# Properties

 Label 5070.2.a.u.1.1 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} +1.00000 q^{45} +8.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +1.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} -4.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} +12.0000 q^{86} +2.00000 q^{87} +14.0000 q^{89} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{93} -4.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +O(q^{100})$$

## 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
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −4.00000 −1.06904
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −4.00000 −0.755929
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ −4.00000 −0.676123
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 8.00000 1.17954
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ 1.00000 0.141421
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ −4.00000 −0.529813
$$58$$ 2.00000 0.262613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 8.00000 1.01600
$$63$$ −4.00000 −0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 8.00000 0.963087
$$70$$ −4.00000 −0.478091
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ −2.00000 −0.216930
$$86$$ 12.0000 1.29399
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 1.00000 0.102062
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 10.0000 0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ −4.00000 −0.377964
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 8.00000 0.746004
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 1.00000 0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ −10.0000 −0.905357
$$123$$ 6.00000 0.541002
$$124$$ 8.00000 0.718421
$$125$$ 1.00000 0.0894427
$$126$$ −4.00000 −0.356348
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ 4.00000 0.345547
$$135$$ 1.00000 0.0860663
$$136$$ −2.00000 −0.171499
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 2.00000 0.166091
$$146$$ 6.00000 0.496564
$$147$$ 9.00000 0.742307
$$148$$ −2.00000 −0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 10.0000 0.793052
$$160$$ 1.00000 0.0790569
$$161$$ −32.0000 −2.52195
$$162$$ 1.00000 0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 0 0
$$170$$ −2.00000 −0.153393
$$171$$ −4.00000 −0.305888
$$172$$ 12.0000 0.914991
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 2.00000 0.151620
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 8.00000 0.589768
$$185$$ −2.00000 −0.147043
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ −4.00000 −0.290191
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 4.00000 0.282138
$$202$$ 10.0000 0.703598
$$203$$ −8.00000 −0.561490
$$204$$ −2.00000 −0.140028
$$205$$ 6.00000 0.419058
$$206$$ −4.00000 −0.278693
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ −4.00000 −0.276026
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 16.0000 1.09630
$$214$$ −12.0000 −0.820303
$$215$$ 12.0000 0.818393
$$216$$ 1.00000 0.0680414
$$217$$ −32.0000 −2.17230
$$218$$ −14.0000 −0.948200
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 1.00000 0.0666667
$$226$$ −10.0000 −0.665190
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 8.00000 0.518563
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 9.00000 0.574989
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ 4.00000 0.253490
$$250$$ 1.00000 0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 0 0
$$254$$ −12.0000 −0.752947
$$255$$ −2.00000 −0.125245
$$256$$ 1.00000 0.0625000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 8.00000 0.494242
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 16.0000 0.981023
$$267$$ 14.0000 0.856786
$$268$$ 4.00000 0.244339
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 8.00000 0.478947
$$280$$ −4.00000 −0.239046
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 16.0000 0.949425
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 2.00000 0.117444
$$291$$ 6.00000 0.351726
$$292$$ 6.00000 0.351123
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ −48.0000 −2.76667
$$302$$ 8.00000 0.460348
$$303$$ 10.0000 0.574485
$$304$$ −4.00000 −0.229416
$$305$$ −10.0000 −0.572598
$$306$$ −2.00000 −0.114332
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 8.00000 0.454369
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ 18.0000 1.01580
$$315$$ −4.00000 −0.225374
$$316$$ −8.00000 −0.450035
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ −12.0000 −0.669775
$$322$$ −32.0000 −1.78329
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ −14.0000 −0.774202
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −2.00000 −0.109599
$$334$$ −16.0000 −0.875481
$$335$$ 4.00000 0.218543
$$336$$ −4.00000 −0.218218
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ −10.0000 −0.543125
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ −8.00000 −0.431959
$$344$$ 12.0000 0.646997
