# Properties

 Label 1050.2.a.c.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{21} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +2.00000 q^{51} +2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +12.0000 q^{59} +14.0000 q^{61} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} -4.00000 q^{86} +2.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} +8.00000 q^{92} +1.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 4.00000 0.852803
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.00000 0.696311
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 2.00000 0.277350
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −4.00000 −0.529813
$$58$$ 2.00000 0.262613
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ −4.00000 −0.455842
$$78$$ 2.00000 0.226455
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 2.00000 0.214423
$$88$$ 4.00000 0.426401
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 1.00000 0.0944911
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000 0.184900
$$118$$ −12.0000 −1.10469
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −14.0000 −1.26750
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 4.00000 0.346844
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −8.00000 −0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ −1.00000 −0.0824786
$$148$$ −6.00000 −0.493197
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ −12.0000 −0.901975
$$178$$ −10.0000 −0.749532
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −14.0000 −1.03491
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 4.00000 0.284268
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ −6.00000 −0.422159
$$203$$ −2.00000 −0.140372
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 8.00000 0.556038
$$208$$ 2.00000 0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 8.00000 0.548151
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −14.0000 −0.948200
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ −6.00000 −0.402694
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 2.00000 0.131306
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ −16.0000 −1.03931
$$238$$ 2.00000 0.129641
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −32.0000 −2.01182
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 4.00000 0.249029
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ −20.0000 −1.23560
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ −10.0000 −0.611990
$$268$$ 12.0000 0.733017
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −2.00000 −0.121046
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ −6.00000 −0.354169
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ −10.0000 −0.585206
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 4.00000 0.232104
$$298$$ 10.0000 0.579284
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 8.00000 0.460348
$$303$$ −6.00000 −0.344691
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ −8.00000 −0.445823
$$323$$ −8.00000 −0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ −14.0000 −0.774202
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 1.00000 0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 4.00000 0.213201
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 2.00000 0.105851
$$358$$ −4.00000 −0.211407
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −6.00000 −0.315353
$$363$$ −5.00000 −0.262432
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 8.00000 0.417029
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 10.0000 0.519174
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 1.00000 0.0514344
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 16.0000 0.818631
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 4.00000 0.203331
$$388$$ −2.00000 −0.101535
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ −1.00000 −0.0505076
$$393$$ −20.0000 −1.00887
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ 24.0000 1.20301
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 24.0000 1.18964
$$408$$ −2.00000 −0.0990148
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ 8.00000 0.394132
$$413$$ 12.0000 0.590481
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 4.00000 0.195881
$$418$$ 16.0000 0.782586
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 14.0000 0.677507
$$428$$ −12.0000 −0.580042
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 32.0000 1.53077
$$438$$ −10.0000 −0.477818
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 4.00000 0.190261
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 10.0000 0.472984
$$448$$ 1.00000 0.0472456
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ −18.0000 −0.846649
$$453$$ 8.00000 0.375873
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ −12.0000 −0.552345
$$473$$ −16.0000 −0.735681
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 10.0000 0.457869
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 14.0000 0.637683
$$483$$ −8.00000 −0.364013
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 4.00000 0.180151
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 12.0000 0.537733
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ −12.0000 −0.535586
$$503$$ −40.0000 −1.78351 −0.891756 0.452517i $$-0.850526\pi$$
−0.891756 + 0.452517i $$0.850526\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ 9.00000 0.399704
$$508$$ 16.0000 0.709885
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 6.00000 0.263625
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 4.00000 0.173422
$$533$$ −12.0000 −0.519778
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ −4.00000 −0.172613
$$538$$ 18.0000 0.776035
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ −6.00000 −0.257485
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 8.00000 0.340503
$$553$$ 16.0000 0.680389
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 6.00000 0.253095
$$563$$ −20.0000 −0.842900 −0.421450 0.906852i $$-0.638479\pi$$
−0.421450 + 0.906852i $$0.638479\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 28.0000 1.17693
$$567$$ 1.00000 0.0419961
$$568$$ 8.00000 0.335673
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 16.0000 0.668410
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 2.00000 0.0831172
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ −2.00000 −0.0829027
$$583$$ −40.0000 −1.65663
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ −6.00000 −0.246598
