# Properties

 Label 5070.2.b.n.1351.1 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ 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: $$0$$ 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 1351.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.n.1351.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} -2.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} +6.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.00000i q^{28} +1.00000 q^{30} +4.00000i q^{31} -1.00000i q^{32} -2.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +6.00000i q^{41} +2.00000 q^{42} +4.00000 q^{43} +1.00000i q^{45} -6.00000i q^{46} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -6.00000 q^{53} -1.00000i q^{54} -2.00000 q^{56} -2.00000i q^{57} -1.00000i q^{60} -10.0000 q^{61} +4.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} -8.00000i q^{67} +6.00000 q^{69} +2.00000i q^{70} +1.00000i q^{72} +8.00000i q^{73} +2.00000 q^{74} -1.00000 q^{75} +2.00000i q^{76} +8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000i q^{83} -2.00000i q^{84} -4.00000i q^{86} +6.00000i q^{89} +1.00000 q^{90} -6.00000 q^{92} +4.00000i q^{93} +2.00000 q^{95} -1.00000i q^{96} -8.00000i q^{97} -3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} - 2q^{12} + 4q^{14} + 2q^{16} + 12q^{23} - 2q^{25} + 2q^{27} + 2q^{30} - 4q^{35} - 2q^{36} - 4q^{38} - 2q^{40} + 4q^{42} + 8q^{43} + 2q^{48} + 6q^{49} - 12q^{53} - 4q^{56} - 20q^{61} + 8q^{62} - 2q^{64} + 12q^{69} + 4q^{74} - 2q^{75} + 16q^{79} + 2q^{81} + 12q^{82} + 2q^{90} - 12q^{92} + 4q^{95} + O(q^{100})$$

## 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 $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ − 2.00000i − 0.458831i −0.973329 0.229416i $$-0.926318\pi$$
0.973329 0.229416i $$-0.0736815\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 2.00000i − 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 4.00000i 0.718421i 0.933257 + 0.359211i $$0.116954\pi$$
−0.933257 + 0.359211i $$0.883046\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −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 $$0.947432\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 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
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 2.00000i 0.239046i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 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
$$76$$ 2.00000i 0.229416i
$$77$$ 0 0
$$78$$ 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
$$82$$ 6.00000 0.662589
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ 0 0
$$86$$ − 4.00000i − 0.431331i
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000i 0.635999i 0.948091 + 0.317999i $$0.103011\pi$$
−0.948091 + 0.317999i $$0.896989\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 8.00000i − 0.812277i −0.913812 0.406138i $$-0.866875\pi$$
0.913812 0.406138i $$-0.133125\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 6.00000i 0.582772i
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 16.0000i 1.53252i 0.642529 + 0.766261i $$0.277885\pi$$
−0.642529 + 0.766261i $$0.722115\pi$$
$$110$$ 0 0
$$111$$ 2.00000i 0.189832i
$$112$$ 2.00000i 0.188982i
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 6.00000i 0.559503i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ 11.0000 1.00000
$$122$$ 10.0000i 0.905357i
$$123$$ 6.00000i 0.541002i
$$124$$ − 4.00000i − 0.359211i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 2.00000 0.178174
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ −8.00000 −0.691095
$$135$$ 1.00000i 0.0860663i
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ 3.00000 0.247436
$$148$$ − 2.00000i − 0.164399i
$$149$$ − 6.00000i − 0.491539i −0.969328 0.245770i $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 20.0000i 1.62758i 0.581161 + 0.813788i $$0.302599\pi$$
−0.581161 + 0.813788i $$0.697401\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 12.0000i 0.945732i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ − 6.00000i − 0.468521i
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 2.00000i − 0.152944i
