# Properties

 Label 5070.2.b.e.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.e.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -8.00000 q^{17} +1.00000i q^{18} -6.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +4.00000 q^{22} -6.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} +1.00000i q^{32} +4.00000i q^{33} -8.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} -2.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} +4.00000i q^{44} -1.00000i q^{45} -6.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +8.00000 q^{51} -10.0000 q^{53} -1.00000i q^{54} -4.00000 q^{55} -2.00000 q^{56} +6.00000i q^{57} -4.00000i q^{58} -4.00000i q^{59} -1.00000i q^{60} -10.0000 q^{61} -2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +8.00000 q^{68} +6.00000 q^{69} -2.00000i q^{70} -8.00000i q^{71} -1.00000i q^{72} +8.00000i q^{73} -2.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} -8.00000 q^{77} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000i q^{83} -2.00000i q^{84} +8.00000i q^{85} +4.00000i q^{86} +4.00000 q^{87} -4.00000 q^{88} +14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} -6.00000 q^{95} -1.00000i q^{96} -16.0000i q^{97} +3.00000i q^{98} -4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} - 16q^{17} + 8q^{22} - 12q^{23} - 2q^{25} - 2q^{27} - 8q^{29} - 2q^{30} - 4q^{35} - 2q^{36} + 12q^{38} - 2q^{40} - 4q^{42} + 8q^{43} - 2q^{48} + 6q^{49} + 16q^{51} - 20q^{53} - 8q^{55} - 4q^{56} - 20q^{61} - 2q^{64} - 8q^{66} + 16q^{68} + 12q^{69} - 4q^{74} + 2q^{75} - 16q^{77} + 16q^{79} + 2q^{81} + 4q^{82} + 8q^{87} - 8q^{88} + 2q^{90} + 12q^{92} - 12q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −8.00000 −1.94029 −0.970143 0.242536i $$-0.922021\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ − 6.00000i − 1.37649i −0.725476 0.688247i $$-0.758380\pi$$
0.725476 0.688247i $$-0.241620\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 2.00000i 0.436436i
$$22$$ 4.00000 0.852803
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000i 0.377964i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ − 8.00000i − 1.37199i
$$35$$ −2.00000 −0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ − 2.00000i − 0.312348i −0.987730 0.156174i $$-0.950084\pi$$
0.987730 0.156174i $$-0.0499160\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000i 0.603023i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 6.00000i − 0.884652i
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 3.00000 0.428571
$$50$$ − 1.00000i − 0.141421i
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ −4.00000 −0.539360
$$56$$ −2.00000 −0.267261
$$57$$ 6.00000i 0.794719i
$$58$$ − 4.00000i − 0.525226i
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 8.00000 0.970143
$$69$$ 6.00000 0.722315
$$70$$ − 2.00000i − 0.239046i
$$71$$ − 8.00000i − 0.949425i −0.880141 0.474713i $$-0.842552\pi$$
0.880141 0.474713i $$-0.157448\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 8.00000i 0.936329i 0.883641 + 0.468165i $$0.155085\pi$$
−0.883641 + 0.468165i $$0.844915\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ 6.00000i 0.688247i
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ 8.00000i 0.867722i
$$86$$ 4.00000i 0.431331i
$$87$$ 4.00000 0.428845
$$88$$ −4.00000 −0.426401
$$89$$ 14.0000i 1.48400i 0.670402 + 0.741999i $$0.266122\pi$$
−0.670402 + 0.741999i $$0.733878\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 16.0000i − 1.62455i −0.583272 0.812277i $$-0.698228\pi$$
0.583272 0.812277i $$-0.301772\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ − 4.00000i − 0.402015i
$$100$$ 1.00000 0.100000
$$101$$ 16.0000 1.59206 0.796030 0.605257i $$-0.206930\pi$$
0.796030 + 0.605257i $$0.206930\pi$$
$$102$$ 8.00000i 0.792118i
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ − 10.0000i − 0.971286i
