# Properties

 Label 4650.2.d.p.3349.1 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ 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 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.p.3349.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -2.00000 q^{21} +4.00000i q^{22} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{58} -6.00000 q^{59} +10.0000 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -10.0000i q^{67} +6.00000i q^{68} -14.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +4.00000 q^{74} -8.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} +8.00000 q^{86} -4.00000i q^{87} -4.00000i q^{88} +8.00000 q^{89} -8.00000 q^{91} +1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 8 q^{29} + 2 q^{31} - 12 q^{34} + 2 q^{36} - 8 q^{39} - 12 q^{41} + 8 q^{44} + 6 q^{49} + 12 q^{51} - 2 q^{54} - 4 q^{56} - 12 q^{59} + 20 q^{61} - 2 q^{64} - 8 q^{66} - 28 q^{71} + 8 q^{74} + 16 q^{79} + 2 q^{81} + 4 q^{84} + 16 q^{86} + 16 q^{89} - 16 q^{91} - 24 q^{94} + 2 q^{96} + 8 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 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
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 4.00000i 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 4.00000i − 0.554700i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 4.00000i 0.525226i
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 10.0000i − 1.22169i −0.791748 0.610847i $$-0.790829\pi$$
0.791748 0.610847i $$-0.209171\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 8.00000i − 0.911685i
$$78$$ 4.00000i 0.452911i
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ − 4.00000i − 0.428845i
$$88$$ − 4.00000i − 0.426401i
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ 1.00000i 0.103695i
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 2.00000i 0.188982i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ − 4.00000i − 0.369800i
$$118$$ 6.00000i 0.552345i
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 14.0000i 1.17485i
$$143$$ − 16.0000i − 1.33799i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 3.00000i 0.247436i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 6.00000i 0.485071i
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 8.00000i − 0.609994i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ − 6.00000i − 0.450988i
$$178$$ − 8.00000i − 0.599625i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ 10.0000i 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 24.0000i 1.75505i
$$188$$ 12.0000i 0.875190i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 10.0000 0.705346
$$202$$ − 14.0000i − 0.985037i
$$203$$ − 8.00000i − 0.561490i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 4.00000i 0.277350i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ − 14.0000i − 0.959264i
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 2.00000i 0.135769i
$$218$$ 18.0000i 1.21911i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 4.00000i 0.268462i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 0 0
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ − 4.00000i − 0.262613i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 8.00000i 0.519656i
$$238$$ − 12.0000i − 0.777844i
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ 1.00000i 0.0635001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ − 14.0000i − 0.864923i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.00000i 0.489592i
$$268$$ 10.0000i 0.610847i
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ − 8.00000i − 0.484182i
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 8.00000i − 0.480673i −0.970690 0.240337i $$-0.922742\pi$$
0.970690 0.240337i $$-0.0772579\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 14.0000i − 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ 14.0000 0.830747
$$285$$ 0 0
$$286$$ −16.0000 −0.946100
$$287$$ − 12.0000i − 0.708338i
$$288$$ 1.00000i 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 4.00000i 0.234082i
$$293$$ − 26.0000i − 1.51894i −0.650545 0.759468i $$-0.725459\pi$$
0.650545 0.759468i $$-0.274541\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ 4.00000i 0.232104i
$$298$$ 14.0000i 0.810998i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 16.0000i 0.920697i
$$303$$ 14.0000i 0.804279i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 8.00000i 0.455842i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 4.00000i 0.226093i 0.993590 + 0.113047i $$0.0360610\pi$$
−0.993590 + 0.113047i $$0.963939\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 26.0000i 1.46031i 0.683284 + 0.730153i $$0.260551\pi$$
−0.683284 + 0.730153i $$0.739449\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 16.0000 0.895828
$$320$$ 0 0
$$321$$ 16.0000 0.893033
