# Properties

 Label 5070.2.a.o.1.1 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} +1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +10.0000 q^{41} -2.00000 q^{42} +5.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -3.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +14.0000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +2.00000 q^{56} -6.00000 q^{57} -1.00000 q^{58} -5.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -3.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} +3.00000 q^{69} -2.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -5.00000 q^{74} -1.00000 q^{75} +6.00000 q^{76} +2.00000 q^{77} +5.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -2.00000 q^{84} -2.00000 q^{85} +5.00000 q^{86} +1.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} -3.00000 q^{92} +3.00000 q^{93} +3.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −2.00000 −0.436436
$$22$$ 1.00000 0.213201
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000 0.377964
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 2.00000 0.342997
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −1.00000 −0.149071
$$46$$ −3.00000 −0.442326
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 1.00000 0.141421
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.00000 −0.134840
$$56$$ 2.00000 0.267261
$$57$$ −6.00000 −0.794719
$$58$$ −1.00000 −0.131306
$$59$$ −5.00000 −0.650945 −0.325472 0.945552i $$-0.605523\pi$$
−0.325472 + 0.945552i $$0.605523\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −3.00000 −0.381000
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 3.00000 0.361158
$$70$$ −2.00000 −0.239046
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −5.00000 −0.581238
$$75$$ −1.00000 −0.115470
$$76$$ 6.00000 0.688247
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ −2.00000 −0.216930
$$86$$ 5.00000 0.539164
$$87$$ 1.00000 0.107211
$$88$$ 1.00000 0.106600
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −3.00000 −0.312772
$$93$$ 3.00000 0.311086
$$94$$ 3.00000 0.309426
$$95$$ −6.00000 −0.615587
$$96$$ −1.00000 −0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 1.00000 0.100504
$$100$$ 1.00000 0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 14.0000 1.35980
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ 5.00000 0.474579
$$112$$ 2.00000 0.188982
$$113$$ 17.0000 1.59923 0.799613 0.600516i $$-0.205038\pi$$
0.799613 + 0.600516i $$0.205038\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 3.00000 0.279751
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ −5.00000 −0.460287
$$119$$ 4.00000 0.366679
$$120$$ 1.00000 0.0912871
$$121$$ −10.0000 −0.909091
$$122$$ −10.0000 −0.905357
$$123$$ −10.0000 −0.901670
$$124$$ −3.00000 −0.269408
$$125$$ −1.00000 −0.0894427
$$126$$ 2.00000 0.178174
$$127$$ −14.0000 −1.24230 −0.621150 0.783692i $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −5.00000 −0.440225
$$130$$ 0 0
$$131$$ 13.0000 1.13582 0.567908 0.823092i $$-0.307753\pi$$
0.567908 + 0.823092i $$0.307753\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 12.0000 1.04053
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 2.00000 0.171499
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 3.00000 0.255377
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −3.00000 −0.252646
$$142$$ 4.00000 0.335673
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 1.00000 0.0830455
$$146$$ −2.00000 −0.165521
$$147$$ 3.00000 0.247436
$$148$$ −5.00000 −0.410997
$$149$$ 11.0000 0.901155 0.450578 0.892737i $$-0.351218\pi$$
0.450578 + 0.892737i $$0.351218\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 2.00000 0.161690
$$154$$ 2.00000 0.161165
$$155$$ 3.00000 0.240966
$$156$$ 0 0
