# Properties

 Label 1950.2.a.u.1.1 Level $1950$ Weight $2$ Character 1950.1 Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.5708283941$$ 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 $$\chi$$ $$=$$ 1950.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} +2.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} +2.00000 q^{44} -6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -2.00000 q^{57} -2.00000 q^{58} -14.0000 q^{59} +10.0000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} +6.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} +2.00000 q^{76} +8.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +8.00000 q^{86} +2.00000 q^{87} +2.00000 q^{88} -18.0000 q^{89} +4.00000 q^{91} -6.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 2.00000 0.426401
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 4.00000 0.755929
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.00000 −0.348155
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 2.00000 0.324443
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 1.00000 0.138675
$$53$$ 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
$$57$$ −2.00000 −0.264906
$$58$$ −2.00000 −0.262613
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 4.00000 0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 8.00000 0.911685
$$78$$ −1.00000 −0.113228
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 2.00000 0.214423
$$88$$ 2.00000 0.213201
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ −6.00000 −0.625543
$$93$$ 4.00000 0.414781
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ −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
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 12.0000 1.14939 0.574696 0.818367i $$-0.305120\pi$$
0.574696 + 0.818367i $$0.305120\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 4.00000 0.377964
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 1.00000 0.0924500
$$118$$ −14.0000 −1.28880
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 10.0000 0.905357
$$123$$ 6.00000 0.541002
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ −18.0000 −1.59724 −0.798621 0.601834i $$-0.794437\pi$$
−0.798621 + 0.601834i $$0.794437\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 8.00000 0.693688
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 6.00000 0.510754
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 8.00000 0.671345
$$143$$ 2.00000 0.167248
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ −9.00000 −0.742307
$$148$$ −6.00000 −0.493197
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 4.00000 0.323381
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ −24.0000 −1.89146
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 8.00000 0.609994
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 14.0000 1.05230
$$178$$ −18.0000 −1.34916
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 4.00000 0.296500
$$183$$ −10.0000 −0.739221
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 8.00000 0.585018
$$188$$ 8.00000 0.583460
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 26.0000 1.85242 0.926212 0.377004i $$-0.123046\pi$$
0.926212 + 0.377004i $$0.123046\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ −14.0000 −0.985037
$$203$$ −8.00000 −0.561490
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ −6.00000 −0.417029
$$208$$ 1.00000 0.0693375
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 10.0000 0.686803
$$213$$ −8.00000 −0.548151
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ −16.0000 −1.08615
$$218$$ 12.0000 0.812743
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 6.00000 0.402694
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ −2.00000 −0.131306
$$233$$ −4.00000 −0.262049 −0.131024 0.991379i $$-0.541827\pi$$
−0.131024 + 0.991379i $$0.541827\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ −14.0000 −0.911322
$$237$$ 8.00000 0.519656
$$238$$ 16.0000 1.03713
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −30.0000 −1.93247 −0.966235 0.257663i $$-0.917048\pi$$
−0.966235 + 0.257663i $$0.917048\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 2.00000 0.127257
$$248$$ −4.00000 −0.254000
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 4.00000 0.251976
$$253$$ −12.0000 −0.754434
$$254$$ −18.0000 −1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −10.0000 −0.616626 −0.308313 0.951285i $$-0.599764\pi$$
−0.308313 + 0.951285i $$0.599764\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ 18.0000 1.10158
$$268$$ 4.00000 0.244339
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 4.00000 0.242536
$$273$$ −4.00000 −0.242091
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 16.0000 0.959616
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ −24.0000 −1.41668
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ 10.0000 0.585206
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 32.0000 1.84445
$$302$$ 8.00000 0.460348
$$303$$ 14.0000 0.804279