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ − 18.0000i − 0.995402i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 4.00000i − 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 4.00000i 0.213201i
$$353$$ − 10.0000i − 0.532246i −0.963939 0.266123i $$-0.914257\pi$$
0.963939 0.266123i $$-0.0857428\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ 0 0
$$356$$ −8.00000 −0.423999
$$357$$ 12.0000i 0.635107i
$$358$$ − 4.00000i − 0.211407i
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 18.0000i − 0.946059i
$$363$$ 5.00000i 0.262432i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ − 1.00000i − 0.0518476i
$$373$$ − 18.0000i − 0.932005i −0.884783 0.466002i $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ − 16.0000i − 0.824042i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 10.0000i 0.511645i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ − 8.00000i − 0.406663i
$$388$$ 2.00000i 0.101535i
$$389$$ −36.0000 −1.82527 −0.912636 0.408773i $$-0.865957\pi$$
−0.912636 + 0.408773i $$0.865957\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 3.00000i 0.151523i
$$393$$ 14.0000i 0.706207i
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 18.0000i − 0.903394i −0.892171 0.451697i $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −4.00000 −0.199750 −0.0998752 0.995000i $$-0.531844\pi$$
−0.0998752 + 0.995000i $$0.531844\pi$$
$$402$$ − 10.0000i − 0.498755i
$$403$$ 4.00000i 0.199254i
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ − 16.0000i − 0.793091i
$$408$$ 6.00000i 0.297044i
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 14.0000i 0.689730i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ − 16.0000i − 0.778868i
$$423$$ 12.0000i 0.583460i
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ −14.0000 −0.678302
$$427$$ 20.0000i 0.967868i
$$428$$ 16.0000i 0.773389i
$$429$$ 16.0000 0.772487
$$430$$ 0 0
$$431$$ −10.0000 −0.481683 −0.240842 0.970564i $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 4.00000i 0.192228i 0.995370 + 0.0961139i $$0.0306413\pi$$
−0.995370 + 0.0961139i $$0.969359\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ − 4.00000i − 0.191127i
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 24.0000i − 1.14156i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ − 14.0000i − 0.662177i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 16.0000i − 0.751746i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4.00000i − 0.187112i −0.995614 0.0935561i $$-0.970177\pi$$
0.995614 0.0935561i $$-0.0298234\pi$$
$$458$$ − 26.0000i − 1.21490i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ − 8.00000i − 0.372194i
$$463$$ − 12.0000i − 0.557687i −0.960337 0.278844i $$-0.910049\pi$$
0.960337 0.278844i $$-0.0899511\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ − 24.0000i − 1.11059i −0.831654 0.555294i $$-0.812606\pi$$
0.831654 0.555294i $$-0.187394\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ − 6.00000i − 0.276172i
$$473$$ − 32.0000i − 1.47136i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 20.0000i 0.914779i
$$479$$ 14.0000 0.639676 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ − 10.0000i − 0.455488i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 28.0000i − 1.26880i −0.773004 0.634401i $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 24.0000i 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ − 28.0000i − 1.25597i
$$498$$ 4.00000i 0.179244i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 28.0000i 1.24970i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 3.00000i − 0.133235i
$$508$$ 12.0000i 0.532414i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 48.0000i 2.11104i
$$518$$ 8.00000i 0.351500i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ − 6.00000i − 0.261364i
$$528$$ − 4.00000i − 0.174078i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ − 24.0000i − 1.03956i
$$534$$ 8.00000 0.346194
$$535$$ 0 0
$$536$$ 10.0000 0.431934
$$537$$ 4.00000i 0.172613i
$$538$$ − 12.0000i − 0.517357i
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 24.0000i 1.03089i
$$543$$ 18.0000i 0.772454i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 46.0000i 1.96682i 0.181402 + 0.983409i $$0.441936\pi$$
−0.181402 + 0.983409i $$0.558064\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 16.0000i 0.680389i
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 8.00000i − 0.337160i −0.985688 0.168580i $$-0.946082\pi$$
0.985688 0.168580i $$-0.0539181\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 2.00000i 0.0839921i
$$568$$ − 14.0000i − 0.587427i
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 16.0000i 0.668994i
$$573$$ − 10.0000i − 0.417756i
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 42.0000i 1.74848i 0.485491 + 0.874241i $$0.338641\pi$$
−0.485491 + 0.874241i $$0.661359\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ − 2.00000i − 0.0829027i
$$583$$ − 8.00000i − 0.331326i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 4.00000i 0.164399i
