# Properties

 Label 2450.4.a.bs.1.1 Level $2450$ Weight $4$ Character 2450.1 Self dual yes Analytic conductor $144.555$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$144.554679514$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.69042$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +18.7617 q^{6} -8.00000 q^{8} +61.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +18.7617 q^{6} -8.00000 q^{8} +61.0000 q^{9} +20.0000 q^{11} -37.5233 q^{12} +65.6658 q^{13} +16.0000 q^{16} +56.2850 q^{17} -122.000 q^{18} -9.38083 q^{19} -40.0000 q^{22} -48.0000 q^{23} +75.0467 q^{24} -131.332 q^{26} -318.948 q^{27} -166.000 q^{29} +206.378 q^{31} -32.0000 q^{32} -187.617 q^{33} -112.570 q^{34} +244.000 q^{36} +78.0000 q^{37} +18.7617 q^{38} -616.000 q^{39} -393.995 q^{41} -436.000 q^{43} +80.0000 q^{44} +96.0000 q^{46} +206.378 q^{47} -150.093 q^{48} -528.000 q^{51} +262.663 q^{52} -62.0000 q^{53} +637.897 q^{54} +88.0000 q^{57} +332.000 q^{58} +666.039 q^{59} -272.044 q^{61} -412.757 q^{62} +64.0000 q^{64} +375.233 q^{66} -580.000 q^{67} +225.140 q^{68} +450.280 q^{69} -544.000 q^{71} -488.000 q^{72} -600.373 q^{73} -156.000 q^{74} -37.5233 q^{76} +1232.00 q^{78} -680.000 q^{79} +1345.00 q^{81} +787.990 q^{82} +196.997 q^{83} +872.000 q^{86} +1557.22 q^{87} -160.000 q^{88} +1500.93 q^{89} -192.000 q^{92} -1936.00 q^{93} -412.757 q^{94} +300.187 q^{96} -656.658 q^{97} +1220.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} + 122q^{9} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} + 122q^{9} + 40q^{11} + 32q^{16} - 244q^{18} - 80q^{22} - 96q^{23} - 332q^{29} - 64q^{32} + 488q^{36} + 156q^{37} - 1232q^{39} - 872q^{43} + 160q^{44} + 192q^{46} - 1056q^{51} - 124q^{53} + 176q^{57} + 664q^{58} + 128q^{64} - 1160q^{67} - 1088q^{71} - 976q^{72} - 312q^{74} + 2464q^{78} - 1360q^{79} + 2690q^{81} + 1744q^{86} - 320q^{88} - 384q^{92} - 3872q^{93} + 2440q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ −9.38083 −1.80534 −0.902671 0.430331i $$-0.858397\pi$$
−0.902671 + 0.430331i $$0.858397\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 18.7617 1.27657
$$7$$ 0 0
$$8$$ −8.00000 −0.353553
$$9$$ 61.0000 2.25926
$$10$$ 0 0
$$11$$ 20.0000 0.548202 0.274101 0.961701i $$-0.411620\pi$$
0.274101 + 0.961701i $$0.411620\pi$$
$$12$$ −37.5233 −0.902671
$$13$$ 65.6658 1.40096 0.700478 0.713674i $$-0.252970\pi$$
0.700478 + 0.713674i $$0.252970\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 56.2850 0.803007 0.401503 0.915858i $$-0.368488\pi$$
0.401503 + 0.915858i $$0.368488\pi$$
$$18$$ −122.000 −1.59754
$$19$$ −9.38083 −0.113269 −0.0566345 0.998395i $$-0.518037\pi$$
−0.0566345 + 0.998395i $$0.518037\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −40.0000 −0.387638
$$23$$ −48.0000 −0.435161 −0.217580 0.976042i $$-0.569816\pi$$
−0.217580 + 0.976042i $$0.569816\pi$$
$$24$$ 75.0467 0.638285
$$25$$ 0 0
$$26$$ −131.332 −0.990625
$$27$$ −318.948 −2.27339
$$28$$ 0 0
$$29$$ −166.000 −1.06295 −0.531473 0.847075i $$-0.678361\pi$$
−0.531473 + 0.847075i $$0.678361\pi$$
$$30$$ 0 0
$$31$$ 206.378 1.19570 0.597849 0.801609i $$-0.296022\pi$$
0.597849 + 0.801609i $$0.296022\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ −187.617 −0.989693
$$34$$ −112.570 −0.567812
$$35$$ 0 0
$$36$$ 244.000 1.12963
$$37$$ 78.0000 0.346571 0.173285 0.984872i $$-0.444562\pi$$
0.173285 + 0.984872i $$0.444562\pi$$
$$38$$ 18.7617 0.0800933
$$39$$ −616.000 −2.52920
$$40$$ 0 0
$$41$$ −393.995 −1.50077 −0.750386 0.661000i $$-0.770132\pi$$
−0.750386 + 0.661000i $$0.770132\pi$$
$$42$$ 0 0
$$43$$ −436.000 −1.54626 −0.773132 0.634245i $$-0.781311\pi$$
−0.773132 + 0.634245i $$0.781311\pi$$
$$44$$ 80.0000 0.274101
$$45$$ 0 0
$$46$$ 96.0000 0.307705
$$47$$ 206.378 0.640497 0.320249 0.947334i $$-0.396234\pi$$
0.320249 + 0.947334i $$0.396234\pi$$
$$48$$ −150.093 −0.451335
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −528.000 −1.44970
$$52$$ 262.663 0.700478
$$53$$ −62.0000 −0.160686 −0.0803430 0.996767i $$-0.525602\pi$$
−0.0803430 + 0.996767i $$0.525602\pi$$
$$54$$ 637.897 1.60753
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 88.0000 0.204489
$$58$$ 332.000 0.751616
$$59$$ 666.039 1.46968 0.734838 0.678243i $$-0.237258\pi$$
0.734838 + 0.678243i $$0.237258\pi$$
$$60$$ 0 0
$$61$$ −272.044 −0.571011 −0.285506 0.958377i $$-0.592162\pi$$
−0.285506 + 0.958377i $$0.592162\pi$$
$$62$$ −412.757 −0.845486
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 375.233 0.699819
$$67$$ −580.000 −1.05759 −0.528793 0.848751i $$-0.677355\pi$$
−0.528793 + 0.848751i $$0.677355\pi$$
$$68$$ 225.140 0.401503
$$69$$ 450.280 0.785613