$$157$$ 25.0000 1.99522 0.997609 0.0691164i $$-0.0220180\pi$$
0.997609 + 0.0691164i $$0.0220180\pi$$
$$158$$ 5.00000 0.397779
$$159$$ −14.0000 −1.11027
$$160$$ −1.00000 −0.0790569
$$161$$ −6.00000 −0.472866
$$162$$ 1.00000 0.0785674
$$163$$ 17.0000 1.33154 0.665771 0.746156i $$-0.268103\pi$$
0.665771 + 0.746156i $$0.268103\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 1.00000 0.0778499
$$166$$ −6.00000 −0.465690
$$167$$ −7.00000 −0.541676 −0.270838 0.962625i $$-0.587301\pi$$
−0.270838 + 0.962625i $$0.587301\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ −2.00000 −0.153393
$$171$$ 6.00000 0.458831
$$172$$ 5.00000 0.381246
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ 1.00000 0.0758098
$$175$$ 2.00000 0.151186
$$176$$ 1.00000 0.0753778
$$177$$ 5.00000 0.375823
$$178$$ 10.0000 0.749532
$$179$$ −7.00000 −0.523205 −0.261602 0.965176i $$-0.584251\pi$$
−0.261602 + 0.965176i $$0.584251\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ −3.00000 −0.221163
$$185$$ 5.00000 0.367607
$$186$$ 3.00000 0.219971
$$187$$ 2.00000 0.146254
$$188$$ 3.00000 0.218797
$$189$$ −2.00000 −0.145479
$$190$$ −6.00000 −0.435286
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 14.0000 0.985037
$$203$$ −2.00000 −0.140372
$$204$$ −2.00000 −0.140028
$$205$$ −10.0000 −0.698430
$$206$$ −6.00000 −0.418040
$$207$$ −3.00000 −0.208514
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 2.00000 0.138013
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 14.0000 0.961524
$$213$$ −4.00000 −0.274075
$$214$$ −6.00000 −0.410152
$$215$$ −5.00000 −0.340997
$$216$$ −1.00000 −0.0680414
$$217$$ −6.00000 −0.407307
$$218$$ −6.00000 −0.406371
$$219$$ 2.00000 0.135147
$$220$$ −1.00000 −0.0674200
$$221$$ 0 0
$$222$$ 5.00000 0.335578
$$223$$ 10.0000 0.669650 0.334825 0.942280i $$-0.391323\pi$$
0.334825 + 0.942280i $$0.391323\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 1.00000 0.0666667
$$226$$ 17.0000 1.13082
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ −6.00000 −0.397360
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 3.00000 0.197814
$$231$$ −2.00000 −0.131590
$$232$$ −1.00000 −0.0656532
$$233$$ 3.00000 0.196537 0.0982683 0.995160i $$-0.468670\pi$$
0.0982683 + 0.995160i $$0.468670\pi$$
$$234$$ 0 0
$$235$$ −3.00000 −0.195698
$$236$$ −5.00000 −0.325472
$$237$$ −5.00000 −0.324785
$$238$$ 4.00000 0.259281
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 3.00000 0.191663
$$246$$ −10.0000 −0.637577
$$247$$ 0 0
$$248$$ −3.00000 −0.190500
$$249$$ 6.00000 0.380235
$$250$$ −1.00000 −0.0632456
$$251$$ 3.00000 0.189358 0.0946792 0.995508i $$-0.469817\pi$$
0.0946792 + 0.995508i $$0.469817\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −3.00000 −0.188608
$$254$$ −14.0000 −0.878438
$$255$$ 2.00000 0.125245
$$256$$ 1.00000 0.0625000
$$257$$ 21.0000 1.30994 0.654972 0.755653i $$-0.272680\pi$$
0.654972 + 0.755653i $$0.272680\pi$$
$$258$$ −5.00000 −0.311286
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 13.0000 0.803143
$$263$$ 23.0000 1.41824 0.709120 0.705087i $$-0.249092\pi$$
0.709120 + 0.705087i $$0.249092\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −14.0000 −0.860013
$$266$$ 12.0000 0.735767
$$267$$ −10.0000 −0.611990
$$268$$ 0 0
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −29.0000 −1.76162 −0.880812 0.473466i $$-0.843003\pi$$
−0.880812 + 0.473466i $$0.843003\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ 1.00000 0.0603023
$$276$$ 3.00000 0.180579
$$277$$ −19.0000 −1.14160 −0.570800 0.821089i $$-0.693367\pi$$
−0.570800 + 0.821089i $$0.693367\pi$$
$$278$$ 10.0000 0.599760
$$279$$ −3.00000 −0.179605
$$280$$ −2.00000 −0.119523
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ −3.00000 −0.178647
