# Properties

 Label 2450.2.a.bj.1.2 Level $2450$ Weight $2$ Character 2450.1 Self dual yes Analytic conductor $19.563$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.41421 q^{12} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -7.07107 q^{19} +2.00000 q^{22} +4.00000 q^{23} -1.41421 q^{24} -5.65685 q^{27} +2.00000 q^{29} +8.48528 q^{31} -1.00000 q^{32} -2.82843 q^{33} -1.41421 q^{34} -1.00000 q^{36} -10.0000 q^{37} +7.07107 q^{38} -9.89949 q^{41} -2.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -2.82843 q^{47} +1.41421 q^{48} +2.00000 q^{51} +2.00000 q^{53} +5.65685 q^{54} -10.0000 q^{57} -2.00000 q^{58} -1.41421 q^{59} +2.82843 q^{61} -8.48528 q^{62} +1.00000 q^{64} +2.82843 q^{66} -12.0000 q^{67} +1.41421 q^{68} +5.65685 q^{69} -12.0000 q^{71} +1.00000 q^{72} +1.41421 q^{73} +10.0000 q^{74} -7.07107 q^{76} -4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{82} -9.89949 q^{83} +2.00000 q^{86} +2.82843 q^{87} +2.00000 q^{88} -7.07107 q^{89} +4.00000 q^{92} +12.0000 q^{93} +2.82843 q^{94} -1.41421 q^{96} -9.89949 q^{97} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 4 q^{11} + 2 q^{16} + 2 q^{18} + 4 q^{22} + 8 q^{23} + 4 q^{29} - 2 q^{32} - 2 q^{36} - 20 q^{37} - 4 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{51} + 4 q^{53} - 20 q^{57} - 4 q^{58} + 2 q^{64} - 24 q^{67} - 24 q^{71} + 2 q^{72} + 20 q^{74} - 8 q^{79} - 10 q^{81} + 4 q^{86} + 4 q^{88} + 8 q^{92} + 24 q^{93} + 4 q^{99} + O(q^{100})$$

## 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
$$3$$ 1.41421 0.816497 0.408248 0.912871i $$-0.366140\pi$$
0.408248 + 0.912871i $$0.366140\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.41421 −0.577350
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.41421 0.408248
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.41421 0.342997 0.171499 0.985184i $$-0.445139\pi$$
0.171499 + 0.985184i $$0.445139\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −7.07107 −1.62221 −0.811107 0.584898i $$-0.801135\pi$$
−0.811107 + 0.584898i $$0.801135\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.41421 −0.288675
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 8.48528 1.52400 0.762001 0.647576i $$-0.224217\pi$$
0.762001 + 0.647576i $$0.224217\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.82843 −0.492366
$$34$$ −1.41421 −0.242536
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 7.07107 1.14708
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.89949 −1.54604 −0.773021 0.634381i $$-0.781255\pi$$
−0.773021 + 0.634381i $$0.781255\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 1.41421 0.204124
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 5.65685 0.769800
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −10.0000 −1.32453
$$58$$ −2.00000 −0.262613
$$59$$ −1.41421 −0.184115 −0.0920575 0.995754i $$-0.529344\pi$$
−0.0920575 + 0.995754i $$0.529344\pi$$
$$60$$ 0 0
$$61$$ 2.82843 0.362143 0.181071 0.983470i $$-0.442043\pi$$
0.181071 + 0.983470i $$0.442043\pi$$
$$62$$ −8.48528 −1.07763
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.82843 0.348155
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 1.41421 0.171499
$$69$$ 5.65685 0.681005
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 1.41421 0.165521 0.0827606 0.996569i $$-0.473626\pi$$
0.0827606 + 0.996569i $$0.473626\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −7.07107 −0.811107
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 9.89949 1.09322
$$83$$ −9.89949 −1.08661 −0.543305 0.839535i $$-0.682827\pi$$
−0.543305 + 0.839535i $$0.682827\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 2.82843 0.303239
$$88$$ 2.00000 0.213201
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 12.0000 1.24434