$$70$$ 0 0
$$71$$ −544.000 −0.909309 −0.454654 0.890668i $$-0.650237\pi$$
−0.454654 + 0.890668i $$0.650237\pi$$
$$72$$ −488.000 −0.798769
$$73$$ −600.373 −0.962580 −0.481290 0.876561i $$-0.659832\pi$$
−0.481290 + 0.876561i $$0.659832\pi$$
$$74$$ −156.000 −0.245063
$$75$$ 0 0
$$76$$ −37.5233 −0.0566345
$$77$$ 0 0
$$78$$ 1232.00 1.78842
$$79$$ −680.000 −0.968430 −0.484215 0.874949i $$-0.660895\pi$$
−0.484215 + 0.874949i $$0.660895\pi$$
$$80$$ 0 0
$$81$$ 1345.00 1.84499
$$82$$ 787.990 1.06121
$$83$$ 196.997 0.260521 0.130261 0.991480i $$-0.458419\pi$$
0.130261 + 0.991480i $$0.458419\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 872.000 1.09337
$$87$$ 1557.22 1.91898
$$88$$ −160.000 −0.193819
$$89$$ 1500.93 1.78762 0.893812 0.448441i $$-0.148021\pi$$
0.893812 + 0.448441i $$0.148021\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −192.000 −0.217580
$$93$$ −1936.00 −2.15864
$$94$$ −412.757 −0.452900
$$95$$ 0 0
$$96$$ 300.187 0.319142
$$97$$ −656.658 −0.687356 −0.343678 0.939088i $$-0.611673\pi$$
−0.343678 + 0.939088i $$0.611673\pi$$
$$98$$ 0 0
$$99$$ 1220.00 1.23853
$$100$$ 0 0
$$101$$ −121.951 −0.120144 −0.0600721 0.998194i $$-0.519133\pi$$
−0.0600721 + 0.998194i $$0.519133\pi$$
$$102$$ 1056.00 1.02509
$$103$$ −1369.60 −1.31020 −0.655101 0.755541i $$-0.727374\pi$$
−0.655101 + 0.755541i $$0.727374\pi$$
$$104$$ −525.327 −0.495313
$$105$$ 0 0
$$106$$ 124.000 0.113622
$$107$$ 260.000 0.234908 0.117454 0.993078i $$-0.462527\pi$$
0.117454 + 0.993078i $$0.462527\pi$$
$$108$$ −1275.79 −1.13670
$$109$$ 1882.00 1.65379 0.826894 0.562358i $$-0.190106\pi$$
0.826894 + 0.562358i $$0.190106\pi$$
$$110$$ 0 0
$$111$$ −731.705 −0.625679
$$112$$ 0 0
$$113$$ 1286.00 1.07059 0.535295 0.844665i $$-0.320200\pi$$
0.535295 + 0.844665i $$0.320200\pi$$
$$114$$ −176.000 −0.144596
$$115$$ 0 0
$$116$$ −664.000 −0.531473
$$117$$ 4005.62 3.16512
$$118$$ −1332.08 −1.03922
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ 544.088 0.403766
$$123$$ 3696.00 2.70941
$$124$$ 825.513 0.597849
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2312.00 −1.61541 −0.807704 0.589588i $$-0.799290\pi$$
−0.807704 + 0.589588i $$0.799290\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 4090.04 2.79154
$$130$$ 0 0
$$131$$ 253.282 0.168927 0.0844633 0.996427i $$-0.473082\pi$$
0.0844633 + 0.996427i $$0.473082\pi$$
$$132$$ −750.467 −0.494846
$$133$$ 0 0
$$134$$ 1160.00 0.747826
$$135$$ 0 0
$$136$$ −450.280 −0.283906
$$137$$ 1114.00 0.694711 0.347356 0.937733i $$-0.387080\pi$$
0.347356 + 0.937733i $$0.387080\pi$$
$$138$$ −900.560 −0.555513
$$139$$ −1378.98 −0.841466 −0.420733 0.907185i $$-0.638227\pi$$
−0.420733 + 0.907185i $$0.638227\pi$$
$$140$$ 0 0
$$141$$ −1936.00 −1.15632
$$142$$ 1088.00 0.642978
$$143$$ 1313.32 0.768007
$$144$$ 976.000 0.564815
$$145$$ 0 0
$$146$$ 1200.75 0.680647
$$147$$ 0 0
$$148$$ 312.000 0.173285
$$149$$ −946.000 −0.520130 −0.260065 0.965591i $$-0.583744\pi$$
−0.260065 + 0.965591i $$0.583744\pi$$
$$150$$ 0 0
$$151$$ 832.000 0.448392 0.224196 0.974544i $$-0.428024\pi$$
0.224196 + 0.974544i $$0.428024\pi$$
$$152$$ 75.0467 0.0400466
$$153$$ 3433.38 1.81420
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2464.00 −1.26460
$$157$$ 2879.92 1.46396 0.731982 0.681324i $$-0.238596\pi$$
0.731982 + 0.681324i $$0.238596\pi$$
$$158$$ 1360.00 0.684783
$$159$$ 581.612 0.290093
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −2690.00 −1.30461
$$163$$ −636.000 −0.305616 −0.152808 0.988256i $$-0.548832\pi$$
−0.152808 + 0.988256i $$0.548832\pi$$
$$164$$ −1575.98 −0.750386
$$165$$ 0 0
$$166$$ −393.995 −0.184216
$$167$$ −656.658 −0.304274 −0.152137 0.988359i $$-0.548615\pi$$
−0.152137 + 0.988359i $$0.548615\pi$$
$$168$$ 0 0
$$169$$ 2115.00 0.962676
$$170$$ 0 0
$$171$$ −572.231 −0.255904
$$172$$ −1744.00 −0.773132
$$173$$ 666.039 0.292705 0.146353 0.989232i $$-0.453247\pi$$
0.146353 + 0.989232i $$0.453247\pi$$
$$174$$ −3114.44 −1.35692
$$175$$ 0 0
$$176$$ 320.000 0.137051
$$177$$ −6248.00 −2.65327
$$178$$ −3001.87 −1.26404
$$179$$ −3228.00 −1.34789 −0.673944 0.738782i $$-0.735401\pi$$
−0.673944 + 0.738782i $$0.735401\pi$$
$$180$$ 0 0
$$181$$ 2823.63 1.15955 0.579776 0.814776i $$-0.303140\pi$$
0.579776 + 0.814776i $$0.303140\pi$$
$$182$$ 0 0
$$183$$ 2552.00 1.03087
$$184$$ 384.000 0.153852
$$185$$ 0 0
$$186$$ 3872.00 1.52639
$$187$$ 1125.70 0.440210
$$188$$ 825.513 0.320249
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2136.00 −0.809191 −0.404596 0.914496i $$-0.632588\pi$$
−0.404596 + 0.914496i $$0.632588\pi$$
$$192$$ −600.373 −0.225668
$$193$$ −1658.00 −0.618370 −0.309185 0.951002i $$-0.600056\pi$$
