\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.0000 1.45010 0.725052 0.688694i $$-0.241816\pi$$
0.725052 + 0.688694i $$0.241816\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −15.0000 −1.29099
$$136$$ 0 0
$$137$$ −21.0000 −1.79415 −0.897076 0.441877i $$-0.854313\pi$$
−0.897076 + 0.441877i $$0.854313\pi$$
$$138$$ 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
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ 18.0000 1.49482
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ −21.0000 −1.68676
$$156$$ 0 0
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0 0
$$165$$ −9.00000 −0.700649
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 9.00000 0.676481
$$178$$ 0 0
$$179$$ 21.0000 1.56961 0.784807 0.619740i $$-0.212762\pi$$
0.784807 + 0.619740i $$0.212762\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 3.00000 0.220564
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ 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
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 0 0
$$195$$ −6.00000 −0.429669
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 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
$$205$$ 18.0000 1.25717
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 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
$$215$$ 12.0000 0.818393
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1.00000 −0.0675737
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ −8.00000 −0.533333
$$226$$ 0 0
$$227$$ 3.00000 0.199117 0.0995585 0.995032i $$-0.468257\pi$$
0.0995585 + 0.995032i $$0.468257\pi$$
$$228$$ 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
$$233$$ −21.0000 −1.37576 −0.687878 0.725826i $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$234$$ 0 0
$$235$$ −27.0000 −1.76129
$$236$$ 0 0
$$237$$ 13.0000 0.844441
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 1.00000 0.0644157 0.0322078 0.999481i $$-0.489746\pi$$
0.0322078 + 0.999481i $$0.489746\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 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
$$253$$ −9.00000 −0.565825
$$254$$ 0 0
$$255$$ −9.00000 −0.563602
$$256$$ 0 0
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 0 0
$$265$$ −9.00000 −0.552866
$$266$$ 0 0
$$267$$ 15.0000 0.917985
$$268$$ 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 $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ −13.0000 −0.781094 −0.390547 0.920583i $$-0.627714\pi$$
−0.390547 + 0.920583i $$0.627714\pi$$
$$278$$ 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
$$283$$ −29.0000 −1.72387 −0.861936 0.507018i $$-0.830748\pi$$
−0.861936 + 0.507018i $$0.830748\pi$$
$$284$$ 0 0
$$285$$ 3.00000 0.177705
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 27.0000 1.57200
$$296$$ 0 0
$$297$$ −15.0000 −0.870388
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 15.0000 0.861727
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ 11.0000 0.625768
$$310$$ 0 0
$$311$$ 27.0000 1.53103 0.765515 0.643418i $$-0.222484\pi$$
0.765515 + 0.643418i $$0.222484\pi$$
$$312$$ 0 0
$$313$$ −23.0000 −1.30004 −0.650018 0.759918i $$-0.725239\pi$$
−0.650018 + 0.759918i $$0.725239\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9.00000 −0.505490 −0.252745 0.967533i $$-0.581333\pi$$
−0.252745 + 0.967533i $$0.581333\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ −15.0000 −0.837218