$$94$$ 2.82843 0.291730
$$95$$ 0 0
$$96$$ −1.41421 −0.144338
$$97$$ −9.89949 −1.00514 −0.502571 0.864536i $$-0.667612\pi$$
−0.502571 + 0.864536i $$0.667612\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 8.48528 0.844317 0.422159 0.906522i $$-0.361273\pi$$
0.422159 + 0.906522i $$0.361273\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −2.82843 −0.278693 −0.139347 0.990244i $$-0.544500\pi$$
−0.139347 + 0.990244i $$0.544500\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ −5.65685 −0.544331
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −14.1421 −1.34231
$$112$$ 0 0
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 10.0000 0.936586
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 1.41421 0.130189
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −2.82843 −0.256074
$$123$$ −14.0000 −1.26234
$$124$$ 8.48528 0.762001
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −2.82843 −0.249029
$$130$$ 0 0
$$131$$ 12.7279 1.11204 0.556022 0.831168i $$-0.312327\pi$$
0.556022 + 0.831168i $$0.312327\pi$$
$$132$$ −2.82843 −0.246183
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −1.41421 −0.121268
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ −5.65685 −0.481543
$$139$$ 9.89949 0.839664 0.419832 0.907602i $$-0.362089\pi$$
0.419832 + 0.907602i $$0.362089\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 12.0000 1.00702
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −1.41421 −0.117041
$$147$$ 0 0
$$148$$ −10.0000 −0.821995
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 7.07107 0.573539
$$153$$ −1.41421 −0.114332
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.3137 0.902932 0.451466 0.892288i $$-0.350901\pi$$
0.451466 + 0.892288i $$0.350901\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 2.82843 0.224309
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 5.00000 0.392837
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ −9.89949 −0.773021
$$165$$ 0 0
$$166$$ 9.89949 0.768350
$$167$$ 19.7990 1.53209 0.766046 0.642786i $$-0.222221\pi$$
0.766046 + 0.642786i $$0.222221\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 7.07107 0.540738
$$172$$ −2.00000 −0.152499
$$173$$ 16.9706 1.29025 0.645124 0.764078i $$-0.276806\pi$$
0.645124 + 0.764078i $$0.276806\pi$$
$$174$$ −2.82843 −0.214423
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ −2.00000 −0.150329
$$178$$ 7.07107 0.529999
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 4.00000 0.295689
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −12.0000 −0.879883
$$187$$ −2.82843 −0.206835
$$188$$ −2.82843 −0.206284
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 1.41421 0.102062
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ 9.89949 0.710742
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 8.48528 0.601506 0.300753 0.953702i $$-0.402762\pi$$
0.300753 + 0.953702i $$0.402762\pi$$
$$200$$ 0 0
$$201$$ −16.9706 −1.19701
$$202$$ −8.48528 −0.597022
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 2.82843 0.197066
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ 14.1421 0.978232
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −16.9706 −1.16280
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 5.65685 0.384900
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 14.1421 0.949158
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ 21.2132 1.40797 0.703985 0.710215i $$-0.251402\pi$$
0.703985 + 0.710215i $$0.251402\pi$$
$$228$$ −10.0000 −0.662266
$$229$$ −16.9706 −1.12145 −0.560723 0.828003i $$-0.689477\pi$$
−0.560723 + 0.828003i $$0.689477\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1.41421 −0.0920575
$$237$$ −5.65685 −0.367452
