# Properties

 Label 5070.2.b.a.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## 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 1351.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.a.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000i q^{11} +1.00000 q^{12} -3.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -5.00000i q^{19} -1.00000i q^{20} -3.00000i q^{21} +1.00000 q^{22} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.00000i q^{28} +1.00000 q^{30} +10.0000i q^{31} +1.00000i q^{32} +1.00000i q^{33} -3.00000 q^{35} -1.00000 q^{36} +1.00000i q^{37} +5.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} +3.00000 q^{42} +2.00000 q^{43} +1.00000i q^{44} +1.00000i q^{45} +4.00000i q^{46} +9.00000i q^{47} -1.00000 q^{48} -2.00000 q^{49} -1.00000i q^{50} -13.0000 q^{53} -1.00000i q^{54} +1.00000 q^{55} +3.00000 q^{56} +5.00000i q^{57} -4.00000i q^{59} +1.00000i q^{60} -2.00000 q^{61} -10.0000 q^{62} +3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -12.0000i q^{67} -4.00000 q^{69} -3.00000i q^{70} -2.00000i q^{71} -1.00000i q^{72} +16.0000i q^{73} -1.00000 q^{74} +1.00000 q^{75} +5.00000i q^{76} +3.00000 q^{77} -10.0000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000i q^{83} +3.00000i q^{84} +2.00000i q^{86} -1.00000 q^{88} -1.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} -10.0000i q^{93} -9.00000 q^{94} +5.00000 q^{95} -1.00000i q^{96} +12.0000i q^{97} -2.00000i q^{98} -1.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 2 q^{22} + 8 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{30} - 6 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} + 6 q^{42} + 4 q^{43} - 2 q^{48} - 4 q^{49} - 26 q^{53} + 2 q^{55} + 6 q^{56} - 4 q^{61} - 20 q^{62} - 2 q^{64} - 2 q^{66} - 8 q^{69} - 2 q^{74} + 2 q^{75} + 6 q^{77} - 20 q^{79} + 2 q^{81} - 12 q^{82} - 2 q^{88} - 2 q^{90} - 8 q^{92} - 18 q^{94} + 10 q^{95} + O(q^{100})$$

## 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
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ − 1.00000i − 0.301511i −0.988571 0.150756i $$-0.951829\pi$$
0.988571 0.150756i $$-0.0481707\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −3.00000 −0.801784
$$15$$ − 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
$$19$$ − 5.00000i − 1.14708i −0.819178 0.573539i $$-0.805570\pi$$
0.819178 0.573539i $$-0.194430\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ − 3.00000i − 0.654654i
$$22$$ 1.00000 0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ − 3.00000i − 0.566947i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 10.0000i 1.79605i 0.439941 + 0.898027i $$0.354999\pi$$
−0.439941 + 0.898027i $$0.645001\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 1.00000i 0.174078i
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ −1.00000 −0.166667
$$37$$ 1.00000i 0.164399i 0.996616 + 0.0821995i $$0.0261945\pi$$
−0.996616 + 0.0821995i $$0.973806\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 3.00000 0.462910
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 1.00000i 0.150756i
$$45$$ 1.00000i 0.149071i
$$46$$ 4.00000i 0.589768i
$$47$$ 9.00000i 1.31278i 0.754420 + 0.656392i $$0.227918\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −2.00000 −0.285714
$$50$$ − 1.00000i − 0.141421i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −13.0000 −1.78569 −0.892844 0.450367i $$-0.851293\pi$$
−0.892844 + 0.450367i $$0.851293\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 1.00000 0.134840
$$56$$ 3.00000 0.400892
$$57$$ 5.00000i 0.662266i
$$58$$ 0 0
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −10.0000 −1.27000
$$63$$ 3.00000i 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ − 3.00000i − 0.358569i
$$71$$ − 2.00000i − 0.237356i −0.992933 0.118678i $$-0.962134\pi$$
0.992933 0.118678i $$-0.0378657\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 16.0000i 1.87266i 0.351123 + 0.936329i $$0.385800\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 1.00000 0.115470
$$76$$ 5.00000i 0.573539i
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 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 $$0.771155\pi$$
$$84$$ 3.00000i 0.327327i
$$85$$ 0 0
$$86$$ 2.00000i 0.215666i
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ − 1.00000i − 0.106000i −0.998595 0.0529999i $$-0.983122\pi$$
0.998595 0.0529999i $$-0.0168783\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ − 10.0000i − 1.03695i