$$322$$ 0 0
$$323$$ −3.00000 −0.166924
$$324$$ 0 0
$$325$$ −8.00000 −0.443760
$$326$$ 0 0
$$327$$ 1.00000 0.0553001
$$328$$ 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
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 21.0000 1.14735
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −21.0000 −1.13721
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 9.00000 0.484544
$$346$$ 0 0
$$347$$ 9.00000 0.483145 0.241573 0.970383i $$-0.422337\pi$$
0.241573 + 0.970383i $$0.422337\pi$$
$$348$$ 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
$$353$$ 21.0000 1.11772 0.558859 0.829263i $$-0.311239\pi$$
0.558859 + 0.829263i $$0.311239\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ −3.00000 −0.157027
$$366$$ 0 0
$$367$$ −5.00000 −0.260998 −0.130499 0.991448i $$-0.541658\pi$$
−0.130499 + 0.991448i $$0.541658\pi$$
$$368$$ 0 0
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −25.0000 −1.29445 −0.647225 0.762299i $$-0.724071\pi$$
−0.647225 + 0.762299i $$0.724071\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 33.0000 1.68622 0.843111 0.537740i $$-0.180722\pi$$
0.843111 + 0.537740i $$0.180722\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000 0.406663
$$388$$ 0 0
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ −9.00000 −0.455150
$$392$$ 0 0
$$393$$ 3.00000 0.151330
$$394$$ 0 0
$$395$$ 39.0000 1.96230
$$396$$ 0 0
$$397$$ 37.0000 1.85698 0.928488 0.371361i $$-0.121109\pi$$
0.928488 + 0.371361i $$0.121109\pi$$
$$398$$ 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
$$403$$ −14.0000 −0.697390
$$404$$ 0 0
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ 3.00000 0.148704
$$408$$ 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
$$415$$ 36.0000 1.76717
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 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
$$423$$ −18.0000 −0.875190
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 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
$$433$$ 10.0000 0.480569 0.240285 0.970702i $$-0.422759\pi$$
0.240285 + 0.970702i $$0.422759\pi$$
$$434$$ 0 0
$$435$$ −18.0000 −0.863034
$$436$$ 0 0
$$437$$ 3.00000 0.143509
$$438$$ 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
$$443$$ −9.00000 −0.427603 −0.213801 0.976877i $$-0.568585\pi$$
−0.213801 + 0.976877i $$0.568585\pi$$
$$444$$ 0 0
$$445$$ 45.0000 2.13320
$$446$$ 0 0
$$447$$ −3.00000 −0.141895
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ 0 0
$$453$$ −17.0000 −0.798730
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.0000 1.07589 0.537947 0.842978i $$-0.319200\pi$$
0.537947 + 0.842978i $$0.319200\pi$$
$$458$$ 0 0
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ 21.0000 0.973852
$$466$$ 0 0
$$467$$ 21.0000 0.971764 0.485882 0.874024i $$-0.338498\pi$$
0.485882 + 0.874024i $$0.338498\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 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
$$485$$ −30.0000 −1.36223
$$486$$ 0 0
$$487$$ −19.0000 −0.860972 −0.430486 0.902597i $$-0.641658\pi$$
−0.430486 + 0.902597i $$0.641658\pi$$
$$488$$ 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
$$493$$ 18.0000 0.810679
$$494$$ 0 0
$$495$$ −18.0000 −0.809040
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 11.0000 0.492428 0.246214 0.969216i $$-0.420813\pi$$