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −21.2132 −1.36646 −0.683231 0.730202i $$-0.739426\pi$$
−0.683231 + 0.730202i $$0.739426\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 9.89949 0.635053
$$244$$ 2.82843 0.181071
$$245$$ 0 0
$$246$$ 14.0000 0.892607
$$247$$ 0 0
$$248$$ −8.48528 −0.538816
$$249$$ −14.0000 −0.887214
$$250$$ 0 0
$$251$$ 9.89949 0.624851 0.312425 0.949942i $$-0.398859\pi$$
0.312425 + 0.949942i $$0.398859\pi$$
$$252$$ 0 0
$$253$$ −8.00000 −0.502956
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.7279 −0.793946 −0.396973 0.917830i $$-0.629939\pi$$
−0.396973 + 0.917830i $$0.629939\pi$$
$$258$$ 2.82843 0.176090
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ −12.7279 −0.786334
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 2.82843 0.174078
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −10.0000 −0.611990
$$268$$ −12.0000 −0.733017
$$269$$ −11.3137 −0.689809 −0.344904 0.938638i $$-0.612089\pi$$
−0.344904 + 0.938638i $$0.612089\pi$$
$$270$$ 0 0
$$271$$ 22.6274 1.37452 0.687259 0.726413i $$-0.258814\pi$$
0.687259 + 0.726413i $$0.258814\pi$$
$$272$$ 1.41421 0.0857493
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 5.65685 0.340503
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −9.89949 −0.593732
$$279$$ −8.48528 −0.508001
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 4.00000 0.238197
$$283$$ 1.41421 0.0840663 0.0420331 0.999116i $$-0.486616\pi$$
0.0420331 + 0.999116i $$0.486616\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −15.0000 −0.882353
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 1.41421 0.0827606
$$293$$ 19.7990 1.15667 0.578335 0.815800i $$-0.303703\pi$$
0.578335 + 0.815800i $$0.303703\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ 11.3137 0.656488
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ 12.0000 0.689382
$$304$$ −7.07107 −0.405554
$$305$$ 0 0
$$306$$ 1.41421 0.0808452
$$307$$ 9.89949 0.564994 0.282497 0.959268i $$-0.408837\pi$$
0.282497 + 0.959268i $$0.408837\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −11.3137 −0.641542 −0.320771 0.947157i $$-0.603942\pi$$
−0.320771 + 0.947157i $$0.603942\pi$$
$$312$$ 0 0
$$313$$ −12.7279 −0.719425 −0.359712 0.933063i $$-0.617125\pi$$
−0.359712 + 0.933063i $$0.617125\pi$$
$$314$$ −11.3137 −0.638470
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ −2.82843 −0.158610
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 5.65685 0.315735
$$322$$ 0 0
$$323$$ −10.0000 −0.556415
$$324$$ −5.00000 −0.277778
$$325$$ 0 0
$$326$$ 10.0000 0.553849
$$327$$ −2.82843 −0.156412
$$328$$ 9.89949 0.546608
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ −9.89949 −0.543305
$$333$$ 10.0000 0.547997
$$334$$ −19.7990 −1.08335
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 16.9706 0.921714
$$340$$ 0 0
$$341$$ −16.9706 −0.919007
$$342$$ −7.07107 −0.382360
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ −16.9706 −0.912343
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ 2.82843 0.151620
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000 0.106600
$$353$$ 1.41421 0.0752710 0.0376355 0.999292i $$-0.488017\pi$$
0.0376355 + 0.999292i $$0.488017\pi$$
$$354$$ 2.00000 0.106299
$$355$$ 0 0
$$356$$ −7.07107 −0.374766
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ 31.0000 1.63158
$$362$$ 0 0
$$363$$ −9.89949 −0.519589
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −4.00000 −0.209083
$$367$$ −28.2843 −1.47643 −0.738213 0.674567i $$-0.764330\pi$$
−0.738213 + 0.674567i $$0.764330\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 9.89949 0.515347
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 12.0000 0.622171