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 12.0000i 1.14939i 0.818367 + 0.574696i $$0.194880\pi$$
−0.818367 + 0.574696i $$0.805120\pi$$
$$110$$ − 4.00000i − 0.381385i
$$111$$ − 2.00000i − 0.189832i
$$112$$ − 2.00000i − 0.188982i
$$113$$ 20.0000 1.88144 0.940721 0.339182i $$-0.110150\pi$$
0.940721 + 0.339182i $$0.110150\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 6.00000i 0.559503i
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 16.0000i 1.46672i
$$120$$ 1.00000 0.0912871
$$121$$ −5.00000 −0.454545
$$122$$ − 10.0000i − 0.905357i
$$123$$ 2.00000i 0.180334i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 2.00000 0.178174
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ −12.0000 −1.04053
$$134$$ −12.0000 −1.03664
$$135$$ 1.00000i 0.0860663i
$$136$$ 8.00000i 0.685994i
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 4.00000i 0.332182i
$$146$$ −8.00000 −0.662085
$$147$$ −3.00000 −0.247436
$$148$$ − 2.00000i − 0.164399i
$$149$$ − 10.0000i − 0.819232i −0.912258 0.409616i $$-0.865663\pi$$
0.912258 0.409616i $$-0.134337\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ −8.00000 −0.646762
$$154$$ − 8.00000i − 0.644658i
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ 10.0000 0.793052
$$160$$ 1.00000 0.0790569
$$161$$ 12.0000i 0.945732i
$$162$$ 1.00000i 0.0785674i
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 2.00000i 0.156174i
$$165$$ 4.00000 0.311400
$$166$$ −12.0000 −0.931381
$$167$$ 4.00000i 0.309529i 0.987951 + 0.154765i $$0.0494619\pi$$
−0.987951 + 0.154765i $$0.950538\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ −8.00000 −0.613572
$$171$$ − 6.00000i − 0.458831i
$$172$$ −4.00000 −0.304997
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 4.00000i 0.303239i
$$175$$ 2.00000i 0.151186i
$$176$$ − 4.00000i − 0.301511i
$$177$$ 4.00000i 0.300658i
$$178$$ −14.0000 −1.04934
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 6.00000i 0.442326i
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 32.0000i 2.34007i
$$188$$ 0 0
$$189$$ 2.00000i 0.145479i
$$190$$ − 6.00000i − 0.435286i
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ − 12.0000i − 0.846415i
$$202$$ 16.0000i 1.12576i
$$203$$ 8.00000i 0.561490i
$$204$$ −8.00000 −0.560112
$$205$$ −2.00000 −0.139686
$$206$$ 12.0000i 0.836080i
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ −24.0000 −1.66011
$$210$$ 2.00000i 0.138013i
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 8.00000i 0.548151i
$$214$$ 12.0000i 0.820303i
$$215$$ − 4.00000i − 0.272798i
$$216$$ 1.00000i 0.0680414i
$$217$$ 0 0
$$218$$ −12.0000 −0.812743
$$219$$ − 8.00000i − 0.540590i
$$220$$ 4.00000 0.269680
$$221$$ 0 0
$$222$$ 2.00000 0.134231
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ 20.0000i 1.33038i
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ − 6.00000i − 0.397360i
$$229$$ − 4.00000i − 0.264327i −0.991228 0.132164i $$-0.957808\pi$$
0.991228 0.132164i $$-0.0421925\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 8.00000 0.526361
$$232$$ 4.00000i 0.262613i
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4.00000i 0.260378i
$$237$$ −8.00000 −0.519656
$$238$$ −16.0000 −1.03713
$$239$$ 16.0000i 1.03495i 0.855697 + 0.517477i $$0.173129\pi$$
−0.855697 + 0.517477i $$0.826871\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ − 2.00000i − 0.128831i −0.997923 0.0644157i $$-0.979482\pi$$
0.997923 0.0644157i $$-0.0205183\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ − 3.00000i − 0.191663i
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ 0 0
$$249$$ − 12.0000i − 0.760469i
$$250$$ −1.00000 −0.0632456
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 24.0000i 1.50887i
$$254$$ − 4.00000i − 0.250982i
$$255$$ − 8.00000i − 0.500979i
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 10.0000i 0.617802i
$$263$$ 2.00000 0.123325 0.0616626 0.998097i $$-0.480360\pi$$