−0.309185 + 0.951002i $$0.600056\pi$$
$$194$$ 1313.32 0.486034
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 978.000 0.353704 0.176852 0.984237i $$-0.443409\pi$$
0.176852 + 0.984237i $$0.443409\pi$$
$$198$$ −2440.00 −0.875774
$$199$$ 4934.32 1.75771 0.878855 0.477088i $$-0.158308\pi$$
0.878855 + 0.477088i $$0.158308\pi$$
$$200$$ 0 0
$$201$$ 5440.88 1.90930
$$202$$ 243.902 0.0849547
$$203$$ 0 0
$$204$$ −2112.00 −0.724851
$$205$$ 0 0
$$206$$ 2739.20 0.926453
$$207$$ −2928.00 −0.983140
$$208$$ 1050.65 0.350239
$$209$$ −187.617 −0.0620943
$$210$$ 0 0
$$211$$ 1556.00 0.507675 0.253838 0.967247i $$-0.418307\pi$$
0.253838 + 0.967247i $$0.418307\pi$$
$$212$$ −248.000 −0.0803430
$$213$$ 5103.17 1.64161
$$214$$ −520.000 −0.166105
$$215$$ 0 0
$$216$$ 2551.59 0.803766
$$217$$ 0 0
$$218$$ −3764.00 −1.16940
$$219$$ 5632.00 1.73779
$$220$$ 0 0
$$221$$ 3696.00 1.12498
$$222$$ 1463.41 0.442422
$$223$$ 2889.30 0.867630 0.433815 0.901002i $$-0.357167\pi$$
0.433815 + 0.901002i $$0.357167\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2572.00 −0.757022
$$227$$ −1979.36 −0.578742 −0.289371 0.957217i $$-0.593446\pi$$
−0.289371 + 0.957217i $$0.593446\pi$$
$$228$$ 352.000 0.102245
$$229$$ −2767.35 −0.798565 −0.399282 0.916828i $$-0.630741\pi$$
−0.399282 + 0.916828i $$0.630741\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1328.00 0.375808
$$233$$ 6490.00 1.82478 0.912391 0.409321i $$-0.134234\pi$$
0.912391 + 0.409321i $$0.134234\pi$$
$$234$$ −8011.23 −2.23808
$$235$$ 0 0
$$236$$ 2664.16 0.734838
$$237$$ 6378.97 1.74835
$$238$$ 0 0
$$239$$ −4296.00 −1.16270 −0.581350 0.813654i $$-0.697475\pi$$
−0.581350 + 0.813654i $$0.697475\pi$$
$$240$$ 0 0
$$241$$ −4521.56 −1.20854 −0.604272 0.796778i $$-0.706536\pi$$
−0.604272 + 0.796778i $$0.706536\pi$$
$$242$$ 1862.00 0.494603
$$243$$ −4005.62 −1.05745
$$244$$ −1088.18 −0.285506
$$245$$ 0 0
$$246$$ −7392.00 −1.91584
$$247$$ −616.000 −0.158685
$$248$$ −1651.03 −0.422743
$$249$$ −1848.00 −0.470330
$$250$$ 0 0
$$251$$ −5581.59 −1.40361 −0.701807 0.712367i $$-0.747623\pi$$
−0.701807 + 0.712367i $$0.747623\pi$$
$$252$$ 0 0
$$253$$ −960.000 −0.238556
$$254$$ 4624.00 1.14227
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −1500.93 −0.364302 −0.182151 0.983271i $$-0.558306\pi$$
−0.182151 + 0.983271i $$0.558306\pi$$
$$258$$ −8180.09 −1.97391
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −10126.0 −2.40147
$$262$$ −506.565 −0.119449
$$263$$ 400.000 0.0937835 0.0468917 0.998900i $$-0.485068\pi$$
0.0468917 + 0.998900i $$0.485068\pi$$
$$264$$ 1500.93 0.349909
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −14080.0 −3.22727
$$268$$ −2320.00 −0.528793
$$269$$ 272.044 0.0616610 0.0308305 0.999525i $$-0.490185\pi$$
0.0308305 + 0.999525i $$0.490185\pi$$
$$270$$ 0 0
$$271$$ −6904.29 −1.54762 −0.773812 0.633416i $$-0.781652\pi$$
−0.773812 + 0.633416i $$0.781652\pi$$
$$272$$ 900.560 0.200752
$$273$$ 0 0
$$274$$ −2228.00 −0.491235
$$275$$ 0 0
$$276$$ 1801.12 0.392807
$$277$$ 6770.00 1.46848 0.734242 0.678888i $$-0.237538\pi$$
0.734242 + 0.678888i $$0.237538\pi$$
$$278$$ 2757.96 0.595006
$$279$$ 12589.1 2.70139
$$280$$ 0 0
$$281$$ 1878.00 0.398691 0.199345 0.979929i $$-0.436118\pi$$
0.199345 + 0.979929i $$0.436118\pi$$
$$282$$ 3872.00 0.817639
$$283$$ 384.614 0.0807878 0.0403939 0.999184i $$-0.487139\pi$$
0.0403939 + 0.999184i $$0.487139\pi$$
$$284$$ −2176.00 −0.454654
$$285$$ 0 0
$$286$$ −2626.63 −0.543063
$$287$$ 0 0
$$288$$ −1952.00 −0.399384
$$289$$ −1745.00 −0.355180
$$290$$ 0 0
$$291$$ 6160.00 1.24091
$$292$$ −2401.49 −0.481290
$$293$$ −3742.95 −0.746299 −0.373149 0.927771i $$-0.621722\pi$$
−0.373149 + 0.927771i $$0.621722\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −624.000 −0.122531
$$297$$ −6378.97 −1.24628
$$298$$ 1892.00 0.367787
$$299$$ −3151.96 −0.609641
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −1664.00 −0.317061
$$303$$ 1144.00 0.216901
$$304$$ −150.093 −0.0283172
$$305$$ 0 0
$$306$$ −6866.77 −1.28283
$$307$$ 722.324 0.134284 0.0671420 0.997743i $$-0.478612\pi$$
0.0671420 + 0.997743i $$0.478612\pi$$
$$308$$ 0 0
$$309$$ 12848.0 2.36536
$$310$$ 0 0
$$311$$ −7279.53 −1.32728 −0.663640 0.748052i $$-0.730989\pi$$
−0.663640 + 0.748052i $$0.730989\pi$$
$$312$$ 4928.00 0.894209
$$313$$ 1519.69 0.274435 0.137218 0.990541i $$-0.456184\pi$$
0.137218 + 0.990541i $$0.456184\pi$$
$$314$$ −5759.83 −1.03518
$$315$$ 0 0
$$316$$ −2720.00 −0.484215
$$317$$ −2358.00 −0.417787 −0.208893 0.977938i $$-0.566986\pi$$
−0.208893 + 0.977938i $$0.566986\pi$$
$$318$$ −1163.22 −0.205127
$$319$$ −3320.00 −0.582709
$$320$$ 0 0
$$321$$ −2439.02 −0.424089
$$322$$ 0 0
$$323$$ −528.000 −0.0909557