$$345$$ 8.00000 0.430706
$$346$$ 2.00000 0.107521
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 14.0000 0.741999
$$357$$ 8.00000 0.423405
$$358$$ 16.0000 0.845626
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ −2.00000 −0.105118
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ −10.0000 −0.522708
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 6.00000 0.312348
$$370$$ −2.00000 −0.103975
$$371$$ −40.0000 −2.07670
$$372$$ 8.00000 0.414781
$$373$$ −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
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −4.00000 −0.205738
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ −12.0000 −0.614779
$$382$$ 8.00000 0.409316
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −18.0000 −0.916176
$$387$$ 12.0000 0.609994
$$388$$ 6.00000 0.304604
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 9.00000 0.454569
$$393$$ 8.00000 0.403547
$$394$$ 6.00000 0.302276
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 16.0000 0.801002
$$400$$ 1.00000 0.0500000
$$401$$ −26.0000 −1.29838 −0.649189 0.760627i $$-0.724892\pi$$
−0.649189 + 0.760627i $$0.724892\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ 1.00000 0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ −2.00000 −0.0990148
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 6.00000 0.296319
$$411$$ −6.00000 −0.295958
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 4.00000 0.196352
$$416$$ 0 0
$$417$$ −20.0000 −0.979404
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ −4.00000 −0.195180
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ −2.00000 −0.0970143
$$426$$ 16.0000 0.775203
$$427$$ 40.0000 1.93574
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 12.0000 0.578691
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 10.0000 0.480569 0.240285 0.970702i $$-0.422759\pi$$
0.240285 + 0.970702i $$0.422759\pi$$
$$434$$ −32.0000 −1.53605
$$435$$ 2.00000 0.0958927
$$436$$ −14.0000 −0.670478
$$437$$ −32.0000 −1.53077
$$438$$ 6.00000 0.286691
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 14.0000 0.663664
$$446$$ −12.0000 −0.568216
$$447$$ −10.0000 −0.472984
$$448$$ −4.00000 −0.188982
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −10.0000 −0.470360
$$453$$ 8.00000 0.375873
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −2.00000 −0.0933520
$$460$$ 8.00000 0.373002
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 8.00000 0.370991
$$466$$ −26.0000 −1.20443
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −8.00000 −0.367452
$$475$$ −4.00000 −0.183533
$$476$$ 8.00000 0.366679
$$477$$ 10.0000 0.457869
$$478$$ −24.0000 −1.09773
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −2.00000 −0.0910975
$$483$$ −32.0000 −1.45605
$$484$$ −11.0000 −0.500000
$$485$$ 6.00000 0.272446
$$486$$ 1.00000 0.0453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 20.0000 0.904431
$$490$$ 9.00000 0.406579
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ −64.0000 −2.87079
$$498$$ 4.00000 0.179244
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ 38.0000 1.68432 0.842160 0.539227i $$-0.181284\pi$$
0.842160 + 0.539227i $$0.181284\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ −24.0000 −1.06170
$$512$$ 1.00000 0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 22.0000 0.970378
$$515$$ −4.00000 −0.176261
$$516$$ 12.0000 0.528271
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ 2.00000 0.0876216 0.0438108 0.999040i $$-0.486050\pi$$
0.0438108 + 0.999040i $$0.486050\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 8.00000 0.349482
$$525$$ −4.00000 −0.174574
$$526$$ 24.0000 1.04645
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 10.0000 0.434372
$$531$$ 0 0
$$532$$ 16.0000 0.693688
$$533$$ 0 0
$$534$$ 14.0000 0.605839
$$535$$ −12.0000 −0.518805
$$536$$ 4.00000 0.172774
$$537$$ 16.0000 0.690451
$$538$$ −14.0000 −0.603583
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 32.0000 1.37452
$$543$$ −2.00000 −0.0858282
$$544$$ −2.00000 −0.0857493
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 8.00000 0.340503
$$553$$ 32.0000 1.36078
$$554$$ 18.0000 0.764747
$$555$$ −2.00000 −0.0848953
$$556$$ −20.0000 −0.848189
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 0 0
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −10.0000 −0.420703
$$566$$ 20.0000 0.840663
$$567$$ −4.00000 −0.167984
$$568$$ 16.0000 0.671345
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 0 0
$$573$$ 8.00000 0.334205
$$574$$ −24.0000 −1.00174
$$575$$ 8.00000 0.333623
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −18.0000 −0.748054
$$580$$ 2.00000 0.0830455
$$581$$ −16.0000 −0.663792
$$582$$ 6.00000 0.248708
$$583$$ 0 0
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 9.00000 0.371154
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ −2.00000 −0.0821995