$$283$$ −13.0000 −0.772770 −0.386385 0.922338i $$-0.626276\pi$$
−0.386385 + 0.922338i $$0.626276\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 6.00000 0.355409
$$286$$ 0 0
$$287$$ 20.0000 1.18056
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 1.00000 0.0587220
$$291$$ 10.0000 0.586210
$$292$$ −2.00000 −0.117041
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 5.00000 0.291111
$$296$$ −5.00000 −0.290619
$$297$$ −1.00000 −0.0580259
$$298$$ 11.0000 0.637213
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 10.0000 0.576390
$$302$$ 24.0000 1.38104
$$303$$ −14.0000 −0.804279
$$304$$ 6.00000 0.344124
$$305$$ 10.0000 0.572598
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 6.00000 0.341328
$$310$$ 3.00000 0.170389
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −12.0000 −0.678280 −0.339140 0.940736i $$-0.610136\pi$$
−0.339140 + 0.940736i $$0.610136\pi$$
$$314$$ 25.0000 1.41083
$$315$$ −2.00000 −0.112687
$$316$$ 5.00000 0.281272
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ −14.0000 −0.785081
$$319$$ −1.00000 −0.0559893
$$320$$ −1.00000 −0.0559017
$$321$$ 6.00000 0.334887
$$322$$ −6.00000 −0.334367
$$323$$ 12.0000 0.667698
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 17.0000 0.941543
$$327$$ 6.00000 0.331801
$$328$$ 10.0000 0.552158
$$329$$ 6.00000 0.330791
$$330$$ 1.00000 0.0550482
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ −5.00000 −0.273998
$$334$$ −7.00000 −0.383023
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ −17.0000 −0.923313
$$340$$ −2.00000 −0.108465
$$341$$ −3.00000 −0.162459
$$342$$ 6.00000 0.324443
$$343$$ −20.0000 −1.07990
$$344$$ 5.00000 0.269582
$$345$$ −3.00000 −0.161515
$$346$$ 4.00000 0.215041
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 1.00000 0.0536056
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 5.00000 0.265747
$$355$$ −4.00000 −0.212298
$$356$$ 10.0000 0.529999
$$357$$ −4.00000 −0.211702
$$358$$ −7.00000 −0.369961
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 17.0000 0.894737
$$362$$ 16.0000 0.840941
$$363$$ 10.0000 0.524864
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 10.0000 0.522708
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 10.0000 0.520579
$$370$$ 5.00000 0.259938
$$371$$ 28.0000 1.45369
$$372$$ 3.00000 0.155543
$$373$$ 37.0000 1.91579 0.957894 0.287123i $$-0.0926989\pi$$
0.957894 + 0.287123i $$0.0926989\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 1.00000 0.0516398
$$376$$ 3.00000 0.154713
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 14.0000 0.717242
$$382$$ 24.0000 1.22795
$$383$$ −27.0000 −1.37964 −0.689818 0.723983i $$-0.742309\pi$$
−0.689818 + 0.723983i $$0.742309\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −2.00000 −0.101929
$$386$$ −4.00000 −0.203595
$$387$$ 5.00000 0.254164
$$388$$ −10.0000 −0.507673
$$389$$ −1.00000 −0.0507020 −0.0253510 0.999679i $$-0.508070\pi$$
−0.0253510 + 0.999679i $$0.508070\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ −3.00000 −0.151523
$$393$$ −13.0000 −0.655763
$$394$$ −12.0000 −0.604551
$$395$$ −5.00000 −0.251577
$$396$$ 1.00000 0.0502519
$$397$$ −13.0000 −0.652451 −0.326226 0.945292i $$-0.605777\pi$$
−0.326226 + 0.945292i $$0.605777\pi$$
$$398$$ 0 0
$$399$$ −12.0000 −0.600751
$$400$$ 1.00000 0.0500000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ −1.00000 −0.0496904
$$406$$ −2.00000 −0.0992583
$$407$$ −5.00000 −0.247841
$$408$$ −2.00000 −0.0990148
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ 9.00000 0.443937
$$412$$ −6.00000 −0.295599
$$413$$ −10.0000 −0.492068
$$414$$ −3.00000 −0.147442
$$415$$ 6.00000 0.294528
$$416$$ 0 0
$$417$$ −10.0000 −0.489702
$$418$$ 6.00000 0.293470
