# Properties

 Label 8470.2.a.bd.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -1.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} +5.00000 q^{39} -1.00000 q^{40} +1.00000 q^{42} +8.00000 q^{43} +2.00000 q^{45} -3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +5.00000 q^{52} +6.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{65} +14.0000 q^{67} +6.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} -2.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} +5.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} +2.00000 q^{90} +5.00000 q^{91} -3.00000 q^{92} -10.0000 q^{93} -6.00000 q^{94} -5.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ −2.00000 −0.666667
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 1.00000 0.288675
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 5.00000 0.980581
$$27$$ −5.00000 −0.962250
$$28$$ 1.00000 0.188982
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ −1.00000 −0.169031
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 5.00000 0.800641
$$40$$ −1.00000 −0.158114
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ −3.00000 −0.442326
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 6.00000 0.840168
$$52$$ 5.00000 0.693375
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 5.00000 0.662266
$$58$$ 0 0
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −10.0000 −1.27000
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ −5.00000 −0.620174
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −3.00000 −0.361158
$$70$$ −1.00000 −0.119523
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −2.00000 −0.235702
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 1.00000 0.115470
$$76$$ 5.00000 0.573539
$$77$$ 0 0
$$78$$ 5.00000 0.566139
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ 1.00000 0.109109
$$85$$ −6.00000 −0.650791
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 5.00000 0.524142
$$92$$ −3.00000 −0.312772
$$93$$ −10.0000 −1.03695
$$94$$ −6.00000 −0.618853
$$95$$ −5.00000 −0.512989
$$96$$ 1.00000 0.102062
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 5.00000 0.490290
$$105$$ −1.00000 −0.0975900
$$106$$ 6.00000 0.582772
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 1.00000 0.0944911
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 5.00000 0.468293
$$115$$ 3.00000 0.279751
$$116$$ 0 0
$$117$$ −10.0000 −0.924500
$$118$$ 9.00000 0.828517
$$119$$ 6.00000 0.550019
$$120$$ −1.00000 −0.0912871
$$121$$ 0 0
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ −10.0000 −0.898027
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ −5.00000 −0.438529
$$131$$ 21.0000 1.83478 0.917389 0.397991i $$-0.130293\pi$$
0.917389 + 0.397991i $$0.130293\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 14.0000 1.20942
$$135$$ 5.00000 0.430331
$$136$$ 6.00000 0.514496
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ −3.00000 −0.255377
$$139$$ −19.0000 −1.61156 −0.805779 0.592216i $$-0.798253\pi$$
−0.805779 + 0.592216i $$0.798253\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 1.00000 0.0824786
$$148$$ 2.00000 0.164399
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −7.00000 −0.569652 −0.284826 0.958579i $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 5.00000 0.405554
$$153$$ −12.0000 −0.970143
$$154$$ 0 0
$$155$$ 10.0000 0.803219
$$156$$ 5.00000 0.400320
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ −1.00000 −0.0795557
$$159$$ 6.00000 0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ −3.00000 −0.236433
$$162$$ 1.00000 0.0785674
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 12.0000 0.923077
$$170$$ −6.00000 −0.460179
$$171$$ −10.0000 −0.764719
$$172$$ 8.00000 0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 9.00000 0.676481
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 5.00000 0.370625
$$183$$ −10.0000 −0.739221