0.0616626 + 0.998097i $$0.480360\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 10.0000i 0.614295i
$$266$$ − 12.0000i − 0.735767i
$$267$$ − 14.0000i − 0.856786i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 16.0000i − 0.971931i −0.873978 0.485965i $$-0.838468\pi$$
0.873978 0.485965i $$-0.161532\pi$$
$$272$$ −8.00000 −0.485071
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 4.00000i 0.241209i
$$276$$ −6.00000 −0.361158
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 0 0
$$280$$ 2.00000i 0.119523i
$$281$$ 6.00000i 0.357930i 0.983855 + 0.178965i $$0.0572749\pi$$
−0.983855 + 0.178965i $$0.942725\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 8.00000i 0.474713i
$$285$$ 6.00000 0.355409
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ 1.00000i 0.0589256i
$$289$$ 47.0000 2.76471
$$290$$ −4.00000 −0.234888
$$291$$ 16.0000i 0.937937i
$$292$$ − 8.00000i − 0.468165i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ −4.00000 −0.232889
$$296$$ 2.00000 0.116248
$$297$$ 4.00000i 0.232104i
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ − 8.00000i − 0.461112i
$$302$$ 0 0
$$303$$ −16.0000 −0.919176
$$304$$ − 6.00000i − 0.344124i
$$305$$ 10.0000i 0.572598i
$$306$$ − 8.00000i − 0.457330i
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 8.00000 0.455842
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 22.0000i 1.24153i
$$315$$ −2.00000 −0.112687
$$316$$ −8.00000 −0.450035
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ 10.0000i 0.560772i
$$319$$ 16.0000i 0.895828i
$$320$$ 1.00000i 0.0559017i
$$321$$ −12.0000 −0.669775
$$322$$ −12.0000 −0.668734
$$323$$ 48.0000i 2.67079i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ − 12.0000i − 0.663602i
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ 4.00000i 0.220193i
$$331$$ 10.0000i 0.549650i 0.961494 + 0.274825i $$0.0886199\pi$$
−0.961494 + 0.274825i $$0.911380\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 2.00000i 0.109599i
$$334$$ −4.00000 −0.218870
$$335$$ 12.0000 0.655630
$$336$$ 2.00000i 0.109109i
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ −20.0000 −1.08625
$$340$$ − 8.00000i − 0.433861i
$$341$$ 0 0
$$342$$ 6.00000 0.324443
$$343$$ − 20.0000i − 1.07990i
$$344$$ − 4.00000i − 0.215666i
$$345$$ − 6.00000i − 0.323029i
$$346$$ − 22.0000i − 1.18273i
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ 28.0000i 1.49881i 0.662114 + 0.749403i $$0.269659\pi$$
−0.662114 + 0.749403i $$0.730341\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ −8.00000 −0.424596
$$356$$ − 14.0000i − 0.741999i
$$357$$ − 16.0000i − 0.846810i
$$358$$ 10.0000i 0.528516i
$$359$$ 24.0000i 1.26667i 0.773877 + 0.633336i $$0.218315\pi$$
−0.773877 + 0.633336i $$0.781685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −17.0000 −0.894737
$$362$$ − 10.0000i − 0.525588i
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 8.00000 0.418739
$$366$$ 10.0000i 0.522708i
$$367$$ −36.0000 −1.87918 −0.939592 0.342296i $$-0.888796\pi$$
−0.939592 + 0.342296i $$0.888796\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ − 2.00000i − 0.104116i
$$370$$ 2.00000i 0.103975i
$$371$$ 20.0000i 1.03835i
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ −32.0000 −1.65468
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ − 18.0000i − 0.924598i −0.886724 0.462299i $$-0.847025\pi$$
0.886724 0.462299i $$-0.152975\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 4.00000 0.204926
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 8.00000i 0.407718i
$$386$$ −4.00000 −0.203595
$$387$$ 4.00000 0.203331
$$388$$ 16.0000i 0.812277i
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ − 3.00000i − 0.151523i
$$393$$ −10.0000 −0.504433
$$394$$ 18.0000 0.906827
$$395$$ − 8.00000i − 0.402524i
$$396$$ 4.00000i 0.201008i
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 0 0
$$399$$ 12.0000 0.600751