$$94$$ −9.00000 −0.928279
$$95$$ 5.00000 0.512989
$$96$$ − 1.00000i − 0.102062i
$$97$$ 12.0000i 1.21842i 0.793011 + 0.609208i $$0.208512\pi$$
−0.793011 + 0.609208i $$0.791488\pi$$
$$98$$ − 2.00000i − 0.202031i
$$99$$ − 1.00000i − 0.100504i
$$100$$ 1.00000 0.100000
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ 9.00000 0.886796 0.443398 0.896325i $$-0.353773\pi$$
0.443398 + 0.896325i $$0.353773\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ − 13.0000i − 1.26267i
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 1.00000i 0.0953463i
$$111$$ − 1.00000i − 0.0949158i
$$112$$ 3.00000i 0.283473i
$$113$$ 16.0000 1.50515 0.752577 0.658505i $$-0.228811\pi$$
0.752577 + 0.658505i $$0.228811\pi$$
$$114$$ −5.00000 −0.468293
$$115$$ 4.00000i 0.373002i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ 10.0000 0.909091
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 6.00000i − 0.541002i
$$124$$ − 10.0000i − 0.898027i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ −3.00000 −0.267261
$$127$$ −5.00000 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ − 1.00000i − 0.0870388i
$$133$$ 15.0000 1.30066
$$134$$ 12.0000 1.03664
$$135$$ − 1.00000i − 0.0860663i
$$136$$ 0 0
$$137$$ − 16.0000i − 1.36697i −0.729964 0.683486i $$-0.760463\pi$$
0.729964 0.683486i $$-0.239537\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 3.00000 0.253546
$$141$$ − 9.00000i − 0.757937i
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −16.0000 −1.32417
$$147$$ 2.00000 0.164957
$$148$$ − 1.00000i − 0.0821995i
$$149$$ − 6.00000i − 0.491539i −0.969328 0.245770i $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 6.00000i 0.488273i 0.969741 + 0.244137i $$0.0785045\pi$$
−0.969741 + 0.244137i $$0.921495\pi$$
$$152$$ −5.00000 −0.405554
$$153$$ 0 0
$$154$$ 3.00000i 0.241747i
$$155$$ −10.0000 −0.803219
$$156$$ 0 0
$$157$$ 1.00000 0.0798087 0.0399043 0.999204i $$-0.487295\pi$$
0.0399043 + 0.999204i $$0.487295\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ 13.0000 1.03097
$$160$$ −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
$$165$$ −1.00000 −0.0778499
$$166$$ −12.0000 −0.931381
$$167$$ 13.0000i 1.00597i 0.864295 + 0.502985i $$0.167765\pi$$
−0.864295 + 0.502985i $$0.832235\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 5.00000i − 0.382360i
$$172$$ −2.00000 −0.152499
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 0 0
$$175$$ − 3.00000i − 0.226779i
$$176$$ − 1.00000i − 0.0753778i
$$177$$ 4.00000i 0.300658i
$$178$$ 1.00000 0.0749532
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ − 4.00000i − 0.294884i
$$185$$ −1.00000 −0.0735215
$$186$$ 10.0000 0.733236
$$187$$ 0 0
$$188$$ − 9.00000i − 0.656392i
$$189$$ − 3.00000i − 0.218218i
$$190$$ 5.00000i 0.362738i
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ − 16.0000i − 1.15171i −0.817554 0.575853i $$-0.804670\pi$$
0.817554 0.575853i $$-0.195330\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ − 15.0000i − 1.06871i −0.845262 0.534353i $$-0.820555\pi$$
0.845262 0.534353i $$-0.179445\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 12.0000i 0.846415i
$$202$$ − 4.00000i − 0.281439i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 9.00000i 0.627060i
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ 3.00000i 0.207020i
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 13.0000 0.892844
$$213$$ 2.00000i 0.137038i
$$214$$ − 6.00000i − 0.410152i
$$215$$ 2.00000i 0.136399i
$$216$$ 1.00000i 0.0680414i
$$217$$ −30.0000 −2.03653
$$218$$ 10.0000 0.677285
$$219$$ − 16.0000i − 1.08118i
$$220$$ −1.00000 −0.0674200
$$221$$ 0 0
$$222$$ 1.00000 0.0671156
$$223$$ 11.0000i 0.736614i 0.929704 + 0.368307i $$0.120063\pi$$
−0.929704 + 0.368307i $$0.879937\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ −1.00000 −0.0666667
$$226$$ 16.0000i 1.06430i
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ − 5.00000i − 0.331133i
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 4.00000i 0.260378i
$$237$$ 10.0000 0.649570
$$238$$ 0 0
$$239$$ − 2.00000i − 0.129369i −0.997906 0.0646846i $$-0.979396\pi$$
0.997906 0.0646846i $$-0.0206041\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 15.0000i − 0.966235i −0.875556 0.483117i $$-0.839504\pi$$
0.875556 0.483117i $$-0.160496\pi$$
$$242$$ 10.0000i 0.642824i
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ − 2.00000i − 0.127775i
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 10.0000 0.635001
$$249$$ − 12.0000i − 0.760469i