$$172$$ −4.00000 −0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ − 2.00000i − 0.151186i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 6.00000i 0.442326i
$$185$$ −2.00000 −0.147043
$$186$$ 4.00000 0.293294
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 2.00000i 0.145479i
$$190$$ − 2.00000i − 0.145095i
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 20.0000i 1.43963i 0.694165 + 0.719816i $$0.255774\pi$$
−0.694165 + 0.719816i $$0.744226\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ − 8.00000i − 0.564276i
$$202$$ − 12.0000i − 0.844317i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 8.00000i 0.557386i
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 2.00000i 0.138013i
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ − 12.0000i − 0.820303i
$$215$$ 4.00000i 0.272798i
$$216$$ 1.00000i 0.0680414i
$$217$$ −8.00000 −0.543075
$$218$$ 16.0000 1.08366
$$219$$ 8.00000i 0.540590i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 2.00000 0.134231
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ 12.0000i 0.798228i
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 2.00000i 0.132453i
$$229$$ − 16.0000i − 1.05731i −0.848837 0.528655i $$-0.822697\pi$$
0.848837 0.528655i $$-0.177303\pi$$
$$230$$ 6.00000 0.395628
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 24.0000i 1.55243i 0.630468 + 0.776215i $$0.282863\pi$$
−0.630468 + 0.776215i $$0.717137\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ 2.00000i 0.128831i 0.997923 + 0.0644157i $$0.0205183\pi$$
−0.997923 + 0.0644157i $$0.979482\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000 0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 3.00000i 0.191663i
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ −4.00000 −0.254000
$$249$$ 12.0000i 0.760469i
$$250$$ −1.00000 −0.0632456
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 0 0
$$254$$ 8.00000i 0.501965i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0000i 1.11204i
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ 0 0
$$265$$ − 6.00000i − 0.368577i
$$266$$ − 4.00000i − 0.245256i
$$267$$ 6.00000i 0.367194i
$$268$$ 8.00000i 0.488678i
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ − 4.00000i − 0.242983i −0.992592 0.121491i $$-0.961232\pi$$
0.992592 0.121491i $$-0.0387677\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 4.00000i 0.239474i
$$280$$ − 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 $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ − 8.00000i − 0.468968i
$$292$$ − 8.00000i − 0.468165i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 8.00000i 0.461112i
$$302$$ 20.0000 1.15087
$$303$$ 12.0000 0.689382
$$304$$ − 2.00000i − 0.114708i
$$305$$ − 10.0000i − 0.572598i
$$306$$ 0 0
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 4.00000i 0.227185i
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ − 2.00000i − 0.112867i
$$315$$ −2.00000 −0.112687
$$316$$ −8.00000 −0.450035
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ − 1.00000i − 0.0559017i
$$321$$ 12.0000 0.669775
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 16.0000i 0.884802i
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 26.0000i − 1.42909i −0.699590 0.714545i $$-0.746634\pi$$
0.699590 0.714545i $$-0.253366\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 2.00000i 0.109599i
$$334$$ 12.0000 0.656611
$$335$$ 8.00000 0.437087
$$336$$ 2.00000i 0.109109i
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −2.00000 −0.108148
$$343$$ 20.0000i 1.07990i
$$344$$ 4.00000i 0.215666i
$$345$$ 6.00000i 0.323029i
$$346$$ − 6.00000i − 0.322562i
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ − 16.0000i − 0.856460i −0.903670 0.428230i $$-0.859137\pi$$
0.903670 0.428230i $$-0.140863\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ − 6.00000i − 0.317999i
$$357$$ 0 0
$$358$$ 18.0000i 0.951330i
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 15.0000 0.789474
$$362$$ − 22.0000i − 1.15629i
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ 10.0000i 0.522708i