$$184$$ −3.00000 −0.221163
$$185$$ −2.00000 −0.147043
$$186$$ −10.0000 −0.733236
$$187$$ 0 0
$$188$$ −6.00000 −0.437595
$$189$$ −5.00000 −0.363696
$$190$$ −5.00000 −0.362738
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 8.00000 0.574367
$$195$$ −5.00000 −0.358057
$$196$$ 1.00000 0.0714286
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 14.0000 0.987484
$$202$$ −3.00000 −0.211079
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 6.00000 0.417029
$$208$$ 5.00000 0.346688
$$209$$ 0 0
$$210$$ −1.00000 −0.0690066
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 6.00000 0.410152
$$215$$ −8.00000 −0.545595
$$216$$ −5.00000 −0.340207
$$217$$ −10.0000 −0.678844
$$218$$ −10.0000 −0.677285
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ 2.00000 0.134231
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −2.00000 −0.133333
$$226$$ 9.00000 0.598671
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 5.00000 0.331133
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −21.0000 −1.37576 −0.687878 0.725826i $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$234$$ −10.0000 −0.653720
$$235$$ 6.00000 0.391397
$$236$$ 9.00000 0.585850
$$237$$ −1.00000 −0.0649570
$$238$$ 6.00000 0.388922
$$239$$ −3.00000 −0.194054 −0.0970269 0.995282i $$-0.530933\pi$$
−0.0970269 + 0.995282i $$0.530933\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ −10.0000 −0.640184
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 25.0000 1.59071
$$248$$ −10.0000 −0.635001
$$249$$ 9.00000 0.570352
$$250$$ −1.00000 −0.0632456
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ −7.00000 −0.439219
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 2.00000 0.124274
$$260$$ −5.00000 −0.310087
$$261$$ 0 0
$$262$$ 21.0000 1.29738
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 5.00000 0.306570
$$267$$ 0 0
$$268$$ 14.0000 0.855186
$$269$$ −21.0000 −1.28039 −0.640196 0.768211i $$-0.721147\pi$$
−0.640196 + 0.768211i $$0.721147\pi$$
$$270$$ 5.00000 0.304290
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 5.00000 0.302614
$$274$$ −3.00000 −0.181237
$$275$$ 0 0
$$276$$ −3.00000 −0.180579
$$277$$ 32.0000 1.92269 0.961347 0.275340i $$-0.0887905\pi$$
0.961347 + 0.275340i $$0.0887905\pi$$
$$278$$ −19.0000 −1.13954
$$279$$ 20.0000 1.19737
$$280$$ −1.00000 −0.0597614
$$281$$ −15.0000 −0.894825 −0.447412 0.894328i $$-0.647654\pi$$
−0.447412 + 0.894328i $$0.647654\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ 29.0000 1.72387 0.861936 0.507018i $$-0.169252\pi$$
0.861936 + 0.507018i $$0.169252\pi$$
$$284$$ 0 0
$$285$$ −5.00000 −0.296174
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −2.00000 −0.117851
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 2.00000 0.117041
$$293$$ 27.0000 1.57736 0.788678 0.614806i $$-0.210766\pi$$
0.788678 + 0.614806i $$0.210766\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −9.00000 −0.524000
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ −15.0000 −0.867472
$$300$$ 1.00000 0.0577350
$$301$$ 8.00000 0.461112
$$302$$ −7.00000 −0.402805
$$303$$ −3.00000 −0.172345
$$304$$ 5.00000 0.286770
$$305$$ 10.0000 0.572598
$$306$$ −12.0000 −0.685994
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 10.0000 0.567962
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 5.00000 0.283069
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 5.00000 0.282166
$$315$$ 2.00000 0.112687
$$316$$ −1.00000 −0.0562544
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 6.00000 0.334887
$$322$$ −3.00000 −0.167183
$$323$$ 30.0000 1.66924
$$324$$ 1.00000 0.0555556
$$325$$ 5.00000 0.277350
$$326$$ 2.00000 0.110770
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 9.00000 0.493939
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −14.0000 −0.764902
$$336$$ 1.00000 0.0545545
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 9.00000 0.488813