$$250$$ 1.00000 0.0632456
$$251$$ −11.0000 −0.694314 −0.347157 0.937807i $$-0.612853\pi$$
−0.347157 + 0.937807i $$0.612853\pi$$
$$252$$ − 3.00000i − 0.188982i
$$253$$ − 4.00000i − 0.251478i
$$254$$ − 5.00000i − 0.313728i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 24.0000 1.49708 0.748539 0.663090i $$-0.230755\pi$$
0.748539 + 0.663090i $$0.230755\pi$$
$$258$$ − 2.00000i − 0.124515i
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 15.0000i − 0.926703i
$$263$$ 3.00000 0.184988 0.0924940 0.995713i $$-0.470516\pi$$
0.0924940 + 0.995713i $$0.470516\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ − 13.0000i − 0.798584i
$$266$$ 15.0000i 0.919709i
$$267$$ 1.00000i 0.0611990i
$$268$$ 12.0000i 0.733017i
$$269$$ −20.0000 −1.21942 −0.609711 0.792624i $$-0.708714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 24.0000i 1.45790i 0.684569 + 0.728948i $$0.259990\pi$$
−0.684569 + 0.728948i $$0.740010\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 16.0000 0.966595
$$275$$ 1.00000i 0.0603023i
$$276$$ 4.00000 0.240772
$$277$$ −23.0000 −1.38194 −0.690968 0.722885i $$-0.742815\pi$$
−0.690968 + 0.722885i $$0.742815\pi$$
$$278$$ − 9.00000i − 0.539784i
$$279$$ 10.0000i 0.598684i
$$280$$ 3.00000i 0.179284i
$$281$$ − 10.0000i − 0.596550i −0.954480 0.298275i $$-0.903589\pi$$
0.954480 0.298275i $$-0.0964112\pi$$
$$282$$ 9.00000 0.535942
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 2.00000i 0.118678i
$$285$$ −5.00000 −0.296174
$$286$$ 0 0
$$287$$ −18.0000 −1.06251
$$288$$ 1.00000i 0.0589256i
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ − 12.0000i − 0.703452i
$$292$$ − 16.0000i − 0.936329i
$$293$$ − 13.0000i − 0.759468i −0.925096 0.379734i $$-0.876015\pi$$
0.925096 0.379734i $$-0.123985\pi$$
$$294$$ 2.00000i 0.116642i
$$295$$ 4.00000 0.232889
$$296$$ 1.00000 0.0581238
$$297$$ 1.00000i 0.0580259i
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 6.00000i 0.345834i
$$302$$ −6.00000 −0.345261
$$303$$ 4.00000 0.229794
$$304$$ − 5.00000i − 0.286770i
$$305$$ − 2.00000i − 0.114520i
$$306$$ 0 0
$$307$$ − 18.0000i − 1.02731i −0.857996 0.513657i $$-0.828290\pi$$
0.857996 0.513657i $$-0.171710\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ −9.00000 −0.511992
$$310$$ − 10.0000i − 0.567962i
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 1.00000i 0.0564333i
$$315$$ −3.00000 −0.169031
$$316$$ 10.0000 0.562544
$$317$$ 19.0000i 1.06715i 0.845754 + 0.533573i $$0.179151\pi$$
−0.845754 + 0.533573i $$0.820849\pi$$
$$318$$ 13.0000i 0.729004i
$$319$$ 0 0
$$320$$ − 1.00000i − 0.0559017i
$$321$$ 6.00000 0.334887
$$322$$ −12.0000 −0.668734
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 10.0000i 0.553001i
$$328$$ 6.00000 0.331295
$$329$$ −27.0000 −1.48856
$$330$$ − 1.00000i − 0.0550482i
$$331$$ 28.0000i 1.53902i 0.638635 + 0.769510i $$0.279499\pi$$
−0.638635 + 0.769510i $$0.720501\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 1.00000i 0.0547997i
$$334$$ −13.0000 −0.711328
$$335$$ 12.0000 0.655630
$$336$$ − 3.00000i − 0.163663i
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 5.00000 0.270369
$$343$$ 15.0000i 0.809924i
$$344$$ − 2.00000i − 0.107833i
$$345$$ − 4.00000i − 0.215353i
$$346$$ 9.00000i 0.483843i
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 0 0
$$349$$ − 36.0000i − 1.92704i −0.267644 0.963518i $$-0.586245\pi$$
0.267644 0.963518i $$-0.413755\pi$$
$$350$$ 3.00000 0.160357
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ − 36.0000i − 1.91609i −0.286623 0.958043i $$-0.592533\pi$$
0.286623 0.958043i $$-0.407467\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 2.00000 0.106149
$$356$$ 1.00000i 0.0529999i
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ − 34.0000i − 1.79445i −0.441572 0.897226i $$-0.645579\pi$$
0.441572 0.897226i $$-0.354421\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −6.00000 −0.315789
$$362$$ − 6.00000i − 0.315353i
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ −16.0000 −0.837478
$$366$$ 2.00000i 0.104542i
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 6.00000i 0.312348i
$$370$$ − 1.00000i − 0.0519875i
$$371$$ − 39.0000i − 2.02478i
$$372$$ 10.0000i 0.518476i
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ 1.00000i 0.0516398i
$$376$$ 9.00000 0.464140
$$377$$ 0 0
$$378$$ 3.00000 0.154303
$$379$$ − 1.00000i − 0.0513665i −0.999670 0.0256833i $$-0.991824\pi$$
0.999670 0.0256833i $$-0.00817614\pi$$
$$380$$ −5.00000 −0.256495