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 6.00000i 0.312348i
$$370$$ 2.00000i 0.103975i
$$371$$ − 12.0000i − 0.623009i
$$372$$ − 4.00000i − 0.207390i
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 2.00000 0.102869
$$379$$ 10.0000i 0.513665i 0.966456 + 0.256833i $$0.0826790\pi$$
−0.966456 + 0.256833i $$0.917321\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −8.00000 −0.409852
$$382$$ − 24.0000i − 1.22795i
$$383$$ 36.0000i 1.83951i 0.392488 + 0.919757i $$0.371614\pi$$
−0.392488 + 0.919757i $$0.628386\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ 20.0000 1.01797
$$387$$ 4.00000 0.203331
$$388$$ 8.00000i 0.406138i
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 3.00000i 0.151523i
$$393$$ −18.0000 −0.907980
$$394$$ −6.00000 −0.302276
$$395$$ 8.00000i 0.402524i
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ − 6.00000i − 0.299626i −0.988714 0.149813i $$-0.952133\pi$$
0.988714 0.149813i $$-0.0478671\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 0 0
$$404$$ −12.0000 −0.597022
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ − 14.0000i − 0.692255i −0.938187 0.346128i $$-0.887496\pi$$
0.938187 0.346128i $$-0.112504\pi$$
$$410$$ 6.00000i 0.296319i
$$411$$ 18.0000i 0.887875i
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ − 6.00000i − 0.294884i
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 4.00000i 0.194948i 0.995238 + 0.0974740i $$0.0310763\pi$$
−0.995238 + 0.0974740i $$0.968924\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 0 0
$$424$$ − 6.00000i − 0.291386i
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 20.0000i − 0.967868i
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ − 24.0000i − 1.15604i −0.816023 0.578020i $$-0.803826\pi$$
0.816023 0.578020i $$-0.196174\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 8.00000i 0.384012i
$$435$$ 0 0
$$436$$ − 16.0000i − 0.766261i
$$437$$ − 12.0000i − 0.574038i
$$438$$ 8.00000 0.382255
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ − 2.00000i − 0.0949158i
$$445$$ −6.00000 −0.284427
$$446$$ −14.0000 −0.662919
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ − 18.0000i − 0.849473i −0.905317 0.424736i $$-0.860367\pi$$
0.905317 0.424736i $$-0.139633\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ 0 0
$$452$$ 12.0000 0.564433
$$453$$ 20.0000i 0.939682i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 2.00000 0.0936586
$$457$$ − 32.0000i − 1.49690i −0.663193 0.748448i $$-0.730799\pi$$
0.663193 0.748448i $$-0.269201\pi$$
$$458$$ −16.0000 −0.747631
$$459$$ 0 0
$$460$$ − 6.00000i − 0.279751i
$$461$$ − 18.0000i − 0.838344i −0.907907 0.419172i $$-0.862320\pi$$
0.907907 0.419172i $$-0.137680\pi$$
$$462$$ 0 0
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 24.0000i 1.11178i
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ − 8.00000i − 0.367452i
$$475$$ 2.00000i 0.0917663i
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 24.0000 1.09773
$$479$$ − 24.0000i − 1.09659i −0.836286 0.548294i $$-0.815277\pi$$
0.836286 0.548294i $$-0.184723\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 2.00000 0.0910975
$$483$$ 12.0000i 0.546019i
$$484$$ −11.0000 −0.500000
$$485$$ 8.00000 0.363261
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 20.0000i 0.904431i
$$490$$ 3.00000 0.135526
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 4.00000i 0.179605i
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ − 14.0000i − 0.626726i −0.949633 0.313363i $$-0.898544\pi$$
0.949633 0.313363i $$-0.101456\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ 12.0000i 0.536120i
$$502$$ 18.0000i 0.803379i
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 12.0000i 0.533993i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ − 18.0000i − 0.797836i −0.916987 0.398918i $$-0.869386\pi$$
0.916987 0.398918i $$-0.130614\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 2.00000i − 0.0883022i
$$514$$ 12.0000i 0.529297i
$$515$$ − 8.00000i − 0.352522i
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 4.00000i 0.175750i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 18.0000 0.786334