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 2.82843 0.146254
$$375$$ 0 0
$$376$$ 2.82843 0.145865
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 0 0
$$381$$ −22.6274 −1.15924
$$382$$ 4.00000 0.204658
$$383$$ 36.7696 1.87884 0.939418 0.342773i $$-0.111366\pi$$
0.939418 + 0.342773i $$0.111366\pi$$
$$384$$ −1.41421 −0.0721688
$$385$$ 0 0
$$386$$ −16.0000 −0.814379
$$387$$ 2.00000 0.101666
$$388$$ −9.89949 −0.502571
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 5.65685 0.286079
$$392$$ 0 0
$$393$$ 18.0000 0.907980
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ −22.6274 −1.13564 −0.567819 0.823154i $$-0.692213\pi$$
−0.567819 + 0.823154i $$0.692213\pi$$
$$398$$ −8.48528 −0.425329
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 16.9706 0.846415
$$403$$ 0 0
$$404$$ 8.48528 0.422159
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ −2.00000 −0.0990148
$$409$$ 38.1838 1.88807 0.944033 0.329851i $$-0.106999\pi$$
0.944033 + 0.329851i $$0.106999\pi$$
$$410$$ 0 0
$$411$$ −16.9706 −0.837096
$$412$$ −2.82843 −0.139347
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 14.0000 0.685583
$$418$$ −14.1421 −0.691714
$$419$$ −9.89949 −0.483622 −0.241811 0.970323i $$-0.577741\pi$$
−0.241811 + 0.970323i $$0.577741\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 2.82843 0.137523
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ 16.9706 0.822226
$$427$$ 0 0
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ −5.65685 −0.272166
$$433$$ 29.6985 1.42722 0.713609 0.700544i $$-0.247059\pi$$
0.713609 + 0.700544i $$0.247059\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ −28.2843 −1.35302
$$438$$ −2.00000 −0.0955637
$$439$$ −16.9706 −0.809961 −0.404980 0.914325i $$-0.632722\pi$$
−0.404980 + 0.914325i $$0.632722\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ −14.1421 −0.671156
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.1421 0.668900
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 19.7990 0.932298
$$452$$ 12.0000 0.564433
$$453$$ −22.6274 −1.06313
$$454$$ −21.2132 −0.995585
$$455$$ 0 0
$$456$$ 10.0000 0.468293
$$457$$ −24.0000 −1.12267 −0.561336 0.827588i $$-0.689713\pi$$
−0.561336 + 0.827588i $$0.689713\pi$$
$$458$$ 16.9706 0.792982
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ 39.5980 1.84426 0.922131 0.386878i $$-0.126447\pi$$
0.922131 + 0.386878i $$0.126447\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ −32.5269 −1.50517 −0.752583 0.658497i $$-0.771192\pi$$
−0.752583 + 0.658497i $$0.771192\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 16.0000 0.737241
$$472$$ 1.41421 0.0650945
$$473$$ 4.00000 0.183920
$$474$$ 5.65685 0.259828
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 12.0000 0.548867
$$479$$ −31.1127 −1.42158 −0.710788 0.703407i $$-0.751661\pi$$
−0.710788 + 0.703407i $$0.751661\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 21.2132 0.966235
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ −9.89949 −0.449050
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ −2.82843 −0.128037
$$489$$ −14.1421 −0.639529
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ −14.0000 −0.631169
$$493$$ 2.82843 0.127386
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.48528 0.381000
$$497$$ 0 0
$$498$$ 14.0000 0.627355
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 28.0000 1.25095
$$502$$ −9.89949 −0.441836
$$503$$ −39.5980 −1.76559 −0.882793 0.469762i $$-0.844340\pi$$
−0.882793 + 0.469762i $$0.844340\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ −18.3848 −0.816497
$$508$$ −16.0000 −0.709885
$$509$$ 22.6274 1.00294 0.501471 0.865174i $$-0.332792\pi$$
0.501471 + 0.865174i $$0.332792\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 40.0000 1.76604