$$381$$ 5.00000 0.256158
$$382$$ − 18.0000i − 0.920960i
$$383$$ − 28.0000i − 1.43073i −0.698749 0.715367i $$-0.746260\pi$$
0.698749 0.715367i $$-0.253740\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 3.00000i 0.152894i
$$386$$ 16.0000 0.814379
$$387$$ 2.00000 0.101666
$$388$$ − 12.0000i − 0.609208i
$$389$$ 16.0000 0.811232 0.405616 0.914044i $$-0.367057\pi$$
0.405616 + 0.914044i $$0.367057\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 2.00000i 0.101015i
$$393$$ 15.0000 0.756650
$$394$$ 15.0000 0.755689
$$395$$ − 10.0000i − 0.503155i
$$396$$ 1.00000i 0.0502519i
$$397$$ 33.0000i 1.65622i 0.560564 + 0.828111i $$0.310584\pi$$
−0.560564 + 0.828111i $$0.689416\pi$$
$$398$$ 2.00000i 0.100251i
$$399$$ −15.0000 −0.750939
$$400$$ −1.00000 −0.0500000
$$401$$ − 25.0000i − 1.24844i −0.781248 0.624220i $$-0.785417\pi$$
0.781248 0.624220i $$-0.214583\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ 0 0
$$404$$ 4.00000 0.199007
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 1.00000 0.0495682
$$408$$ 0 0
$$409$$ 17.0000i 0.840596i 0.907386 + 0.420298i $$0.138074\pi$$
−0.907386 + 0.420298i $$0.861926\pi$$
$$410$$ − 6.00000i − 0.296319i
$$411$$ 16.0000i 0.789222i
$$412$$ −9.00000 −0.443398
$$413$$ 12.0000 0.590481
$$414$$ 4.00000i 0.196589i
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 9.00000 0.440732
$$418$$ − 5.00000i − 0.244558i
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ −3.00000 −0.146385
$$421$$ 20.0000i 0.974740i 0.873195 + 0.487370i $$0.162044\pi$$
−0.873195 + 0.487370i $$0.837956\pi$$
$$422$$ − 15.0000i − 0.730189i
$$423$$ 9.00000i 0.437595i
$$424$$ 13.0000i 0.631336i
$$425$$ 0 0
$$426$$ −2.00000 −0.0969003
$$427$$ − 6.00000i − 0.290360i
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ 36.0000i 1.73406i 0.498257 + 0.867029i $$0.333974\pi$$
−0.498257 + 0.867029i $$0.666026\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ − 30.0000i − 1.44005i
$$435$$ 0 0
$$436$$ 10.0000i 0.478913i
$$437$$ − 20.0000i − 0.956730i
$$438$$ 16.0000 0.764510
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ − 1.00000i − 0.0476731i
$$441$$ −2.00000 −0.0952381
$$442$$ 0 0
$$443$$ −18.0000 −0.855206 −0.427603 0.903967i $$-0.640642\pi$$
−0.427603 + 0.903967i $$0.640642\pi$$
$$444$$ 1.00000i 0.0474579i
$$445$$ 1.00000 0.0474045
$$446$$ −11.0000 −0.520865
$$447$$ 6.00000i 0.283790i
$$448$$ − 3.00000i − 0.141737i
$$449$$ 15.0000i 0.707894i 0.935266 + 0.353947i $$0.115161\pi$$
−0.935266 + 0.353947i $$0.884839\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 6.00000 0.282529
$$452$$ −16.0000 −0.752577
$$453$$ − 6.00000i − 0.281905i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 0 0
$$460$$ − 4.00000i − 0.186501i
$$461$$ − 12.0000i − 0.558896i −0.960161 0.279448i $$-0.909849\pi$$
0.960161 0.279448i $$-0.0901514\pi$$
$$462$$ − 3.00000i − 0.139573i
$$463$$ 16.0000i 0.743583i 0.928316 + 0.371792i $$0.121256\pi$$
−0.928316 + 0.371792i $$0.878744\pi$$
$$464$$ 0 0
$$465$$ 10.0000 0.463739
$$466$$ − 10.0000i − 0.463241i
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 36.0000 1.66233
$$470$$ − 9.00000i − 0.415139i
$$471$$ −1.00000 −0.0460776
$$472$$ −4.00000 −0.184115
$$473$$ − 2.00000i − 0.0919601i
$$474$$ 10.0000i 0.459315i
$$475$$ 5.00000i 0.229416i
$$476$$ 0 0
$$477$$ −13.0000 −0.595229
$$478$$ 2.00000 0.0914779
$$479$$ 8.00000i 0.365529i 0.983157 + 0.182765i $$0.0585046\pi$$
−0.983157 + 0.182765i $$0.941495\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 15.0000 0.683231
$$483$$ − 12.0000i − 0.546019i
$$484$$ −10.0000 −0.454545
$$485$$ −12.0000 −0.544892
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 35.0000i 1.58600i 0.609221 + 0.793001i $$0.291482\pi$$
−0.609221 + 0.793001i $$0.708518\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ − 20.0000i − 0.904431i
$$490$$ 2.00000 0.0903508
$$491$$ −25.0000 −1.12823 −0.564117 0.825695i $$-0.690783\pi$$
−0.564117 + 0.825695i $$0.690783\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 1.00000 0.0449467
$$496$$ 10.0000i 0.449013i
$$497$$ 6.00000 0.269137
$$498$$ 12.0000 0.537733
$$499$$ 20.0000i 0.895323i 0.894203 + 0.447661i $$0.147743\pi$$
−0.894203 + 0.447661i $$0.852257\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ − 13.0000i − 0.580797i
$$502$$ − 11.0000i − 0.490954i
$$503$$ −1.00000 −0.0445878 −0.0222939 0.999751i $$-0.507097\pi$$
−0.0222939 + 0.999751i $$0.507097\pi$$
$$504$$ 3.00000 0.133631
$$505$$ − 4.00000i − 0.177998i