$$340$$ −6.00000 −0.325396
$$341$$ 0 0
$$342$$ −10.0000 −0.540738
$$343$$ 1.00000 0.0539949
$$344$$ 8.00000 0.431331
$$345$$ 3.00000 0.161515
$$346$$ 6.00000 0.322562
$$347$$ −6.00000 −0.322097 −0.161048 0.986947i $$-0.551488\pi$$
−0.161048 + 0.986947i $$0.551488\pi$$
$$348$$ 0 0
$$349$$ 11.0000 0.588817 0.294408 0.955680i $$-0.404877\pi$$
0.294408 + 0.955680i $$0.404877\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −25.0000 −1.33440
$$352$$ 0 0
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 9.00000 0.478345
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 6.00000 0.317554
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 2.00000 0.105409
$$361$$ 6.00000 0.315789
$$362$$ −25.0000 −1.31397
$$363$$ 0 0
$$364$$ 5.00000 0.262071
$$365$$ −2.00000 −0.104685
$$366$$ −10.0000 −0.522708
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 0 0
$$370$$ −2.00000 −0.103975
$$371$$ 6.00000 0.311504
$$372$$ −10.0000 −0.518476
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ −6.00000 −0.309426
$$377$$ 0 0
$$378$$ −5.00000 −0.257172
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ −5.00000 −0.256495
$$381$$ −7.00000 −0.358621
$$382$$ 15.0000 0.767467
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 11.0000 0.559885
$$387$$ −16.0000 −0.813326
$$388$$ 8.00000 0.406138
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ −5.00000 −0.253185
$$391$$ −18.0000 −0.910299
$$392$$ 1.00000 0.0505076
$$393$$ 21.0000 1.05931
$$394$$ 12.0000 0.604551
$$395$$ 1.00000 0.0503155
$$396$$ 0 0
$$397$$ 38.0000 1.90717 0.953583 0.301131i $$-0.0973643\pi$$
0.953583 + 0.301131i $$0.0973643\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 5.00000 0.250313
$$400$$ 1.00000 0.0500000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 14.0000 0.698257
$$403$$ −50.0000 −2.49068
$$404$$ −3.00000 −0.149256
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ −28.0000 −1.38451 −0.692255 0.721653i $$-0.743383\pi$$
−0.692255 + 0.721653i $$0.743383\pi$$
$$410$$ 0 0
$$411$$ −3.00000 −0.147979
$$412$$ −4.00000 −0.197066
$$413$$ 9.00000 0.442861
$$414$$ 6.00000 0.294884
$$415$$ −9.00000 −0.441793
$$416$$ 5.00000 0.245145
$$417$$ −19.0000 −0.930434
$$418$$ 0 0
$$419$$ 21.0000 1.02592 0.512959 0.858413i $$-0.328549\pi$$
0.512959 + 0.858413i $$0.328549\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ −40.0000 −1.94948 −0.974740 0.223341i $$-0.928304\pi$$
−0.974740 + 0.223341i $$0.928304\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 12.0000 0.583460
$$424$$ 6.00000 0.291386
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ −10.0000 −0.483934
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ −3.00000 −0.144505 −0.0722525 0.997386i $$-0.523019\pi$$
−0.0722525 + 0.997386i $$0.523019\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ −15.0000 −0.717547
$$438$$ 2.00000 0.0955637
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 30.0000 1.42695
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −10.0000 −0.473514
$$447$$ −6.00000 −0.283790
$$448$$ 1.00000 0.0472456
$$449$$ −3.00000 −0.141579 −0.0707894 0.997491i $$-0.522552\pi$$
−0.0707894 + 0.997491i $$0.522552\pi$$
$$450$$ −2.00000 −0.0942809
$$451$$ 0 0
$$452$$ 9.00000 0.423324
$$453$$ −7.00000 −0.328889
$$454$$ 24.0000 1.12638
$$455$$ −5.00000 −0.234404
$$456$$ 5.00000 0.234146
$$457$$ −1.00000 −0.0467780 −0.0233890 0.999726i $$-0.507446\pi$$
−0.0233890 + 0.999726i $$0.507446\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ −30.0000 −1.40028
$$460$$ 3.00000 0.139876
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ −7.00000 −0.325318 −0.162659 0.986682i $$-0.552007\pi$$
−0.162659 + 0.986682i $$0.552007\pi$$
$$464$$ 0 0
$$465$$ 10.0000 0.463739
$$466$$ −21.0000 −0.972806
$$467$$ −3.00000 −0.138823 −0.0694117 0.997588i $$-0.522112\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ −10.0000 −0.462250