$$400$$ −1.00000 −0.0500000
$$401$$ − 30.0000i − 1.49813i −0.662497 0.749064i $$-0.730503\pi$$
0.662497 0.749064i $$-0.269497\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 0 0
$$404$$ −16.0000 −0.796030
$$405$$ − 1.00000i − 0.0496904i
$$406$$ −8.00000 −0.397033
$$407$$ 8.00000 0.396545
$$408$$ − 8.00000i − 0.396059i
$$409$$ − 26.0000i − 1.28562i −0.766027 0.642809i $$-0.777769\pi$$
0.766027 0.642809i $$-0.222231\pi$$
$$410$$ − 2.00000i − 0.0987730i
$$411$$ − 6.00000i − 0.295958i
$$412$$ −12.0000 −0.591198
$$413$$ −8.00000 −0.393654
$$414$$ − 6.00000i − 0.294884i
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ − 24.0000i − 1.17388i
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ − 8.00000i − 0.389896i −0.980814 0.194948i $$-0.937546\pi$$
0.980814 0.194948i $$-0.0624538\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 0 0
$$424$$ 10.0000i 0.485643i
$$425$$ 8.00000 0.388057
$$426$$ −8.00000 −0.387601
$$427$$ 20.0000i 0.967868i
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ − 4.00000i − 0.191785i
$$436$$ − 12.0000i − 0.574696i
$$437$$ 36.0000i 1.72211i
$$438$$ 8.00000 0.382255
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 4.00000i 0.190693i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 2.00000i 0.0949158i
$$445$$ 14.0000 0.663664
$$446$$ 2.00000 0.0947027
$$447$$ 10.0000i 0.472984i
$$448$$ 2.00000i 0.0944911i
$$449$$ 6.00000i 0.283158i 0.989927 + 0.141579i $$0.0452178\pi$$
−0.989927 + 0.141579i $$0.954782\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ −8.00000 −0.376705
$$452$$ −20.0000 −0.940721
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 8.00000 0.373408
$$460$$ − 6.00000i − 0.279751i
$$461$$ − 6.00000i − 0.279448i −0.990190 0.139724i $$-0.955378\pi$$
0.990190 0.139724i $$-0.0446215\pi$$
$$462$$ 8.00000i 0.372194i
$$463$$ 26.0000i 1.20832i 0.796862 + 0.604161i $$0.206492\pi$$
−0.796862 + 0.604161i $$0.793508\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ − 24.0000i − 1.11178i
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ −4.00000 −0.184115
$$473$$ − 16.0000i − 0.735681i
$$474$$ − 8.00000i − 0.367452i
$$475$$ 6.00000i 0.275299i
$$476$$ − 16.0000i − 0.733359i
$$477$$ −10.0000 −0.457869
$$478$$ −16.0000 −0.731823
$$479$$ − 32.0000i − 1.46212i −0.682315 0.731059i $$-0.739027\pi$$
0.682315 0.731059i $$-0.260973\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 2.00000 0.0910975
$$483$$ − 12.0000i − 0.546019i
$$484$$ 5.00000 0.227273
$$485$$ −16.0000 −0.726523
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 26.0000i 1.17817i 0.808070 + 0.589086i $$0.200512\pi$$
−0.808070 + 0.589086i $$0.799488\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ − 16.0000i − 0.723545i
$$490$$ 3.00000 0.135526
$$491$$ −42.0000 −1.89543 −0.947717 0.319113i $$-0.896615\pi$$
−0.947717 + 0.319113i $$0.896615\pi$$
$$492$$ − 2.00000i − 0.0901670i
$$493$$ 32.0000 1.44121
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ −16.0000 −0.717698
$$498$$ 12.0000 0.537733
$$499$$ 38.0000i 1.70111i 0.525883 + 0.850557i $$0.323735\pi$$
−0.525883 + 0.850557i $$0.676265\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ − 4.00000i − 0.178707i
$$502$$ − 6.00000i − 0.267793i
$$503$$ 10.0000 0.445878 0.222939 0.974832i $$-0.428435\pi$$
0.222939 + 0.974832i $$0.428435\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ − 16.0000i − 0.711991i
$$506$$ −24.0000 −1.06693
$$507$$ 0 0
$$508$$ 4.00000 0.177471
$$509$$ 18.0000i 0.797836i 0.916987 + 0.398918i $$0.130614\pi$$
−0.916987 + 0.398918i $$0.869386\pi$$
$$510$$ 8.00000 0.354246
$$511$$ 16.0000 0.707798
$$512$$ 1.00000i 0.0441942i
$$513$$ 6.00000i 0.264906i
$$514$$ − 12.0000i − 0.529297i
$$515$$ − 12.0000i − 0.528783i
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 4.00000i 0.175750i
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ − 2.00000i − 0.0872872i