$$506$$ 4.00000 0.177822
$$507$$ 0 0
$$508$$ 5.00000 0.221839
$$509$$ 18.0000i 0.797836i 0.916987 + 0.398918i $$0.130614\pi$$
−0.916987 + 0.398918i $$0.869386\pi$$
$$510$$ 0 0
$$511$$ −48.0000 −2.12339
$$512$$ 1.00000i 0.0441942i
$$513$$ 5.00000i 0.220755i
$$514$$ 24.0000i 1.05859i
$$515$$ 9.00000i 0.396587i
$$516$$ 2.00000 0.0880451
$$517$$ 9.00000 0.395820
$$518$$ − 3.00000i − 0.131812i
$$519$$ −9.00000 −0.395056
$$520$$ 0 0
$$521$$ 33.0000 1.44576 0.722878 0.690976i $$-0.242819\pi$$
0.722878 + 0.690976i $$0.242819\pi$$
$$522$$ 0 0
$$523$$ 6.00000 0.262362 0.131181 0.991358i $$-0.458123\pi$$
0.131181 + 0.991358i $$0.458123\pi$$
$$524$$ 15.0000 0.655278
$$525$$ 3.00000i 0.130931i
$$526$$ 3.00000i 0.130806i
$$527$$ 0 0
$$528$$ 1.00000i 0.0435194i
$$529$$ −7.00000 −0.304348
$$530$$ 13.0000 0.564684
$$531$$ − 4.00000i − 0.173585i
$$532$$ −15.0000 −0.650332
$$533$$ 0 0
$$534$$ −1.00000 −0.0432742
$$535$$ − 6.00000i − 0.259403i
$$536$$ −12.0000 −0.518321
$$537$$ 12.0000 0.517838
$$538$$ − 20.0000i − 0.862261i
$$539$$ 2.00000i 0.0861461i
$$540$$ 1.00000i 0.0430331i
$$541$$ 2.00000i 0.0859867i 0.999075 + 0.0429934i $$0.0136894\pi$$
−0.999075 + 0.0429934i $$0.986311\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 6.00000 0.257485
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 34.0000 1.45374 0.726868 0.686778i $$-0.240975\pi$$
0.726868 + 0.686778i $$0.240975\pi$$
$$548$$ 16.0000i 0.683486i
$$549$$ −2.00000 −0.0853579
$$550$$ −1.00000 −0.0426401
$$551$$ 0 0
$$552$$ 4.00000i 0.170251i
$$553$$ − 30.0000i − 1.27573i
$$554$$ − 23.0000i − 0.977176i
$$555$$ 1.00000 0.0424476
$$556$$ 9.00000 0.381685
$$557$$ 3.00000i 0.127114i 0.997978 + 0.0635570i $$0.0202445\pi$$
−0.997978 + 0.0635570i $$0.979756\pi$$
$$558$$ −10.0000 −0.423334
$$559$$ 0 0
$$560$$ −3.00000 −0.126773
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 9.00000i 0.378968i
$$565$$ 16.0000i 0.673125i
$$566$$ − 2.00000i − 0.0840663i
$$567$$ 3.00000i 0.125988i
$$568$$ −2.00000 −0.0839181
$$569$$ 11.0000 0.461144 0.230572 0.973055i $$-0.425940\pi$$
0.230572 + 0.973055i $$0.425940\pi$$
$$570$$ − 5.00000i − 0.209427i
$$571$$ −7.00000 −0.292941 −0.146470 0.989215i $$-0.546791\pi$$
−0.146470 + 0.989215i $$0.546791\pi$$
$$572$$ 0 0
$$573$$ 18.0000 0.751961
$$574$$ − 18.0000i − 0.751305i
$$575$$ −4.00000 −0.166812
$$576$$ −1.00000 −0.0416667
$$577$$ 18.0000i 0.749350i 0.927156 + 0.374675i $$0.122246\pi$$
−0.927156 + 0.374675i $$0.877754\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ 16.0000i 0.664937i
$$580$$ 0 0
$$581$$ −36.0000 −1.49353
$$582$$ 12.0000 0.497416
$$583$$ 13.0000i 0.538405i
$$584$$ 16.0000 0.662085
$$585$$ 0 0
$$586$$ 13.0000 0.537025
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 50.0000 2.06021
$$590$$ 4.00000i 0.164677i
$$591$$ 15.0000i 0.617018i
$$592$$ 1.00000i 0.0410997i
$$593$$ − 28.0000i − 1.14982i −0.818216 0.574911i $$-0.805037\pi$$
0.818216 0.574911i $$-0.194963\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 6.00000i 0.245770i
$$597$$ −2.00000 −0.0818546
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −27.0000 −1.10135 −0.550676 0.834719i $$-0.685630\pi$$
−0.550676 + 0.834719i $$0.685630\pi$$
$$602$$ −6.00000 −0.244542
$$603$$ − 12.0000i − 0.488678i
$$604$$ − 6.00000i − 0.244137i
$$605$$ 10.0000i 0.406558i
$$606$$ 4.00000i 0.162489i
$$607$$ 37.0000 1.50178 0.750892 0.660425i $$-0.229624\pi$$
0.750892 + 0.660425i $$0.229624\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 23.0000i 0.928961i 0.885583 + 0.464481i $$0.153759\pi$$
−0.885583 + 0.464481i $$0.846241\pi$$
$$614$$ 18.0000 0.726421
$$615$$ 6.00000 0.241943
$$616$$ − 3.00000i − 0.120873i
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ − 9.00000i − 0.362033i
$$619$$ 31.0000i 1.24600i 0.782224 + 0.622998i $$0.214085\pi$$
−0.782224 + 0.622998i $$0.785915\pi$$
$$620$$ 10.0000 0.401610
$$621$$ −4.00000 −0.160514
$$622$$ 12.0000i 0.481156i
$$623$$ 3.00000 0.120192
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 34.0000i − 1.35891i
$$627$$ 5.00000 0.199681
$$628$$ −1.00000 −0.0399043
$$629$$ 0 0
$$630$$ − 3.00000i − 0.119523i
$$631$$ 4.00000i 0.159237i 0.996825 + 0.0796187i $$0.0253703\pi$$
−0.996825 + 0.0796187i $$0.974630\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ 15.0000 0.596196
$$634$$ −19.0000 −0.754586
$$635$$ − 5.00000i − 0.198419i
$$636$$ −13.0000 −0.515484
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 2.00000i − 0.0791188i