$$514$$ 12.7279 0.561405
$$515$$ 0 0
$$516$$ −2.82843 −0.124515
$$517$$ 5.65685 0.248788
$$518$$ 0 0
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ −1.41421 −0.0619578 −0.0309789 0.999520i $$-0.509862\pi$$
−0.0309789 + 0.999520i $$0.509862\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −12.7279 −0.556553 −0.278277 0.960501i $$-0.589763\pi$$
−0.278277 + 0.960501i $$0.589763\pi$$
$$524$$ 12.7279 0.556022
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 12.0000 0.522728
$$528$$ −2.82843 −0.123091
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 1.41421 0.0613716
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 16.9706 0.732334
$$538$$ 11.3137 0.487769
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ −22.6274 −0.971931
$$543$$ 0 0
$$544$$ −1.41421 −0.0606339
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −2.82843 −0.120714
$$550$$ 0 0
$$551$$ −14.1421 −0.602475
$$552$$ −5.65685 −0.240772
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 9.89949 0.419832
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 8.48528 0.359211
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ −16.0000 −0.674919
$$563$$ 1.41421 0.0596020 0.0298010 0.999556i $$-0.490513\pi$$
0.0298010 + 0.999556i $$0.490513\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −1.41421 −0.0594438
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −2.00000 −0.0836974 −0.0418487 0.999124i $$-0.513325\pi$$
−0.0418487 + 0.999124i $$0.513325\pi$$
$$572$$ 0 0
$$573$$ −5.65685 −0.236318
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ 21.2132 0.883117 0.441559 0.897232i $$-0.354426\pi$$
0.441559 + 0.897232i $$0.354426\pi$$
$$578$$ 15.0000 0.623918
$$579$$ 22.6274 0.940363
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 14.0000 0.580319
$$583$$ −4.00000 −0.165663
$$584$$ −1.41421 −0.0585206
$$585$$ 0 0
$$586$$ −19.7990 −0.817889
$$587$$ 29.6985 1.22579 0.612894 0.790165i $$-0.290005\pi$$
0.612894 + 0.790165i $$0.290005\pi$$
$$588$$ 0 0
$$589$$ −60.0000 −2.47226
$$590$$ 0 0
$$591$$ −2.82843 −0.116346
$$592$$ −10.0000 −0.410997
$$593$$ 7.07107 0.290374 0.145187 0.989404i $$-0.453622\pi$$
0.145187 + 0.989404i $$0.453622\pi$$
$$594$$ −11.3137 −0.464207
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 12.0000 0.491127
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ 29.6985 1.21143 0.605713 0.795683i $$-0.292888\pi$$
0.605713 + 0.795683i $$0.292888\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −12.0000 −0.487467
$$607$$ 16.9706 0.688814 0.344407 0.938820i $$-0.388080\pi$$
0.344407 + 0.938820i $$0.388080\pi$$
$$608$$ 7.07107 0.286770
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −1.41421 −0.0571662
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ −9.89949 −0.399511
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 18.3848 0.738947 0.369473 0.929241i $$-0.379538\pi$$
0.369473 + 0.929241i $$0.379538\pi$$
$$620$$ 0 0
$$621$$ −22.6274 −0.908007
$$622$$ 11.3137 0.453638
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 12.7279 0.508710
$$627$$ 20.0000 0.798723
$$628$$ 11.3137 0.451466
$$629$$ −14.1421 −0.563884
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 4.00000 0.159111
$$633$$ −16.9706 −0.674519
$$634$$ 10.0000 0.397151
$$635$$ 0 0
$$636$$ 2.82843 0.112154
$$637$$ 0 0
$$638$$ 4.00000 0.158362
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ −5.65685 −0.223258
$$643$$ −9.89949 −0.390398 −0.195199 0.980764i $$-0.562535\pi$$
−0.195199 + 0.980764i $$0.562535\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 10.0000 0.393445
$$647$$ −8.48528 −0.333591 −0.166795 0.985992i $$-0.553342\pi$$
−0.166795 + 0.985992i $$0.553342\pi$$
$$648$$ 5.00000 0.196419
$$649$$ 2.82843 0.111025