0.246214 + 0.969216i $$0.420813\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 45.0000 2.00247
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 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
$$513$$ 5.00000 0.220755
$$514$$ 0 0
$$515$$ 33.0000 1.45415
$$516$$ 0 0
$$517$$ −27.0000 −1.18746
$$518$$ 0 0
$$519$$ −9.00000 −0.395056
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ 1.00000 0.0437269 0.0218635 0.999761i $$-0.493040\pi$$
0.0218635 + 0.999761i $$0.493040\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −21.0000 −0.914774
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 18.0000 0.781133
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ −45.0000 −1.94552
$$536$$ 0 0
$$537$$ −21.0000 −0.906217
$$538$$ 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
$$543$$ −10.0000 −0.429141
$$544$$ 0 0
$$545$$ 3.00000 0.128506
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −3.00000 −0.127343
$$556$$ 0 0
$$557$$ −33.0000 −1.39825 −0.699127 0.714997i $$-0.746428\pi$$
−0.699127 + 0.714997i $$0.746428\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ −9.00000 −0.379305 −0.189652 0.981851i $$-0.560736\pi$$
−0.189652 + 0.981851i $$0.560736\pi$$
$$564$$ 0 0
$$565$$ −18.0000 −0.757266
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −9.00000 −0.377300 −0.188650 0.982044i $$-0.560411\pi$$
−0.188650 + 0.982044i $$0.560411\pi$$
$$570$$ 0 0
$$571$$ 29.0000 1.21361 0.606806 0.794850i $$-0.292450\pi$$
0.606806 + 0.794850i $$0.292450\pi$$
$$572$$ 0 0
$$573$$ 9.00000 0.375980
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ 1.00000 0.0416305 0.0208153 0.999783i $$-0.493374\pi$$
0.0208153 + 0.999783i $$0.493374\pi$$
$$578$$ 0 0
$$579$$ −11.0000 −0.457144
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 7.00000 0.288430
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 0 0
$$593$$ 21.0000 0.862367 0.431183 0.902264i $$-0.358096\pi$$
0.431183 + 0.902264i $$0.358096\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −7.00000 −0.286491
$$598$$ 0 0
$$599$$ −27.0000 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 0 0
$$603$$ 14.0000 0.570124
$$604$$ 0 0
$$605$$ 6.00000 0.243935
$$606$$ 0 0
$$607$$ −47.0000 −1.90767 −0.953836 0.300329i $$-0.902903\pi$$
−0.953836 + 0.300329i $$0.902903\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ 0 0
$$613$$ −25.0000 −1.00974 −0.504870 0.863195i $$-0.668460\pi$$
−0.504870 + 0.863195i $$0.668460\pi$$
$$614$$ 0 0
$$615$$ −18.0000 −0.725830
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 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
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 3.00000 0.119808
$$628$$ 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
$$633$$ 4.00000 0.158986
$$634$$ 0 0
$$635$$ −24.0000 −0.952411
$$636$$ 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
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ −12.0000 −0.472500
$$646$$ 0 0
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ 0 0
$$649$$ 27.0000 1.05984
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 39.0000 1.52619 0.763094 0.646288i $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ 9.00000 0.351659
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −11.0000 −0.427850 −0.213925 0.976850i $$-0.568625\pi$$
−0.213925 + 0.976850i $$0.568625\pi$$
$$662$$ 0 0