$$526$$ 2.00000i 0.0872041i
$$527$$ 0 0
$$528$$ 4.00000i 0.174078i
$$529$$ 13.0000 0.565217
$$530$$ −10.0000 −0.434372
$$531$$ − 4.00000i − 0.173585i
$$532$$ 12.0000 0.520266
$$533$$ 0 0
$$534$$ 14.0000 0.605839
$$535$$ − 12.0000i − 0.518805i
$$536$$ 12.0000 0.518321
$$537$$ −10.0000 −0.431532
$$538$$ − 24.0000i − 1.03471i
$$539$$ − 12.0000i − 0.516877i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ 8.00000i 0.343947i 0.985102 + 0.171973i $$0.0550143\pi$$
−0.985102 + 0.171973i $$0.944986\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 10.0000 0.429141
$$544$$ − 8.00000i − 0.342997i
$$545$$ 12.0000 0.514024
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −10.0000 −0.426790
$$550$$ −4.00000 −0.170561
$$551$$ 24.0000i 1.02243i
$$552$$ − 6.00000i − 0.255377i
$$553$$ − 16.0000i − 0.680389i
$$554$$ 10.0000i 0.424859i
$$555$$ −2.00000 −0.0848953
$$556$$ 8.00000 0.339276
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ − 32.0000i − 1.35104i
$$562$$ −6.00000 −0.253095
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ − 20.0000i − 0.841406i
$$566$$ 4.00000i 0.168133i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ −8.00000 −0.335673
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 6.00000i 0.251312i
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ − 4.00000i − 0.166957i
$$575$$ 6.00000 0.250217
$$576$$ −1.00000 −0.0416667
$$577$$ 12.0000i 0.499567i 0.968302 + 0.249783i $$0.0803594\pi$$
−0.968302 + 0.249783i $$0.919641\pi$$
$$578$$ 47.0000i 1.95494i
$$579$$ − 4.00000i − 0.166234i
$$580$$ − 4.00000i − 0.166091i
$$581$$ 24.0000 0.995688
$$582$$ −16.0000 −0.663221
$$583$$ 40.0000i 1.65663i
$$584$$ 8.00000 0.331042
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ 36.0000i 1.48588i 0.669359 + 0.742940i $$0.266569\pi$$
−0.669359 + 0.742940i $$0.733431\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ − 4.00000i − 0.164677i
$$591$$ 18.0000i 0.740421i
$$592$$ 2.00000i 0.0821995i
$$593$$ − 22.0000i − 0.903432i −0.892162 0.451716i $$-0.850812\pi$$
0.892162 0.451716i $$-0.149188\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 16.0000 0.655936
$$596$$ 10.0000i 0.409616i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 5.00000i 0.203279i
$$606$$ − 16.0000i − 0.649956i
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 6.00000 0.243332
$$609$$ − 8.00000i − 0.324176i
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ 8.00000 0.323381
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 2.00000 0.0806478
$$616$$ 8.00000i 0.322329i
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ − 12.0000i − 0.482711i
$$619$$ 10.0000i 0.401934i 0.979598 + 0.200967i $$0.0644084\pi$$
−0.979598 + 0.200967i $$0.935592\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ − 24.0000i − 0.962312i
$$623$$ 28.0000 1.12180
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 6.00000i 0.239808i
$$627$$ 24.0000 0.958468
$$628$$ −22.0000 −0.877896
$$629$$ − 16.0000i − 0.637962i
$$630$$ − 2.00000i − 0.0796819i
$$631$$ − 12.0000i − 0.477712i −0.971055 0.238856i $$-0.923228\pi$$
0.971055 0.238856i $$-0.0767725\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ 20.0000 0.794929
$$634$$ 30.0000 1.19145
$$635$$ 4.00000i 0.158735i
$$636$$ −10.0000 −0.396526
$$637$$ 0 0
$$638$$ −16.0000 −0.633446
$$639$$ − 8.00000i − 0.316475i
$$640$$ −1.00000 −0.0395285
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 44.0000i 1.73519i 0.497271 + 0.867595i $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ − 12.0000i − 0.472866i
$$645$$ 4.00000i 0.157500i
$$646$$ −48.0000 −1.88853
$$647$$ 30.0000 1.17942 0.589711 0.807614i $$-0.299242\pi$$
0.589711 + 0.807614i $$0.299242\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 16.0000i − 0.626608i
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ 12.0000 0.469237
$$655$$ − 10.0000i − 0.390732i