$$324$$ 5380.00 0.922497
$$325$$ 0 0
$$326$$ 1272.00 0.216103
$$327$$ −17654.7 −2.98565
$$328$$ 3151.96 0.530603
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2372.00 0.393888 0.196944 0.980415i $$-0.436898\pi$$
0.196944 + 0.980415i $$0.436898\pi$$
$$332$$ 787.990 0.130261
$$333$$ 4758.00 0.782993
$$334$$ 1313.32 0.215154
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 250.000 0.0404106 0.0202053 0.999796i $$-0.493568\pi$$
0.0202053 + 0.999796i $$0.493568\pi$$
$$338$$ −4230.00 −0.680715
$$339$$ −12063.7 −1.93278
$$340$$ 0 0
$$341$$ 4127.57 0.655485
$$342$$ 1144.46 0.180951
$$343$$ 0 0
$$344$$ 3488.00 0.546687
$$345$$ 0 0
$$346$$ −1332.08 −0.206974
$$347$$ −9540.00 −1.47589 −0.737945 0.674861i $$-0.764204\pi$$
−0.737945 + 0.674861i $$0.764204\pi$$
$$348$$ 6228.87 0.959490
$$349$$ 5712.93 0.876235 0.438117 0.898918i $$-0.355645\pi$$
0.438117 + 0.898918i $$0.355645\pi$$
$$350$$ 0 0
$$351$$ −20944.0 −3.18492
$$352$$ −640.000 −0.0969094
$$353$$ 4390.23 0.661950 0.330975 0.943640i $$-0.392622\pi$$
0.330975 + 0.943640i $$0.392622\pi$$
$$354$$ 12496.0 1.87614
$$355$$ 0 0
$$356$$ 6003.73 0.893812
$$357$$ 0 0
$$358$$ 6456.00 0.953101
$$359$$ 1840.00 0.270506 0.135253 0.990811i $$-0.456815\pi$$
0.135253 + 0.990811i $$0.456815\pi$$
$$360$$ 0 0
$$361$$ −6771.00 −0.987170
$$362$$ −5647.26 −0.819927
$$363$$ 8733.55 1.26279
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −5104.00 −0.728935
$$367$$ −2964.34 −0.421628 −0.210814 0.977526i $$-0.567611\pi$$
−0.210814 + 0.977526i $$0.567611\pi$$
$$368$$ −768.000 −0.108790
$$369$$ −24033.7 −3.39063
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −7744.00 −1.07932
$$373$$ −3982.00 −0.552762 −0.276381 0.961048i $$-0.589135\pi$$
−0.276381 + 0.961048i $$0.589135\pi$$
$$374$$ −2251.40 −0.311276
$$375$$ 0 0
$$376$$ −1651.03 −0.226450
$$377$$ −10900.5 −1.48914
$$378$$ 0 0
$$379$$ 2676.00 0.362683 0.181342 0.983420i $$-0.441956\pi$$
0.181342 + 0.983420i $$0.441956\pi$$
$$380$$ 0 0
$$381$$ 21688.5 2.91636
$$382$$ 4272.00 0.572185
$$383$$ 7035.62 0.938652 0.469326 0.883025i $$-0.344497\pi$$
0.469326 + 0.883025i $$0.344497\pi$$
$$384$$ 1200.75 0.159571
$$385$$ 0 0
$$386$$ 3316.00 0.437254
$$387$$ −26596.0 −3.49341
$$388$$ −2626.63 −0.343678
$$389$$ 8658.00 1.12848 0.564239 0.825611i $$-0.309170\pi$$
0.564239 + 0.825611i $$0.309170\pi$$
$$390$$ 0 0
$$391$$ −2701.68 −0.349437
$$392$$ 0 0
$$393$$ −2376.00 −0.304970
$$394$$ −1956.00 −0.250106
$$395$$ 0 0
$$396$$ 4880.00 0.619266
$$397$$ −9052.50 −1.14441 −0.572207 0.820109i $$-0.693912\pi$$
−0.572207 + 0.820109i $$0.693912\pi$$
$$398$$ −9868.63 −1.24289
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5706.00 −0.710584 −0.355292 0.934755i $$-0.615619\pi$$
−0.355292 + 0.934755i $$0.615619\pi$$
$$402$$ −10881.8 −1.35008
$$403$$ 13552.0 1.67512
$$404$$ −487.803 −0.0600721
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1560.00 0.189991
$$408$$ 4224.00 0.512547
$$409$$ −2420.25 −0.292601 −0.146301 0.989240i $$-0.546737\pi$$
−0.146301 + 0.989240i $$0.546737\pi$$
$$410$$ 0 0
$$411$$ −10450.2 −1.25419
$$412$$ −5478.41 −0.655101
$$413$$ 0 0
$$414$$ 5856.00 0.695185
$$415$$ 0 0
$$416$$ −2101.31 −0.247656
$$417$$ 12936.0 1.51913
$$418$$ 375.233 0.0439073
$$419$$ 1510.31 0.176095 0.0880473 0.996116i $$-0.471937\pi$$
0.0880473 + 0.996116i $$0.471937\pi$$
$$420$$ 0 0
$$421$$ −16770.0 −1.94138 −0.970689 0.240341i $$-0.922741\pi$$
−0.970689 + 0.240341i $$0.922741\pi$$
$$422$$ −3112.00 −0.358981
$$423$$ 12589.1 1.44705
$$424$$ 496.000 0.0568111
$$425$$ 0 0
$$426$$ −10206.3 −1.16080
$$427$$ 0 0
$$428$$ 1040.00 0.117454
$$429$$ −12320.0 −1.38652
$$430$$ 0 0
$$431$$ 1336.00 0.149311 0.0746553 0.997209i $$-0.476214\pi$$
0.0746553 + 0.997209i $$0.476214\pi$$
$$432$$ −5103.17 −0.568348
$$433$$ −11163.2 −1.23896 −0.619479 0.785013i $$-0.712656\pi$$
−0.619479 + 0.785013i $$0.712656\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7528.00 0.826894
$$437$$ 450.280 0.0492902
$$438$$ −11264.0 −1.22880
$$439$$ 3602.24 0.391630 0.195815 0.980641i $$-0.437265\pi$$
0.195815 + 0.980641i $$0.437265\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −7392.00 −0.795479
$$443$$ −6348.00 −0.680818 −0.340409 0.940277i $$-0.610566\pi$$
−0.340409 + 0.940277i $$0.610566\pi$$
$$444$$ −2926.82 −0.312839
$$445$$ 0 0
$$446$$ −5778.59 −0.613507
$$447$$ 8874.27 0.939012
$$448$$ 0 0
$$449$$ 7170.00 0.753615 0.376808 0.926292i $$-0.377022\pi$$
0.376808 + 0.926292i $$0.377022\pi$$
$$450$$ 0 0
$$451$$ −7879.90 −0.822727
$$452$$ 5144.00 0.535295
$$453$$ −7804.85 −0.809501
$$454$$ 3958.71 0.409232
$$455$$ 0 0
$$456$$ −704.000 −0.0722979