$$663$$ −6.00000 −0.233021
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −18.0000 −0.696963
$$668$$ 0 0
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −3.00000 −0.115814
$$672$$ 0 0
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 0 0
$$675$$ 20.0000 0.769800
$$676$$ 0 0
$$677$$ −27.0000 −1.03769 −0.518847 0.854867i $$-0.673639\pi$$
−0.518847 + 0.854867i $$0.673639\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −3.00000 −0.114960
$$682$$ 0 0
$$683$$ 21.0000 0.803543 0.401771 0.915740i $$-0.368395\pi$$
0.401771 + 0.915740i $$0.368395\pi$$
$$684$$ 0 0
$$685$$ 63.0000 2.40711
$$686$$ 0 0
$$687$$ 11.0000 0.419676
$$688$$ 0 0
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 13.0000 0.494543 0.247272 0.968946i $$-0.420466\pi$$
0.247272 + 0.968946i $$0.420466\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 60.0000 2.27593
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 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
$$703$$ −1.00000 −0.0377157
$$704$$ 0 0
$$705$$ 27.0000 1.01688
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.00000 −0.0375558 −0.0187779 0.999824i $$-0.505978\pi$$
−0.0187779 + 0.999824i $$0.505978\pi$$
$$710$$ 0 0
$$711$$ 26.0000 0.975076
$$712$$ 0 0
$$713$$ 21.0000 0.786456
$$714$$ 0 0
$$715$$ −18.0000 −0.673162
$$716$$ 0 0
$$717$$ 12.0000 0.448148
$$718$$ 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
$$723$$ −1.00000 −0.0371904
$$724$$ 0 0
$$725$$ −24.0000 −0.891338
$$726$$ 0 0
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ 25.0000 0.923396 0.461698 0.887037i $$-0.347240\pi$$
0.461698 + 0.887037i $$0.347240\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.0000 0.773545
$$738$$ 0 0
$$739$$ −19.0000 −0.698926 −0.349463 0.936950i $$-0.613636\pi$$
−0.349463 + 0.936950i $$0.613636\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ −9.00000 −0.329734
$$746$$ 0 0
$$747$$ 24.0000 0.878114
$$748$$ 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
$$755$$ −51.0000 −1.85608
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 9.00000 0.326679
$$760$$ 0 0
$$761$$ −3.00000 −0.108750 −0.0543750 0.998521i $$-0.517317\pi$$
−0.0543750 + 0.998521i $$0.517317\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −18.0000 −0.650791
$$766$$ 0 0
$$767$$ 18.0000 0.649942
$$768$$ 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
$$773$$ 33.0000 1.18693 0.593464 0.804861i $$-0.297760\pi$$
0.593464 + 0.804861i $$0.297760\pi$$
$$774$$ 0 0
$$775$$ 28.0000 1.00579
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −30.0000 −1.07211
$$784$$ 0 0
$$785$$ −39.0000 −1.39197
$$786$$ 0 0
$$787$$ 31.0000 1.10503 0.552515 0.833503i $$-0.313668\pi$$
0.552515 + 0.833503i $$0.313668\pi$$
$$788$$ 0 0
$$789$$ 3.00000 0.106803
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 0 0
$$795$$ 9.00000 0.319197
$$796$$ 0 0
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ −27.0000 −0.955191
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ −3.00000 −0.105868
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3.00000 0.105605
$$808$$ 0 0
$$809$$ −33.0000 −1.16022 −0.580109 0.814539i $$-0.696990\pi$$
−0.580109 + 0.814539i $$0.696990\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ 0 0
$$813$$ 11.0000 0.385787
$$814$$ 0 0
$$815$$ −33.0000 −1.15594
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 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
