# Properties

 Label 1950.2.a.g.1.1 Level $1950$ Weight $2$ Character 1950.1 Self dual yes Analytic conductor $15.571$ Analytic rank $1$ 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: $$1$$ 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 $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −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 $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −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 $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −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 $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −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 $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 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 $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ −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 $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −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 $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ −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 $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −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 $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −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 $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 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 $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ −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 $$-0.205563\pi$$
0.798621 + 0.601834i $$0.205563\pi$$
$$128$$ −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 $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ −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 $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 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 $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −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 $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ 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 $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ −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 $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 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 $$-0.458173\pi$$
0.131024 + 0.991379i $$0.458173\pi$$
$$234$$ 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 $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 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 $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ −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 $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −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 $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 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 $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ −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 $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ −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 $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 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 $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ −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 $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ −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 $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 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 $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 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 $$-0.483511\pi$$
0.0517780 + 0.998659i $$0.483511\pi$$
$$374$$ 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 $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ −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 $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ −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 $$-0.560864\pi$$
−0.190046 + 0.981775i $$0.560864\pi$$
$$444$$ 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 $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 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 $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −4.00000 −0.185296
$$467$$ 16.0000 0.740392 0.370196 0.928954i $$-0.379291\pi$$
0.370196 + 0.928954i $$0.379291\pi$$
$$468$$ −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 $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ −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 $$-0.631438\pi$$
−0.401290 + 0.915951i $$0.631438\pi$$
$$504$$ 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 $$-0.555963\pi$$
−0.174908 + 0.984585i $$0.555963\pi$$
$$524$$ 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 $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ −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 $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 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 $$-0.390534\pi$$
0.337160 + 0.941447i $$0.390534\pi$$
$$564$$ −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 $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ 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 $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 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 $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ −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 $$-0.619033\pi$$
−0.365299 + 0.930890i $$0.619033\pi$$
$$608$$ −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 $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 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 $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ −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 $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ −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 $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ −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 $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 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 $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ −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 $$-0.488240\pi$$
0.0369358 + 0.999318i $$0.488240\pi$$
$$734$$ 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 $$-0.476624\pi$$
0.0733729 + 0.997305i $$0.476624\pi$$
$$744$$ 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 $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ −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 $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 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 $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −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 $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 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 $$-0.649700\pi$$
−0.453152 + 0.891434i $$0.649700\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ −56.0000 −1.94849
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 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 $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 40.0000 1.36637 0.683187 0.730243i $$-0.260593\pi$$
0.683187 + 0.730243i $$0.260593\pi$$
$$858$$ 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 $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −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 $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ −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 $$-0.799269\pi$$
−0.807664 + 0.589643i $$0.799269\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 8.00000 0.268765
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ −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 $$-0.321714\pi$$
0.531271 + 0.847202i $$0.321714\pi$$
$$908$$ −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 $$-0.584167\pi$$
−0.261349 + 0.965244i $$0.584167\pi$$
$$938$$ −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 $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −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 $$-0.216520\pi$$
0.777436 + 0.628962i $$0.216520\pi$$
$$954$$ 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 $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ 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 $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ −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 $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 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 $$-0.705524\pi$$
−0.601736 + 0.798695i $$0.705524\pi$$
$$998$$ −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.g.1.1 1
3.2 odd 2 5850.2.a.bd.1.1 1
5.2 odd 4 390.2.e.b.79.1 2
5.3 odd 4 390.2.e.b.79.2 yes 2
5.4 even 2 1950.2.a.u.1.1 1
15.2 even 4 1170.2.e.b.469.2 2
15.8 even 4 1170.2.e.b.469.1 2
15.14 odd 2 5850.2.a.x.1.1 1
20.3 even 4 3120.2.l.h.1249.1 2
20.7 even 4 3120.2.l.h.1249.2 2

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