# Properties

 Label 3234.2.a.bf.1.1 Level $3234$ Weight $2$ Character 3234.1 Self dual yes Analytic conductor $25.824$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2700.1 Defining polynomial: $$x^{3} - 15 x - 20$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.80560$$ of defining polynomial Character $$\chi$$ $$=$$ 3234.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80560 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80560 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.80560 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.80560 q^{15} +1.00000 q^{16} +7.96523 q^{17} +1.00000 q^{18} -6.48261 q^{19} -2.80560 q^{20} -1.00000 q^{22} -5.28822 q^{23} -1.00000 q^{24} +2.87141 q^{25} -1.00000 q^{27} +5.12859 q^{29} +2.80560 q^{30} -8.96523 q^{31} +1.00000 q^{32} +1.00000 q^{33} +7.96523 q^{34} +1.00000 q^{36} +10.4826 q^{37} -6.48261 q^{38} -2.80560 q^{40} -5.09382 q^{41} +11.4478 q^{43} -1.00000 q^{44} -2.80560 q^{45} -5.28822 q^{46} +0.322990 q^{47} -1.00000 q^{48} +2.87141 q^{50} -7.96523 q^{51} +8.00000 q^{53} -1.00000 q^{54} +2.80560 q^{55} +6.48261 q^{57} +5.12859 q^{58} -0.871407 q^{59} +2.80560 q^{60} +2.15962 q^{61} -8.96523 q^{62} +1.00000 q^{64} +1.00000 q^{66} -7.09382 q^{67} +7.96523 q^{68} +5.28822 q^{69} -7.44784 q^{71} +1.00000 q^{72} +12.5764 q^{73} +10.4826 q^{74} -2.87141 q^{75} -6.48261 q^{76} +6.80560 q^{79} -2.80560 q^{80} +1.00000 q^{81} -5.09382 q^{82} +15.0938 q^{83} -22.3473 q^{85} +11.4478 q^{86} -5.12859 q^{87} -1.00000 q^{88} +1.61121 q^{89} -2.80560 q^{90} -5.28822 q^{92} +8.96523 q^{93} +0.322990 q^{94} +18.1876 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
$$5$$ −2.80560 −1.25470 −0.627352 0.778736i $$-0.715861\pi$$
−0.627352 + 0.778736i $$0.715861\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −2.80560 −0.887210
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 2.80560 0.724404
$$16$$ 1.00000 0.250000
$$17$$ 7.96523 1.93185 0.965926 0.258820i $$-0.0833337\pi$$
0.965926 + 0.258820i $$0.0833337\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −6.48261 −1.48721 −0.743607 0.668617i $$-0.766886\pi$$
−0.743607 + 0.668617i $$0.766886\pi$$
$$20$$ −2.80560 −0.627352
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −5.28822 −1.10267 −0.551335 0.834284i $$-0.685881\pi$$
−0.551335 + 0.834284i $$0.685881\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 2.87141 0.574281
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.12859 0.952356 0.476178 0.879349i $$-0.342022\pi$$
0.476178 + 0.879349i $$0.342022\pi$$
$$30$$ 2.80560 0.512231
$$31$$ −8.96523 −1.61020 −0.805101 0.593138i $$-0.797889\pi$$
−0.805101 + 0.593138i $$0.797889\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ 7.96523 1.36602
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.4826 1.72333 0.861665 0.507477i $$-0.169422\pi$$
0.861665 + 0.507477i $$0.169422\pi$$
$$38$$ −6.48261 −1.05162
$$39$$ 0 0
$$40$$ −2.80560 −0.443605
$$41$$ −5.09382 −0.795521 −0.397760 0.917489i $$-0.630212\pi$$
−0.397760 + 0.917489i $$0.630212\pi$$
$$42$$ 0 0
$$43$$ 11.4478 1.74578 0.872890 0.487918i $$-0.162243\pi$$
0.872890 + 0.487918i $$0.162243\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −2.80560 −0.418235
$$46$$ −5.28822 −0.779705
$$47$$ 0.322990 0.0471129 0.0235565 0.999723i $$-0.492501\pi$$
0.0235565 + 0.999723i $$0.492501\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 2.87141 0.406078
$$51$$ −7.96523 −1.11535
$$52$$ 0 0
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 2.80560 0.378307
$$56$$ 0 0
$$57$$ 6.48261 0.858643
$$58$$ 5.12859 0.673417
$$59$$ −0.871407 −0.113448 −0.0567238 0.998390i $$-0.518065\pi$$
−0.0567238 + 0.998390i $$0.518065\pi$$
$$60$$ 2.80560 0.362202
$$61$$ 2.15962 0.276511 0.138256 0.990397i $$-0.455850\pi$$
0.138256 + 0.990397i $$0.455850\pi$$
$$62$$ −8.96523 −1.13858
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ −7.09382 −0.866648 −0.433324 0.901238i $$-0.642659\pi$$
−0.433324 + 0.901238i $$0.642659\pi$$
$$68$$ 7.96523 0.965926
$$69$$ 5.28822 0.636626
$$70$$ 0 0
$$71$$ −7.44784 −0.883896 −0.441948 0.897041i $$-0.645712\pi$$
−0.441948 + 0.897041i $$0.645712\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 12.5764 1.47196 0.735980 0.677003i $$-0.236722\pi$$
0.735980 + 0.677003i $$0.236722\pi$$
$$74$$ 10.4826 1.21858
$$75$$ −2.87141 −0.331562
$$76$$ −6.48261 −0.743607
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.80560 0.765690 0.382845 0.923813i $$-0.374944\pi$$
0.382845 + 0.923813i $$0.374944\pi$$
$$80$$ −2.80560 −0.313676
$$81$$ 1.00000 0.111111
$$82$$ −5.09382 −0.562518
$$83$$ 15.0938 1.65676 0.828381 0.560165i $$-0.189262\pi$$
0.828381 + 0.560165i $$0.189262\pi$$
$$84$$ 0 0
$$85$$ −22.3473 −2.42390