$$593$$ 46.0000i 1.88899i 0.328521 + 0.944497i $$0.393450\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 14.0000 0.573462
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ 10.0000i 0.407231i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 14.0000 0.568711
$$607$$ 14.0000i 0.568242i 0.958788 + 0.284121i $$0.0917018\pi$$
−0.958788 + 0.284121i $$0.908298\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ − 6.00000i − 0.242536i
$$613$$ 24.0000i 0.969351i 0.874694 + 0.484675i $$0.161062\pi$$
−0.874694 + 0.484675i $$0.838938\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ 26.0000i 1.04672i 0.852111 + 0.523360i $$0.175322\pi$$
−0.852111 + 0.523360i $$0.824678\pi$$
$$618$$ − 14.0000i − 0.563163i
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 6.00000i − 0.240578i
$$623$$ 16.0000i 0.641026i
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 4.00000 0.159872
$$627$$ 0 0
$$628$$ 6.00000i 0.239426i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 16.0000i 0.635943i
$$634$$ 26.0000 1.03259
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ 12.0000i 0.475457i
$$638$$ − 16.0000i − 0.633446i
$$639$$ 14.0000 0.553831
$$640$$ 0 0
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ − 16.0000i − 0.631470i
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ 6.00000i 0.234978i
$$653$$ 22.0000i 0.860927i 0.902608 + 0.430463i $$0.141650\pi$$
−0.902608 + 0.430463i $$0.858350\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 4.00000i 0.156055i
$$658$$ − 24.0000i − 0.935617i
$$659$$ 22.0000 0.856998 0.428499 0.903542i $$-0.359042\pi$$
0.428499 + 0.903542i $$0.359042\pi$$
$$660$$ 0 0
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ 28.0000i 1.08825i
$$663$$ 24.0000i 0.932083i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 2.00000i 0.0771517i
$$673$$ 16.0000i 0.616755i 0.951264 + 0.308377i $$0.0997859\pi$$
−0.951264 + 0.308377i $$0.900214\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 34.0000i − 1.30673i −0.757045 0.653363i $$-0.773358\pi$$
0.757045 0.653363i $$-0.226642\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 4.00000i 0.153168i
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 26.0000i 0.991962i
$$688$$ 8.00000i 0.304997i
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 8.00000i 0.303895i
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ 36.0000i 1.36360i
$$698$$ 2.00000i 0.0757011i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 0 0
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −10.0000 −0.376355
$$707$$ 28.0000i 1.05305i
$$708$$ 6.00000i 0.225494i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 8.00000i 0.299813i
$$713$$ 0 0
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ − 20.0000i − 0.746914i
$$718$$ 30.0000i 1.11959i
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 19.0000i 0.707107i
$$723$$ 10.0000i 0.371904i
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 30.0000i − 1.11264i −0.830969 0.556319i $$-0.812213\pi$$
0.830969 0.556319i $$-0.187787\pi$$
$$728$$ − 8.00000i − 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ − 10.0000i − 0.369611i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 40.0000i 1.47342i
$$738$$ − 6.00000i − 0.220863i
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.00000i 0.146845i
$$743$$ − 8.00000i − 0.293492i −0.989174 0.146746i $$-0.953120\pi$$
0.989174 0.146746i $$-0.0468799\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ −18.0000 −0.659027
$$747$$ − 4.00000i − 0.146352i
$$748$$ − 24.0000i − 0.877527i
$$749$$ 32.0000 1.16925
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ − 28.0000i − 1.02038i
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ − 12.0000i − 0.436147i −0.975932 0.218074i $$-0.930023\pi$$
0.975932 0.218074i $$-0.0699773\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ − 12.0000i − 0.434714i
$$763$$ − 36.0000i − 1.30329i
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ − 24.0000i − 0.866590i
$$768$$ 1.00000i 0.0360844i
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ − 8.00000i − 0.286998i
$$778$$ 36.0000i 1.29066i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 56.0000 2.00384
$$782$$ 0 0
$$783$$ 4.00000i 0.142948i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ − 2.00000i − 0.0712470i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 4.00000i 0.142134i
$$793$$ 40.0000i 1.42044i
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 4.00000i 0.141245i
$$803$$ 16.0000i 0.564628i
$$804$$ −10.0000 −0.352673
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 12.0000i 0.422420i
$$808$$ 14.0000i 0.492518i
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 0 0
$$811$$ −40.0000 −1.40459 −0.702295 0.711886i $$-0.747841\pi$$
−0.702295 + 0.711886i $$0.747841\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ − 24.0000i − 0.841717i