$$457$$ −6866.00 −0.702796 −0.351398 0.936226i $$-0.614294\pi$$
−0.351398 + 0.936226i $$0.614294\pi$$
$$458$$ 5534.69 0.564671
$$459$$ −17952.0 −1.82555
$$460$$ 0 0
$$461$$ −1378.98 −0.139318 −0.0696590 0.997571i $$-0.522191\pi$$
−0.0696590 + 0.997571i $$0.522191\pi$$
$$462$$ 0 0
$$463$$ −2648.00 −0.265795 −0.132897 0.991130i $$-0.542428\pi$$
−0.132897 + 0.991130i $$0.542428\pi$$
$$464$$ −2656.00 −0.265736
$$465$$ 0 0
$$466$$ −12980.0 −1.29032
$$467$$ −12335.8 −1.22234 −0.611170 0.791500i $$-0.709301\pi$$
−0.611170 + 0.791500i $$0.709301\pi$$
$$468$$ 16022.5 1.58256
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −27016.0 −2.64295
$$472$$ −5328.31 −0.519609
$$473$$ −8720.00 −0.847666
$$474$$ −12757.9 −1.23627
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −3782.00 −0.363031
$$478$$ 8592.00 0.822153
$$479$$ 13339.5 1.27244 0.636221 0.771507i $$-0.280497\pi$$
0.636221 + 0.771507i $$0.280497\pi$$
$$480$$ 0 0
$$481$$ 5121.93 0.485530
$$482$$ 9043.12 0.854570
$$483$$ 0 0
$$484$$ −3724.00 −0.349737
$$485$$ 0 0
$$486$$ 8011.23 0.747730
$$487$$ −13936.0 −1.29672 −0.648358 0.761336i $$-0.724544\pi$$
−0.648358 + 0.761336i $$0.724544\pi$$
$$488$$ 2176.35 0.201883
$$489$$ 5966.21 0.551741
$$490$$ 0 0
$$491$$ −12276.0 −1.12833 −0.564163 0.825663i $$-0.690801\pi$$
−0.564163 + 0.825663i $$0.690801\pi$$
$$492$$ 14784.0 1.35470
$$493$$ −9343.31 −0.853553
$$494$$ 1232.00 0.112207
$$495$$ 0 0
$$496$$ 3302.05 0.298924
$$497$$ 0 0
$$498$$ 3696.00 0.332574
$$499$$ −2220.00 −0.199160 −0.0995800 0.995030i $$-0.531750\pi$$
−0.0995800 + 0.995030i $$0.531750\pi$$
$$500$$ 0 0
$$501$$ 6160.00 0.549318
$$502$$ 11163.2 0.992505
$$503$$ −11294.5 −1.00119 −0.500594 0.865682i $$-0.666885\pi$$
−0.500594 + 0.865682i $$0.666885\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1920.00 0.168685
$$507$$ −19840.5 −1.73796
$$508$$ −9248.00 −0.807704
$$509$$ 15881.7 1.38300 0.691499 0.722377i $$-0.256951\pi$$
0.691499 + 0.722377i $$0.256951\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −512.000 −0.0441942
$$513$$ 2992.00 0.257505
$$514$$ 3001.87 0.257600
$$515$$ 0 0
$$516$$ 16360.2 1.39577
$$517$$ 4127.57 0.351122
$$518$$ 0 0
$$519$$ −6248.00 −0.528433
$$520$$ 0 0
$$521$$ −11613.5 −0.976575 −0.488287 0.872683i $$-0.662378\pi$$
−0.488287 + 0.872683i $$0.662378\pi$$
$$522$$ 20252.0 1.69810
$$523$$ 12617.2 1.05490 0.527450 0.849586i $$-0.323148\pi$$
0.527450 + 0.849586i $$0.323148\pi$$
$$524$$ 1013.13 0.0844633
$$525$$ 0 0
$$526$$ −800.000 −0.0663149
$$527$$ 11616.0 0.960154
$$528$$ −3001.87 −0.247423
$$529$$ −9863.00 −0.810635
$$530$$ 0 0
$$531$$ 40628.4 3.32038
$$532$$ 0 0
$$533$$ −25872.0 −2.10252
$$534$$ 28160.0 2.28203
$$535$$ 0 0
$$536$$ 4640.00 0.373913
$$537$$ 30281.3 2.43340
$$538$$ −544.088 −0.0436009
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1798.00 0.142887 0.0714437 0.997445i $$-0.477239\pi$$
0.0714437 + 0.997445i $$0.477239\pi$$
$$542$$ 13808.6 1.09433
$$543$$ −26488.0 −2.09339
$$544$$ −1801.12 −0.141953
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1276.00 −0.0997401 −0.0498700 0.998756i $$-0.515881\pi$$
−0.0498700 + 0.998756i $$0.515881\pi$$
$$548$$ 4456.00 0.347356
$$549$$ −16594.7 −1.29006
$$550$$ 0 0
$$551$$ 1557.22 0.120399
$$552$$ −3602.24 −0.277756
$$553$$ 0 0
$$554$$ −13540.0 −1.03837
$$555$$ 0 0
$$556$$ −5515.93 −0.420733
$$557$$ −2694.00 −0.204934 −0.102467 0.994736i $$-0.532674\pi$$
−0.102467 + 0.994736i $$0.532674\pi$$
$$558$$ −25178.2 −1.91017
$$559$$ −28630.3 −2.16625
$$560$$ 0 0
$$561$$ −10560.0 −0.794730
$$562$$ −3756.00 −0.281917
$$563$$ −15769.2 −1.18045 −0.590223 0.807240i $$-0.700960\pi$$
−0.590223 + 0.807240i $$0.700960\pi$$
$$564$$ −7744.00 −0.578158
$$565$$ 0 0
$$566$$ −769.228 −0.0571256
$$567$$ 0 0
$$568$$ 4352.00 0.321489
$$569$$ 12606.0 0.928772 0.464386 0.885633i $$-0.346275\pi$$
0.464386 + 0.885633i $$0.346275\pi$$
$$570$$ 0 0
$$571$$ 6852.00 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 5253.27 0.384004
$$573$$ 20037.5 1.46087
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 3904.00 0.282407
$$577$$ 14371.4 1.03690 0.518449 0.855108i $$-0.326509\pi$$
0.518449 + 0.855108i $$0.326509\pi$$
$$578$$ 3490.00 0.251150
$$579$$ 15553.4 1.11637
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −12320.0 −0.877458
$$583$$ −1240.00 −0.0880884
$$584$$ 4802.99 0.340324
$$585$$ 0 0
$$586$$ 7485.90 0.527713
$$587$$ 18977.4 1.33438 0.667191 0.744887i $$-0.267497\pi$$
0.667191 + 0.744887i $$0.267497\pi$$
$$588$$ 0 0
$$589$$ −1936.00 −0.135435
$$590$$ 0 0
$$591$$ −9174.45 −0.638556
$$592$$ 1248.00 0.0866427
$$593$$ 8217.61 0.569067 0.284534 0.958666i $$-0.408161\pi$$