$$86$$ 11.4478 1.23445
$$87$$ −5.12859 −0.549843
$$88$$ −1.00000 −0.106600
$$89$$ 1.61121 0.170787 0.0853937 0.996347i $$-0.472785\pi$$
0.0853937 + 0.996347i $$0.472785\pi$$
$$90$$ −2.80560 −0.295737
$$91$$ 0 0
$$92$$ −5.28822 −0.551335
$$93$$ 8.96523 0.929651
$$94$$ 0.322990 0.0333139
$$95$$ 18.1876 1.86601
$$96$$ −1.00000 −0.102062
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 2.87141 0.287141
$$101$$ 10.7398 1.06865 0.534325 0.845279i $$-0.320566\pi$$
0.534325 + 0.845279i $$0.320566\pi$$
$$102$$ −7.96523 −0.788675
$$103$$ 2.96523 0.292172 0.146086 0.989272i $$-0.453332\pi$$
0.146086 + 0.989272i $$0.453332\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ 4.83663 0.467575 0.233787 0.972288i $$-0.424888\pi$$
0.233787 + 0.972288i $$0.424888\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −10.1596 −0.973115 −0.486558 0.873648i $$-0.661748\pi$$
−0.486558 + 0.873648i $$0.661748\pi$$
$$110$$ 2.80560 0.267504
$$111$$ −10.4826 −0.994966
$$112$$ 0 0
$$113$$ 10.9652 1.03152 0.515761 0.856733i $$-0.327509\pi$$
0.515761 + 0.856733i $$0.327509\pi$$
$$114$$ 6.48261 0.607152
$$115$$ 14.8366 1.38352
$$116$$ 5.12859 0.476178
$$117$$ 0 0
$$118$$ −0.871407 −0.0802195
$$119$$ 0 0
$$120$$ 2.80560 0.256115
$$121$$ 1.00000 0.0909091
$$122$$ 2.15962 0.195523
$$123$$ 5.09382 0.459294
$$124$$ −8.96523 −0.805101
$$125$$ 5.97199 0.534151
$$126$$ 0 0
$$127$$ −9.28822 −0.824196 −0.412098 0.911140i $$-0.635204\pi$$
−0.412098 + 0.911140i $$0.635204\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −11.4478 −1.00793
$$130$$ 0 0
$$131$$ 12.9652 1.13278 0.566389 0.824138i $$-0.308340\pi$$
0.566389 + 0.824138i $$0.308340\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ −7.09382 −0.612813
$$135$$ 2.80560 0.241468
$$136$$ 7.96523 0.683012
$$137$$ 5.61121 0.479398 0.239699 0.970847i $$-0.422951\pi$$
0.239699 + 0.970847i $$0.422951\pi$$
$$138$$ 5.28822 0.450163
$$139$$ 13.1286 1.11355 0.556776 0.830662i $$-0.312038\pi$$
0.556776 + 0.830662i $$0.312038\pi$$
$$140$$ 0 0
$$141$$ −0.322990 −0.0272007
$$142$$ −7.44784 −0.625009
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −14.3888 −1.19492
$$146$$ 12.5764 1.04083
$$147$$ 0 0
$$148$$ 10.4826 0.861665
$$149$$ 16.7398 1.37138 0.685689 0.727895i $$-0.259501\pi$$
0.685689 + 0.727895i $$0.259501\pi$$
$$150$$ −2.87141 −0.234449
$$151$$ 10.8994 0.886982 0.443491 0.896279i $$-0.353740\pi$$
0.443491 + 0.896279i $$0.353740\pi$$
$$152$$ −6.48261 −0.525809
$$153$$ 7.96523 0.643950
$$154$$ 0 0
$$155$$ 25.1529 2.02033
$$156$$ 0 0
$$157$$ 1.77457 0.141626 0.0708132 0.997490i $$-0.477441\pi$$
0.0708132 + 0.997490i $$0.477441\pi$$
$$158$$ 6.80560 0.541425
$$159$$ −8.00000 −0.634441
$$160$$ −2.80560 −0.221802
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −4.05904 −0.317929 −0.158964 0.987284i $$-0.550816\pi$$
−0.158964 + 0.987284i $$0.550816\pi$$
$$164$$ −5.09382 −0.397760
$$165$$ −2.80560 −0.218416
$$166$$ 15.0938 1.17151
$$167$$ 13.8684 1.07317 0.536584 0.843847i $$-0.319714\pi$$
0.536584 + 0.843847i $$0.319714\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ −22.3473 −1.71396
$$171$$ −6.48261 −0.495738
$$172$$ 11.4478 0.872890
$$173$$ 16.9652 1.28984 0.644921 0.764249i $$-0.276890\pi$$
0.644921 + 0.764249i $$0.276890\pi$$
$$174$$ −5.12859 −0.388798
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0.871407 0.0654990
$$178$$ 1.61121 0.120765
$$179$$ −12.6703 −0.947019 −0.473509 0.880789i $$-0.657013\pi$$
−0.473509 + 0.880789i $$0.657013\pi$$
$$180$$ −2.80560 −0.209117
$$181$$ −2.57643 −0.191505 −0.0957523 0.995405i $$-0.530526\pi$$
−0.0957523 + 0.995405i $$0.530526\pi$$
$$182$$ 0 0
$$183$$ −2.15962 −0.159644
$$184$$ −5.28822 −0.389852
$$185$$ −29.4100 −2.16227
$$186$$ 8.96523 0.657362
$$187$$ −7.96523 −0.582475
$$188$$ 0.322990 0.0235565
$$189$$ 0 0
$$190$$ 18.1876 1.31947
$$191$$ −2.38879 −0.172847 −0.0864235 0.996258i $$-0.527544\pi$$
−0.0864235 + 0.996258i $$0.527544\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −14.5764 −1.04923 −0.524617 0.851338i $$-0.675792\pi$$
−0.524617 + 0.851338i $$0.675792\pi$$
$$194$$ 7.00000 0.502571
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −12.4826 −0.889349 −0.444675 0.895692i $$-0.646681\pi$$
−0.444675 + 0.895692i $$0.646681\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −14.1876 −1.00573 −0.502867 0.864364i $$-0.667722\pi$$
−0.502867 + 0.864364i $$0.667722\pi$$
$$200$$ 2.87141 0.203039
$$201$$ 7.09382 0.500359
$$202$$ 10.7398 0.755650
$$203$$ 0 0
$$204$$ −7.96523 −0.557677
$$205$$ 14.2912 0.998143
$$206$$ 2.96523 0.206597
$$207$$ −5.28822 −0.367556
$$208$$ 0 0
$$209$$ 6.48261 0.448412
$$210$$ 0 0
$$211$$ −20.8336 −1.43425 −0.717123 0.696947i $$-0.754541\pi$$
−0.717123 + 0.696947i $$0.754541\pi$$
$$212$$ 8.00000 0.549442