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 8.00000 0.455842
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ −24.0000 −1.33747
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −12.0000 −0.663602
$$328$$ −6.00000 −0.331295
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 18.0000 0.989369 0.494685 0.869072i $$-0.335284\pi$$
0.494685 + 0.869072i $$0.335284\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ −32.0000 −1.74315 −0.871576 0.490261i $$-0.836901\pi$$
−0.871576 + 0.490261i $$0.836901\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 2.00000 0.108148
$$343$$ 8.00000 0.431959
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 10.0000 0.537603
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 2.00000 0.106600
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 14.0000 0.744092
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ −16.0000 −0.846810
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 6.00000 0.315353
$$363$$ 7.00000 0.367405
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 40.0000 2.07670
$$372$$ 4.00000 0.207390
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −2.00000 −0.103005
$$378$$ −4.00000 −0.205738
$$379$$ 26.0000 1.33553 0.667765 0.744372i $$-0.267251\pi$$
0.667765 + 0.744372i $$0.267251\pi$$
$$380$$ 0 0
$$381$$ 18.0000 0.922168
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 8.00000 0.406663
$$388$$ 6.00000 0.304604
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ 26.0000 1.30986
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 24.0000 1.20301
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ −4.00000 −0.199254
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ −12.0000 −0.594818
$$408$$ −4.00000 −0.198030
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 6.00000 0.295599
$$413$$ −56.0000 −2.75558
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ −16.0000 −0.783523
$$418$$ 4.00000 0.195646
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ 28.0000 1.36464 0.682318 0.731055i $$-0.260972\pi$$
0.682318 + 0.731055i $$0.260972\pi$$
$$422$$ −28.0000 −1.36302
$$423$$ 8.00000 0.388973
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 40.0000 1.93574
$$428$$ −8.00000 −0.386695
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ −12.0000 −0.574038
$$438$$ −10.0000 −0.477818
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 4.00000 0.190261
$$443$$ 8.00000 0.380091 0.190046 0.981775i $$-0.439136\pi$$
0.190046 + 0.981775i $$0.439136\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 4.00000 0.188982
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ −12.0000 −0.564433
$$453$$ −8.00000 −0.375873
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ −16.0000 −0.747631
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −4.00000 −0.186299 −0.0931493 0.995652i $$-0.529693\pi$$
−0.0931493 + 0.995652i $$0.529693\pi$$
$$462$$ −8.00000 −0.372194
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −4.00000 −0.185296
$$467$$ −16.0000 −0.740392 −0.370196 0.928954i $$-0.620709\pi$$
−0.370196 + 0.928954i $$0.620709\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −14.0000 −0.644402
$$473$$ 16.0000 0.735681
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 16.0000 0.733359
$$477$$ 10.0000 0.457869
$$478$$ −12.0000 −0.548867
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ −30.0000 −1.36646
$$483$$ 24.0000 1.09204
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −40.0000 −1.81257 −0.906287 0.422664i $$-0.861095\pi$$
−0.906287 + 0.422664i $$0.861095\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −8.00000 −0.360302
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 32.0000 1.43540
$$498$$ −12.0000 −0.537733
$$499$$ 30.0000 1.34298 0.671492 0.741012i $$-0.265654\pi$$
0.671492 + 0.741012i $$0.265654\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ 18.0000 0.802580 0.401290 0.915951i $$-0.368562\pi$$
0.401290 + 0.915951i $$0.368562\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ −12.0000 −0.533465
$$507$$ −1.00000 −0.0444116
$$508$$ −18.0000 −0.798621
$$509$$ −16.0000 −0.709188 −0.354594 0.935020i $$-0.615381\pi$$
−0.354594 + 0.935020i $$0.615381\pi$$
$$510$$ 0 0
$$511$$ 40.0000 1.76950
$$512$$ 1.00000 0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 16.0000 0.703679
$$518$$ −24.0000 −1.05450
$$519$$ −10.0000 −0.438951
$$520$$ 0 0
$$521$$ 38.0000 1.66481 0.832405 0.554168i $$-0.186963\pi$$
0.832405 + 0.554168i $$0.186963\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −10.0000 −0.436021
$$527$$ −16.0000 −0.696971
$$528$$ −2.00000 −0.0870388
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −14.0000 −0.607548
$$532$$ 8.00000 0.346844
$$533$$ −6.00000 −0.259889
$$534$$ 18.0000 0.778936
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −14.0000 −0.603583
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −6.00000 −0.257485
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 6.00000 0.255377
$$553$$ −32.0000 −1.36078
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ −18.0000 −0.759284