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 28.0000 1.36464 0.682318 0.731055i $$-0.260972\pi$$
0.682318 + 0.731055i $$0.260972\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 3.00000 0.145865
$$424$$ 14.0000 0.679900
$$425$$ 2.00000 0.0970143
$$426$$ −4.00000 −0.193801
$$427$$ −20.0000 −0.967868
$$428$$ −6.00000 −0.290021
$$429$$ 0 0
$$430$$ −5.00000 −0.241121
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ −6.00000 −0.288009
$$435$$ −1.00000 −0.0479463
$$436$$ −6.00000 −0.287348
$$437$$ −18.0000 −0.861057
$$438$$ 2.00000 0.0955637
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 5.00000 0.237289
$$445$$ −10.0000 −0.474045
$$446$$ 10.0000 0.473514
$$447$$ −11.0000 −0.520282
$$448$$ 2.00000 0.0944911
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 10.0000 0.470882
$$452$$ 17.0000 0.799613
$$453$$ −24.0000 −1.12762
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ −6.00000 −0.280976
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ 22.0000 1.02799
$$459$$ −2.00000 −0.0933520
$$460$$ 3.00000 0.139876
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ −2.00000 −0.0930484
$$463$$ 10.0000 0.464739 0.232370 0.972628i $$-0.425352\pi$$
0.232370 + 0.972628i $$0.425352\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ −3.00000 −0.139122
$$466$$ 3.00000 0.138972
$$467$$ −2.00000 −0.0925490 −0.0462745 0.998929i $$-0.514735\pi$$
−0.0462745 + 0.998929i $$0.514735\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −3.00000 −0.138380
$$471$$ −25.0000 −1.15194
$$472$$ −5.00000 −0.230144
$$473$$ 5.00000 0.229900
$$474$$ −5.00000 −0.229658
$$475$$ 6.00000 0.275299
$$476$$ 4.00000 0.183340
$$477$$ 14.0000 0.641016
$$478$$ −8.00000 −0.365911
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −7.00000 −0.318841
$$483$$ 6.00000 0.273009
$$484$$ −10.0000 −0.454545
$$485$$ 10.0000 0.454077
$$486$$ −1.00000 −0.0453609
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ −17.0000 −0.768767
$$490$$ 3.00000 0.135526
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ −2.00000 −0.0900755
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ −3.00000 −0.134704
$$497$$ 8.00000 0.358849
$$498$$ 6.00000 0.268866
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 7.00000 0.312737
$$502$$ 3.00000 0.133897
$$503$$ 40.0000 1.78351 0.891756 0.452517i $$-0.149474\pi$$
0.891756 + 0.452517i $$0.149474\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ −14.0000 −0.622992
$$506$$ −3.00000 −0.133366
$$507$$ 0 0
$$508$$ −14.0000 −0.621150
$$509$$ −9.00000 −0.398918 −0.199459 0.979906i $$-0.563918\pi$$
−0.199459 + 0.979906i $$0.563918\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000 0.0441942
$$513$$ −6.00000 −0.264906
$$514$$ 21.0000 0.926270
$$515$$ 6.00000 0.264392
$$516$$ −5.00000 −0.220113
$$517$$ 3.00000 0.131940
$$518$$ −10.0000 −0.439375
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ −1.00000 −0.0437688
$$523$$ 33.0000 1.44299 0.721495 0.692420i $$-0.243455\pi$$
0.721495 + 0.692420i $$0.243455\pi$$
$$524$$ 13.0000 0.567908
$$525$$ −2.00000 −0.0872872
$$526$$ 23.0000 1.00285
$$527$$ −6.00000 −0.261364
$$528$$ −1.00000 −0.0435194
$$529$$ −14.0000 −0.608696
$$530$$ −14.0000 −0.608121
$$531$$ −5.00000 −0.216982
$$532$$ 12.0000 0.520266
$$533$$ 0 0
$$534$$ −10.0000 −0.432742
$$535$$ 6.00000 0.259403
$$536$$ 0 0
$$537$$ 7.00000 0.302072
$$538$$ 18.0000 0.776035
$$539$$ −3.00000 −0.129219
$$540$$ 1.00000 0.0430331
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ −29.0000 −1.24566
$$543$$ −16.0000 −0.686626
$$544$$ 2.00000 0.0857493
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −9.00000 −0.384461
$$549$$ −10.0000 −0.426790
$$550$$ 1.00000 0.0426401
$$551$$ −6.00000 −0.255609
$$552$$ 3.00000 0.127688
$$553$$ 10.0000 0.425243