$$469$$ 14.0000 0.646460
$$470$$ 6.00000 0.276759
$$471$$ 5.00000 0.230388
$$472$$ 9.00000 0.414259
$$473$$ 0 0
$$474$$ −1.00000 −0.0459315
$$475$$ 5.00000 0.229416
$$476$$ 6.00000 0.275010
$$477$$ −12.0000 −0.549442
$$478$$ −3.00000 −0.137217
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 10.0000 0.455961
$$482$$ −4.00000 −0.182195
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −8.00000 −0.363261
$$486$$ 16.0000 0.725775
$$487$$ −13.0000 −0.589086 −0.294543 0.955638i $$-0.595167\pi$$
−0.294543 + 0.955638i $$0.595167\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 2.00000 0.0904431
$$490$$ −1.00000 −0.0451754
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 25.0000 1.12480
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ 0 0
$$498$$ 9.00000 0.403300
$$499$$ −34.0000 −1.52205 −0.761025 0.648723i $$-0.775303\pi$$
−0.761025 + 0.648723i $$0.775303\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 24.0000 1.07117
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 3.00000 0.133498
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ −7.00000 −0.310575
$$509$$ 3.00000 0.132973 0.0664863 0.997787i $$-0.478821\pi$$
0.0664863 + 0.997787i $$0.478821\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ 2.00000 0.0884748
$$512$$ 1.00000 0.0441942
$$513$$ −25.0000 −1.10378
$$514$$ 6.00000 0.264649
$$515$$ 4.00000 0.176261
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 2.00000 0.0878750
$$519$$ 6.00000 0.263371
$$520$$ −5.00000 −0.219265
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 0 0
$$523$$ 17.0000 0.743358 0.371679 0.928361i $$-0.378782\pi$$
0.371679 + 0.928361i $$0.378782\pi$$
$$524$$ 21.0000 0.917389
$$525$$ 1.00000 0.0436436
$$526$$ −9.00000 −0.392419
$$527$$ −60.0000 −2.61364
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ −6.00000 −0.260623
$$531$$ −18.0000 −0.781133
$$532$$ 5.00000 0.216777
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −6.00000 −0.259403
$$536$$ 14.0000 0.604708
$$537$$ 0 0
$$538$$ −21.0000 −0.905374
$$539$$ 0 0
$$540$$ 5.00000 0.215166
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 20.0000 0.859074
$$543$$ −25.0000 −1.07285
$$544$$ 6.00000 0.257248
$$545$$ 10.0000 0.428353
$$546$$ 5.00000 0.213980
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −3.00000 −0.128154
$$549$$ 20.0000 0.853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −3.00000 −0.127688
$$553$$ −1.00000 −0.0425243
$$554$$ 32.0000 1.35955
$$555$$ −2.00000 −0.0848953
$$556$$ −19.0000 −0.805779
$$557$$ 42.0000 1.77960 0.889799 0.456354i $$-0.150845\pi$$
0.889799 + 0.456354i $$0.150845\pi$$
$$558$$ 20.0000 0.846668
$$559$$ 40.0000 1.69182
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −15.0000 −0.632737
$$563$$ 9.00000 0.379305 0.189652 0.981851i $$-0.439264\pi$$
0.189652 + 0.981851i $$0.439264\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ −9.00000 −0.378633
$$566$$ 29.0000 1.21896
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −21.0000 −0.880366 −0.440183 0.897908i $$-0.645086\pi$$
−0.440183 + 0.897908i $$0.645086\pi$$
$$570$$ −5.00000 −0.209427
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 15.0000 0.626634
$$574$$ 0 0
$$575$$ −3.00000 −0.125109
$$576$$ −2.00000 −0.0833333
$$577$$ 8.00000 0.333044 0.166522 0.986038i $$-0.446746\pi$$
0.166522 + 0.986038i $$0.446746\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ 9.00000 0.373383
$$582$$ 8.00000 0.331611
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 10.0000 0.413449
$$586$$ 27.0000 1.11536
$$587$$ 15.0000 0.619116 0.309558 0.950881i $$-0.399819\pi$$
0.309558 + 0.950881i $$0.399819\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −50.0000 −2.06021
$$590$$ −9.00000 −0.370524
$$591$$ 12.0000 0.493614
$$592$$ 2.00000 0.0821995
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ −6.00000 −0.245976
$$596$$ −6.00000 −0.245770
$$597$$ 8.00000 0.327418