$$656$$ − 2.00000i − 0.0780869i
$$657$$ 8.00000i 0.312110i
$$658$$ 0 0
$$659$$ 2.00000 0.0779089 0.0389545 0.999241i $$-0.487597\pi$$
0.0389545 + 0.999241i $$0.487597\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ − 48.0000i − 1.86698i −0.358599 0.933492i $$-0.616745\pi$$
0.358599 0.933492i $$-0.383255\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 12.0000i 0.465340i
$$666$$ −2.00000 −0.0774984
$$667$$ 24.0000 0.929284
$$668$$ − 4.00000i − 0.154765i
$$669$$ 2.00000i 0.0773245i
$$670$$ 12.0000i 0.463600i
$$671$$ 40.0000i 1.54418i
$$672$$ −2.00000 −0.0771517
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ − 14.0000i − 0.539260i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ − 20.0000i − 0.768095i
$$679$$ −32.0000 −1.22805
$$680$$ 8.00000 0.306786
$$681$$ 4.00000i 0.153280i
$$682$$ 0 0
$$683$$ 44.0000i 1.68361i 0.539779 + 0.841807i $$0.318508\pi$$
−0.539779 + 0.841807i $$0.681492\pi$$
$$684$$ 6.00000i 0.229416i
$$685$$ 6.00000 0.229248
$$686$$ 20.0000 0.763604
$$687$$ 4.00000i 0.152610i
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 6.00000 0.228416
$$691$$ − 14.0000i − 0.532585i −0.963892 0.266293i $$-0.914201\pi$$
0.963892 0.266293i $$-0.0857987\pi$$
$$692$$ 22.0000 0.836315
$$693$$ −8.00000 −0.303895
$$694$$ − 24.0000i − 0.911028i
$$695$$ 8.00000i 0.303457i
$$696$$ − 4.00000i − 0.151620i
$$697$$ 16.0000i 0.606043i
$$698$$ −28.0000 −1.05982
$$699$$ 24.0000 0.907763
$$700$$ − 2.00000i − 0.0755929i
$$701$$ −32.0000 −1.20862 −0.604312 0.796748i $$-0.706552\pi$$
−0.604312 + 0.796748i $$0.706552\pi$$
$$702$$ 0 0
$$703$$ 12.0000 0.452589
$$704$$ 4.00000i 0.150756i
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ − 32.0000i − 1.20348i
$$708$$ − 4.00000i − 0.150329i
$$709$$ 4.00000i 0.150223i 0.997175 + 0.0751116i $$0.0239313\pi$$
−0.997175 + 0.0751116i $$0.976069\pi$$
$$710$$ − 8.00000i − 0.300235i
$$711$$ 8.00000 0.300023
$$712$$ 14.0000 0.524672
$$713$$ 0 0
$$714$$ 16.0000 0.598785
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ − 16.0000i − 0.597531i
$$718$$ −24.0000 −0.895672
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 24.0000i − 0.893807i
$$722$$ − 17.0000i − 0.632674i
$$723$$ 2.00000i 0.0743808i
$$724$$ 10.0000 0.371647
$$725$$ 4.00000 0.148556
$$726$$ 5.00000i 0.185567i
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.00000i 0.296093i
$$731$$ −32.0000 −1.18356
$$732$$ −10.0000 −0.369611
$$733$$ 38.0000i 1.40356i 0.712393 + 0.701781i $$0.247612\pi$$
−0.712393 + 0.701781i $$0.752388\pi$$
$$734$$ − 36.0000i − 1.32878i
$$735$$ 3.00000i 0.110657i
$$736$$ − 6.00000i − 0.221163i
$$737$$ 48.0000 1.76810
$$738$$ 2.00000 0.0736210
$$739$$ − 30.0000i − 1.10357i −0.833987 0.551784i $$-0.813947\pi$$
0.833987 0.551784i $$-0.186053\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ 0 0
$$742$$ −20.0000 −0.734223
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ − 2.00000i − 0.0732252i
$$747$$ 12.0000i 0.439057i
$$748$$ − 32.0000i − 1.17004i
$$749$$ − 24.0000i − 0.876941i
$$750$$ 1.00000 0.0365148
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ 6.00000 0.218652
$$754$$ 0 0
$$755$$ 0 0
$$756$$ − 2.00000i − 0.0727393i
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ 18.0000 0.653789
$$759$$ − 24.0000i − 0.871145i
$$760$$ 6.00000i 0.217643i
$$761$$ − 22.0000i − 0.797499i −0.917060 0.398750i $$-0.869444\pi$$
0.917060 0.398750i $$-0.130556\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ 24.0000 0.868858
$$764$$ 0 0
$$765$$ 8.00000i 0.289241i
$$766$$ 12.0000 0.433578
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 10.0000i 0.360609i 0.983611 + 0.180305i $$0.0577084\pi$$
−0.983611 + 0.180305i $$0.942292\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ 12.0000 0.432169
$$772$$ − 4.00000i − 0.143963i
$$773$$ 38.0000i 1.36677i 0.730061 + 0.683383i $$0.239492\pi$$
−0.730061 + 0.683383i $$0.760508\pi$$