$$640$$ 1.00000 0.0395285
$$641$$ 9.00000 0.355479 0.177739 0.984078i $$-0.443122\pi$$
0.177739 + 0.984078i $$0.443122\pi$$
$$642$$ 6.00000i 0.236801i
$$643$$ − 44.0000i − 1.73519i −0.497271 0.867595i $$-0.665665\pi$$
0.497271 0.867595i $$-0.334335\pi$$
$$644$$ − 12.0000i − 0.472866i
$$645$$ − 2.00000i − 0.0787499i
$$646$$ 0 0
$$647$$ −9.00000 −0.353827 −0.176913 0.984226i $$-0.556611\pi$$
−0.176913 + 0.984226i $$0.556611\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −4.00000 −0.157014
$$650$$ 0 0
$$651$$ 30.0000 1.17579
$$652$$ − 20.0000i − 0.783260i
$$653$$ 41.0000 1.60445 0.802227 0.597019i $$-0.203648\pi$$
0.802227 + 0.597019i $$0.203648\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ − 15.0000i − 0.586098i
$$656$$ 6.00000i 0.234261i
$$657$$ 16.0000i 0.624219i
$$658$$ − 27.0000i − 1.05257i
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ − 14.0000i − 0.544537i −0.962221 0.272268i $$-0.912226\pi$$
0.962221 0.272268i $$-0.0877739\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 15.0000i 0.581675i
$$666$$ −1.00000 −0.0387492
$$667$$ 0 0
$$668$$ − 13.0000i − 0.502985i
$$669$$ − 11.0000i − 0.425285i
$$670$$ 12.0000i 0.463600i
$$671$$ 2.00000i 0.0772091i
$$672$$ 3.00000 0.115728
$$673$$ 32.0000 1.23351 0.616755 0.787155i $$-0.288447\pi$$
0.616755 + 0.787155i $$0.288447\pi$$
$$674$$ − 2.00000i − 0.0770371i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ − 16.0000i − 0.614476i
$$679$$ −36.0000 −1.38155
$$680$$ 0 0
$$681$$ 20.0000i 0.766402i
$$682$$ 10.0000i 0.382920i
$$683$$ 26.0000i 0.994862i 0.867503 + 0.497431i $$0.165723\pi$$
−0.867503 + 0.497431i $$0.834277\pi$$
$$684$$ 5.00000i 0.191180i
$$685$$ 16.0000 0.611329
$$686$$ −15.0000 −0.572703
$$687$$ − 10.0000i − 0.381524i
$$688$$ 2.00000 0.0762493
$$689$$ 0 0
$$690$$ 4.00000 0.152277
$$691$$ 33.0000i 1.25538i 0.778464 + 0.627690i $$0.215999\pi$$
−0.778464 + 0.627690i $$0.784001\pi$$
$$692$$ −9.00000 −0.342129
$$693$$ 3.00000 0.113961
$$694$$ − 28.0000i − 1.06287i
$$695$$ − 9.00000i − 0.341389i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 36.0000 1.36262
$$699$$ 10.0000 0.378235
$$700$$ 3.00000i 0.113389i
$$701$$ −4.00000 −0.151078 −0.0755390 0.997143i $$-0.524068\pi$$
−0.0755390 + 0.997143i $$0.524068\pi$$
$$702$$ 0 0
$$703$$ 5.00000 0.188579
$$704$$ 1.00000i 0.0376889i
$$705$$ 9.00000 0.338960
$$706$$ 36.0000 1.35488
$$707$$ − 12.0000i − 0.451306i
$$708$$ − 4.00000i − 0.150329i
$$709$$ − 4.00000i − 0.150223i −0.997175 0.0751116i $$-0.976069\pi$$
0.997175 0.0751116i $$-0.0239313\pi$$
$$710$$ 2.00000i 0.0750587i
$$711$$ −10.0000 −0.375029
$$712$$ −1.00000 −0.0374766
$$713$$ 40.0000i 1.49801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 2.00000i 0.0746914i
$$718$$ 34.0000 1.26887
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ 27.0000i 1.00553i
$$722$$ − 6.00000i − 0.223297i
$$723$$ 15.0000i 0.557856i
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ − 10.0000i − 0.371135i
$$727$$ 37.0000 1.37225 0.686127 0.727482i $$-0.259309\pi$$
0.686127 + 0.727482i $$0.259309\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 16.0000i − 0.592187i
$$731$$ 0 0
$$732$$ −2.00000 −0.0739221
$$733$$ − 5.00000i − 0.184679i −0.995728 0.0923396i $$-0.970565\pi$$
0.995728 0.0923396i $$-0.0294345\pi$$
$$734$$ 0 0
$$735$$ 2.00000i 0.0737711i
$$736$$ 4.00000i 0.147442i
$$737$$ −12.0000 −0.442026
$$738$$ −6.00000 −0.220863
$$739$$ − 35.0000i − 1.28750i −0.765238 0.643748i $$-0.777379\pi$$
0.765238 0.643748i $$-0.222621\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ 0 0
$$742$$ 39.0000 1.43174
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ −10.0000 −0.366618
$$745$$ 6.00000 0.219823
$$746$$ 10.0000i 0.366126i
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ − 18.0000i − 0.657706i
$$750$$ −1.00000 −0.0365148
$$751$$ −44.0000 −1.60558 −0.802791 0.596260i $$-0.796653\pi$$
−0.802791 + 0.596260i $$0.796653\pi$$
$$752$$ 9.00000i 0.328196i
$$753$$ 11.0000 0.400862
$$754$$ 0 0
$$755$$ −6.00000 −0.218362
$$756$$ 3.00000i 0.109109i
$$757$$ 39.0000 1.41748 0.708740 0.705470i $$-0.249264\pi$$
0.708740 + 0.705470i $$0.249264\pi$$
$$758$$ 1.00000 0.0363216
$$759$$ 4.00000i 0.145191i
$$760$$ − 5.00000i − 0.181369i
$$761$$ − 45.0000i − 1.63125i −0.578582 0.815624i $$-0.696394\pi$$
0.578582 0.815624i $$-0.303606\pi$$
$$762$$ 5.00000i 0.181131i
$$763$$ 30.0000 1.08607
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 28.0000 1.01168