$$554$$ −19.0000 −0.807233
$$555$$ −5.00000 −0.212238
$$556$$ 10.0000 0.424094
$$557$$ −34.0000 −1.44063 −0.720313 0.693649i $$-0.756002\pi$$
−0.720313 + 0.693649i $$0.756002\pi$$
$$558$$ −3.00000 −0.127000
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ −2.00000 −0.0844401
$$562$$ −30.0000 −1.26547
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ −17.0000 −0.715195
$$566$$ −13.0000 −0.546431
$$567$$ 2.00000 0.0839921
$$568$$ 4.00000 0.167836
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 6.00000 0.251312
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 20.0000 0.834784
$$575$$ −3.00000 −0.125109
$$576$$ 1.00000 0.0416667
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 4.00000 0.166234
$$580$$ 1.00000 0.0415227
$$581$$ −12.0000 −0.497844
$$582$$ 10.0000 0.414513
$$583$$ 14.0000 0.579821
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 3.00000 0.123718
$$589$$ −18.0000 −0.741677
$$590$$ 5.00000 0.205847
$$591$$ 12.0000 0.493614
$$592$$ −5.00000 −0.205499
$$593$$ −35.0000 −1.43728 −0.718639 0.695383i $$-0.755235\pi$$
−0.718639 + 0.695383i $$0.755235\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ −4.00000 −0.163984
$$596$$ 11.0000 0.450578
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −11.0000 −0.448699 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$602$$ 10.0000 0.407570
$$603$$ 0 0
$$604$$ 24.0000 0.976546
$$605$$ 10.0000 0.406558
$$606$$ −14.0000 −0.568711
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 2.00000 0.0810441
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ 2.00000 0.0808452
$$613$$ −37.0000 −1.49442 −0.747208 0.664590i $$-0.768606\pi$$
−0.747208 + 0.664590i $$0.768606\pi$$
$$614$$ 0 0
$$615$$ 10.0000 0.403239
$$616$$ 2.00000 0.0805823
$$617$$ 3.00000 0.120775 0.0603877 0.998175i $$-0.480766\pi$$
0.0603877 + 0.998175i $$0.480766\pi$$
$$618$$ 6.00000 0.241355
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 3.00000 0.120483
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 20.0000 0.801283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −12.0000 −0.479616
$$627$$ −6.00000 −0.239617
$$628$$ 25.0000 0.997609
$$629$$ −10.0000 −0.398726
$$630$$ −2.00000 −0.0796819
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ 5.00000 0.198889
$$633$$ 8.00000 0.317971
$$634$$ 12.0000 0.476581
$$635$$ 14.0000 0.555573
$$636$$ −14.0000 −0.555136
$$637$$ 0 0
$$638$$ −1.00000 −0.0395904
$$639$$ 4.00000 0.158238
$$640$$ −1.00000 −0.0395285
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 6.00000 0.236801
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 5.00000 0.196875
$$646$$ 12.0000 0.472134
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −5.00000 −0.196267
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ 17.0000 0.665771
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 6.00000 0.234619
$$655$$ −13.0000 −0.507952
$$656$$ 10.0000 0.390434
$$657$$ −2.00000 −0.0780274
$$658$$ 6.00000 0.233904
$$659$$ −13.0000 −0.506408 −0.253204 0.967413i $$-0.581484\pi$$
−0.253204 + 0.967413i $$0.581484\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −12.0000 −0.466746 −0.233373 0.972387i $$-0.574976\pi$$
−0.233373 + 0.972387i $$0.574976\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ −12.0000 −0.465340
$$666$$ −5.00000 −0.193746
$$667$$ 3.00000 0.116160
$$668$$ −7.00000 −0.270838
$$669$$ −10.0000 −0.386622
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ −2.00000 −0.0771517
$$673$$ −16.0000 −0.616755 −0.308377 0.951264i $$-0.599786\pi$$
−0.308377 + 0.951264i $$0.599786\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ −17.0000 −0.652881
$$679$$ −20.0000 −0.767530
$$680$$ −2.00000 −0.0766965
$$681$$ −20.0000 −0.766402
$$682$$ −3.00000 −0.114876