$$213$$ 7.44784 0.510318
$$214$$ 4.83663 0.330625
$$215$$ −32.1181 −2.19044
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −10.1596 −0.688096
$$219$$ −12.5764 −0.849836
$$220$$ 2.80560 0.189154
$$221$$ 0 0
$$222$$ −10.4826 −0.703547
$$223$$ 12.5764 0.842180 0.421090 0.907019i $$-0.361648\pi$$
0.421090 + 0.907019i $$0.361648\pi$$
$$224$$ 0 0
$$225$$ 2.87141 0.191427
$$226$$ 10.9652 0.729396
$$227$$ 20.0590 1.33137 0.665683 0.746235i $$-0.268140\pi$$
0.665683 + 0.746235i $$0.268140\pi$$
$$228$$ 6.48261 0.429322
$$229$$ −22.5764 −1.49189 −0.745946 0.666006i $$-0.768002\pi$$
−0.745946 + 0.666006i $$0.768002\pi$$
$$230$$ 14.8366 0.978299
$$231$$ 0 0
$$232$$ 5.12859 0.336709
$$233$$ −0.742815 −0.0486634 −0.0243317 0.999704i $$-0.507746\pi$$
−0.0243317 + 0.999704i $$0.507746\pi$$
$$234$$ 0 0
$$235$$ −0.906181 −0.0591128
$$236$$ −0.871407 −0.0567238
$$237$$ −6.80560 −0.442071
$$238$$ 0 0
$$239$$ 4.31925 0.279389 0.139694 0.990195i $$-0.455388\pi$$
0.139694 + 0.990195i $$0.455388\pi$$
$$240$$ 2.80560 0.181101
$$241$$ −18.1876 −1.17157 −0.585784 0.810467i $$-0.699213\pi$$
−0.585784 + 0.810467i $$0.699213\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 2.15962 0.138256
$$245$$ 0 0
$$246$$ 5.09382 0.324770
$$247$$ 0 0
$$248$$ −8.96523 −0.569292
$$249$$ −15.0938 −0.956532
$$250$$ 5.97199 0.377702
$$251$$ 28.6703 1.80965 0.904825 0.425784i $$-0.140001\pi$$
0.904825 + 0.425784i $$0.140001\pi$$
$$252$$ 0 0
$$253$$ 5.28822 0.332467
$$254$$ −9.28822 −0.582794
$$255$$ 22.3473 1.39944
$$256$$ 1.00000 0.0625000
$$257$$ −13.6112 −0.849044 −0.424522 0.905418i $$-0.639558\pi$$
−0.424522 + 0.905418i $$0.639558\pi$$
$$258$$ −11.4478 −0.712711
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 5.12859 0.317452
$$262$$ 12.9652 0.800994
$$263$$ −3.61121 −0.222676 −0.111338 0.993783i $$-0.535514\pi$$
−0.111338 + 0.993783i $$0.535514\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −22.4448 −1.37877
$$266$$ 0 0
$$267$$ −1.61121 −0.0986042
$$268$$ −7.09382 −0.433324
$$269$$ −9.38203 −0.572033 −0.286016 0.958225i $$-0.592331\pi$$
−0.286016 + 0.958225i $$0.592331\pi$$
$$270$$ 2.80560 0.170744
$$271$$ 5.61121 0.340856 0.170428 0.985370i $$-0.445485\pi$$
0.170428 + 0.985370i $$0.445485\pi$$
$$272$$ 7.96523 0.482963
$$273$$ 0 0
$$274$$ 5.61121 0.338985
$$275$$ −2.87141 −0.173152
$$276$$ 5.28822 0.318313
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 13.1286 0.787401
$$279$$ −8.96523 −0.536734
$$280$$ 0 0
$$281$$ 24.1529 1.44084 0.720420 0.693539i $$-0.243949\pi$$
0.720420 + 0.693539i $$0.243949\pi$$
$$282$$ −0.322990 −0.0192338
$$283$$ 7.54166 0.448305 0.224152 0.974554i $$-0.428039\pi$$
0.224152 + 0.974554i $$0.428039\pi$$
$$284$$ −7.44784 −0.441948
$$285$$ −18.1876 −1.07734
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 46.4448 2.73205
$$290$$ −14.3888 −0.844939
$$291$$ −7.00000 −0.410347
$$292$$ 12.5764 0.735980
$$293$$ 7.90618 0.461884 0.230942 0.972968i $$-0.425819\pi$$
0.230942 + 0.972968i $$0.425819\pi$$
$$294$$ 0 0
$$295$$ 2.44482 0.142343
$$296$$ 10.4826 0.609289
$$297$$ 1.00000 0.0580259
$$298$$ 16.7398 0.969710
$$299$$ 0 0
$$300$$ −2.87141 −0.165781
$$301$$ 0 0
$$302$$ 10.8994 0.627191
$$303$$ −10.7398 −0.616985
$$304$$ −6.48261 −0.371803
$$305$$ −6.05904 −0.346940
$$306$$ 7.96523 0.455342
$$307$$ 12.8957 0.735995 0.367998 0.929827i $$-0.380043\pi$$
0.367998 + 0.929827i $$0.380043\pi$$
$$308$$ 0 0
$$309$$ −2.96523 −0.168686
$$310$$ 25.1529 1.42859
$$311$$ 10.8994 0.618049 0.309025 0.951054i $$-0.399997\pi$$
0.309025 + 0.951054i $$0.399997\pi$$
$$312$$ 0 0
$$313$$ −9.12859 −0.515979 −0.257989 0.966148i $$-0.583060\pi$$
−0.257989 + 0.966148i $$0.583060\pi$$
$$314$$ 1.77457 0.100145
$$315$$ 0 0
$$316$$ 6.80560 0.382845
$$317$$ −9.51364 −0.534339 −0.267170 0.963649i $$-0.586088\pi$$
−0.267170 + 0.963649i $$0.586088\pi$$
$$318$$ −8.00000 −0.448618
$$319$$ −5.12859 −0.287146
$$320$$ −2.80560 −0.156838
$$321$$ −4.83663 −0.269955
$$322$$ 0 0
$$323$$ −51.6355 −2.87307
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.05904 −0.224810
$$327$$ 10.1596 0.561828
$$328$$ −5.09382 −0.281259
$$329$$ 0 0
$$330$$ −2.80560 −0.154443
$$331$$ 1.09382 0.0601217 0.0300609 0.999548i $$-0.490430\pi$$
0.0300609 + 0.999548i $$0.490430\pi$$
$$332$$ 15.0938 0.828381
$$333$$ 10.4826 0.574444
$$334$$ 13.8684 0.758845
$$335$$ 19.9024 1.08739
$$336$$ 0 0
$$337$$ −25.7988 −1.40535 −0.702676 0.711510i $$-0.748012\pi$$
−0.702676 + 0.711510i $$0.748012\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ −10.9652 −0.595549
$$340$$ −22.3473 −1.21195
$$341$$ 8.96523 0.485494
$$342$$ −6.48261 −0.350540
$$343$$ 0 0
$$344$$ 11.4478 0.617226
$$345$$ −14.8366 −0.798777
$$346$$ 16.9652 0.912056