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ −10.0000 −0.409616
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ −48.0000 −1.95633
$$603$$ 4.00000 0.162893
$$604$$ 8.00000 0.325515
$$605$$ −11.0000 −0.447214
$$606$$ 10.0000 0.406222
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ −8.00000 −0.324176
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 6.00000 0.241943
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ −56.0000 −2.24359
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 18.0000 0.719425
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ 4.00000 0.159490
$$630$$ −4.00000 −0.159364
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −12.0000 −0.476957
$$634$$ 14.0000 0.556011
$$635$$ −12.0000 −0.476205
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 16.0000 0.632950
$$640$$ 1.00000 0.0395285
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ −32.0000 −1.26098
$$645$$ 12.0000 0.472500
$$646$$ 8.00000 0.314756
$$647$$ −48.0000 −1.88707 −0.943537 0.331266i $$-0.892524\pi$$
−0.943537 + 0.331266i $$0.892524\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ 20.0000 0.783260
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 8.00000 0.312586
$$656$$ 6.00000 0.234261
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 16.0000 0.620453
$$666$$ −2.00000 −0.0774984
$$667$$ 16.0000 0.619522
$$668$$ −16.0000 −0.619059
$$669$$ −12.0000 −0.463947
$$670$$ 4.00000 0.154533
$$671$$ 0 0
$$672$$ −4.00000 −0.154303
$$673$$ −38.0000 −1.46479 −0.732396 0.680879i $$-0.761598\pi$$
−0.732396 + 0.680879i $$0.761598\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 26.0000 0.999261 0.499631 0.866239i $$-0.333469\pi$$
0.499631 + 0.866239i $$0.333469\pi$$
$$678$$ −10.0000 −0.384048
$$679$$ −24.0000 −0.921035
$$680$$ −2.00000 −0.0766965
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −6.00000 −0.229248
$$686$$ −8.00000 −0.305441
$$687$$ −14.0000 −0.534133
$$688$$ 12.0000 0.457496
$$689$$ 0 0
$$690$$ 8.00000 0.304555
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ −20.0000 −0.758643
$$696$$ 2.00000 0.0758098
$$697$$ −12.0000 −0.454532
$$698$$ −14.0000 −0.529908
$$699$$ −26.0000 −0.983410
$$700$$ −4.00000 −0.151186
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ −40.0000 −1.50435
$$708$$ 0 0
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 16.0000 0.600469
$$711$$ −8.00000 −0.300023
$$712$$ 14.0000 0.524672
$$713$$ 64.0000 2.39682
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ 16.0000 0.597948
$$717$$ −24.0000 −0.896296
$$718$$ −24.0000 −0.895672
$$719$$ 16.0000 0.596699 0.298350 0.954457i $$-0.403564\pi$$
0.298350 + 0.954457i $$0.403564\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 16.0000 0.595871
$$722$$ −3.00000 −0.111648
$$723$$ −2.00000 −0.0743808
$$724$$ −2.00000 −0.0743294
$$725$$ 2.00000 0.0742781
$$726$$ −11.0000 −0.408248
$$727$$ 52.0000 1.92857 0.964287 0.264861i $$-0.0853260\pi$$
0.964287 + 0.264861i $$0.0853260\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 6.00000 0.222070
$$731$$ −24.0000 −0.887672
$$732$$ −10.0000 −0.369611
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 9.00000 0.331970
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ 0 0
$$742$$ −40.0000 −1.46845
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 8.00000 0.293294
$$745$$ −10.0000 −0.366372
$$746$$ −22.0000 −0.805477
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 48.0000 1.75388
$$750$$ 1.00000 0.0365148
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ −4.00000 −0.145479
$$757$$ 50.0000 1.81728 0.908640 0.417579i $$-0.137121\pi$$
0.908640 + 0.417579i $$0.137121\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ −4.00000 −0.145095
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ 56.0000 2.02734
$$764$$ 8.00000 0.289430
$$765$$ −2.00000 −0.0723102
$$766$$ 16.0000 0.578103
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 22.0000 0.792311
$$772$$ −18.0000 −0.647834
$$773$$ −10.0000 −0.359675 −0.179838 0.983696i $$-0.557557\pi$$
−0.179838 + 0.983696i $$0.557557\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 8.00000 0.287368
$$776$$ 6.00000 0.215387
$$777$$ 8.00000 0.286998
$$778$$ −14.0000 −0.501924
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −16.0000 −0.572159
$$783$$ 2.00000 0.0714742
$$784$$ 9.00000 0.321429
$$785$$ 18.0000 0.642448
$$786$$ 8.00000 0.285351
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 24.0000 0.854423
$$790$$ −8.00000 −0.284627
$$791$$ 40.0000 1.42224
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −10.0000 −0.354887
$$795$$ 10.0000 0.354663
$$796$$ −8.00000 −0.283552
$$797$$ −46.0000 −1.62940 −0.814702 0.579880i $$-0.803099\pi$$
−0.814702 + 0.579880i $$0.803099\pi$$
$$798$$ 16.0000 0.566394
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 14.0000 0.494666
$$802$$ −26.0000 −0.918092
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ −32.0000 −1.12785