0.284534 + 0.958666i $$0.408161\pi$$
$$594$$ 12757.9 0.881253
$$595$$ 0 0
$$596$$ −3784.00 −0.260065
$$597$$ −46288.0 −3.17327
$$598$$ 6303.92 0.431081
$$599$$ −19104.0 −1.30312 −0.651559 0.758598i $$-0.725885\pi$$
−0.651559 + 0.758598i $$0.725885\pi$$
$$600$$ 0 0
$$601$$ −21538.4 −1.46185 −0.730923 0.682460i $$-0.760910\pi$$
−0.730923 + 0.682460i $$0.760910\pi$$
$$602$$ 0 0
$$603$$ −35380.0 −2.38936
$$604$$ 3328.00 0.224196
$$605$$ 0 0
$$606$$ −2288.00 −0.153372
$$607$$ 13733.5 0.918331 0.459166 0.888351i $$-0.348148\pi$$
0.459166 + 0.888351i $$0.348148\pi$$
$$608$$ 300.187 0.0200233
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 13552.0 0.897308
$$612$$ 13733.5 0.907100
$$613$$ −28034.0 −1.84712 −0.923558 0.383458i $$-0.874733\pi$$
−0.923558 + 0.383458i $$0.874733\pi$$
$$614$$ −1444.65 −0.0949532
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8258.00 0.538824 0.269412 0.963025i $$-0.413171\pi$$
0.269412 + 0.963025i $$0.413171\pi$$
$$618$$ −25696.0 −1.67256
$$619$$ 5131.31 0.333191 0.166595 0.986025i $$-0.446723\pi$$
0.166595 + 0.986025i $$0.446723\pi$$
$$620$$ 0 0
$$621$$ 15309.5 0.989291
$$622$$ 14559.1 0.938529
$$623$$ 0 0
$$624$$ −9856.00 −0.632301
$$625$$ 0 0
$$626$$ −3039.39 −0.194055
$$627$$ 1760.00 0.112101
$$628$$ 11519.7 0.731982
$$629$$ 4390.23 0.278299
$$630$$ 0 0
$$631$$ 912.000 0.0575375 0.0287687 0.999586i $$-0.490841\pi$$
0.0287687 + 0.999586i $$0.490841\pi$$
$$632$$ 5440.00 0.342392
$$633$$ −14596.6 −0.916527
$$634$$ 4716.00 0.295420
$$635$$ 0 0
$$636$$ 2326.45 0.145047
$$637$$ 0 0
$$638$$ 6640.00 0.412038
$$639$$ −33184.0 −2.05436
$$640$$ 0 0
$$641$$ −890.000 −0.0548407 −0.0274203 0.999624i $$-0.508729\pi$$
−0.0274203 + 0.999624i $$0.508729\pi$$
$$642$$ 4878.03 0.299876
$$643$$ −29352.6 −1.80024 −0.900120 0.435642i $$-0.856521\pi$$
−0.900120 + 0.435642i $$0.856521\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 1056.00 0.0643154
$$647$$ 11876.1 0.721637 0.360818 0.932636i $$-0.382497\pi$$
0.360818 + 0.932636i $$0.382497\pi$$
$$648$$ −10760.0 −0.652304
$$649$$ 13320.8 0.805680
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −2544.00 −0.152808
$$653$$ 21526.0 1.29001 0.645006 0.764178i $$-0.276855\pi$$
0.645006 + 0.764178i $$0.276855\pi$$
$$654$$ 35309.4 2.11118
$$655$$ 0 0
$$656$$ −6303.92 −0.375193
$$657$$ −36622.8 −2.17472
$$658$$ 0 0
$$659$$ 23452.0 1.38628 0.693141 0.720802i $$-0.256226\pi$$
0.693141 + 0.720802i $$0.256226\pi$$
$$660$$ 0 0
$$661$$ 26669.7 1.56934 0.784668 0.619916i $$-0.212833\pi$$
0.784668 + 0.619916i $$0.212833\pi$$
$$662$$ −4744.00 −0.278521
$$663$$ −34671.6 −2.03097
$$664$$ −1575.98 −0.0921082
$$665$$ 0 0
$$666$$ −9516.00 −0.553660
$$667$$ 7968.00 0.462552
$$668$$ −2626.63 −0.152137
$$669$$ −27104.0 −1.56637
$$670$$ 0 0
$$671$$ −5440.88 −0.313030
$$672$$ 0 0
$$673$$ 13858.0 0.793739 0.396870 0.917875i $$-0.370096\pi$$
0.396870 + 0.917875i $$0.370096\pi$$
$$674$$ −500.000 −0.0285746
$$675$$ 0 0
$$676$$ 8460.00 0.481338
$$677$$ 32448.3 1.84208 0.921041 0.389466i $$-0.127340\pi$$
0.921041 + 0.389466i $$0.127340\pi$$
$$678$$ 24127.5 1.36668
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18568.0 1.04483
$$682$$ −8255.13 −0.463498
$$683$$ 27812.0 1.55812 0.779060 0.626949i $$-0.215696\pi$$
0.779060 + 0.626949i $$0.215696\pi$$
$$684$$ −2288.92 −0.127952
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 25960.0 1.44168
$$688$$ −6976.00 −0.386566
$$689$$ −4071.28 −0.225114
$$690$$ 0 0
$$691$$ 1303.94 0.0717859 0.0358929 0.999356i $$-0.488572\pi$$
0.0358929 + 0.999356i $$0.488572\pi$$
$$692$$ 2664.16 0.146353
$$693$$ 0 0
$$694$$ 19080.0 1.04361
$$695$$ 0 0
$$696$$ −12457.7 −0.678462
$$697$$ −22176.0 −1.20513
$$698$$ −11425.9 −0.619592
$$699$$ −60881.6 −3.29435
$$700$$ 0 0
$$701$$ 22906.0 1.23416 0.617081 0.786900i $$-0.288315\pi$$
0.617081 + 0.786900i $$0.288315\pi$$
$$702$$ 41888.0 2.25208
$$703$$ −731.705 −0.0392557
$$704$$ 1280.00 0.0685253
$$705$$ 0 0
$$706$$ −8780.46 −0.468069
$$707$$ 0 0
$$708$$ −24992.0 −1.32663
$$709$$ −15086.0 −0.799107 −0.399553 0.916710i $$-0.630835\pi$$
−0.399553 + 0.916710i $$0.630835\pi$$
$$710$$ 0 0
$$711$$ −41480.0 −2.18793
$$712$$ −12007.5 −0.632021
$$713$$ −9906.16 −0.520321
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12912.0 −0.673944
$$717$$ 40300.1 2.09907
$$718$$ −3680.00 −0.191276
$$719$$ 20544.0 1.06559 0.532797 0.846243i $$-0.321141\pi$$
0.532797 + 0.846243i $$0.321141\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 13542.0 0.698035
$$723$$ 42416.0 2.18184
$$724$$ 11294.5 0.579776
$$725$$ 0 0
$$726$$ −17467.1 −0.892927
$$727$$ −7223.24 −0.368494 −0.184247 0.982880i $$-0.558985\pi$$