$$563$$ −16.0000 −0.674320 −0.337160 0.941447i $$-0.609466\pi$$
−0.337160 + 0.941447i $$0.609466\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 4.00000 0.167984
$$568$$ 8.00000 0.335673
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 26.0000 1.08239 0.541197 0.840896i $$-0.317971\pi$$
0.541197 + 0.840896i $$0.317971\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ −6.00000 −0.248708
$$583$$ 20.0000 0.828315
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −26.0000 −1.06950
$$592$$ −6.00000 −0.246598
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −24.0000 −0.982255
$$598$$ −6.00000 −0.245358
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 32.0000 1.30422
$$603$$ 4.00000 0.162893
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 14.0000 0.568711
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 4.00000 0.161690
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ −34.0000 −1.36879 −0.684394 0.729112i $$-0.739933\pi$$
−0.684394 + 0.729112i $$0.739933\pi$$
$$618$$ −6.00000 −0.241355
$$619$$ 42.0000 1.68812 0.844061 0.536247i $$-0.180158\pi$$
0.844061 + 0.536247i $$0.180158\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ 8.00000 0.320771
$$623$$ −72.0000 −2.88462
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 8.00000 0.319744
$$627$$ −4.00000 −0.159745
$$628$$ −10.0000 −0.399043
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 28.0000 1.11290
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ 9.00000 0.356593
$$638$$ −4.00000 −0.158362
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 8.00000 0.315735
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −28.0000 −1.09910
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ −4.00000 −0.156652
$$653$$ −26.0000 −1.01746 −0.508729 0.860927i $$-0.669885\pi$$
−0.508729 + 0.860927i $$0.669885\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 10.0000 0.390137
$$658$$ 32.0000 1.24749
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 24.0000 0.933492 0.466746 0.884391i $$-0.345426\pi$$
0.466746 + 0.884391i $$0.345426\pi$$
$$662$$ 18.0000 0.699590
$$663$$ −4.00000 −0.155347
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 12.0000 0.464642
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 20.0000 0.772091
$$672$$ −4.00000 −0.154303
$$673$$ 12.0000 0.462566 0.231283 0.972887i $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 12.0000 0.460857
$$679$$ 24.0000 0.921035
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ −8.00000 −0.306336
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 16.0000 0.610438
$$688$$ 8.00000 0.304997
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ 10.0000 0.380143
$$693$$ 8.00000 0.303895
$$694$$ −24.0000 −0.911028
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ −24.0000 −0.909065
$$698$$ −28.0000 −1.05982
$$699$$ 4.00000 0.151294
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −12.0000 −0.452589
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −26.0000 −0.978523
$$707$$ −56.0000 −2.10610
$$708$$ 14.0000 0.526152
$$709$$ −32.0000 −1.20179 −0.600893 0.799330i $$-0.705188\pi$$
−0.600893 + 0.799330i $$0.705188\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −18.0000 −0.674579
$$713$$ 24.0000 0.898807
$$714$$ −16.0000 −0.598785
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0000 0.448148
$$718$$ 20.0000 0.746393
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 24.0000 0.893807
$$722$$ −15.0000 −0.558242
$$723$$ 30.0000 1.11571
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ −10.0000 −0.369611
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 22.0000 0.812035
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 8.00000 0.294684
$$738$$ −6.00000 −0.220863
$$739$$ −2.00000 −0.0735712 −0.0367856 0.999323i $$-0.511712\pi$$
−0.0367856 + 0.999323i $$0.511712\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 40.0000 1.46845
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −2.00000 −0.0732252
$$747$$ 12.0000 0.439057
$$748$$ 8.00000 0.292509
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 12.0000 0.437304
$$754$$ −2.00000 −0.0728357
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 26.0000 0.944363
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 18.0000 0.652071
$$763$$ 48.0000 1.73772
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ −14.0000 −0.505511
$$768$$ −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
$$771$$ 12.0000 0.432169
$$772$$ 6.00000 0.215945
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 24.0000 0.860995
$$778$$ −26.0000 −0.932145
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 16.0000 0.572525
$$782$$ −24.0000 −0.858238
$$783$$ 2.00000 0.0714742
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 26.0000 0.926212
$$789$$ 10.0000 0.356009
$$790$$ 0 0
$$791$$ −48.0000 −1.70668
$$792$$ 2.00000 0.0710669
$$793$$ 10.0000 0.355110
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ −8.00000 −0.283197
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ −18.0000 −0.635602