$$598$$ −15.0000 −0.613396
$$599$$ 3.00000 0.122577 0.0612883 0.998120i $$-0.480479\pi$$
0.0612883 + 0.998120i $$0.480479\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −28.0000 −1.14025
$$604$$ −7.00000 −0.284826
$$605$$ 0 0
$$606$$ −3.00000 −0.121867
$$607$$ −34.0000 −1.38002 −0.690009 0.723801i $$-0.742393\pi$$
−0.690009 + 0.723801i $$0.742393\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 10.0000 0.404888
$$611$$ −30.0000 −1.21367
$$612$$ −12.0000 −0.485071
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ 23.0000 0.924448 0.462224 0.886763i $$-0.347052\pi$$
0.462224 + 0.886763i $$0.347052\pi$$
$$620$$ 10.0000 0.401610
$$621$$ 15.0000 0.601929
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 5.00000 0.200160
$$625$$ 1.00000 0.0400000
$$626$$ −34.0000 −1.35891
$$627$$ 0 0
$$628$$ 5.00000 0.199522
$$629$$ 12.0000 0.478471
$$630$$ 2.00000 0.0796819
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ −1.00000 −0.0397779
$$633$$ 20.0000 0.794929
$$634$$ 30.0000 1.19145
$$635$$ 7.00000 0.277787
$$636$$ 6.00000 0.237915
$$637$$ 5.00000 0.198107
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −21.0000 −0.829450 −0.414725 0.909947i $$-0.636122\pi$$
−0.414725 + 0.909947i $$0.636122\pi$$
$$642$$ 6.00000 0.236801
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ −8.00000 −0.315000
$$646$$ 30.0000 1.18033
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 5.00000 0.196116
$$651$$ −10.0000 −0.391931
$$652$$ 2.00000 0.0783260
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ −21.0000 −0.820538
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ −6.00000 −0.233904
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ 5.00000 0.194477 0.0972387 0.995261i $$-0.468999\pi$$
0.0972387 + 0.995261i $$0.468999\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 30.0000 1.16510
$$664$$ 9.00000 0.349268
$$665$$ −5.00000 −0.193892
$$666$$ −4.00000 −0.154997
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −10.0000 −0.386622
$$670$$ −14.0000 −0.540867
$$671$$ 0 0
$$672$$ 1.00000 0.0385758
$$673$$ 17.0000 0.655302 0.327651 0.944799i $$-0.393743\pi$$
0.327651 + 0.944799i $$0.393743\pi$$
$$674$$ 23.0000 0.885927
$$675$$ −5.00000 −0.192450
$$676$$ 12.0000 0.461538
$$677$$ −3.00000 −0.115299 −0.0576497 0.998337i $$-0.518361\pi$$
−0.0576497 + 0.998337i $$0.518361\pi$$
$$678$$ 9.00000 0.345643
$$679$$ 8.00000 0.307012
$$680$$ −6.00000 −0.230089
$$681$$ 24.0000 0.919682
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ −10.0000 −0.382360
$$685$$ 3.00000 0.114624
$$686$$ 1.00000 0.0381802
$$687$$ −22.0000 −0.839352
$$688$$ 8.00000 0.304997
$$689$$ 30.0000 1.14291
$$690$$ 3.00000 0.114208
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −6.00000 −0.227757
$$695$$ 19.0000 0.720711
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 11.0000 0.416356
$$699$$ −21.0000 −0.794293
$$700$$ 1.00000 0.0377964
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ −25.0000 −0.943564
$$703$$ 10.0000 0.377157
$$704$$ 0 0
$$705$$ 6.00000 0.225973
$$706$$ 30.0000 1.12906
$$707$$ −3.00000 −0.112827
$$708$$ 9.00000 0.338241
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 2.00000 0.0750059
$$712$$ 0 0
$$713$$ 30.0000 1.12351
$$714$$ 6.00000 0.224544
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −3.00000 −0.112037
$$718$$ −24.0000 −0.895672
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ −4.00000 −0.148968
$$722$$ 6.00000 0.223297
$$723$$ −4.00000 −0.148762
$$724$$ −25.0000 −0.929118
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 5.00000 0.185312
$$729$$ 13.0000 0.481481
$$730$$ −2.00000 −0.0740233
$$731$$ 48.0000 1.77534
$$732$$ −10.0000 −0.369611
$$733$$ −13.0000 −0.480166 −0.240083 0.970752i $$-0.577175\pi$$
−0.240083 + 0.970752i $$0.577175\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ −1.00000 −0.0368856
$$736$$ −3.00000 −0.110581