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 2.82843 0.110600
$$655$$ 0 0
$$656$$ −9.89949 −0.386510
$$657$$ −1.41421 −0.0551737
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ 8.48528 0.330039 0.165020 0.986290i $$-0.447231\pi$$
0.165020 + 0.986290i $$0.447231\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ 9.89949 0.384175
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 8.00000 0.309761
$$668$$ 19.7990 0.766046
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −5.65685 −0.218380
$$672$$ 0 0
$$673$$ 12.0000 0.462566 0.231283 0.972887i $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 16.9706 0.652232 0.326116 0.945330i $$-0.394260\pi$$
0.326116 + 0.945330i $$0.394260\pi$$
$$678$$ −16.9706 −0.651751
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 30.0000 1.14960
$$682$$ 16.9706 0.649836
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 7.07107 0.270369
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −24.0000 −0.915657
$$688$$ −2.00000 −0.0762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 12.7279 0.484193 0.242096 0.970252i $$-0.422165\pi$$
0.242096 + 0.970252i $$0.422165\pi$$
$$692$$ 16.9706 0.645124
$$693$$ 0 0
$$694$$ −30.0000 −1.13878
$$695$$ 0 0
$$696$$ −2.82843 −0.107211
$$697$$ −14.0000 −0.530288
$$698$$ 0 0
$$699$$ −33.9411 −1.28377
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ 70.7107 2.66690
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −1.41421 −0.0532246
$$707$$ 0 0
$$708$$ −2.00000 −0.0751646
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 7.07107 0.264999
$$713$$ 33.9411 1.27111
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −16.9706 −0.633777
$$718$$ 32.0000 1.19423
$$719$$ 2.82843 0.105483 0.0527413 0.998608i $$-0.483204\pi$$
0.0527413 + 0.998608i $$0.483204\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −31.0000 −1.15370
$$723$$ −30.0000 −1.11571
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 9.89949 0.367405
$$727$$ 19.7990 0.734304 0.367152 0.930161i $$-0.380333\pi$$
0.367152 + 0.930161i $$0.380333\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 0 0
$$731$$ −2.82843 −0.104613
$$732$$ 4.00000 0.147844
$$733$$ −42.4264 −1.56706 −0.783528 0.621357i $$-0.786582\pi$$
−0.783528 + 0.621357i $$0.786582\pi$$
$$734$$ 28.2843 1.04399
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 24.0000 0.884051
$$738$$ −9.89949 −0.364405
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 9.89949 0.362204
$$748$$ −2.82843 −0.103418
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −2.82843 −0.103142
$$753$$ 14.0000 0.510188
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 26.0000 0.944363
$$759$$ −11.3137 −0.410662
$$760$$ 0 0
$$761$$ −7.07107 −0.256326 −0.128163 0.991753i $$-0.540908\pi$$
−0.128163 + 0.991753i $$0.540908\pi$$
$$762$$ 22.6274 0.819705
$$763$$ 0 0
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −36.7696 −1.32854
$$767$$ 0 0
$$768$$ 1.41421 0.0510310
$$769$$ 29.6985 1.07095 0.535477 0.844550i $$-0.320132\pi$$
0.535477 + 0.844550i $$0.320132\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 16.0000 0.575853
$$773$$ −48.0833 −1.72943 −0.864717 0.502259i $$-0.832502\pi$$
−0.864717 + 0.502259i $$0.832502\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ 9.89949 0.355371
$$777$$ 0 0
$$778$$ −26.0000 −0.932145
$$779$$ 70.0000 2.50801
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ −5.65685 −0.202289
$$783$$ −11.3137 −0.404319
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 1.41421 0.0504113 0.0252056 0.999682i $$-0.491976\pi$$
0.0252056 + 0.999682i $$0.491976\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ −16.9706 −0.604168