$$803$$ 20.0000 0.705785
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 14.0000 0.492823
$$808$$ −14.0000 −0.492518
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ −16.0000 −0.561144
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 16.0000 0.559769
$$818$$ −2.00000 −0.0699284
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ −12.0000 −0.418803 −0.209401 0.977830i $$-0.567152\pi$$
−0.209401 + 0.977830i $$0.567152\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 26.0000 0.906303 0.453152 0.891434i $$-0.350300\pi$$
0.453152 + 0.891434i $$0.350300\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ −56.0000 −1.94849
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 42.0000 1.45872 0.729360 0.684130i $$-0.239818\pi$$
0.729360 + 0.684130i $$0.239818\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 1.00000 0.0346688
$$833$$ 36.0000 1.24733
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 4.00000 0.138260
$$838$$ 16.0000 0.552711
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 28.0000 0.964944
$$843$$ 18.0000 0.619953
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ −28.0000 −0.962091
$$848$$ 10.0000 0.343401
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 36.0000 1.23406
$$852$$ −8.00000 −0.274075
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ −40.0000 −1.36637 −0.683187 0.730243i $$-0.739407\pi$$
−0.683187 + 0.730243i $$0.739407\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ −8.00000 −0.272481
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ −16.0000 −0.543075
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 12.0000 0.406371
$$873$$ 6.00000 0.203069
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ −30.0000 −1.01303 −0.506514 0.862232i $$-0.669066\pi$$
−0.506514 + 0.862232i $$0.669066\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 9.00000 0.303046
$$883$$ 48.0000 1.61533 0.807664 0.589643i $$-0.200731\pi$$
0.807664 + 0.589643i $$0.200731\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 8.00000 0.268765
$$887$$ −6.00000 −0.201460 −0.100730 0.994914i $$-0.532118\pi$$
−0.100730 + 0.994914i $$0.532118\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −72.0000 −2.41480
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ −16.0000 −0.535720
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 6.00000 0.200334
$$898$$ 6.00000 0.200223
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 40.0000 1.33259
$$902$$ −12.0000 −0.399556
$$903$$ −32.0000 −1.06489
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ 24.0000 0.794284
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ 0 0
$$918$$ −4.00000 −0.132020
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ −4.00000 −0.131733
$$923$$ 8.00000 0.263323
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 6.00000 0.197066
$$928$$ −2.00000 −0.0656532
$$929$$ 46.0000 1.50921 0.754606 0.656179i $$-0.227828\pi$$
0.754606 + 0.656179i $$0.227828\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ −4.00000 −0.131024
$$933$$ −8.00000 −0.261908
$$934$$ −16.0000 −0.523536
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −8.00000 −0.261070
$$940$$ 0 0
$$941$$ 40.0000 1.30396 0.651981 0.758235i $$-0.273938\pi$$
0.651981 + 0.758235i $$0.273938\pi$$
$$942$$ 10.0000 0.325818
$$943$$ 36.0000 1.17232
$$944$$ −14.0000 −0.455661
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 16.0000 0.518563
$$953$$ −48.0000 −1.55487 −0.777436 0.628962i $$-0.783480\pi$$
−0.777436 + 0.628962i $$0.783480\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 4.00000 0.129302
$$958$$ −12.0000 −0.387702
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −6.00000 −0.193448
$$963$$ −8.00000 −0.257796
$$964$$ −30.0000 −0.966235
$$965$$ 0 0
$$966$$ 24.0000 0.772187
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 64.0000 2.05175
$$974$$ −40.0000 −1.28168
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −36.0000 −1.15056
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ 8.00000 0.255290
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ −32.0000 −1.01857
$$988$$ 2.00000 0.0636285
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ −18.0000 −0.571213
$$994$$ 32.0000 1.01498
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 30.0000 0.949633
$$999$$ 6.00000 0.189832
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 1950.2.a.u.1.1 1
3.2 odd 2 5850.2.a.x.1.1 1
5.2 odd 4 390.2.e.b.79.2 yes 2
5.3 odd 4 390.2.e.b.79.1 2
5.4 even 2 1950.2.a.g.1.1 1
15.2 even 4 1170.2.e.b.469.1 2
15.8 even 4 1170.2.e.b.469.2 2
15.14 odd 2 5850.2.a.bd.1.1 1
20.3 even 4 3120.2.l.h.1249.2 2
20.7 even 4 3120.2.l.h.1249.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.b.79.1 2 5.3 odd 4
390.2.e.b.79.2 yes 2 5.2 odd 4
1170.2.e.b.469.1 2 15.2 even 4
1170.2.e.b.469.2 2 15.8 even 4
1950.2.a.g.1.1 1 5.4 even 2
1950.2.a.u.1.1 1 1.1 even 1 trivial
3120.2.l.h.1249.1 2 20.7 even 4
3120.2.l.h.1249.2 2 20.3 even 4
5850.2.a.x.1.1 1 3.2 odd 2
5850.2.a.bd.1.1 1 15.14 odd 2