$$823$$ 5.00000 0.174289 0.0871445 0.996196i $$-0.472226\pi$$
0.0871445 + 0.996196i $$0.472226\pi$$
$$824$$ 0 0
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 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
$$835$$ −36.0000 −1.24583
$$836$$ 0 0
$$837$$ 35.0000 1.20978
$$838$$ 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
$$843$$ −30.0000 −1.03325
$$844$$ 0 0
$$845$$ 27.0000 0.928828
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 29.0000 0.995277
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 0 0
$$857$$ −15.0000 −0.512390 −0.256195 0.966625i $$-0.582469\pi$$
−0.256195 + 0.966625i $$0.582469\pi$$
$$858$$ 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
$$863$$ 51.0000 1.73606 0.868030 0.496512i $$-0.165386\pi$$
0.868030 + 0.496512i $$0.165386\pi$$
$$864$$ 0 0
$$865$$ −27.0000 −0.918028
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ 39.0000 1.32298
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 0 0
$$873$$ −20.0000 −0.676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 0 0
$$885$$ −27.0000 −0.907595
$$886$$ 0 0
$$887$$ −39.0000 −1.30949 −0.654746 0.755849i $$-0.727224\pi$$
−0.654746 + 0.755849i $$0.727224\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 0 0
$$893$$ 9.00000 0.301174
$$894$$ 0 0
$$895$$ −63.0000 −2.10586
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 0 0
$$899$$ −42.0000 −1.40078
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −30.0000 −0.997234
$$906$$ 0 0
$$907$$ 41.0000 1.36138 0.680691 0.732570i $$-0.261680\pi$$
0.680691 + 0.732570i $$0.261680\pi$$
$$908$$ 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
$$913$$ 36.0000 1.19143
$$914$$ 0 0
$$915$$ 3.00000 0.0991769
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1.00000 −0.0329870 −0.0164935 0.999864i $$-0.505250\pi$$
−0.0164935 + 0.999864i $$0.505250\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ 22.0000 0.722575
$$928$$ 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
$$933$$ −27.0000 −0.883940
$$934$$ 0 0
$$935$$ −27.0000 −0.882994
$$936$$ 0 0
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ 0 0
$$939$$ 23.0000 0.750577
$$940$$ 0 0
$$941$$ −27.0000 −0.880175 −0.440087 0.897955i $$-0.645053\pi$$
−0.440087 + 0.897955i $$0.645053\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −45.0000 −1.46230 −0.731152 0.682215i $$-0.761017\pi$$
−0.731152 + 0.682215i $$0.761017\pi$$
$$948$$ 0 0
$$949$$ −2.00000 −0.0649227
$$950$$ 0 0
$$951$$ 9.00000 0.291845
$$952$$ 0 0
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ 27.0000 0.873699
$$956$$ 0 0
$$957$$ −18.0000 −0.581857
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ −30.0000 −0.966736
$$964$$ 0 0
$$965$$ −33.0000 −1.06231
$$966$$ 0 0
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ 0 0
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ −51.0000 −1.63667 −0.818334 0.574743i $$-0.805102\pi$$
−0.818334 + 0.574743i $$0.805102\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 8.00000 0.256205
$$976$$ 0 0
$$977$$ 27.0000 0.863807 0.431903 0.901920i $$-0.357842\pi$$
0.431903 + 0.901920i $$0.357842\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ −33.0000 −1.05254 −0.526268 0.850319i $$-0.676409\pi$$
−0.526268 + 0.850319i $$0.676409\pi$$
$$984$$ 0 0
$$985$$ −54.0000 −1.72058