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ − 26.0000i − 0.937584i −0.883309 0.468792i $$-0.844689\pi$$
0.883309 0.468792i $$-0.155311\pi$$
$$770$$ −3.00000 −0.108112
$$771$$ −24.0000 −0.864339
$$772$$ 16.0000i 0.575853i
$$773$$ − 37.0000i − 1.33080i −0.746488 0.665399i $$-0.768262\pi$$
0.746488 0.665399i $$-0.231738\pi$$
$$774$$ 2.00000i 0.0718885i
$$775$$ − 10.0000i − 0.359211i
$$776$$ 12.0000 0.430775
$$777$$ 3.00000 0.107624
$$778$$ 16.0000i 0.573628i
$$779$$ 30.0000 1.07486
$$780$$ 0 0
$$781$$ −2.00000 −0.0715656
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −2.00000 −0.0714286
$$785$$ 1.00000i 0.0356915i
$$786$$ 15.0000i 0.535032i
$$787$$ − 16.0000i − 0.570338i −0.958477 0.285169i $$-0.907950\pi$$
0.958477 0.285169i $$-0.0920498\pi$$
$$788$$ 15.0000i 0.534353i
$$789$$ −3.00000 −0.106803
$$790$$ 10.0000 0.355784
$$791$$ 48.0000i 1.70668i
$$792$$ −1.00000 −0.0355335
$$793$$ 0 0
$$794$$ −33.0000 −1.17113
$$795$$ 13.0000i 0.461062i
$$796$$ −2.00000 −0.0708881
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ − 15.0000i − 0.530994i
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 1.00000i − 0.0353333i
$$802$$ 25.0000 0.882781
$$803$$ 16.0000 0.564628
$$804$$ − 12.0000i − 0.423207i
$$805$$ −12.0000 −0.422944
$$806$$ 0 0
$$807$$ 20.0000 0.704033
$$808$$ 4.00000i 0.140720i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 33.0000i 1.15879i 0.815048 + 0.579393i $$0.196710\pi$$
−0.815048 + 0.579393i $$0.803290\pi$$
$$812$$ 0 0
$$813$$ − 24.0000i − 0.841717i
$$814$$ 1.00000i 0.0350500i
$$815$$ −20.0000 −0.700569
$$816$$ 0 0
$$817$$ − 10.0000i − 0.349856i
$$818$$ −17.0000 −0.594391
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 18.0000i 0.628204i 0.949389 + 0.314102i $$0.101703\pi$$
−0.949389 + 0.314102i $$0.898297\pi$$
$$822$$ −16.0000 −0.558064
$$823$$ 5.00000 0.174289 0.0871445 0.996196i $$-0.472226\pi$$
0.0871445 + 0.996196i $$0.472226\pi$$
$$824$$ − 9.00000i − 0.313530i
$$825$$ − 1.00000i − 0.0348155i
$$826$$ 12.0000i 0.417533i
$$827$$ 30.0000i 1.04320i 0.853189 + 0.521601i $$0.174665\pi$$
−0.853189 + 0.521601i $$0.825335\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 44.0000 1.52818 0.764092 0.645108i $$-0.223188\pi$$
0.764092 + 0.645108i $$0.223188\pi$$
$$830$$ − 12.0000i − 0.416526i
$$831$$ 23.0000 0.797861
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 9.00000i 0.311645i
$$835$$ −13.0000 −0.449884
$$836$$ 5.00000 0.172929
$$837$$ − 10.0000i − 0.345651i
$$838$$ − 28.0000i − 0.967244i
$$839$$ 54.0000i 1.86429i 0.362089 + 0.932144i $$0.382064\pi$$
−0.362089 + 0.932144i $$0.617936\pi$$
$$840$$ − 3.00000i − 0.103510i
$$841$$ −29.0000 −1.00000
$$842$$ −20.0000 −0.689246
$$843$$ 10.0000i 0.344418i
$$844$$ 15.0000 0.516321
$$845$$ 0 0
$$846$$ −9.00000 −0.309426
$$847$$ 30.0000i 1.03081i
$$848$$ −13.0000 −0.446422
$$849$$ 2.00000 0.0686398
$$850$$ 0 0
$$851$$ 4.00000i 0.137118i
$$852$$ − 2.00000i − 0.0685189i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 5.00000 0.170996
$$856$$ 6.00000i 0.205076i
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 19.0000 0.648272 0.324136 0.946011i $$-0.394927\pi$$
0.324136 + 0.946011i $$0.394927\pi$$
$$860$$ − 2.00000i − 0.0681994i
$$861$$ 18.0000 0.613438
$$862$$ −36.0000 −1.22616
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 9.00000i 0.306009i
$$866$$ − 16.0000i − 0.543702i
$$867$$ 17.0000 0.577350
$$868$$ 30.0000 1.01827
$$869$$ 10.0000i 0.339227i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −10.0000 −0.338643
$$873$$ 12.0000i 0.406138i
$$874$$ 20.0000 0.676510
$$875$$ 3.00000 0.101419
$$876$$ 16.0000i 0.540590i
$$877$$ 38.0000i 1.28317i 0.767052 + 0.641584i $$0.221723\pi$$
−0.767052 + 0.641584i $$0.778277\pi$$
$$878$$ 10.0000i 0.337484i
$$879$$ 13.0000i 0.438479i
$$880$$ 1.00000 0.0337100
$$881$$ −5.00000 −0.168454 −0.0842271 0.996447i $$-0.526842\pi$$
−0.0842271 + 0.996447i $$0.526842\pi$$
$$882$$ − 2.00000i − 0.0673435i
$$883$$ −42.0000 −1.41341 −0.706706 0.707507i $$-0.749820\pi$$
−0.706706 + 0.707507i $$0.749820\pi$$
$$884$$ 0 0
$$885$$ −4.00000 −0.134459
$$886$$ − 18.0000i − 0.604722i
$$887$$ −13.0000 −0.436497 −0.218249 0.975893i $$-0.570034\pi$$
−0.218249 + 0.975893i $$0.570034\pi$$
$$888$$ −1.00000 −0.0335578
$$889$$ − 15.0000i − 0.503084i
$$890$$ 1.00000i 0.0335201i
$$891$$ − 1.00000i − 0.0335013i
$$892$$ − 11.0000i − 0.368307i
$$893$$ 45.0000 1.50587
$$894$$ −6.00000 −0.200670
$$895$$ − 12.0000i − 0.401116i