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 6.00000 0.229416
$$685$$ 9.00000 0.343872
$$686$$ −20.0000 −0.763604
$$687$$ −22.0000 −0.839352
$$688$$ 5.00000 0.190623
$$689$$ 0 0
$$690$$ −3.00000 −0.114208
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ 4.00000 0.152057
$$693$$ 2.00000 0.0759737
$$694$$ −18.0000 −0.683271
$$695$$ −10.0000 −0.379322
$$696$$ 1.00000 0.0379049
$$697$$ 20.0000 0.757554
$$698$$ 8.00000 0.302804
$$699$$ −3.00000 −0.113470
$$700$$ 2.00000 0.0755929
$$701$$ 23.0000 0.868698 0.434349 0.900745i $$-0.356978\pi$$
0.434349 + 0.900745i $$0.356978\pi$$
$$702$$ 0 0
$$703$$ −30.0000 −1.13147
$$704$$ 1.00000 0.0376889
$$705$$ 3.00000 0.112987
$$706$$ −14.0000 −0.526897
$$707$$ 28.0000 1.05305
$$708$$ 5.00000 0.187912
$$709$$ −16.0000 −0.600893 −0.300446 0.953799i $$-0.597136\pi$$
−0.300446 + 0.953799i $$0.597136\pi$$
$$710$$ −4.00000 −0.150117
$$711$$ 5.00000 0.187515
$$712$$ 10.0000 0.374766
$$713$$ 9.00000 0.337053
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ −7.00000 −0.261602
$$717$$ 8.00000 0.298765
$$718$$ 12.0000 0.447836
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −12.0000 −0.446903
$$722$$ 17.0000 0.632674
$$723$$ 7.00000 0.260333
$$724$$ 16.0000 0.594635
$$725$$ −1.00000 −0.0371391
$$726$$ 10.0000 0.371135
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000 0.0740233
$$731$$ 10.0000 0.369863
$$732$$ 10.0000 0.369611
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 36.0000 1.32878
$$735$$ −3.00000 −0.110657
$$736$$ −3.00000 −0.110581
$$737$$ 0 0
$$738$$ 10.0000 0.368105
$$739$$ 24.0000 0.882854 0.441427 0.897297i $$-0.354472\pi$$
0.441427 + 0.897297i $$0.354472\pi$$
$$740$$ 5.00000 0.183804
$$741$$ 0 0
$$742$$ 28.0000 1.02791
$$743$$ −15.0000 −0.550297 −0.275148 0.961402i $$-0.588727\pi$$
−0.275148 + 0.961402i $$0.588727\pi$$
$$744$$ 3.00000 0.109985
$$745$$ −11.0000 −0.403009
$$746$$ 37.0000 1.35467
$$747$$ −6.00000 −0.219529
$$748$$ 2.00000 0.0731272
$$749$$ −12.0000 −0.438470
$$750$$ 1.00000 0.0365148
$$751$$ 41.0000 1.49611 0.748056 0.663636i $$-0.230988\pi$$
0.748056 + 0.663636i $$0.230988\pi$$
$$752$$ 3.00000 0.109399
$$753$$ −3.00000 −0.109326
$$754$$ 0 0
$$755$$ −24.0000 −0.873449
$$756$$ −2.00000 −0.0727393
$$757$$ −54.0000 −1.96266 −0.981332 0.192323i $$-0.938398\pi$$
−0.981332 + 0.192323i $$0.938398\pi$$
$$758$$ −30.0000 −1.08965
$$759$$ 3.00000 0.108893
$$760$$ −6.00000 −0.217643
$$761$$ −20.0000 −0.724999 −0.362500 0.931984i $$-0.618077\pi$$
−0.362500 + 0.931984i $$0.618077\pi$$
$$762$$ 14.0000 0.507166
$$763$$ −12.0000 −0.434429
$$764$$ 24.0000 0.868290
$$765$$ −2.00000 −0.0723102
$$766$$ −27.0000 −0.975550
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −29.0000 −1.04577 −0.522883 0.852404i $$-0.675144\pi$$
−0.522883 + 0.852404i $$0.675144\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ −21.0000 −0.756297
$$772$$ −4.00000 −0.143963
$$773$$ −16.0000 −0.575480 −0.287740 0.957709i $$-0.592904\pi$$
−0.287740 + 0.957709i $$0.592904\pi$$
$$774$$ 5.00000 0.179721
$$775$$ −3.00000 −0.107763
$$776$$ −10.0000 −0.358979
$$777$$ 10.0000 0.358748
$$778$$ −1.00000 −0.0358517
$$779$$ 60.0000 2.14972
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ −6.00000 −0.214560
$$783$$ 1.00000 0.0357371
$$784$$ −3.00000 −0.107143
$$785$$ −25.0000 −0.892288
$$786$$ −13.0000 −0.463695
$$787$$ 25.0000 0.891154 0.445577 0.895244i $$-0.352999\pi$$
0.445577 + 0.895244i $$0.352999\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ −23.0000 −0.818822
$$790$$ −5.00000 −0.177892
$$791$$ 34.0000 1.20890
$$792$$ 1.00000 0.0355335
$$793$$ 0 0
$$794$$ −13.0000 −0.461353
$$795$$ 14.0000 0.496529
$$796$$ 0 0
$$797$$ −52.0000 −1.84193 −0.920967 0.389640i $$-0.872599\pi$$