$$525$$ − 2.00000i − 0.0872872i
$$526$$ − 6.00000i − 0.261612i
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ 0 0
$$534$$ 6.00000 0.259645
$$535$$ 12.0000i 0.518805i
$$536$$ 8.00000 0.345547
$$537$$ −18.0000 −0.776757
$$538$$ − 12.0000i − 0.517357i
$$539$$ 0 0
$$540$$ − 1.00000i − 0.0430331i
$$541$$ 20.0000i 0.859867i 0.902861 + 0.429934i $$0.141463\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ 22.0000 0.944110
$$544$$ 0 0
$$545$$ −16.0000 −0.685365
$$546$$ 0 0
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ − 18.0000i − 0.768922i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 6.00000i 0.255377i
$$553$$ 16.0000i 0.680389i
$$554$$ − 22.0000i − 0.934690i
$$555$$ −2.00000 −0.0848953
$$556$$ −8.00000 −0.339276
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ − 12.0000i − 0.504844i
$$566$$ − 4.00000i − 0.168133i
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ − 2.00000i − 0.0837708i
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ 24.0000 1.00261
$$574$$ 12.0000i 0.500870i
$$575$$ −6.00000 −0.250217
$$576$$ −1.00000 −0.0416667
$$577$$ 4.00000i 0.166522i 0.996528 + 0.0832611i $$0.0265335\pi$$
−0.996528 + 0.0832611i $$0.973466\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ 20.0000i 0.831172i
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ −8.00000 −0.331611
$$583$$ 0 0
$$584$$ −8.00000 −0.331042
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ − 6.00000i − 0.246807i
$$592$$ 2.00000i 0.0821995i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000i 0.245770i
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 8.00000 0.326056
$$603$$ − 8.00000i − 0.325785i
$$604$$ − 20.0000i − 0.813788i
$$605$$ 11.0000i 0.447214i
$$606$$ − 12.0000i − 0.487467i
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ 14.0000i 0.562708i 0.959604 + 0.281354i $$0.0907834\pi$$
−0.959604 + 0.281354i $$0.909217\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 6.00000 0.240772
$$622$$ 0 0
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 14.0000i − 0.559553i
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 0 0
$$630$$ 2.00000i 0.0796819i
$$631$$ 8.00000i 0.318475i 0.987240 + 0.159237i $$0.0509036\pi$$
−0.987240 + 0.159237i $$0.949096\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ −4.00000 −0.158986
$$634$$ 30.0000 1.19145
$$635$$ − 8.00000i − 0.317470i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ − 12.0000i − 0.472866i
$$645$$ 4.00000i 0.157500i
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ − 20.0000i − 0.783260i
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 16.0000 0.625650
$$655$$ − 18.0000i − 0.703318i
$$656$$ 6.00000i 0.234261i
$$657$$ 8.00000i 0.312110i
$$658$$ 0 0
$$659$$ −18.0000 −0.701180 −0.350590 0.936529i $$-0.614019\pi$$
−0.350590 + 0.936529i $$0.614019\pi$$
$$660$$ 0 0
$$661$$ − 28.0000i − 1.08907i −0.838737 0.544537i $$-0.816705\pi$$
0.838737 0.544537i $$-0.183295\pi$$
$$662$$ −26.0000 −1.01052
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 4.00000i 0.155113i
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ − 12.0000i − 0.464294i
$$669$$ − 14.0000i − 0.541271i
$$670$$ − 8.00000i − 0.309067i
$$671$$ 0 0
$$672$$ 2.00000 0.0771517
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ − 34.0000i − 1.30963i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ − 12.0000i − 0.459841i
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 2.00000i 0.0764719i
$$685$$ −18.0000 −0.687745
$$686$$ 20.0000 0.763604
$$687$$ − 16.0000i − 0.610438i
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 6.00000 0.228416
$$691$$ − 2.00000i − 0.0760836i −0.999276 0.0380418i $$-0.987888\pi$$
0.999276 0.0380418i $$-0.0121120\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ − 24.0000i − 0.911028i
$$695$$ 8.00000i 0.303457i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −16.0000 −0.605609
$$699$$ −24.0000 −0.907763
$$700$$ 2.00000i 0.0755929i
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ 8.00000i 0.300446i 0.988652 + 0.150223i $$0.0479992\pi$$
−0.988652 + 0.150223i $$0.952001\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ −6.00000 −0.224860