−0.184247 + 0.982880i $$0.558985\pi$$
$$728$$ 0 0
$$729$$ 1261.00 0.0640654
$$730$$ 0 0
$$731$$ −24540.3 −1.24166
$$732$$ 10208.0 0.515435
$$733$$ 29427.7 1.48286 0.741430 0.671031i $$-0.234148\pi$$
0.741430 + 0.671031i $$0.234148\pi$$
$$734$$ 5928.69 0.298136
$$735$$ 0 0
$$736$$ 1536.00 0.0769262
$$737$$ −11600.0 −0.579771
$$738$$ 48067.4 2.39754
$$739$$ 32668.0 1.62613 0.813066 0.582171i $$-0.197797\pi$$
0.813066 + 0.582171i $$0.197797\pi$$
$$740$$ 0 0
$$741$$ 5778.59 0.286480
$$742$$ 0 0
$$743$$ 37056.0 1.82968 0.914840 0.403816i $$-0.132316\pi$$
0.914840 + 0.403816i $$0.132316\pi$$
$$744$$ 15488.0 0.763196
$$745$$ 0 0
$$746$$ 7964.00 0.390862
$$747$$ 12016.8 0.588586
$$748$$ 4502.80 0.220105
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −19608.0 −0.952738 −0.476369 0.879246i $$-0.658047\pi$$
−0.476369 + 0.879246i $$0.658047\pi$$
$$752$$ 3302.05 0.160124
$$753$$ 52360.0 2.53400
$$754$$ 21801.1 1.05298
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −19378.0 −0.930390 −0.465195 0.885208i $$-0.654016\pi$$
−0.465195 + 0.885208i $$0.654016\pi$$
$$758$$ −5352.00 −0.256456
$$759$$ 9005.60 0.430675
$$760$$ 0 0
$$761$$ 13977.4 0.665810 0.332905 0.942960i $$-0.391971\pi$$
0.332905 + 0.942960i $$0.391971\pi$$
$$762$$ −43377.0 −2.06218
$$763$$ 0 0
$$764$$ −8544.00 −0.404596
$$765$$ 0 0
$$766$$ −14071.2 −0.663727
$$767$$ 43736.0 2.05895
$$768$$ −2401.49 −0.112834
$$769$$ −8536.56 −0.400307 −0.200154 0.979765i $$-0.564144\pi$$
−0.200154 + 0.979765i $$0.564144\pi$$
$$770$$ 0 0
$$771$$ 14080.0 0.657690
$$772$$ −6632.00 −0.309185
$$773$$ −29296.3 −1.36315 −0.681576 0.731748i $$-0.738705\pi$$
−0.681576 + 0.731748i $$0.738705\pi$$
$$774$$ 53192.0 2.47022
$$775$$ 0 0
$$776$$ 5253.27 0.243017
$$777$$ 0 0
$$778$$ −17316.0 −0.797955
$$779$$ 3696.00 0.169991
$$780$$ 0 0
$$781$$ −10880.0 −0.498485
$$782$$ 5403.36 0.247089
$$783$$ 52945.4 2.41649
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 4752.00 0.215647
$$787$$ 13780.4 0.624167 0.312084 0.950055i $$-0.398973\pi$$
0.312084 + 0.950055i $$0.398973\pi$$
$$788$$ 3912.00 0.176852
$$789$$ −3752.33 −0.169311
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −9760.00 −0.437887
$$793$$ −17864.0 −0.799961
$$794$$ 18105.0 0.809222
$$795$$ 0 0
$$796$$ 19737.3 0.878855
$$797$$ −34868.6 −1.54970 −0.774848 0.632148i $$-0.782174\pi$$
−0.774848 + 0.632148i $$0.782174\pi$$
$$798$$ 0 0
$$799$$ 11616.0 0.514324
$$800$$ 0 0
$$801$$ 91556.9 4.03871
$$802$$ 11412.0 0.502459
$$803$$ −12007.5 −0.527689
$$804$$ 21763.5 0.954652
$$805$$ 0 0
$$806$$ −27104.0 −1.18449
$$807$$ −2552.00 −0.111319
$$808$$ 975.606 0.0424774
$$809$$ 14034.0 0.609900 0.304950 0.952368i $$-0.401360\pi$$
0.304950 + 0.952368i $$0.401360\pi$$
$$810$$ 0 0
$$811$$ 6632.25 0.287164 0.143582 0.989638i $$-0.454138\pi$$
0.143582 + 0.989638i $$0.454138\pi$$
$$812$$ 0 0
$$813$$ 64768.0 2.79399
$$814$$ −3120.00 −0.134344
$$815$$ 0 0
$$816$$ −8448.00 −0.362425
$$817$$ 4090.04 0.175144
$$818$$ 4840.51 0.206900
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 28622.0 1.21670 0.608352 0.793667i $$-0.291831\pi$$
0.608352 + 0.793667i $$0.291831\pi$$
$$822$$ 20900.5 0.886847
$$823$$ −24688.0 −1.04565 −0.522825 0.852440i $$-0.675122\pi$$
−0.522825 + 0.852440i $$0.675122\pi$$
$$824$$ 10956.8 0.463226
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30756.0 1.29322 0.646609 0.762822i $$-0.276187\pi$$
0.646609 + 0.762822i $$0.276187\pi$$
$$828$$ −11712.0 −0.491570
$$829$$ −23236.3 −0.973499 −0.486750 0.873542i $$-0.661818\pi$$
−0.486750 + 0.873542i $$0.661818\pi$$
$$830$$ 0 0
$$831$$ −63508.2 −2.65111
$$832$$ 4202.61 0.175119
$$833$$ 0 0
$$834$$ −25872.0 −1.07419
$$835$$ 0 0
$$836$$ −750.467 −0.0310472
$$837$$ −65824.0 −2.71829
$$838$$ −3020.63 −0.124518
$$839$$ 24033.7 0.988957 0.494479 0.869190i $$-0.335359\pi$$
0.494479 + 0.869190i $$0.335359\pi$$
$$840$$ 0 0
$$841$$ 3167.00 0.129854
$$842$$ 33540.0 1.37276
$$843$$ −17617.2 −0.719773
$$844$$ 6224.00 0.253838
$$845$$ 0 0
$$846$$ −25178.2 −1.02322
$$847$$ 0 0
$$848$$ −992.000 −0.0401715
$$849$$ −3608.00 −0.145850
$$850$$ 0 0
$$851$$ −3744.00 −0.150814
$$852$$ 20412.7 0.820807
$$853$$ −23574.0 −0.946260 −0.473130 0.880993i $$-0.656876\pi$$
−0.473130 + 0.880993i $$0.656876\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −2080.00 −0.0830525
$$857$$ 24484.0 0.975912 0.487956 0.872868i $$-0.337743\pi$$
0.487956 + 0.872868i $$0.337743\pi$$
$$858$$ 24640.0 0.980415
$$859$$ 32954.9 1.30897 0.654485 0.756075i $$-0.272885\pi$$
0.654485 + 0.756075i $$0.272885\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −2672.00 −0.105579