−0.920967 + 0.389640i $$0.872599\pi$$
$$798$$ −12.0000 −0.424795
$$799$$ 6.00000 0.212265
$$800$$ 1.00000 0.0353553
$$801$$ 10.0000 0.353333
$$802$$ −12.0000 −0.423735
$$803$$ −2.00000 −0.0705785
$$804$$ 0 0
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ −18.0000 −0.633630
$$808$$ 14.0000 0.492518
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 29.0000 1.01707
$$814$$ −5.00000 −0.175250
$$815$$ −17.0000 −0.595484
$$816$$ −2.00000 −0.0700140
$$817$$ 30.0000 1.04957
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ 23.0000 0.802706 0.401353 0.915924i $$-0.368540\pi$$
0.401353 + 0.915924i $$0.368540\pi$$
$$822$$ 9.00000 0.313911
$$823$$ −22.0000 −0.766872 −0.383436 0.923567i $$-0.625259\pi$$
−0.383436 + 0.923567i $$0.625259\pi$$
$$824$$ −6.00000 −0.209020
$$825$$ −1.00000 −0.0348155
$$826$$ −10.0000 −0.347945
$$827$$ −18.0000 −0.625921 −0.312961 0.949766i $$-0.601321\pi$$
−0.312961 + 0.949766i $$0.601321\pi$$
$$828$$ −3.00000 −0.104257
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 6.00000 0.208263
$$831$$ 19.0000 0.659103
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ −10.0000 −0.346272
$$835$$ 7.00000 0.242245
$$836$$ 6.00000 0.207514
$$837$$ 3.00000 0.103695
$$838$$ 4.00000 0.138178
$$839$$ 38.0000 1.31191 0.655953 0.754802i $$-0.272267\pi$$
0.655953 + 0.754802i $$0.272267\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ −28.0000 −0.965517
$$842$$ 28.0000 0.964944
$$843$$ 30.0000 1.03325
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ −20.0000 −0.687208
$$848$$ 14.0000 0.480762
$$849$$ 13.0000 0.446159
$$850$$ 2.00000 0.0685994
$$851$$ 15.0000 0.514193
$$852$$ −4.00000 −0.137038
$$853$$ 7.00000 0.239675 0.119838 0.992793i $$-0.461763\pi$$
0.119838 + 0.992793i $$0.461763\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ −6.00000 −0.205196
$$856$$ −6.00000 −0.205076
$$857$$ −35.0000 −1.19558 −0.597789 0.801654i $$-0.703954\pi$$
−0.597789 + 0.801654i $$0.703954\pi$$
$$858$$ 0 0
$$859$$ 18.0000 0.614152 0.307076 0.951685i $$-0.400649\pi$$
0.307076 + 0.951685i $$0.400649\pi$$
$$860$$ −5.00000 −0.170499
$$861$$ −20.0000 −0.681598
$$862$$ 12.0000 0.408722
$$863$$ −51.0000 −1.73606 −0.868030 0.496512i $$-0.834614\pi$$
−0.868030 + 0.496512i $$0.834614\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −4.00000 −0.136004
$$866$$ 16.0000 0.543702
$$867$$ 13.0000 0.441503
$$868$$ −6.00000 −0.203653
$$869$$ 5.00000 0.169613
$$870$$ −1.00000 −0.0339032
$$871$$ 0 0
$$872$$ −6.00000 −0.203186
$$873$$ −10.0000 −0.338449
$$874$$ −18.0000 −0.608859
$$875$$ −2.00000 −0.0676123
$$876$$ 2.00000 0.0675737
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 14.0000 0.472208
$$880$$ −1.00000 −0.0337100
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −1.00000 −0.0336527 −0.0168263 0.999858i $$-0.505356\pi$$
−0.0168263 + 0.999858i $$0.505356\pi$$
$$884$$ 0 0
$$885$$ −5.00000 −0.168073
$$886$$ 16.0000 0.537531
$$887$$ 1.00000 0.0335767 0.0167884 0.999859i $$-0.494656\pi$$
0.0167884 + 0.999859i $$0.494656\pi$$
$$888$$ 5.00000 0.167789
$$889$$ −28.0000 −0.939090
$$890$$ −10.0000 −0.335201
$$891$$ 1.00000 0.0335013
$$892$$ 10.0000 0.334825
$$893$$ 18.0000 0.602347
$$894$$ −11.0000 −0.367895
$$895$$ 7.00000 0.233984
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ 3.00000 0.100056
$$900$$ 1.00000 0.0333333
$$901$$ 28.0000 0.932815
$$902$$ 10.0000 0.332964
$$903$$ −10.0000 −0.332779
$$904$$ 17.0000 0.565412
$$905$$ −16.0000 −0.531858
$$906$$ −24.0000 −0.797347
$$907$$ −53.0000 −1.75984 −0.879918 0.475125i $$-0.842403\pi$$
−0.879918 + 0.475125i $$0.842403\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ 34.0000 1.12647 0.563235 0.826297i $$-0.309557\pi$$