$$347$$ 20.7671 1.11484 0.557418 0.830232i $$-0.311792\pi$$
0.557418 + 0.830232i $$0.311792\pi$$
$$348$$ −5.12859 −0.274921
$$349$$ −9.51364 −0.509254 −0.254627 0.967039i $$-0.581953\pi$$
−0.254627 + 0.967039i $$0.581953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ −12.5764 −0.669376 −0.334688 0.942329i $$-0.608631\pi$$
−0.334688 + 0.942329i $$0.608631\pi$$
$$354$$ 0.871407 0.0463148
$$355$$ 20.8957 1.10903
$$356$$ 1.61121 0.0853937
$$357$$ 0 0
$$358$$ −12.6703 −0.669644
$$359$$ −0.0695483 −0.00367062 −0.00183531 0.999998i $$-0.500584\pi$$
−0.00183531 + 0.999998i $$0.500584\pi$$
$$360$$ −2.80560 −0.147868
$$361$$ 23.0243 1.21180
$$362$$ −2.57643 −0.135414
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −35.2845 −1.84687
$$366$$ −2.15962 −0.112885
$$367$$ −31.5417 −1.64646 −0.823231 0.567707i $$-0.807831\pi$$
−0.823231 + 0.567707i $$0.807831\pi$$
$$368$$ −5.28822 −0.275667
$$369$$ −5.09382 −0.265174
$$370$$ −29.4100 −1.52896
$$371$$ 0 0
$$372$$ 8.96523 0.464825
$$373$$ 15.3125 0.792850 0.396425 0.918067i $$-0.370251\pi$$
0.396425 + 0.918067i $$0.370251\pi$$
$$374$$ −7.96523 −0.411872
$$375$$ −5.97199 −0.308392
$$376$$ 0.322990 0.0166569
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.0590 0.824898 0.412449 0.910981i $$-0.364674\pi$$
0.412449 + 0.910981i $$0.364674\pi$$
$$380$$ 18.1876 0.933006
$$381$$ 9.28822 0.475850
$$382$$ −2.38879 −0.122221
$$383$$ −0.739798 −0.0378019 −0.0189010 0.999821i $$-0.506017\pi$$
−0.0189010 + 0.999821i $$0.506017\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.5764 −0.741921
$$387$$ 11.4478 0.581926
$$388$$ 7.00000 0.355371
$$389$$ 3.58319 0.181675 0.0908375 0.995866i $$-0.471046\pi$$
0.0908375 + 0.995866i $$0.471046\pi$$
$$390$$ 0 0
$$391$$ −42.1218 −2.13019
$$392$$ 0 0
$$393$$ −12.9652 −0.654009
$$394$$ −12.4826 −0.628865
$$395$$ −19.0938 −0.960714
$$396$$ −1.00000 −0.0502519
$$397$$ −25.1286 −1.26117 −0.630584 0.776121i $$-0.717185\pi$$
−0.630584 + 0.776121i $$0.717185\pi$$
$$398$$ −14.1876 −0.711162
$$399$$ 0 0
$$400$$ 2.87141 0.143570
$$401$$ −3.86839 −0.193178 −0.0965891 0.995324i $$-0.530793\pi$$
−0.0965891 + 0.995324i $$0.530793\pi$$
$$402$$ 7.09382 0.353808
$$403$$ 0 0
$$404$$ 10.7398 0.534325
$$405$$ −2.80560 −0.139412
$$406$$ 0 0
$$407$$ −10.4826 −0.519604
$$408$$ −7.96523 −0.394337
$$409$$ 4.64598 0.229729 0.114864 0.993381i $$-0.463357\pi$$
0.114864 + 0.993381i $$0.463357\pi$$
$$410$$ 14.2912 0.705794
$$411$$ −5.61121 −0.276780
$$412$$ 2.96523 0.146086
$$413$$ 0 0
$$414$$ −5.28822 −0.259902
$$415$$ −42.3473 −2.07875
$$416$$ 0 0
$$417$$ −13.1286 −0.642910
$$418$$ 6.48261 0.317075
$$419$$ −19.5174 −0.953487 −0.476743 0.879043i $$-0.658183\pi$$
−0.476743 + 0.879043i $$0.658183\pi$$
$$420$$ 0 0
$$421$$ 15.5174 0.756271 0.378136 0.925750i $$-0.376565\pi$$
0.378136 + 0.925750i $$0.376565\pi$$
$$422$$ −20.8336 −1.01416
$$423$$ 0.322990 0.0157043
$$424$$ 8.00000 0.388514
$$425$$ 22.8714 1.10943
$$426$$ 7.44784 0.360849
$$427$$ 0 0
$$428$$ 4.83663 0.233787
$$429$$ 0 0
$$430$$ −32.1181 −1.54887
$$431$$ −14.3888 −0.693084 −0.346542 0.938034i $$-0.612644\pi$$
−0.346542 + 0.938034i $$0.612644\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 2.03477 0.0977850 0.0488925 0.998804i $$-0.484431\pi$$
0.0488925 + 0.998804i $$0.484431\pi$$
$$434$$ 0 0
$$435$$ 14.3888 0.689890
$$436$$ −10.1596 −0.486558
$$437$$ 34.2815 1.63990
$$438$$ −12.5764 −0.600925
$$439$$ 3.22616 0.153976 0.0769880 0.997032i $$-0.475470\pi$$
0.0769880 + 0.997032i $$0.475470\pi$$
$$440$$ 2.80560 0.133752
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −22.0938 −1.04971 −0.524854 0.851192i $$-0.675880\pi$$
−0.524854 + 0.851192i $$0.675880\pi$$
$$444$$ −10.4826 −0.497483
$$445$$ −4.52040 −0.214288
$$446$$ 12.5764 0.595511
$$447$$ −16.7398 −0.791765
$$448$$ 0 0
$$449$$ 22.2497 1.05003 0.525014 0.851094i $$-0.324060\pi$$
0.525014 + 0.851094i $$0.324060\pi$$
$$450$$ 2.87141 0.135359
$$451$$ 5.09382 0.239859
$$452$$ 10.9652 0.515761
$$453$$ −10.8994 −0.512099
$$454$$ 20.0590 0.941418
$$455$$ 0 0
$$456$$ 6.48261 0.303576
$$457$$ −18.6385 −0.871872 −0.435936 0.899978i $$-0.643583\pi$$
−0.435936 + 0.899978i $$0.643583\pi$$
$$458$$ −22.5764 −1.05493
$$459$$ −7.96523 −0.371785
$$460$$ 14.8366 0.691762
$$461$$ −20.9895 −0.977578 −0.488789 0.872402i $$-0.662561\pi$$
−0.488789 + 0.872402i $$0.662561\pi$$
$$462$$ 0 0
$$463$$ −5.35402 −0.248822 −0.124411 0.992231i $$-0.539704\pi$$
−0.124411 + 0.992231i $$0.539704\pi$$
$$464$$ 5.12859 0.238089
$$465$$ −25.1529 −1.16644
$$466$$ −0.742815 −0.0344102
$$467$$ 26.0243 1.20426 0.602130 0.798398i $$-0.294319\pi$$
0.602130 + 0.798398i $$0.294319\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −0.906181 −0.0417990
$$471$$ −1.77457 −0.0817680
$$472$$ −0.871407 −0.0401098