$$713$$ 24.0000i 0.898807i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 18.0000 0.672692
$$717$$ 24.0000i 0.896296i
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ − 16.0000i − 0.595871i
$$722$$ − 15.0000i − 0.558242i
$$723$$ 2.00000i 0.0743808i
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ − 11.0000i − 0.408248i
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.00000i 0.296093i
$$731$$ 0 0
$$732$$ 10.0000 0.369611
$$733$$ − 26.0000i − 0.960332i −0.877178 0.480166i $$-0.840576\pi$$
0.877178 0.480166i $$-0.159424\pi$$
$$734$$ 16.0000i 0.590571i
$$735$$ 3.00000i 0.110657i
$$736$$ − 6.00000i − 0.221163i
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ − 34.0000i − 1.25071i −0.780340 0.625355i $$-0.784954\pi$$
0.780340 0.625355i $$-0.215046\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 6.00000 0.219823
$$746$$ 22.0000i 0.805477i
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 24.0000i 0.876941i
$$750$$ −1.00000 −0.0365148
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ −18.0000 −0.655956
$$754$$ 0 0
$$755$$ −20.0000 −0.727875
$$756$$ − 2.00000i − 0.0727393i
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ 10.0000 0.363216
$$759$$ 0 0
$$760$$ 2.00000i 0.0725476i
$$761$$ − 54.0000i − 1.95750i −0.205061 0.978749i $$-0.565739\pi$$
0.205061 0.978749i $$-0.434261\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ −32.0000 −1.15848
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 22.0000i 0.793340i 0.917961 + 0.396670i $$0.129834\pi$$
−0.917961 + 0.396670i $$0.870166\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ − 20.0000i − 0.719816i
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ − 4.00000i − 0.143777i
$$775$$ − 4.00000i − 0.143684i
$$776$$ 8.00000 0.287183
$$777$$ −4.00000 −0.143499
$$778$$ 12.0000i 0.430221i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 2.00000i 0.0713831i
$$786$$ 18.0000i 0.642039i
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 6.00000 0.213606
$$790$$ 8.00000 0.284627
$$791$$ − 24.0000i − 0.853342i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −22.0000 −0.780751
$$795$$ − 6.00000i − 0.212798i
$$796$$ −16.0000 −0.567105
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ 0 0
$$800$$ 1.00000i 0.0353553i
$$801$$ 6.00000i 0.212000i
$$802$$ −6.00000 −0.211867
$$803$$ 0 0
$$804$$ 8.00000i 0.282138i
$$805$$ −12.0000 −0.422944
$$806$$ 0 0
$$807$$ 12.0000 0.422420
$$808$$ 12.0000i 0.422159i
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 22.0000i 0.772524i 0.922389 + 0.386262i $$0.126234\pi$$
−0.922389 + 0.386262i $$0.873766\pi$$
$$812$$ 0 0
$$813$$ − 4.00000i − 0.140286i
$$814$$ 0 0
$$815$$ −20.0000 −0.700569
$$816$$ 0 0
$$817$$ − 8.00000i − 0.279885i
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ − 30.0000i − 1.04701i −0.852023 0.523504i $$-0.824625\pi$$
0.852023 0.523504i $$-0.175375\pi$$
$$822$$ 18.0000 0.627822
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ − 8.00000i − 0.278693i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 12.0000i 0.416526i
$$831$$ 22.0000 0.763172
$$832$$ 0 0
$$833$$ 0 0
$$834$$ − 8.00000i − 0.277017i
$$835$$ −12.0000 −0.415277
$$836$$ 0 0
$$837$$ 4.00000i 0.138260i
$$838$$ 18.0000i 0.621800i
$$839$$ − 24.0000i − 0.828572i −0.910147 0.414286i $$-0.864031\pi$$
0.910147 0.414286i $$-0.135969\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −29.0000 −1.00000
$$842$$ 4.00000 0.137849
$$843$$ 6.00000i 0.206651i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000i 0.755929i
$$848$$ −6.00000 −0.206041
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 12.0000i 0.411355i
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 2.00000 0.0683986
$$856$$ 12.0000i 0.410152i
$$857$$ −12.0000 −0.409912 −0.204956 0.978771i $$-0.565705\pi$$
−0.204956 + 0.978771i $$0.565705\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ − 4.00000i − 0.136399i
$$861$$ −12.0000 −0.408959
$$862$$ −24.0000 −0.817443
$$863$$ − 36.0000i − 1.22545i −0.790295 0.612727i $$-0.790072\pi$$