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −2.00000 −0.0710669
$$793$$ 0 0
$$794$$ 22.6274 0.803017
$$795$$ 0 0
$$796$$ 8.48528 0.300753
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ 7.07107 0.249844
$$802$$ 18.0000 0.635602
$$803$$ −2.82843 −0.0998130
$$804$$ −16.9706 −0.598506
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −16.0000 −0.563227
$$808$$ −8.48528 −0.298511
$$809$$ −16.0000 −0.562530 −0.281265 0.959630i $$-0.590754\pi$$
−0.281265 + 0.959630i $$0.590754\pi$$
$$810$$ 0 0
$$811$$ −29.6985 −1.04285 −0.521427 0.853296i $$-0.674600\pi$$
−0.521427 + 0.853296i $$0.674600\pi$$
$$812$$ 0 0
$$813$$ 32.0000 1.12229
$$814$$ −20.0000 −0.701000
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 14.1421 0.494771
$$818$$ −38.1838 −1.33506
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 16.9706 0.591916
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 2.82843 0.0985329
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −31.1127 −1.08059 −0.540294 0.841476i $$-0.681687\pi$$
−0.540294 + 0.841476i $$0.681687\pi$$
$$830$$ 0 0
$$831$$ 2.82843 0.0981170
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −14.0000 −0.484780
$$835$$ 0 0
$$836$$ 14.1421 0.489116
$$837$$ −48.0000 −1.65912
$$838$$ 9.89949 0.341972
$$839$$ 19.7990 0.683537 0.341769 0.939784i $$-0.388974\pi$$
0.341769 + 0.939784i $$0.388974\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −30.0000 −1.03387
$$843$$ 22.6274 0.779330
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −2.82843 −0.0972433
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 2.00000 0.0686398
$$850$$ 0 0
$$851$$ −40.0000 −1.37118
$$852$$ −16.9706 −0.581402
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ −18.3848 −0.628012 −0.314006 0.949421i $$-0.601671\pi$$
−0.314006 + 0.949421i $$0.601671\pi$$
$$858$$ 0 0
$$859$$ −26.8701 −0.916795 −0.458397 0.888747i $$-0.651576\pi$$
−0.458397 + 0.888747i $$0.651576\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 5.65685 0.192450
$$865$$ 0 0
$$866$$ −29.6985 −1.00920
$$867$$ −21.2132 −0.720438
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ 9.89949 0.335047
$$874$$ 28.2843 0.956730
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 46.0000 1.55331 0.776655 0.629926i $$-0.216915\pi$$
0.776655 + 0.629926i $$0.216915\pi$$
$$878$$ 16.9706 0.572729
$$879$$ 28.0000 0.944417
$$880$$ 0 0
$$881$$ 29.6985 1.00057 0.500284 0.865862i $$-0.333229\pi$$
0.500284 + 0.865862i $$0.333229\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 36.7696 1.23460 0.617300 0.786728i $$-0.288226\pi$$
0.617300 + 0.786728i $$0.288226\pi$$
$$888$$ 14.1421 0.474579
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 10.0000 0.335013
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ −14.1421 −0.472984
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 16.9706 0.566000
$$900$$ 0 0
$$901$$ 2.82843 0.0942286
$$902$$ −19.7990 −0.659234
$$903$$ 0 0
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ 22.6274 0.751746
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 21.2132 0.703985
$$909$$ −8.48528 −0.281439
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ −10.0000 −0.331133
$$913$$ 19.7990 0.655251
$$914$$ 24.0000 0.793849
$$915$$ 0 0
$$916$$ −16.9706 −0.560723
$$917$$ 0 0
$$918$$ 8.00000 0.264039
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 14.0000 0.461316
$$922$$ −39.5980 −1.30409
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.0000 0.525793
$$927$$ 2.82843 0.0928977
$$928$$ −2.00000 −0.0656532
$$929$$ 32.5269 1.06717 0.533587 0.845745i $$-0.320844\pi$$
0.533587 + 0.845745i $$0.320844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −24.0000 −0.786146
$$933$$ −16.0000 −0.523816
$$934$$ 32.5269 1.06431