$$473$$ −11.4478 −0.526372
$$474$$ −6.80560 −0.312592
$$475$$ −18.6142 −0.854079
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 4.31925 0.197558
$$479$$ −3.61121 −0.165000 −0.0825001 0.996591i $$-0.526290\pi$$
−0.0825001 + 0.996591i $$0.526290\pi$$
$$480$$ 2.80560 0.128058
$$481$$ 0 0
$$482$$ −18.1876 −0.828424
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −19.6392 −0.891771
$$486$$ −1.00000 −0.0453609
$$487$$ −8.06206 −0.365327 −0.182663 0.983176i $$-0.558472\pi$$
−0.182663 + 0.983176i $$0.558472\pi$$
$$488$$ 2.15962 0.0977615
$$489$$ 4.05904 0.183556
$$490$$ 0 0
$$491$$ 20.9062 0.943483 0.471741 0.881737i $$-0.343626\pi$$
0.471741 + 0.881737i $$0.343626\pi$$
$$492$$ 5.09382 0.229647
$$493$$ 40.8504 1.83981
$$494$$ 0 0
$$495$$ 2.80560 0.126102
$$496$$ −8.96523 −0.402551
$$497$$ 0 0
$$498$$ −15.0938 −0.676370
$$499$$ 4.96523 0.222274 0.111137 0.993805i $$-0.464551\pi$$
0.111137 + 0.993805i $$0.464551\pi$$
$$500$$ 5.97199 0.267075
$$501$$ −13.8684 −0.619594
$$502$$ 28.6703 1.27962
$$503$$ −17.7988 −0.793611 −0.396806 0.917903i $$-0.629881\pi$$
−0.396806 + 0.917903i $$0.629881\pi$$
$$504$$ 0 0
$$505$$ −30.1316 −1.34084
$$506$$ 5.28822 0.235090
$$507$$ 13.0000 0.577350
$$508$$ −9.28822 −0.412098
$$509$$ 4.51437 0.200096 0.100048 0.994983i $$-0.468100\pi$$
0.100048 + 0.994983i $$0.468100\pi$$
$$510$$ 22.3473 0.989553
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 6.48261 0.286214
$$514$$ −13.6112 −0.600365
$$515$$ −8.31925 −0.366590
$$516$$ −11.4478 −0.503963
$$517$$ −0.322990 −0.0142051
$$518$$ 0 0
$$519$$ −16.9652 −0.744691
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 5.12859 0.224472
$$523$$ 6.90317 0.301854 0.150927 0.988545i $$-0.451774\pi$$
0.150927 + 0.988545i $$0.451774\pi$$
$$524$$ 12.9652 0.566389
$$525$$ 0 0
$$526$$ −3.61121 −0.157456
$$527$$ −71.4100 −3.11067
$$528$$ 1.00000 0.0435194
$$529$$ 4.96523 0.215879
$$530$$ −22.4448 −0.974941
$$531$$ −0.871407 −0.0378159
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −1.61121 −0.0697237
$$535$$ −13.5697 −0.586668
$$536$$ −7.09382 −0.306406
$$537$$ 12.6703 0.546762
$$538$$ −9.38203 −0.404488
$$539$$ 0 0
$$540$$ 2.80560 0.120734
$$541$$ 2.80560 0.120622 0.0603111 0.998180i $$-0.480791\pi$$
0.0603111 + 0.998180i $$0.480791\pi$$
$$542$$ 5.61121 0.241022
$$543$$ 2.57643 0.110565
$$544$$ 7.96523 0.341506
$$545$$ 28.5039 1.22097
$$546$$ 0 0
$$547$$ 32.9274 1.40788 0.703938 0.710262i $$-0.251423\pi$$
0.703938 + 0.710262i $$0.251423\pi$$
$$548$$ 5.61121 0.239699
$$549$$ 2.15962 0.0921705
$$550$$ −2.87141 −0.122437
$$551$$ −33.2467 −1.41636
$$552$$ 5.28822 0.225081
$$553$$ 0 0
$$554$$ −16.0000 −0.679775
$$555$$ 29.4100 1.24839
$$556$$ 13.1286 0.556776
$$557$$ 23.7050 1.00441 0.502207 0.864747i $$-0.332522\pi$$
0.502207 + 0.864747i $$0.332522\pi$$
$$558$$ −8.96523 −0.379528
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 7.96523 0.336292
$$562$$ 24.1529 1.01883
$$563$$ 30.4448 1.28310 0.641548 0.767083i $$-0.278292\pi$$
0.641548 + 0.767083i $$0.278292\pi$$
$$564$$ −0.322990 −0.0136003
$$565$$ −30.7641 −1.29425
$$566$$ 7.54166 0.317000
$$567$$ 0 0
$$568$$ −7.44784 −0.312504
$$569$$ 22.2815 0.934087 0.467044 0.884234i $$-0.345319\pi$$
0.467044 + 0.884234i $$0.345319\pi$$
$$570$$ −18.1876 −0.761796
$$571$$ −2.87141 −0.120165 −0.0600823 0.998193i $$-0.519136\pi$$
−0.0600823 + 0.998193i $$0.519136\pi$$
$$572$$ 0 0
$$573$$ 2.38879 0.0997933
$$574$$ 0 0
$$575$$ −15.1846 −0.633242
$$576$$ 1.00000 0.0416667
$$577$$ −15.0243 −0.625469 −0.312734 0.949841i $$-0.601245\pi$$
−0.312734 + 0.949841i $$0.601245\pi$$
$$578$$ 46.4448 1.93185
$$579$$ 14.5764 0.605776
$$580$$ −14.3888 −0.597462
$$581$$ 0 0
$$582$$ −7.00000 −0.290159
$$583$$ −8.00000 −0.331326
$$584$$ 12.5764 0.520416
$$585$$ 0 0
$$586$$ 7.90618 0.326601
$$587$$ −42.5069 −1.75445 −0.877223 0.480082i $$-0.840607\pi$$
−0.877223 + 0.480082i $$0.840607\pi$$
$$588$$ 0 0
$$589$$ 58.1181 2.39471
$$590$$ 2.44482 0.100652
$$591$$ 12.4826 0.513466
$$592$$ 10.4826 0.430833
$$593$$ 15.3858 0.631818 0.315909 0.948789i $$-0.397691\pi$$
0.315909 + 0.948789i $$0.397691\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ 16.7398 0.685689
$$597$$ 14.1876 0.580661
$$598$$ 0 0
$$599$$ −39.6392 −1.61961 −0.809807 0.586696i $$-0.800428\pi$$
−0.809807 + 0.586696i $$0.800428\pi$$
$$600$$ −2.87141 −0.117225
$$601$$ 33.3405 1.35999 0.679994 0.733218i $$-0.261983\pi$$
0.679994 + 0.733218i $$0.261983\pi$$
$$602$$ 0 0
$$603$$ −7.09382 −0.288883
$$604$$ 10.8994 0.443491
$$605$$ −2.80560 −0.114064
$$606$$ −10.7398 −0.436274
$$607$$ 15.9585 0.647734 0.323867 0.946103i $$-0.395017\pi$$
0.323867 + 0.946103i $$0.395017\pi$$
$$608$$ −6.48261 −0.262905
$$609$$ 0 0
$$610$$ −6.05904 −0.245324
$$611$$ 0 0