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 10.0000 0.351799
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 20.0000 0.700569
$$816$$ −2.00000 −0.0700140
$$817$$ −48.0000 −1.67931
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 4.00000 0.138842
$$831$$ 18.0000 0.624413
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ −20.0000 −0.692543
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ −24.0000 −0.829066
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ −4.00000 −0.138013
$$841$$ −25.0000 −0.862069
$$842$$ −30.0000 −1.03387
$$843$$ −18.0000 −0.619953
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 44.0000 1.51186
$$848$$ 10.0000 0.343401
$$849$$ 20.0000 0.686398
$$850$$ −2.00000 −0.0685994
$$851$$ −16.0000 −0.548473
$$852$$ 16.0000 0.548151
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 40.0000 1.36877
$$855$$ −4.00000 −0.136797
$$856$$ −12.0000 −0.410152
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 12.0000 0.409197
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 2.00000 0.0680020
$$866$$ 10.0000 0.339814
$$867$$ −13.0000 −0.441503
$$868$$ −32.0000 −1.08615
$$869$$ 0 0
$$870$$ 2.00000 0.0678064
$$871$$ 0 0
$$872$$ −14.0000 −0.474100
$$873$$ 6.00000 0.203069
$$874$$ −32.0000 −1.08242
$$875$$ −4.00000 −0.135225
$$876$$ 6.00000 0.202721
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ 40.0000 1.34993
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 48.0000 1.60987
$$890$$ 14.0000 0.469281
$$891$$ 0 0
$$892$$ −12.0000 −0.401790
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 16.0000 0.534821
$$896$$ −4.00000 −0.133631
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 16.0000 0.533630
$$900$$ 1.00000 0.0333333
$$901$$ −20.0000 −0.666297
$$902$$ 0 0
$$903$$ −48.0000 −1.59734
$$904$$ −10.0000 −0.332595
$$905$$ −2.00000 −0.0664822
$$906$$ 8.00000 0.265782
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ −18.0000 −0.595387
$$915$$ −10.0000 −0.330590
$$916$$ −14.0000 −0.462573
$$917$$ −32.0000 −1.05673
$$918$$ −2.00000 −0.0660098
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 12.0000 0.395413
$$922$$ −2.00000 −0.0658665
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −20.0000 −0.657241
$$927$$ −4.00000 −0.131377
$$928$$ 2.00000 0.0656532
$$929$$ −26.0000 −0.853032 −0.426516 0.904480i $$-0.640259\pi$$
−0.426516 + 0.904480i $$0.640259\pi$$
$$930$$ 8.00000 0.262330
$$931$$ −36.0000 −1.17985
$$932$$ −26.0000 −0.851658
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −6.00000 −0.196011 −0.0980057 0.995186i $$-0.531246\pi$$
−0.0980057 + 0.995186i $$0.531246\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ −4.00000 −0.129777
$$951$$ 14.0000 0.453981
$$952$$ 8.00000 0.259281
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 8.00000 0.258874
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ −8.00000 −0.258468
$$959$$ 24.0000 0.775000
$$960$$ 1.00000 0.0322749
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ −2.00000 −0.0644157
$$965$$ −18.0000 −0.579441
$$966$$ −32.0000 −1.02958
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 8.00000 0.256997
$$970$$ 6.00000 0.192648
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 80.0000 2.56468
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −38.0000 −1.21573 −0.607864 0.794041i $$-0.707973\pi$$
−0.607864 + 0.794041i $$0.707973\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 0 0
$$980$$ 9.00000 0.287494
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ −56.0000 −1.78612 −0.893061 0.449935i $$-0.851447\pi$$
−0.893061 + 0.449935i $$0.851447\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 6.00000 0.191176
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 96.0000 3.05262
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 20.0000 0.634681
$$994$$ −64.0000 −2.02996
$$995$$ −8.00000 −0.253617
$$996$$ 4.00000 0.126745
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 44.0000 1.39280
$$999$$ −2.00000 −0.0632772
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 5070.2.a.u.1.1 1
13.5 odd 4 5070.2.b.i.1351.1 2
13.8 odd 4 5070.2.b.i.1351.2 2
13.12 even 2 390.2.a.c.1.1 1
39.38 odd 2 1170.2.a.n.1.1 1
52.51 odd 2 3120.2.a.a.1.1 1
65.12 odd 4 1950.2.e.e.1249.1 2
65.38 odd 4 1950.2.e.e.1249.2 2
65.64 even 2 1950.2.a.n.1.1 1
156.155 even 2 9360.2.a.bc.1.1 1
195.38 even 4 5850.2.e.m.5149.1 2
195.77 even 4 5850.2.e.m.5149.2 2
195.194 odd 2 5850.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.c.1.1 1 13.12 even 2
1170.2.a.n.1.1 1 39.38 odd 2
1950.2.a.n.1.1 1 65.64 even 2
1950.2.e.e.1249.1 2 65.12 odd 4
1950.2.e.e.1249.2 2 65.38 odd 4
3120.2.a.a.1.1 1 52.51 odd 2
5070.2.a.u.1.1 1 1.1 even 1 trivial
5070.2.b.i.1351.1 2 13.5 odd 4
5070.2.b.i.1351.2 2 13.8 odd 4
5850.2.a.c.1.1 1 195.194 odd 2
5850.2.e.m.5149.1 2 195.38 even 4
5850.2.e.m.5149.2 2 195.77 even 4
9360.2.a.bc.1.1 1 156.155 even 2