0.563235 + 0.826297i $$0.309557\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ −6.00000 −0.198571
$$914$$ 14.0000 0.463079
$$915$$ −10.0000 −0.330590
$$916$$ 22.0000 0.726900
$$917$$ 26.0000 0.858596
$$918$$ −2.00000 −0.0660098
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 3.00000 0.0989071
$$921$$ 0 0
$$922$$ 27.0000 0.889198
$$923$$ 0 0
$$924$$ −2.00000 −0.0657952
$$925$$ −5.00000 −0.164399
$$926$$ 10.0000 0.328620
$$927$$ −6.00000 −0.197066
$$928$$ −1.00000 −0.0328266
$$929$$ 24.0000 0.787414 0.393707 0.919236i $$-0.371192\pi$$
0.393707 + 0.919236i $$0.371192\pi$$
$$930$$ −3.00000 −0.0983739
$$931$$ −18.0000 −0.589926
$$932$$ 3.00000 0.0982683
$$933$$ 0 0
$$934$$ −2.00000 −0.0654420
$$935$$ −2.00000 −0.0654070
$$936$$ 0 0
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ 0 0
$$939$$ 12.0000 0.391605
$$940$$ −3.00000 −0.0978492
$$941$$ 26.0000 0.847576 0.423788 0.905761i $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ −25.0000 −0.814544
$$943$$ −30.0000 −0.976934
$$944$$ −5.00000 −0.162736
$$945$$ 2.00000 0.0650600
$$946$$ 5.00000 0.162564
$$947$$ 52.0000 1.68977 0.844886 0.534946i $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ −5.00000 −0.162392
$$949$$ 0 0
$$950$$ 6.00000 0.194666
$$951$$ −12.0000 −0.389127
$$952$$ 4.00000 0.129641
$$953$$ 15.0000 0.485898 0.242949 0.970039i $$-0.421885\pi$$
0.242949 + 0.970039i $$0.421885\pi$$
$$954$$ 14.0000 0.453267
$$955$$ −24.0000 −0.776622
$$956$$ −8.00000 −0.258738
$$957$$ 1.00000 0.0323254
$$958$$ −4.00000 −0.129234
$$959$$ −18.0000 −0.581250
$$960$$ 1.00000 0.0322749
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ −7.00000 −0.225455
$$965$$ 4.00000 0.128765
$$966$$ 6.00000 0.193047
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ −12.0000 −0.385496
$$970$$ 10.0000 0.321081
$$971$$ −40.0000 −1.28366 −0.641831 0.766846i $$-0.721825\pi$$
−0.641831 + 0.766846i $$0.721825\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 20.0000 0.641171
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −39.0000 −1.24772 −0.623860 0.781536i $$-0.714437\pi$$
−0.623860 + 0.781536i $$0.714437\pi$$
$$978$$ −17.0000 −0.543600
$$979$$ 10.0000 0.319601
$$980$$ 3.00000 0.0958315
$$981$$ −6.00000 −0.191565
$$982$$ 24.0000 0.765871
$$983$$ −25.0000 −0.797376 −0.398688 0.917087i $$-0.630534\pi$$
−0.398688 + 0.917087i $$0.630534\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 12.0000 0.382352
$$986$$ −2.00000 −0.0636930
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ −15.0000 −0.476972
$$990$$ −1.00000 −0.0317821
$$991$$ 39.0000 1.23888 0.619438 0.785046i $$-0.287361\pi$$
0.619438 + 0.785046i $$0.287361\pi$$
$$992$$ −3.00000 −0.0952501
$$993$$ −4.00000 −0.126936
$$994$$ 8.00000 0.253745
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ 5.00000 0.158193
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.a.o.1.1 1
13.3 even 3 390.2.i.a.61.1 2
13.5 odd 4 5070.2.b.g.1351.1 2
13.8 odd 4 5070.2.b.g.1351.2 2
13.9 even 3 390.2.i.a.211.1 yes 2
13.12 even 2 5070.2.a.f.1.1 1
39.29 odd 6 1170.2.i.k.451.1 2
39.35 odd 6 1170.2.i.k.991.1 2
65.3 odd 12 1950.2.z.h.1699.2 4
65.9 even 6 1950.2.i.s.601.1 2
65.22 odd 12 1950.2.z.h.1849.2 4
65.29 even 6 1950.2.i.s.451.1 2
65.42 odd 12 1950.2.z.h.1699.1 4
65.48 odd 12 1950.2.z.h.1849.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.a.61.1 2 13.3 even 3
390.2.i.a.211.1 yes 2 13.9 even 3
1170.2.i.k.451.1 2 39.29 odd 6
1170.2.i.k.991.1 2 39.35 odd 6
1950.2.i.s.451.1 2 65.29 even 6
1950.2.i.s.601.1 2 65.9 even 6
1950.2.z.h.1699.1 4 65.42 odd 12
1950.2.z.h.1699.2 4 65.3 odd 12
1950.2.z.h.1849.1 4 65.48 odd 12
1950.2.z.h.1849.2 4 65.22 odd 12
5070.2.a.f.1.1 1 13.12 even 2
5070.2.a.o.1.1 1 1.1 even 1 trivial
5070.2.b.g.1351.1 2 13.5 odd 4
5070.2.b.g.1351.2 2 13.8 odd 4