$$612$$ 7.96523 0.321975
$$613$$ 46.6665 1.88484 0.942421 0.334428i $$-0.108543\pi$$
0.942421 + 0.334428i $$0.108543\pi$$
$$614$$ 12.8957 0.520427
$$615$$ −14.2912 −0.576278
$$616$$ 0 0
$$617$$ −9.28447 −0.373779 −0.186889 0.982381i $$-0.559841\pi$$
−0.186889 + 0.982381i $$0.559841\pi$$
$$618$$ −2.96523 −0.119279
$$619$$ 33.9895 1.36615 0.683077 0.730347i $$-0.260642\pi$$
0.683077 + 0.730347i $$0.260642\pi$$
$$620$$ 25.1529 1.01016
$$621$$ 5.28822 0.212209
$$622$$ 10.8994 0.437027
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.1121 −1.24448
$$626$$ −9.12859 −0.364852
$$627$$ −6.48261 −0.258891
$$628$$ 1.77457 0.0708132
$$629$$ 83.4964 3.32922
$$630$$ 0 0
$$631$$ −14.6460 −0.583047 −0.291524 0.956564i $$-0.594162\pi$$
−0.291524 + 0.956564i $$0.594162\pi$$
$$632$$ 6.80560 0.270712
$$633$$ 20.8336 0.828062
$$634$$ −9.51364 −0.377835
$$635$$ 26.0590 1.03412
$$636$$ −8.00000 −0.317221
$$637$$ 0 0
$$638$$ −5.12859 −0.203043
$$639$$ −7.44784 −0.294632
$$640$$ −2.80560 −0.110901
$$641$$ 20.9652 0.828077 0.414038 0.910259i $$-0.364118\pi$$
0.414038 + 0.910259i $$0.364118\pi$$
$$642$$ −4.83663 −0.190887
$$643$$ −23.6733 −0.933582 −0.466791 0.884368i $$-0.654590\pi$$
−0.466791 + 0.884368i $$0.654590\pi$$
$$644$$ 0 0
$$645$$ 32.1181 1.26465
$$646$$ −51.6355 −2.03157
$$647$$ −12.4168 −0.488155 −0.244078 0.969756i $$-0.578485\pi$$
−0.244078 + 0.969756i $$0.578485\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0.871407 0.0342057
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.05904 −0.158964
$$653$$ −0.0975625 −0.00381792 −0.00190896 0.999998i $$-0.500608\pi$$
−0.00190896 + 0.999998i $$0.500608\pi$$
$$654$$ 10.1596 0.397273
$$655$$ −36.3753 −1.42130
$$656$$ −5.09382 −0.198880
$$657$$ 12.5764 0.490653
$$658$$ 0 0
$$659$$ −35.0243 −1.36435 −0.682176 0.731188i $$-0.738966\pi$$
−0.682176 + 0.731188i $$0.738966\pi$$
$$660$$ −2.80560 −0.109208
$$661$$ 29.6355 1.15269 0.576343 0.817208i $$-0.304479\pi$$
0.576343 + 0.817208i $$0.304479\pi$$
$$662$$ 1.09382 0.0425125
$$663$$ 0 0
$$664$$ 15.0938 0.585754
$$665$$ 0 0
$$666$$ 10.4826 0.406193
$$667$$ −27.1211 −1.05013
$$668$$ 13.8684 0.536584
$$669$$ −12.5764 −0.486233
$$670$$ 19.9024 0.768898
$$671$$ −2.15962 −0.0833713
$$672$$ 0 0
$$673$$ 16.8957 0.651281 0.325640 0.945494i $$-0.394420\pi$$
0.325640 + 0.945494i $$0.394420\pi$$
$$674$$ −25.7988 −0.993734
$$675$$ −2.87141 −0.110521
$$676$$ −13.0000 −0.500000
$$677$$ 13.1907 0.506958 0.253479 0.967341i $$-0.418425\pi$$
0.253479 + 0.967341i $$0.418425\pi$$
$$678$$ −10.9652 −0.421117
$$679$$ 0 0
$$680$$ −22.3473 −0.856978
$$681$$ −20.0590 −0.768664
$$682$$ 8.96523 0.343296
$$683$$ 9.83663 0.376388 0.188194 0.982132i $$-0.439737\pi$$
0.188194 + 0.982132i $$0.439737\pi$$
$$684$$ −6.48261 −0.247869
$$685$$ −15.7428 −0.601502
$$686$$ 0 0
$$687$$ 22.5764 0.861345
$$688$$ 11.4478 0.436445
$$689$$ 0 0
$$690$$ −14.8366 −0.564821
$$691$$ −49.9895 −1.90169 −0.950845 0.309667i $$-0.899782\pi$$
−0.950845 + 0.309667i $$0.899782\pi$$
$$692$$ 16.9652 0.644921
$$693$$ 0 0
$$694$$ 20.7671 0.788308
$$695$$ −36.8336 −1.39718
$$696$$ −5.12859 −0.194399
$$697$$ −40.5734 −1.53683
$$698$$ −9.51364 −0.360097
$$699$$ 0.742815 0.0280958
$$700$$ 0 0
$$701$$ −17.7050 −0.668710 −0.334355 0.942447i $$-0.608518\pi$$
−0.334355 + 0.942447i $$0.608518\pi$$
$$702$$ 0 0
$$703$$ −67.9547 −2.56296
$$704$$ −1.00000 −0.0376889
$$705$$ 0.906181 0.0341288
$$706$$ −12.5764 −0.473320
$$707$$ 0 0
$$708$$ 0.871407 0.0327495
$$709$$ −31.9547 −1.20008 −0.600042 0.799968i $$-0.704850\pi$$
−0.600042 + 0.799968i $$0.704850\pi$$
$$710$$ 20.8957 0.784201
$$711$$ 6.80560 0.255230
$$712$$ 1.61121 0.0603825
$$713$$ 47.4100 1.77552
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.6703 −0.473509
$$717$$ −4.31925 −0.161305
$$718$$ −0.0695483 −0.00259552
$$719$$ 21.7951 0.812820 0.406410 0.913691i $$-0.366780\pi$$
0.406410 + 0.913691i $$0.366780\pi$$
$$720$$ −2.80560 −0.104559
$$721$$ 0 0
$$722$$ 23.0243 0.856875
$$723$$ 18.1876 0.676406
$$724$$ −2.57643 −0.0957523
$$725$$ 14.7263 0.546920
$$726$$ −1.00000 −0.0371135
$$727$$ −4.84714 −0.179770 −0.0898852 0.995952i $$-0.528650\pi$$
−0.0898852 + 0.995952i $$0.528650\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −35.2845 −1.30594
$$731$$ 91.1846 3.37259
$$732$$ −2.15962 −0.0798220
$$733$$ 12.2292 0.451695 0.225847 0.974163i $$-0.427485\pi$$
0.225847 + 0.974163i $$0.427485\pi$$
$$734$$ −31.5417 −1.16422
$$735$$ 0 0
$$736$$ −5.28822 −0.194926
$$737$$ 7.09382 0.261304
$$738$$ −5.09382 −0.187506
$$739$$ −1.03477 −0.0380648 −0.0190324 0.999819i $$-0.506059\pi$$
−0.0190324 + 0.999819i $$0.506059\pi$$
$$740$$ −29.4100 −1.08113
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −23.9925 −0.880200 −0.440100 0.897949i $$-0.645057\pi$$