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −22.0000 −0.809283 −0.404642 0.914475i $$-0.632604\pi$$
−0.404642 + 0.914475i $$0.632604\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ 25.0000 0.918398
$$742$$ 6.00000 0.220267
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ −10.0000 −0.366618
$$745$$ 6.00000 0.219823
$$746$$ −22.0000 −0.805477
$$747$$ −18.0000 −0.658586
$$748$$ 0 0
$$749$$ 6.00000 0.219235
$$750$$ −1.00000 −0.0365148
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ 7.00000 0.254756
$$756$$ −5.00000 −0.181848
$$757$$ 20.0000 0.726912 0.363456 0.931611i $$-0.381597\pi$$
0.363456 + 0.931611i $$0.381597\pi$$
$$758$$ −34.0000 −1.23494
$$759$$ 0 0
$$760$$ −5.00000 −0.181369
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ −7.00000 −0.253583
$$763$$ −10.0000 −0.362024
$$764$$ 15.0000 0.542681
$$765$$ 12.0000 0.433861
$$766$$ 12.0000 0.433578
$$767$$ 45.0000 1.62486
$$768$$ 1.00000 0.0360844
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 11.0000 0.395899
$$773$$ 39.0000 1.40273 0.701366 0.712801i $$-0.252574\pi$$
0.701366 + 0.712801i $$0.252574\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ −10.0000 −0.359211
$$776$$ 8.00000 0.287183
$$777$$ 2.00000 0.0717496
$$778$$ 30.0000 1.07555
$$779$$ 0 0
$$780$$ −5.00000 −0.179029
$$781$$ 0 0
$$782$$ −18.0000 −0.643679
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −5.00000 −0.178458
$$786$$ 21.0000 0.749045
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 12.0000 0.427482
$$789$$ −9.00000 −0.320408
$$790$$ 1.00000 0.0355784
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ −50.0000 −1.77555
$$794$$ 38.0000 1.34857
$$795$$ −6.00000 −0.212798
$$796$$ 8.00000 0.283552
$$797$$ 33.0000 1.16892 0.584460 0.811423i $$-0.301306\pi$$
0.584460 + 0.811423i $$0.301306\pi$$
$$798$$ 5.00000 0.176998
$$799$$ −36.0000 −1.27359
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −30.0000 −1.05934
$$803$$ 0 0
$$804$$ 14.0000 0.493742
$$805$$ 3.00000 0.105736
$$806$$ −50.0000 −1.76117
$$807$$ −21.0000 −0.739235
$$808$$ −3.00000 −0.105540
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 0 0
$$813$$ 20.0000 0.701431
$$814$$ 0 0
$$815$$ −2.00000 −0.0700569
$$816$$ 6.00000 0.210042
$$817$$ 40.0000 1.39942
$$818$$ −28.0000 −0.978997
$$819$$ −10.0000 −0.349428
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ −3.00000 −0.104637
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 9.00000 0.313150
$$827$$ −30.0000 −1.04320 −0.521601 0.853189i $$-0.674665\pi$$
−0.521601 + 0.853189i $$0.674665\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 17.0000 0.590434 0.295217 0.955430i $$-0.404608\pi$$
0.295217 + 0.955430i $$0.404608\pi$$
$$830$$ −9.00000 −0.312395
$$831$$ 32.0000 1.11007
$$832$$ 5.00000 0.173344
$$833$$ 6.00000 0.207888
$$834$$ −19.0000 −0.657916
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 50.0000 1.72825
$$838$$ 21.0000 0.725433
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ −1.00000 −0.0345033
$$841$$ −29.0000 −1.00000
$$842$$ −40.0000 −1.37849
$$843$$ −15.0000 −0.516627
$$844$$ 20.0000 0.688428
$$845$$ −12.0000 −0.412813
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 29.0000 0.995277
$$850$$ 6.00000 0.205798
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ −19.0000 −0.650548 −0.325274 0.945620i $$-0.605456\pi$$
−0.325274 + 0.945620i $$0.605456\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 10.0000 0.341993
$$856$$ 6.00000 0.205076
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −3.00000 −0.102180
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ −6.00000 −0.204006
$$866$$ −4.00000 −0.135926
$$867$$ 19.0000 0.645274
$$868$$ −10.0000 −0.339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 70.0000 2.37186
$$872$$ −10.0000 −0.338643
$$873$$ −16.0000 −0.541518
$$874$$ −15.0000 −0.507383
$$875$$ −1.00000 −0.0338062
$$876$$ 2.00000 0.0675737