$$896$$ 3.00000 0.100223
$$897$$ 0 0
$$898$$ −15.0000 −0.500556
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 6.00000i 0.199778i
$$903$$ − 6.00000i − 0.199667i
$$904$$ − 16.0000i − 0.532152i
$$905$$ − 6.00000i − 0.199447i
$$906$$ 6.00000 0.199337
$$907$$ −42.0000 −1.39459 −0.697294 0.716786i $$-0.745613\pi$$
−0.697294 + 0.716786i $$0.745613\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ −4.00000 −0.132672
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 5.00000i 0.165567i
$$913$$ 12.0000 0.397142
$$914$$ −22.0000 −0.727695
$$915$$ 2.00000i 0.0661180i
$$916$$ − 10.0000i − 0.330409i
$$917$$ − 45.0000i − 1.48603i
$$918$$ 0 0
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 18.0000i 0.593120i
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ 3.00000 0.0986928
$$925$$ − 1.00000i − 0.0328798i
$$926$$ −16.0000 −0.525793
$$927$$ 9.00000 0.295599
$$928$$ 0 0
$$929$$ 34.0000i 1.11550i 0.830008 + 0.557752i $$0.188336\pi$$
−0.830008 + 0.557752i $$0.811664\pi$$
$$930$$ 10.0000i 0.327913i
$$931$$ 10.0000i 0.327737i
$$932$$ 10.0000 0.327561
$$933$$ −12.0000 −0.392862
$$934$$ 36.0000i 1.17796i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 36.0000i 1.17544i
$$939$$ 34.0000 1.10955
$$940$$ 9.00000 0.293548
$$941$$ 24.0000i 0.782378i 0.920310 + 0.391189i $$0.127936\pi$$
−0.920310 + 0.391189i $$0.872064\pi$$
$$942$$ − 1.00000i − 0.0325818i
$$943$$ 24.0000i 0.781548i
$$944$$ − 4.00000i − 0.130189i
$$945$$ 3.00000 0.0975900
$$946$$ 2.00000 0.0650256
$$947$$ 38.0000i 1.23483i 0.786636 + 0.617417i $$0.211821\pi$$
−0.786636 + 0.617417i $$0.788179\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 0 0
$$950$$ −5.00000 −0.162221
$$951$$ − 19.0000i − 0.616117i
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ − 13.0000i − 0.420891i
$$955$$ − 18.0000i − 0.582466i
$$956$$ 2.00000i 0.0646846i
$$957$$ 0 0
$$958$$ −8.00000 −0.258468
$$959$$ 48.0000 1.55000
$$960$$ 1.00000i 0.0322749i
$$961$$ −69.0000 −2.22581
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 15.0000i 0.483117i
$$965$$ 16.0000 0.515058
$$966$$ 12.0000 0.386094
$$967$$ 7.00000i 0.225105i 0.993646 + 0.112552i $$0.0359026\pi$$
−0.993646 + 0.112552i $$0.964097\pi$$
$$968$$ − 10.0000i − 0.321412i
$$969$$ 0 0
$$970$$ − 12.0000i − 0.385297i
$$971$$ 27.0000 0.866471 0.433236 0.901281i $$-0.357372\pi$$
0.433236 + 0.901281i $$0.357372\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ − 27.0000i − 0.865580i
$$974$$ −35.0000 −1.12147
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 48.0000i 1.53566i 0.640656 + 0.767828i $$0.278662\pi$$
−0.640656 + 0.767828i $$0.721338\pi$$
$$978$$ 20.0000 0.639529
$$979$$ −1.00000 −0.0319601
$$980$$ 2.00000i 0.0638877i
$$981$$ − 10.0000i − 0.319275i
$$982$$ − 25.0000i − 0.797782i
$$983$$ − 53.0000i − 1.69044i −0.534421 0.845219i $$-0.679470\pi$$
0.534421 0.845219i $$-0.320530\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 15.0000 0.477940
$$986$$ 0 0
$$987$$ 27.0000 0.859419
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 1.00000i 0.0317821i
$$991$$ 38.0000 1.20711 0.603555 0.797321i $$-0.293750\pi$$
0.603555 + 0.797321i $$0.293750\pi$$
$$992$$ −10.0000 −0.317500
$$993$$ − 28.0000i − 0.888553i
$$994$$ 6.00000i 0.190308i
$$995$$ 2.00000i 0.0634043i
$$996$$ 12.0000i 0.380235i
$$997$$ −7.00000 −0.221692 −0.110846 0.993838i $$-0.535356\pi$$
−0.110846 + 0.993838i $$0.535356\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ − 1.00000i − 0.0316386i
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.a.1351.2 2
13.2 odd 12 390.2.i.b.61.1 2
13.5 odd 4 5070.2.a.q.1.1 1
13.6 odd 12 390.2.i.b.211.1 yes 2
13.8 odd 4 5070.2.a.c.1.1 1
13.12 even 2 inner 5070.2.b.a.1351.1 2
39.2 even 12 1170.2.i.j.451.1 2
39.32 even 12 1170.2.i.j.991.1 2
65.2 even 12 1950.2.z.i.1699.1 4
65.19 odd 12 1950.2.i.o.601.1 2
65.28 even 12 1950.2.z.i.1699.2 4
65.32 even 12 1950.2.z.i.1849.2 4
65.54 odd 12 1950.2.i.o.451.1 2
65.58 even 12 1950.2.z.i.1849.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.b.61.1 2 13.2 odd 12
390.2.i.b.211.1 yes 2 13.6 odd 12
1170.2.i.j.451.1 2 39.2 even 12
1170.2.i.j.991.1 2 39.32 even 12
1950.2.i.o.451.1 2 65.54 odd 12
1950.2.i.o.601.1 2 65.19 odd 12
1950.2.z.i.1699.1 4 65.2 even 12
1950.2.z.i.1699.2 4 65.28 even 12
1950.2.z.i.1849.1 4 65.58 even 12
1950.2.z.i.1849.2 4 65.32 even 12
5070.2.a.c.1.1 1 13.8 odd 4
5070.2.a.q.1.1 1 13.5 odd 4
5070.2.b.a.1351.1 2 13.12 even 2 inner
5070.2.b.a.1351.2 2 1.1 even 1 trivial