−0.440100 + 0.897949i $$0.645057\pi$$
$$744$$ 8.96523 0.328681
$$745$$ −46.9652 −1.72067
$$746$$ 15.3125 0.560630
$$747$$ 15.0938 0.552254
$$748$$ −7.96523 −0.291238
$$749$$ 0 0
$$750$$ −5.97199 −0.218066
$$751$$ 20.5069 0.748307 0.374153 0.927367i $$-0.377933\pi$$
0.374153 + 0.927367i $$0.377933\pi$$
$$752$$ 0.322990 0.0117782
$$753$$ −28.6703 −1.04480
$$754$$ 0 0
$$755$$ −30.5794 −1.11290
$$756$$ 0 0
$$757$$ −16.8019 −0.610674 −0.305337 0.952244i $$-0.598769\pi$$
−0.305337 + 0.952244i $$0.598769\pi$$
$$758$$ 16.0590 0.583291
$$759$$ −5.28822 −0.191950
$$760$$ 18.1876 0.659735
$$761$$ −14.5099 −0.525983 −0.262992 0.964798i $$-0.584709\pi$$
−0.262992 + 0.964798i $$0.584709\pi$$
$$762$$ 9.28822 0.336477
$$763$$ 0 0
$$764$$ −2.38879 −0.0864235
$$765$$ −22.3473 −0.807967
$$766$$ −0.739798 −0.0267300
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 3.35402 0.120949 0.0604745 0.998170i $$-0.480739\pi$$
0.0604745 + 0.998170i $$0.480739\pi$$
$$770$$ 0 0
$$771$$ 13.6112 0.490196
$$772$$ −14.5764 −0.524617
$$773$$ 1.84038 0.0661938 0.0330969 0.999452i $$-0.489463\pi$$
0.0330969 + 0.999452i $$0.489463\pi$$
$$774$$ 11.4478 0.411484
$$775$$ −25.7428 −0.924709
$$776$$ 7.00000 0.251285
$$777$$ 0 0
$$778$$ 3.58319 0.128464
$$779$$ 33.0213 1.18311
$$780$$ 0 0
$$781$$ 7.44784 0.266505
$$782$$ −42.1218 −1.50627
$$783$$ −5.12859 −0.183281
$$784$$ 0 0
$$785$$ −4.97875 −0.177699
$$786$$ −12.9652 −0.462454
$$787$$ 11.5869 0.413030 0.206515 0.978443i $$-0.433788\pi$$
0.206515 + 0.978443i $$0.433788\pi$$
$$788$$ −12.4826 −0.444675
$$789$$ 3.61121 0.128562
$$790$$ −19.0938 −0.679328
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ 0 0
$$794$$ −25.1286 −0.891780
$$795$$ 22.4448 0.796036
$$796$$ −14.1876 −0.502867
$$797$$ −48.4653 −1.71673 −0.858365 0.513039i $$-0.828520\pi$$
−0.858365 + 0.513039i $$0.828520\pi$$
$$798$$ 0 0
$$799$$ 2.57269 0.0910151
$$800$$ 2.87141 0.101520
$$801$$ 1.61121 0.0569292
$$802$$ −3.86839 −0.136598
$$803$$ −12.5764 −0.443813
$$804$$ 7.09382 0.250180
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 9.38203 0.330263
$$808$$ 10.7398 0.377825
$$809$$ −2.76708 −0.0972855 −0.0486428 0.998816i $$-0.515490\pi$$
−0.0486428 + 0.998816i $$0.515490\pi$$
$$810$$ −2.80560 −0.0985788
$$811$$ 4.89568 0.171910 0.0859552 0.996299i $$-0.472606\pi$$
0.0859552 + 0.996299i $$0.472606\pi$$
$$812$$ 0 0
$$813$$ −5.61121 −0.196794
$$814$$ −10.4826 −0.367415
$$815$$ 11.3881 0.398907
$$816$$ −7.96523 −0.278839
$$817$$ −74.2119 −2.59635
$$818$$ 4.64598 0.162443
$$819$$ 0 0
$$820$$ 14.2912 0.499071
$$821$$ 3.54915 0.123866 0.0619330 0.998080i $$-0.480273\pi$$
0.0619330 + 0.998080i $$0.480273\pi$$
$$822$$ −5.61121 −0.195713
$$823$$ 32.5069 1.13312 0.566559 0.824021i $$-0.308274\pi$$
0.566559 + 0.824021i $$0.308274\pi$$
$$824$$ 2.96523 0.103299
$$825$$ 2.87141 0.0999696
$$826$$ 0 0
$$827$$ 19.5447 0.679635 0.339817 0.940491i $$-0.389635\pi$$
0.339817 + 0.940491i $$0.389635\pi$$
$$828$$ −5.28822 −0.183778
$$829$$ 25.1907 0.874908 0.437454 0.899241i $$-0.355880\pi$$
0.437454 + 0.899241i $$0.355880\pi$$
$$830$$ −42.3473 −1.46989
$$831$$ 16.0000 0.555034
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −13.1286 −0.454606
$$835$$ −38.9092 −1.34651
$$836$$ 6.48261 0.224206
$$837$$ 8.96523 0.309884
$$838$$ −19.5174 −0.674217
$$839$$ −50.4033 −1.74011 −0.870057 0.492950i $$-0.835918\pi$$
−0.870057 + 0.492950i $$0.835918\pi$$
$$840$$ 0 0
$$841$$ −2.69754 −0.0930185
$$842$$ 15.5174 0.534764
$$843$$ −24.1529 −0.831869
$$844$$ −20.8336 −0.717123
$$845$$ 36.4728 1.25470
$$846$$ 0.322990 0.0111046
$$847$$ 0 0
$$848$$ 8.00000 0.274721
$$849$$ −7.54166 −0.258829
$$850$$ 22.8714 0.784483
$$851$$ −55.4343 −1.90026
$$852$$ 7.44784 0.255159
$$853$$ −16.8752 −0.577794 −0.288897 0.957360i $$-0.593289\pi$$
−0.288897 + 0.957360i $$0.593289\pi$$
$$854$$ 0 0
$$855$$ 18.1876 0.622004
$$856$$ 4.83663 0.165313
$$857$$ 43.3057 1.47930 0.739648 0.672994i $$-0.234992\pi$$
0.739648 + 0.672994i $$0.234992\pi$$
$$858$$ 0 0
$$859$$ 3.61422 0.123316 0.0616578 0.998097i $$-0.480361\pi$$
0.0616578 + 0.998097i $$0.480361\pi$$
$$860$$ −32.1181 −1.09522
$$861$$ 0 0
$$862$$ −14.3888 −0.490084
$$863$$ 4.41681 0.150350 0.0751750 0.997170i $$-0.476048\pi$$
0.0751750 + 0.997170i $$0.476048\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −47.5977 −1.61837
$$866$$ 2.03477 0.0691444
$$867$$ −46.4448 −1.57735
$$868$$ 0 0
$$869$$ −6.80560 −0.230864
$$870$$ 14.3888 0.487826
$$871$$ 0 0
$$872$$ −10.1596 −0.344048
$$873$$ 7.00000 0.236914
$$874$$ 34.2815 1.15959
$$875$$ 0 0
$$876$$ −12.5764 −0.424918
$$877$$ 44.2777 1.49515 0.747576 0.664176i $$-0.231218\pi$$
0.747576 + 0.664176i $$0.231218\pi$$
$$878$$ 3.22616 0.108877
$$879$$ −7.90618 −0.266669