$$877$$ −40.0000 −1.35070 −0.675352 0.737496i $$-0.736008\pi$$
−0.675352 + 0.737496i $$0.736008\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 27.0000 0.910687
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ −2.00000 −0.0673435
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 30.0000 1.00901
$$885$$ −9.00000 −0.302532
$$886$$ −24.0000 −0.806296
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ −7.00000 −0.234772
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −10.0000 −0.334825
$$893$$ −30.0000 −1.00391
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ −15.0000 −0.500835
$$898$$ −3.00000 −0.100111
$$899$$ 0 0
$$900$$ −2.00000 −0.0666667
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 9.00000 0.299336
$$905$$ 25.0000 0.831028
$$906$$ −7.00000 −0.232559
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 6.00000 0.199007
$$910$$ −5.00000 −0.165748
$$911$$ 45.0000 1.49092 0.745458 0.666552i $$-0.232231\pi$$
0.745458 + 0.666552i $$0.232231\pi$$
$$912$$ 5.00000 0.165567
$$913$$ 0 0
$$914$$ −1.00000 −0.0330771
$$915$$ 10.0000 0.330590
$$916$$ −22.0000 −0.726900
$$917$$ 21.0000 0.693481
$$918$$ −30.0000 −0.990148
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 3.00000 0.0989071
$$921$$ −4.00000 −0.131804
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ −7.00000 −0.230034
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 10.0000 0.327913
$$931$$ 5.00000 0.163868
$$932$$ −21.0000 −0.687878
$$933$$ 0 0
$$934$$ −3.00000 −0.0981630
$$935$$ 0 0
$$936$$ −10.0000 −0.326860
$$937$$ −52.0000 −1.69877 −0.849383 0.527777i $$-0.823026\pi$$
−0.849383 + 0.527777i $$0.823026\pi$$
$$938$$ 14.0000 0.457116
$$939$$ −34.0000 −1.10955
$$940$$ 6.00000 0.195698
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 5.00000 0.162909
$$943$$ 0 0
$$944$$ 9.00000 0.292925
$$945$$ 5.00000 0.162650
$$946$$ 0 0
$$947$$ 6.00000 0.194974 0.0974869 0.995237i $$-0.468920\pi$$
0.0974869 + 0.995237i $$0.468920\pi$$
$$948$$ −1.00000 −0.0324785
$$949$$ 10.0000 0.324614
$$950$$ 5.00000 0.162221
$$951$$ 30.0000 0.972817
$$952$$ 6.00000 0.194461
$$953$$ −57.0000 −1.84641 −0.923206 0.384307i $$-0.874441\pi$$
−0.923206 + 0.384307i $$0.874441\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ −15.0000 −0.485389
$$956$$ −3.00000 −0.0970269
$$957$$ 0 0
$$958$$ −36.0000 −1.16311
$$959$$ −3.00000 −0.0968751
$$960$$ −1.00000 −0.0322749
$$961$$ 69.0000 2.22581
$$962$$ 10.0000 0.322413
$$963$$ −12.0000 −0.386695
$$964$$ −4.00000 −0.128831
$$965$$ −11.0000 −0.354103
$$966$$ −3.00000 −0.0965234
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ 30.0000 0.963739
$$970$$ −8.00000 −0.256865
$$971$$ −9.00000 −0.288824 −0.144412 0.989518i $$-0.546129\pi$$
−0.144412 + 0.989518i $$0.546129\pi$$
$$972$$ 16.0000 0.513200
$$973$$ −19.0000 −0.609112
$$974$$ −13.0000 −0.416547
$$975$$ 5.00000 0.160128
$$976$$ −10.0000 −0.320092
$$977$$ −27.0000 −0.863807 −0.431903 0.901920i $$-0.642158\pi$$
−0.431903 + 0.901920i $$0.642158\pi$$
$$978$$ 2.00000 0.0639529
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ 20.0000 0.638551
$$982$$ −6.00000 −0.191468
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ 0 0
$$985$$ −12.0000 −0.382352
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 25.0000 0.795356
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 35.0000 1.11181 0.555906 0.831245i $$-0.312372\pi$$
0.555906 + 0.831245i $$0.312372\pi$$
$$992$$ −10.0000 −0.317500
$$993$$ 8.00000 0.253872
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 9.00000 0.285176
$$997$$ −25.0000 −0.791758 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$998$$ −34.0000 −1.07625
$$999$$ −10.0000 −0.316386
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 8470.2.a.bd.1.1 yes 1
11.10 odd 2 8470.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.k.1.1 1 11.10 odd 2
8470.2.a.bd.1.1 yes 1 1.1 even 1 trivial