$$986$$ 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
$$993$$ 13.0000 0.412543
$$994$$ 0 0
$$995$$ −21.0000 −0.665745
$$996$$ 0 0
$$997$$ −59.0000 −1.86855 −0.934274 0.356555i $$-0.883951\pi$$
−0.934274 + 0.356555i $$0.883951\pi$$
$$998$$ 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 196.2.a.a.1.1 1
3.2 odd 2 1764.2.a.j.1.1 1
4.3 odd 2 784.2.a.g.1.1 1
5.2 odd 4 4900.2.e.h.2549.2 2
5.3 odd 4 4900.2.e.h.2549.1 2
5.4 even 2 4900.2.a.n.1.1 1
7.2 even 3 196.2.e.a.165.1 2
7.3 odd 6 28.2.e.a.9.1 2
7.4 even 3 196.2.e.a.177.1 2
7.5 odd 6 28.2.e.a.25.1 yes 2
7.6 odd 2 196.2.a.b.1.1 1
8.3 odd 2 3136.2.a.k.1.1 1
8.5 even 2 3136.2.a.v.1.1 1
12.11 even 2 7056.2.a.bw.1.1 1
21.2 odd 6 1764.2.k.b.361.1 2
21.5 even 6 252.2.k.c.109.1 2
21.11 odd 6 1764.2.k.b.1549.1 2
21.17 even 6 252.2.k.c.37.1 2
21.20 even 2 1764.2.a.a.1.1 1
28.3 even 6 112.2.i.b.65.1 2
28.11 odd 6 784.2.i.d.177.1 2
28.19 even 6 112.2.i.b.81.1 2
28.23 odd 6 784.2.i.d.753.1 2
28.27 even 2 784.2.a.d.1.1 1
35.3 even 12 700.2.r.b.149.2 4
35.12 even 12 700.2.r.b.249.2 4
35.13 even 4 4900.2.e.i.2549.2 2
35.17 even 12 700.2.r.b.149.1 4
35.19 odd 6 700.2.i.c.501.1 2
35.24 odd 6 700.2.i.c.401.1 2
35.27 even 4 4900.2.e.i.2549.1 2
35.33 even 12 700.2.r.b.249.1 4
35.34 odd 2 4900.2.a.g.1.1 1
56.3 even 6 448.2.i.c.65.1 2
56.5 odd 6 448.2.i.e.193.1 2
56.13 odd 2 3136.2.a.h.1.1 1
56.19 even 6 448.2.i.c.193.1 2
56.27 even 2 3136.2.a.s.1.1 1
56.45 odd 6 448.2.i.e.65.1 2
63.5 even 6 2268.2.i.h.865.1 2
63.31 odd 6 2268.2.l.h.541.1 2
63.38 even 6 2268.2.i.h.2053.1 2
63.40 odd 6 2268.2.i.a.865.1 2
63.47 even 6 2268.2.l.a.109.1 2
63.52 odd 6 2268.2.i.a.2053.1 2
63.59 even 6 2268.2.l.a.541.1 2
63.61 odd 6 2268.2.l.h.109.1 2
84.47 odd 6 1008.2.s.p.865.1 2
84.59 odd 6 1008.2.s.p.289.1 2
84.83 odd 2 7056.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.e.a.9.1 2 7.3 odd 6
28.2.e.a.25.1 yes 2 7.5 odd 6
112.2.i.b.65.1 2 28.3 even 6
112.2.i.b.81.1 2 28.19 even 6
196.2.a.a.1.1 1 1.1 even 1 trivial
196.2.a.b.1.1 1 7.6 odd 2
196.2.e.a.165.1 2 7.2 even 3
196.2.e.a.177.1 2 7.4 even 3
252.2.k.c.37.1 2 21.17 even 6
252.2.k.c.109.1 2 21.5 even 6
448.2.i.c.65.1 2 56.3 even 6
448.2.i.c.193.1 2 56.19 even 6
448.2.i.e.65.1 2 56.45 odd 6
448.2.i.e.193.1 2 56.5 odd 6
700.2.i.c.401.1 2 35.24 odd 6
700.2.i.c.501.1 2 35.19 odd 6
700.2.r.b.149.1 4 35.17 even 12
700.2.r.b.149.2 4 35.3 even 12
700.2.r.b.249.1 4 35.33 even 12
700.2.r.b.249.2 4 35.12 even 12
784.2.a.d.1.1 1 28.27 even 2
784.2.a.g.1.1 1 4.3 odd 2
784.2.i.d.177.1 2 28.11 odd 6
784.2.i.d.753.1 2 28.23 odd 6
1008.2.s.p.289.1 2 84.59 odd 6
1008.2.s.p.865.1 2 84.47 odd 6
1764.2.a.a.1.1 1 21.20 even 2
1764.2.a.j.1.1 1 3.2 odd 2
1764.2.k.b.361.1 2 21.2 odd 6
1764.2.k.b.1549.1 2 21.11 odd 6
2268.2.i.a.865.1 2 63.40 odd 6
2268.2.i.a.2053.1 2 63.52 odd 6
2268.2.i.h.865.1 2 63.5 even 6
2268.2.i.h.2053.1 2 63.38 even 6
2268.2.l.a.109.1 2 63.47 even 6
2268.2.l.a.541.1 2 63.59 even 6
2268.2.l.h.109.1 2 63.61 odd 6
2268.2.l.h.541.1 2 63.31 odd 6
3136.2.a.h.1.1 1 56.13 odd 2
3136.2.a.k.1.1 1 8.3 odd 2
3136.2.a.s.1.1 1 56.27 even 2
3136.2.a.v.1.1 1 8.5 even 2
4900.2.a.g.1.1 1 35.34 odd 2
4900.2.a.n.1.1 1 5.4 even 2
4900.2.e.h.2549.1 2 5.3 odd 4
4900.2.e.h.2549.2 2 5.2 odd 4
4900.2.e.i.2549.1 2 35.27 even 4
4900.2.e.i.2549.2 2 35.13 even 4
7056.2.a.f.1.1 1 84.83 odd 2
7056.2.a.bw.1.1 1 12.11 even 2