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 24.0000 0.982255
$$598$$ −16.0000 −0.654289
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ 12.0000 0.488678
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 8.00000 0.320771
$$623$$ 10.0000 0.400642
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 16.0000 0.638978
$$628$$ 18.0000 0.718278
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ 12.0000 0.476957
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ 2.00000 0.0792429
$$638$$ −8.00000 −0.316723
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −20.0000 −0.783260
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 4.00000 0.155347
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ −16.0000 −0.619522
$$668$$ 8.00000 0.309529
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 1.00000 0.0385758
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 26.0000 0.999261 0.499631 0.866239i $$-0.333469\pi$$
0.499631 + 0.866239i $$0.333469\pi$$
$$678$$ −18.0000 −0.691286
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 10.0000 0.381524
$$688$$ 4.00000 0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −4.00000 −0.151947
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 12.0000 0.454532
$$698$$ −14.0000 −0.529908
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −24.0000 −0.905177
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 6.00000 0.225653
$$708$$ −12.0000 −0.450988
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ −10.0000 −0.374766
$$713$$ 0 0
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 3.00000 0.111648
$$723$$ 14.0000 0.520666
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ −14.0000 −0.517455
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ −48.0000 −1.76810
$$738$$ 6.00000 0.220863
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ −10.0000 −0.367112
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ 12.0000 0.439057
$$748$$ 8.00000 0.292509
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 14.0000 0.506834
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 24.0000 0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ −2.00000 −0.0719816
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 6.00000 0.215249
$$778$$ 26.0000 0.932145
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 16.0000 0.572159
$$783$$ 2.00000 0.0714742
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ −20.0000 −0.712923 −0.356462 0.934310i $$-0.616017\pi$$
−0.356462 + 0.934310i $$0.616017\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 4.00000 0.142134
$$793$$ 28.0000 0.994309
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 14.0000 0.494357
$$803$$ 40.0000 1.41157
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ −6.00000 −0.211079
$$809$$ −22.0000 −0.773479 −0.386739 0.922189i $$-0.626399\pi$$
−0.386739 + 0.922189i $$0.626399\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ −16.0000 −0.561144
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 16.0000 0.559769
$$818$$ 38.0000 1.32864
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ −10.0000 −0.348790
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 0 0
$$831$$ 22.0000 0.763172
$$832$$ 2.00000 0.0693375
$$833$$ −2.00000 −0.0692959
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ −20.0000 −0.690889
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000 0.896019
$$843$$ 6.00000 0.206651
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ 10.0000 0.343401
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ −48.0000 −1.64542
$$852$$ 8.00000 0.274075
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −14.0000 −0.479070
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ −14.0000 −0.474100
$$873$$ −2.00000 −0.0676897
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ 8.00000 0.269987
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ −16.0000 −0.534224
$$898$$ −2.00000 −0.0667409
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ −24.0000 −0.799113
$$903$$ −4.00000 −0.133112
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −48.0000 −1.58857
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 20.0000 0.660458
$$918$$ −2.00000 −0.0660098
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ −14.0000 −0.461065
$$923$$ −16.0000 −0.526646
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 2.00000 0.0656532
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 22.0000 0.720634
$$933$$ 8.00000 0.261908
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 18.0000 0.586472
$$943$$ −48.0000 −1.56310
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 2.00000 0.0648204
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −8.00000 −0.258603
$$958$$ −32.0000 −1.03387
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 12.0000 0.386896
$$963$$ −12.0000 −0.386695
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −4.00000 −0.128234
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 36.0000 1.14881
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ −12.0000 −0.380808
$$994$$ 8.00000 0.253745
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 12.0000 0.379853
$$999$$ 6.00000 0.189832
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 1050.2.a.c.1.1 1
3.2 odd 2 3150.2.a.bp.1.1 1
4.3 odd 2 8400.2.a.ce.1.1 1
5.2 odd 4 1050.2.g.g.799.1 2
5.3 odd 4 1050.2.g.g.799.2 2
5.4 even 2 210.2.a.e.1.1 1
7.6 odd 2 7350.2.a.w.1.1 1
15.2 even 4 3150.2.g.q.2899.2 2
15.8 even 4 3150.2.g.q.2899.1 2
15.14 odd 2 630.2.a.a.1.1 1
20.19 odd 2 1680.2.a.j.1.1 1
35.4 even 6 1470.2.i.a.961.1 2
35.9 even 6 1470.2.i.a.361.1 2
35.19 odd 6 1470.2.i.j.361.1 2
35.24 odd 6 1470.2.i.j.961.1 2
35.34 odd 2 1470.2.a.j.1.1 1
40.19 odd 2 6720.2.a.bq.1.1 1
40.29 even 2 6720.2.a.j.1.1 1
60.59 even 2 5040.2.a.k.1.1 1
105.104 even 2 4410.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.e.1.1 1 5.4 even 2
630.2.a.a.1.1 1 15.14 odd 2
1050.2.a.c.1.1 1 1.1 even 1 trivial
1050.2.g.g.799.1 2 5.2 odd 4
1050.2.g.g.799.2 2 5.3 odd 4
1470.2.a.j.1.1 1 35.34 odd 2
1470.2.i.a.361.1 2 35.9 even 6
1470.2.i.a.961.1 2 35.4 even 6
1470.2.i.j.361.1 2 35.19 odd 6
1470.2.i.j.961.1 2 35.24 odd 6
1680.2.a.j.1.1 1 20.19 odd 2
3150.2.a.bp.1.1 1 3.2 odd 2
3150.2.g.q.2899.1 2 15.8 even 4
3150.2.g.q.2899.2 2 15.2 even 4
4410.2.a.t.1.1 1 105.104 even 2
5040.2.a.k.1.1 1 60.59 even 2
6720.2.a.j.1.1 1 40.29 even 2
6720.2.a.bq.1.1 1 40.19 odd 2
7350.2.a.w.1.1 1 7.6 odd 2
8400.2.a.ce.1.1 1 4.3 odd 2