$$880$$ 2.80560 0.0945769
$$881$$ −18.1876 −0.612757 −0.306379 0.951910i $$-0.599117\pi$$
−0.306379 + 0.951910i $$0.599117\pi$$
$$882$$ 0 0
$$883$$ −4.57945 −0.154111 −0.0770553 0.997027i $$-0.524552\pi$$
−0.0770553 + 0.997027i $$0.524552\pi$$
$$884$$ 0 0
$$885$$ −2.44482 −0.0821818
$$886$$ −22.0938 −0.742256
$$887$$ −0.576432 −0.0193547 −0.00967734 0.999953i $$-0.503080\pi$$
−0.00967734 + 0.999953i $$0.503080\pi$$
$$888$$ −10.4826 −0.351773
$$889$$ 0 0
$$890$$ −4.52040 −0.151524
$$891$$ −1.00000 −0.0335013
$$892$$ 12.5764 0.421090
$$893$$ −2.09382 −0.0700670
$$894$$ −16.7398 −0.559863
$$895$$ 35.5477 1.18823
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 22.2497 0.742482
$$899$$ −45.9790 −1.53349
$$900$$ 2.87141 0.0957136
$$901$$ 63.7218 2.12288
$$902$$ 5.09382 0.169606
$$903$$ 0 0
$$904$$ 10.9652 0.364698
$$905$$ 7.22844 0.240282
$$906$$ −10.8994 −0.362109
$$907$$ −22.0590 −0.732459 −0.366229 0.930525i $$-0.619351\pi$$
−0.366229 + 0.930525i $$0.619351\pi$$
$$908$$ 20.0590 0.665683
$$909$$ 10.7398 0.356217
$$910$$ 0 0
$$911$$ 9.93420 0.329135 0.164567 0.986366i $$-0.447377\pi$$
0.164567 + 0.986366i $$0.447377\pi$$
$$912$$ 6.48261 0.214661
$$913$$ −15.0938 −0.499532
$$914$$ −18.6385 −0.616507
$$915$$ 6.05904 0.200306
$$916$$ −22.5764 −0.745946
$$917$$ 0 0
$$918$$ −7.96523 −0.262892
$$919$$ 44.6347 1.47236 0.736182 0.676783i $$-0.236627\pi$$
0.736182 + 0.676783i $$0.236627\pi$$
$$920$$ 14.8366 0.489149
$$921$$ −12.8957 −0.424927
$$922$$ −20.9895 −0.691252
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 30.0999 0.989677
$$926$$ −5.35402 −0.175944
$$927$$ 2.96523 0.0973908
$$928$$ 5.12859 0.168354
$$929$$ −42.5764 −1.39689 −0.698444 0.715665i $$-0.746124\pi$$
−0.698444 + 0.715665i $$0.746124\pi$$
$$930$$ −25.1529 −0.824795
$$931$$ 0 0
$$932$$ −0.742815 −0.0243317
$$933$$ −10.8994 −0.356831
$$934$$ 26.0243 0.851540
$$935$$ 22.3473 0.730834
$$936$$ 0 0
$$937$$ −36.2572 −1.18447 −0.592235 0.805765i $$-0.701754\pi$$
−0.592235 + 0.805765i $$0.701754\pi$$
$$938$$ 0 0
$$939$$ 9.12859 0.297900
$$940$$ −0.906181 −0.0295564
$$941$$ 2.68377 0.0874884 0.0437442 0.999043i $$-0.486071\pi$$
0.0437442 + 0.999043i $$0.486071\pi$$
$$942$$ −1.77457 −0.0578187
$$943$$ 26.9372 0.877196
$$944$$ −0.871407 −0.0283619
$$945$$ 0 0
$$946$$ −11.4478 −0.372201
$$947$$ 49.5039 1.60866 0.804330 0.594183i $$-0.202525\pi$$
0.804330 + 0.594183i $$0.202525\pi$$
$$948$$ −6.80560 −0.221036
$$949$$ 0 0
$$950$$ −18.6142 −0.603925
$$951$$ 9.51364 0.308501
$$952$$ 0 0
$$953$$ 15.8019 0.511872 0.255936 0.966694i $$-0.417616\pi$$
0.255936 + 0.966694i $$0.417616\pi$$
$$954$$ 8.00000 0.259010
$$955$$ 6.70201 0.216872
$$956$$ 4.31925 0.139694
$$957$$ 5.12859 0.165784
$$958$$ −3.61121 −0.116673
$$959$$ 0 0
$$960$$ 2.80560 0.0905504
$$961$$ 49.3753 1.59275
$$962$$ 0 0
$$963$$ 4.83663 0.155858
$$964$$ −18.1876 −0.585784
$$965$$ 40.8957 1.31648
$$966$$ 0 0
$$967$$ −12.6497 −0.406788 −0.203394 0.979097i $$-0.565197\pi$$
−0.203394 + 0.979097i $$0.565197\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 51.6355 1.65877
$$970$$ −19.6392 −0.630577
$$971$$ 49.5342 1.58963 0.794814 0.606854i $$-0.207569\pi$$
0.794814 + 0.606854i $$0.207569\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −8.06206 −0.258325
$$975$$ 0 0
$$976$$ 2.15962 0.0691278
$$977$$ 34.4448 1.10199 0.550994 0.834509i $$-0.314249\pi$$
0.550994 + 0.834509i $$0.314249\pi$$
$$978$$ 4.05904 0.129794
$$979$$ −1.61121 −0.0514944
$$980$$ 0 0
$$981$$ −10.1596 −0.324372
$$982$$ 20.9062 0.667143
$$983$$ −3.35776 −0.107096 −0.0535480 0.998565i $$-0.517053\pi$$
−0.0535480 + 0.998565i $$0.517053\pi$$
$$984$$ 5.09382 0.162385
$$985$$ 35.0213 1.11587
$$986$$ 40.8504 1.30094
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −60.5386 −1.92502
$$990$$ 2.80560 0.0891679
$$991$$ −45.8474 −1.45639 −0.728195 0.685370i $$-0.759641\pi$$
−0.728195 + 0.685370i $$0.759641\pi$$
$$992$$ −8.96523 −0.284646
$$993$$ −1.09382 −0.0347113
$$994$$ 0 0
$$995$$ 39.8049 1.26190
$$996$$ −15.0938 −0.478266
$$997$$ −43.8609 −1.38909 −0.694544 0.719450i $$-0.744394\pi$$
−0.694544 + 0.719450i $$0.744394\pi$$
$$998$$ 4.96523 0.157171
$$999$$ −10.4826 −0.331655
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 3234.2.a.bf.1.1 3
3.2 odd 2 9702.2.a.dv.1.3 3
7.2 even 3 462.2.i.g.67.3 6
7.4 even 3 462.2.i.g.331.3 yes 6
7.6 odd 2 3234.2.a.bh.1.3 3
21.2 odd 6 1386.2.k.v.991.1 6
21.11 odd 6 1386.2.k.v.793.1 6
21.20 even 2 9702.2.a.dw.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.g.67.3 6 7.2 even 3
462.2.i.g.331.3 yes 6 7.4 even 3
1386.2.k.v.793.1 6 21.11 odd 6
1386.2.k.v.991.1 6 21.2 odd 6
3234.2.a.bf.1.1 3 1.1 even 1 trivial
3234.2.a.bh.1.3 3 7.6 odd 2
9702.2.a.dv.1.3 3 3.2 odd 2
9702.2.a.dw.1.1 3 21.20 even 2