$$774$$ 4.00000i 0.143777i
$$775$$ 0 0
$$776$$ −16.0000 −0.574367
$$777$$ −4.00000 −0.143499
$$778$$ − 8.00000i − 0.286814i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 48.0000i 1.71648i
$$783$$ 4.00000 0.142948
$$784$$ 3.00000 0.107143
$$785$$ − 22.0000i − 0.785214i
$$786$$ − 10.0000i − 0.356688i
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ −2.00000 −0.0712019
$$790$$ 8.00000 0.284627
$$791$$ − 40.0000i − 1.42224i
$$792$$ −4.00000 −0.142134
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ − 10.0000i − 0.354663i
$$796$$ 0 0
$$797$$ 10.0000 0.354218 0.177109 0.984191i $$-0.443325\pi$$
0.177109 + 0.984191i $$0.443325\pi$$
$$798$$ 12.0000i 0.424795i
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ 14.0000i 0.494666i
$$802$$ 30.0000 1.05934
$$803$$ 32.0000 1.12926
$$804$$ 12.0000i 0.423207i
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 24.0000 0.844840
$$808$$ − 16.0000i − 0.562878i
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ − 38.0000i − 1.33436i −0.744896 0.667180i $$-0.767501\pi$$
0.744896 0.667180i $$-0.232499\pi$$
$$812$$ − 8.00000i − 0.280745i
$$813$$ 16.0000i 0.561144i
$$814$$ 8.00000i 0.280400i
$$815$$ 16.0000 0.560456
$$816$$ 8.00000 0.280056
$$817$$ − 24.0000i − 0.839654i
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 2.00000 0.0698430
$$821$$ − 10.0000i − 0.349002i −0.984657 0.174501i $$-0.944169\pi$$
0.984657 0.174501i $$-0.0558313\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 52.0000 1.81261 0.906303 0.422628i $$-0.138892\pi$$
0.906303 + 0.422628i $$0.138892\pi$$
$$824$$ − 12.0000i − 0.418040i
$$825$$ − 4.00000i − 0.139262i
$$826$$ − 8.00000i − 0.278356i
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 12.0000i 0.416526i
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ −24.0000 −0.831551
$$834$$ 8.00000i 0.277017i
$$835$$ 4.00000 0.138426
$$836$$ 24.0000 0.830057
$$837$$ 0 0
$$838$$ 10.0000i 0.345444i
$$839$$ 40.0000i 1.38095i 0.723355 + 0.690477i $$0.242599\pi$$
−0.723355 + 0.690477i $$0.757401\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −13.0000 −0.448276
$$842$$ 8.00000 0.275698
$$843$$ − 6.00000i − 0.206651i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000i 0.343604i
$$848$$ −10.0000 −0.343401
$$849$$ −4.00000 −0.137280
$$850$$ 8.00000i 0.274398i
$$851$$ − 12.0000i − 0.411355i
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 46.0000i − 1.57501i −0.616308 0.787505i $$-0.711372\pi$$
0.616308 0.787505i $$-0.288628\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ −6.00000 −0.205196
$$856$$ − 12.0000i − 0.410152i
$$857$$ −4.00000 −0.136637 −0.0683187 0.997664i $$-0.521763\pi$$
−0.0683187 + 0.997664i $$0.521763\pi$$
$$858$$ 0 0
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ 4.00000i 0.136399i
$$861$$ 4.00000 0.136320
$$862$$ 0 0
$$863$$ − 12.0000i − 0.408485i −0.978920 0.204242i $$-0.934527\pi$$
0.978920 0.204242i $$-0.0654731\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 22.0000i 0.748022i
$$866$$ 2.00000i 0.0679628i
$$867$$ −47.0000 −1.59620
$$868$$ 0 0
$$869$$ − 32.0000i − 1.08553i
$$870$$ 4.00000 0.135613
$$871$$ 0 0
$$872$$ 12.0000 0.406371
$$873$$ − 16.0000i − 0.541518i
$$874$$ −36.0000 −1.21772
$$875$$ 2.00000 0.0676123
$$876$$ 8.00000i 0.270295i
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ 32.0000i 1.07995i
$$879$$ − 26.0000i − 0.876958i
$$880$$ −4.00000 −0.134840
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 16.0000i 0.537531i
$$887$$ −2.00000 −0.0671534 −0.0335767 0.999436i $$-0.510690\pi$$
−0.0335767 + 0.999436i $$0.510690\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 8.00000i 0.268311i
$$890$$ 14.0000i 0.469281i
$$891$$ − 4.00000i − 0.134005i
$$892$$ 2.00000i 0.0669650i
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ − 10.0000i − 0.334263i
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 80.0000 2.66519