0.790295 0.612727i $$-0.209928\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 6.00000i 0.204006i
$$866$$ − 34.0000i − 1.15537i
$$867$$ −17.0000 −0.577350
$$868$$ 8.00000 0.271538
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −16.0000 −0.541828
$$873$$ − 8.00000i − 0.270759i
$$874$$ −12.0000 −0.405906
$$875$$ 2.00000 0.0676123
$$876$$ − 8.00000i − 0.270295i
$$877$$ 34.0000i 1.14810i 0.818821 + 0.574049i $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 6.00000i 0.202375i
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 24.0000i 0.806296i
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ − 16.0000i − 0.536623i
$$890$$ 6.00000i 0.201120i
$$891$$ 0 0
$$892$$ 14.0000i 0.468755i
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ − 18.0000i − 0.601674i
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ − 12.0000i − 0.399114i
$$905$$ 22.0000i 0.731305i
$$906$$ 20.0000 0.664455
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ − 2.00000i − 0.0662266i
$$913$$ 0 0
$$914$$ −32.0000 −1.05847
$$915$$ − 10.0000i − 0.330590i
$$916$$ 16.0000i 0.528655i
$$917$$ − 36.0000i − 1.18882i
$$918$$ 0 0
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ − 16.0000i − 0.527218i
$$922$$ −18.0000 −0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 14.0000 0.460069
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ − 30.0000i − 0.984268i −0.870519 0.492134i $$-0.836217\pi$$
0.870519 0.492134i $$-0.163783\pi$$
$$930$$ 4.00000i 0.131165i
$$931$$ − 6.00000i − 0.196642i
$$932$$ 24.0000 0.786146
$$933$$ 0 0
$$934$$ − 12.0000i − 0.392652i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ − 16.0000i − 0.522419i
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ − 6.00000i − 0.195594i −0.995206 0.0977972i $$-0.968820\pi$$
0.995206 0.0977972i $$-0.0311797\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 36.0000i 1.17232i
$$944$$ 0 0
$$945$$ −2.00000 −0.0650600
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ 2.00000 0.0648886
$$951$$ 30.0000i 0.972817i
$$952$$ 0 0
$$953$$ −48.0000 −1.55487 −0.777436 0.628962i $$-0.783480\pi$$
−0.777436 + 0.628962i $$0.783480\pi$$
$$954$$ 6.00000i 0.194257i
$$955$$ 24.0000i 0.776622i
$$956$$ − 24.0000i − 0.776215i
$$957$$ 0 0
$$958$$ −24.0000 −0.775405
$$959$$ −36.0000 −1.16250
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ − 2.00000i − 0.0644157i
$$965$$ −20.0000 −0.643823
$$966$$ 12.0000 0.386094
$$967$$ − 50.0000i − 1.60789i −0.594703 0.803946i $$-0.702730\pi$$
0.594703 0.803946i $$-0.297270\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 0 0
$$970$$ − 8.00000i − 0.256865i
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 16.0000i 0.512936i
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ − 6.00000i − 0.191957i −0.995383 0.0959785i $$-0.969402\pi$$
0.995383 0.0959785i $$-0.0305980\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 0 0
$$980$$ − 3.00000i − 0.0958315i
$$981$$ 16.0000i 0.510841i
$$982$$ 6.00000i 0.191468i
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 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
$$993$$ − 26.0000i − 0.825085i
$$994$$ 0 0
$$995$$ 16.0000i 0.507234i
$$996$$ − 12.0000i − 0.380235i
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ 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.n.1351.1 2
13.5 odd 4 5070.2.a.k.1.1 1
13.8 odd 4 390.2.a.g.1.1 1
13.12 even 2 inner 5070.2.b.n.1351.2 2
39.8 even 4 1170.2.a.g.1.1 1
52.47 even 4 3120.2.a.b.1.1 1
65.8 even 4 1950.2.e.k.1249.1 2
65.34 odd 4 1950.2.a.b.1.1 1
65.47 even 4 1950.2.e.k.1249.2 2
156.47 odd 4 9360.2.a.bg.1.1 1
195.8 odd 4 5850.2.e.r.5149.2 2
195.47 odd 4 5850.2.e.r.5149.1 2
195.164 even 4 5850.2.a.bk.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.g.1.1 1 13.8 odd 4
1170.2.a.g.1.1 1 39.8 even 4
1950.2.a.b.1.1 1 65.34 odd 4
1950.2.e.k.1249.1 2 65.8 even 4
1950.2.e.k.1249.2 2 65.47 even 4
3120.2.a.b.1.1 1 52.47 even 4
5070.2.a.k.1.1 1 13.5 odd 4
5070.2.b.n.1351.1 2 1.1 even 1 trivial
5070.2.b.n.1351.2 2 13.12 even 2 inner
5850.2.a.bk.1.1 1 195.164 even 4
5850.2.e.r.5149.1 2 195.47 odd 4
5850.2.e.r.5149.2 2 195.8 odd 4
9360.2.a.bg.1.1 1 156.47 odd 4