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −9.89949 −0.323402 −0.161701 0.986840i $$-0.551698\pi$$
−0.161701 + 0.986840i $$0.551698\pi$$
$$938$$ 0 0
$$939$$ −18.0000 −0.587408
$$940$$ 0 0
$$941$$ −31.1127 −1.01424 −0.507122 0.861874i $$-0.669291\pi$$
−0.507122 + 0.861874i $$0.669291\pi$$
$$942$$ −16.0000 −0.521308
$$943$$ −39.5980 −1.28949
$$944$$ −1.41421 −0.0460287
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ −5.65685 −0.183726
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −14.1421 −0.458590
$$952$$ 0 0
$$953$$ 26.0000 0.842223 0.421111 0.907009i $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ −5.65685 −0.182860
$$958$$ 31.1127 1.00521
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ −21.2132 −0.683231
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12.0000 0.385894 0.192947 0.981209i $$-0.438195\pi$$
0.192947 + 0.981209i $$0.438195\pi$$
$$968$$ 7.00000 0.224989
$$969$$ −14.1421 −0.454311
$$970$$ 0 0
$$971$$ 32.5269 1.04384 0.521919 0.852995i $$-0.325216\pi$$
0.521919 + 0.852995i $$0.325216\pi$$
$$972$$ 9.89949 0.317526
$$973$$ 0 0
$$974$$ 12.0000 0.384505
$$975$$ 0 0
$$976$$ 2.82843 0.0905357
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 14.1421 0.452216
$$979$$ 14.1421 0.451985
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 12.0000 0.382935
$$983$$ −48.0833 −1.53362 −0.766809 0.641875i $$-0.778157\pi$$
−0.766809 + 0.641875i $$0.778157\pi$$
$$984$$ 14.0000 0.446304
$$985$$ 0 0
$$986$$ −2.82843 −0.0900755
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ −8.48528 −0.269408
$$993$$ 14.1421 0.448787
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ 31.1127 0.985349 0.492675 0.870214i $$-0.336019\pi$$
0.492675 + 0.870214i $$0.336019\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 56.5685 1.78975
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 2450.2.a.bj.1.2 2
5.2 odd 4 2450.2.c.v.99.1 4
5.3 odd 4 2450.2.c.v.99.4 4
5.4 even 2 98.2.a.b.1.1 2
7.6 odd 2 inner 2450.2.a.bj.1.1 2
15.14 odd 2 882.2.a.n.1.1 2
20.19 odd 2 784.2.a.l.1.2 2
35.4 even 6 98.2.c.c.79.2 4
35.9 even 6 98.2.c.c.67.2 4
35.13 even 4 2450.2.c.v.99.3 4
35.19 odd 6 98.2.c.c.67.1 4
35.24 odd 6 98.2.c.c.79.1 4
35.27 even 4 2450.2.c.v.99.2 4
35.34 odd 2 98.2.a.b.1.2 yes 2
40.19 odd 2 3136.2.a.bm.1.1 2
40.29 even 2 3136.2.a.bn.1.2 2
60.59 even 2 7056.2.a.cl.1.1 2
105.44 odd 6 882.2.g.l.361.2 4
105.59 even 6 882.2.g.l.667.1 4
105.74 odd 6 882.2.g.l.667.2 4
105.89 even 6 882.2.g.l.361.1 4
105.104 even 2 882.2.a.n.1.2 2
140.19 even 6 784.2.i.m.753.2 4
140.39 odd 6 784.2.i.m.177.1 4
140.59 even 6 784.2.i.m.177.2 4
140.79 odd 6 784.2.i.m.753.1 4
140.139 even 2 784.2.a.l.1.1 2
280.69 odd 2 3136.2.a.bn.1.1 2
280.139 even 2 3136.2.a.bm.1.2 2
420.419 odd 2 7056.2.a.cl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 5.4 even 2
98.2.a.b.1.2 yes 2 35.34 odd 2
98.2.c.c.67.1 4 35.19 odd 6
98.2.c.c.67.2 4 35.9 even 6
98.2.c.c.79.1 4 35.24 odd 6
98.2.c.c.79.2 4 35.4 even 6
784.2.a.l.1.1 2 140.139 even 2
784.2.a.l.1.2 2 20.19 odd 2
784.2.i.m.177.1 4 140.39 odd 6
784.2.i.m.177.2 4 140.59 even 6
784.2.i.m.753.1 4 140.79 odd 6
784.2.i.m.753.2 4 140.19 even 6
882.2.a.n.1.1 2 15.14 odd 2
882.2.a.n.1.2 2 105.104 even 2
882.2.g.l.361.1 4 105.89 even 6
882.2.g.l.361.2 4 105.44 odd 6
882.2.g.l.667.1 4 105.59 even 6
882.2.g.l.667.2 4 105.74 odd 6
2450.2.a.bj.1.1 2 7.6 odd 2 inner
2450.2.a.bj.1.2 2 1.1 even 1 trivial
2450.2.c.v.99.1 4 5.2 odd 4
2450.2.c.v.99.2 4 35.27 even 4
2450.2.c.v.99.3 4 35.13 even 4
2450.2.c.v.99.4 4 5.3 odd 4
3136.2.a.bm.1.1 2 40.19 odd 2
3136.2.a.bm.1.2 2 280.139 even 2
3136.2.a.bn.1.1 2 280.69 odd 2
3136.2.a.bn.1.2 2 40.29 even 2
7056.2.a.cl.1.1 2 60.59 even 2
7056.2.a.cl.1.2 2 420.419 odd 2