$$902$$ − 8.00000i − 0.266371i
$$903$$ 8.00000i 0.266223i
$$904$$ − 20.0000i − 0.665190i
$$905$$ 10.0000i 0.332411i
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ 16.0000 0.530687
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 6.00000i 0.198680i
$$913$$ 48.0000 1.58857
$$914$$ −8.00000 −0.264616
$$915$$ − 10.0000i − 0.330590i
$$916$$ 4.00000i 0.132164i
$$917$$ − 20.0000i − 0.660458i
$$918$$ 8.00000i 0.264039i
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 6.00000 0.197814
$$921$$ 12.0000i 0.395413i
$$922$$ 6.00000 0.197599
$$923$$ 0 0
$$924$$ −8.00000 −0.263181
$$925$$ − 2.00000i − 0.0657596i
$$926$$ −26.0000 −0.854413
$$927$$ 12.0000 0.394132
$$928$$ − 4.00000i − 0.131306i
$$929$$ − 30.0000i − 0.984268i −0.870519 0.492134i $$-0.836217\pi$$
0.870519 0.492134i $$-0.163783\pi$$
$$930$$ 0 0
$$931$$ − 18.0000i − 0.589926i
$$932$$ 24.0000 0.786146
$$933$$ 24.0000 0.785725
$$934$$ − 28.0000i − 0.916188i
$$935$$ 32.0000 1.04651
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ 24.0000i 0.783628i
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ − 34.0000i − 1.10837i −0.832394 0.554184i $$-0.813030\pi$$
0.832394 0.554184i $$-0.186970\pi$$
$$942$$ − 22.0000i − 0.716799i
$$943$$ 12.0000i 0.390774i
$$944$$ − 4.00000i − 0.130189i
$$945$$ 2.00000 0.0650600
$$946$$ 16.0000 0.520205
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ −6.00000 −0.194666
$$951$$ 30.0000i 0.972817i
$$952$$ 16.0000 0.518563
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ − 10.0000i − 0.323762i
$$955$$ 0 0
$$956$$ − 16.0000i − 0.517477i
$$957$$ − 16.0000i − 0.517207i
$$958$$ 32.0000 1.03387
$$959$$ 12.0000 0.387500
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 31.0000 1.00000
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 2.00000i 0.0644157i
$$965$$ 4.00000 0.128765
$$966$$ 12.0000 0.386094
$$967$$ − 14.0000i − 0.450210i −0.974335 0.225105i $$-0.927728\pi$$
0.974335 0.225105i $$-0.0722725\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ − 48.0000i − 1.54198i
$$970$$ − 16.0000i − 0.513729i
$$971$$ 38.0000 1.21948 0.609739 0.792602i $$-0.291274\pi$$
0.609739 + 0.792602i $$0.291274\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 16.0000i 0.512936i
$$974$$ −26.0000 −0.833094
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ 16.0000 0.511624
$$979$$ 56.0000 1.78977
$$980$$ 3.00000i 0.0958315i
$$981$$ 12.0000i 0.383131i
$$982$$ − 42.0000i − 1.34027i
$$983$$ 52.0000i 1.65854i 0.558846 + 0.829271i $$0.311244\pi$$
−0.558846 + 0.829271i $$0.688756\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ −18.0000 −0.573528
$$986$$ 32.0000i 1.01909i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ − 4.00000i − 0.127128i
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ − 10.0000i − 0.317340i
$$994$$ − 16.0000i − 0.507489i
$$995$$ 0 0
$$996$$ 12.0000i 0.380235i
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ −38.0000 −1.20287
$$999$$ − 2.00000i − 0.0632772i
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.b.e.1351.2 2
13.5 odd 4 390.2.a.e.1.1 1
13.8 odd 4 5070.2.a.e.1.1 1
13.12 even 2 inner 5070.2.b.e.1351.1 2
39.5 even 4 1170.2.a.e.1.1 1
52.31 even 4 3120.2.a.o.1.1 1
65.18 even 4 1950.2.e.f.1249.1 2
65.44 odd 4 1950.2.a.h.1.1 1
65.57 even 4 1950.2.e.f.1249.2 2
156.83 odd 4 9360.2.a.bh.1.1 1
195.44 even 4 5850.2.a.bi.1.1 1
195.83 odd 4 5850.2.e.i.5149.2 2
195.122 odd 4 5850.2.e.i.5149.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 13.5 odd 4
1170.2.a.e.1.1 1 39.5 even 4
1950.2.a.h.1.1 1 65.44 odd 4
1950.2.e.f.1249.1 2 65.18 even 4
1950.2.e.f.1249.2 2 65.57 even 4
3120.2.a.o.1.1 1 52.31 even 4
5070.2.a.e.1.1 1 13.8 odd 4
5070.2.b.e.1351.1 2 13.12 even 2 inner
5070.2.b.e.1351.2 2 1.1 even 1 trivial
5850.2.a.bi.1.1 1 195.44 even 4
5850.2.e.i.5149.1 2 195.122 odd 4
5850.2.e.i.5149.2 2 195.83 odd 4
9360.2.a.bh.1.1 1 156.83 odd 4