$$814$$ −16.0000 −0.560800
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 0 0
$$818$$ 6.00000i 0.209785i
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ − 18.0000i − 0.627822i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 4.00000i − 0.139094i −0.997579 0.0695468i $$-0.977845\pi$$
0.997579 0.0695468i $$-0.0221553\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ − 4.00000i − 0.138675i
$$833$$ − 18.0000i − 0.623663i
$$834$$ 12.0000 0.415526
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 1.00000i − 0.0345651i
$$838$$ − 30.0000i − 1.03633i
$$839$$ 34.0000 1.17381 0.586905 0.809656i $$-0.300346\pi$$
0.586905 + 0.809656i $$0.300346\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 30.0000i 1.03387i
$$843$$ 6.00000i 0.206651i
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 10.0000i 0.343604i
$$848$$ 2.00000i 0.0686803i
$$849$$ 14.0000 0.480479
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 14.0000i 0.479632i
$$853$$ − 30.0000i − 1.02718i −0.858036 0.513590i $$-0.828315\pi$$
0.858036 0.513590i $$-0.171685\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ 16.0000 0.546869
$$857$$ 42.0000i 1.43469i 0.696717 + 0.717346i $$0.254643\pi$$
−0.696717 + 0.717346i $$0.745357\pi$$
$$858$$ − 16.0000i − 0.546231i
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ 10.0000i 0.340601i
$$863$$ 24.0000i 0.816970i 0.912765 + 0.408485i $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 4.00000 0.135926
$$867$$ − 19.0000i − 0.645274i
$$868$$ − 2.00000i − 0.0678844i
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 40.0000 1.35535
$$872$$ − 18.0000i − 0.609557i
$$873$$ 2.00000i 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ − 18.0000i − 0.607817i −0.952701 0.303908i $$-0.901708\pi$$
0.952701 0.303908i $$-0.0982917\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ −56.0000 −1.88669 −0.943344 0.331816i $$-0.892339\pi$$
−0.943344 + 0.331816i $$0.892339\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ − 4.00000i − 0.134231i
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ −14.0000 −0.468230
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ 20.0000i 0.667409i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 24.0000i − 0.799113i
$$903$$ − 16.0000i − 0.532447i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 58.0000i 1.92586i 0.269754 + 0.962929i $$0.413058\pi$$
−0.269754 + 0.962929i $$0.586942\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ − 16.0000i − 0.529523i
$$914$$ −4.00000 −0.132308
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 28.0000i 0.924641i
$$918$$ 6.00000i 0.198030i
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ − 24.0000i − 0.790398i
$$923$$ − 56.0000i − 1.84326i
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ −12.0000 −0.394344
$$927$$ 14.0000i 0.459820i
$$928$$ 4.00000i 0.131306i
$$929$$ −24.0000 −0.787414 −0.393707 0.919236i $$-0.628808\pi$$
−0.393707 + 0.919236i $$0.628808\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000i 0.196537i
$$933$$ 6.00000i 0.196431i
$$934$$ −24.0000 −0.785304
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ − 20.0000i − 0.653023i
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ − 6.00000i − 0.195491i
$$943$$ 0 0
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −32.0000 −1.04041
$$947$$ − 28.0000i − 0.909878i −0.890523 0.454939i $$-0.849661\pi$$
0.890523 0.454939i $$-0.150339\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ −26.0000 −0.843108
$$952$$ 12.0000i 0.388922i
$$953$$ 50.0000i 1.61966i 0.586665 + 0.809829i $$0.300440\pi$$
−0.586665 + 0.809829i $$0.699560\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ 20.0000 0.646846
$$957$$ 16.0000i 0.517207i
$$958$$ − 14.0000i − 0.452319i
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 16.0000i 0.515861i
$$963$$ 16.0000i 0.515593i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 24.0000i 0.769405i
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ − 6.00000i − 0.191859i
$$979$$ −32.0000 −1.02272
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ 0 0
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ 24.0000i 0.763928i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ − 28.0000i − 0.888553i
$$994$$ −28.0000 −0.888106
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 18.0000i 0.570066i 0.958518 + 0.285033i $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 4.00000 0.126554
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 4650.2.d.p.3349.1 2
5.2 odd 4 4650.2.a.bl.1.1 1
5.3 odd 4 930.2.a.d.1.1 1
5.4 even 2 inner 4650.2.d.p.3349.2 2
15.8 even 4 2790.2.a.t.1.1 1
20.3 even 4 7440.2.a.y.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.d.1.1 1 5.3 odd 4
2790.2.a.t.1.1 1 15.8 even 4
4650.2.a.bl.1.1 1 5.2 odd 4
4650.2.d.p.3349.1 2 1.1 even 1 trivial
4650.2.d.p.3349.2 2 5.4 even 2 inner
7440.2.a.y.1.1 1 20.3 even 4