$$863$$ −40872.0 −1.61217 −0.806083 0.591803i $$-0.798416\pi$$
−0.806083 + 0.591803i $$0.798416\pi$$
$$864$$ 10206.3 0.401883
$$865$$ 0 0
$$866$$ 22326.4 0.876075
$$867$$ 16369.6 0.641222
$$868$$ 0 0
$$869$$ −13600.0 −0.530896
$$870$$ 0 0
$$871$$ −38086.2 −1.48163
$$872$$ −15056.0 −0.584702
$$873$$ −40056.2 −1.55292
$$874$$ −900.560 −0.0348534
$$875$$ 0 0
$$876$$ 22528.0 0.868893
$$877$$ 12006.0 0.462273 0.231137 0.972921i $$-0.425756\pi$$
0.231137 + 0.972921i $$0.425756\pi$$
$$878$$ −7204.48 −0.276924
$$879$$ 35112.0 1.34732
$$880$$ 0 0
$$881$$ −35722.2 −1.36607 −0.683037 0.730383i $$-0.739341\pi$$
−0.683037 + 0.730383i $$0.739341\pi$$
$$882$$ 0 0
$$883$$ −19588.0 −0.746533 −0.373267 0.927724i $$-0.621762\pi$$
−0.373267 + 0.927724i $$0.621762\pi$$
$$884$$ 14784.0 0.562488
$$885$$ 0 0
$$886$$ 12696.0 0.481411
$$887$$ −40243.8 −1.52340 −0.761699 0.647931i $$-0.775634\pi$$
−0.761699 + 0.647931i $$0.775634\pi$$
$$888$$ 5853.64 0.221211
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 26900.0 1.01143
$$892$$ 11557.2 0.433815
$$893$$ −1936.00 −0.0725485
$$894$$ −17748.5 −0.663982
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 29568.0 1.10061
$$898$$ −14340.0 −0.532886
$$899$$ −34258.8 −1.27096
$$900$$ 0 0
$$901$$ −3489.67 −0.129032
$$902$$ 15759.8 0.581756
$$903$$ 0 0
$$904$$ −10288.0 −0.378511
$$905$$ 0 0
$$906$$ 15609.7 0.572404
$$907$$ −15868.0 −0.580913 −0.290457 0.956888i $$-0.593807\pi$$
−0.290457 + 0.956888i $$0.593807\pi$$
$$908$$ −7917.42 −0.289371
$$909$$ −7439.00 −0.271437
$$910$$ 0 0
$$911$$ 39832.0 1.44862 0.724310 0.689474i $$-0.242158\pi$$
0.724310 + 0.689474i $$0.242158\pi$$
$$912$$ 1408.00 0.0511223
$$913$$ 3939.95 0.142818
$$914$$ 13732.0 0.496952
$$915$$ 0 0
$$916$$ −11069.4 −0.399282
$$917$$ 0 0
$$918$$ 35904.0 1.29086
$$919$$ −30528.0 −1.09578 −0.547892 0.836549i $$-0.684570\pi$$
−0.547892 + 0.836549i $$0.684570\pi$$
$$920$$ 0 0
$$921$$ −6776.00 −0.242429
$$922$$ 2757.96 0.0985127
$$923$$ −35722.2 −1.27390
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 5296.00 0.187945
$$927$$ −83545.7 −2.96009
$$928$$ 5312.00 0.187904
$$929$$ 16604.1 0.586396 0.293198 0.956052i $$-0.405280\pi$$
0.293198 + 0.956052i $$0.405280\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 25960.0 0.912391
$$933$$ 68288.0 2.39619
$$934$$ 24671.6 0.864324
$$935$$ 0 0
$$936$$ −32044.9 −1.11904
$$937$$ 29943.6 1.04399 0.521993 0.852950i $$-0.325189\pi$$
0.521993 + 0.852950i $$0.325189\pi$$
$$938$$ 0 0
$$939$$ −14256.0 −0.495449
$$940$$ 0 0
$$941$$ −5375.22 −0.186214 −0.0931068 0.995656i $$-0.529680\pi$$
−0.0931068 + 0.995656i $$0.529680\pi$$
$$942$$ 54032.0 1.86885
$$943$$ 18911.8 0.653077
$$944$$ 10656.6 0.367419
$$945$$ 0 0
$$946$$ 17440.0 0.599390
$$947$$ −45212.0 −1.55142 −0.775709 0.631091i $$-0.782607\pi$$
−0.775709 + 0.631091i $$0.782607\pi$$
$$948$$ 25515.9 0.874174
$$949$$ −39424.0 −1.34853
$$950$$ 0 0
$$951$$ 22120.0 0.754248
$$952$$ 0 0
$$953$$ −34218.0 −1.16310 −0.581548 0.813512i $$-0.697553\pi$$
−0.581548 + 0.813512i $$0.697553\pi$$
$$954$$ 7564.00 0.256702
$$955$$ 0 0
$$956$$ −17184.0 −0.581350
$$957$$ 31144.4 1.05199
$$958$$ −26679.1 −0.899752
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 12801.0 0.429694
$$962$$ −10243.9 −0.343322
$$963$$ 15860.0 0.530718
$$964$$ −18086.2 −0.604272
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −14464.0 −0.481004 −0.240502 0.970649i $$-0.577312\pi$$
−0.240502 + 0.970649i $$0.577312\pi$$
$$968$$ 7448.00 0.247301
$$969$$ 4953.08 0.164206
$$970$$ 0 0
$$971$$ −37832.9 −1.25038 −0.625188 0.780474i $$-0.714978\pi$$
−0.625188 + 0.780474i $$0.714978\pi$$
$$972$$ −16022.5 −0.528725
$$973$$ 0 0
$$974$$ 27872.0 0.916916
$$975$$ 0 0
$$976$$ −4352.71 −0.142753
$$977$$ −42062.0 −1.37736 −0.688681 0.725065i $$-0.741810\pi$$
−0.688681 + 0.725065i $$0.741810\pi$$
$$978$$ −11932.4 −0.390140
$$979$$ 30018.7 0.979980
$$980$$ 0 0
$$981$$ 114802. 3.73634
$$982$$ 24552.0 0.797847
$$983$$ −43020.5 −1.39587 −0.697935 0.716161i $$-0.745898\pi$$
−0.697935 + 0.716161i $$0.745898\pi$$
$$984$$ −29568.0 −0.957920
$$985$$ 0 0
$$986$$ 18686.6 0.603553
$$987$$ 0 0
$$988$$ −2464.00 −0.0793424
$$989$$ 20928.0 0.672873
$$990$$ 0 0
$$991$$ 21272.0 0.681864 0.340932 0.940088i $$-0.389257\pi$$
0.340932 + 0.940088i $$0.389257\pi$$
$$992$$ −6604.11 −0.211372
$$993$$ −22251.3 −0.711102
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −7392.00 −0.235165
$$997$$ 121.951 0.00387384 0.00193692 0.999998i $$-0.499383\pi$$
0.00193692 + 0.999998i $$0.499383\pi$$
$$998$$ 4440.00 0.140827
$$999$$ −24878.0 −0.787892
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 2450.4.a.bs.1.1 2
5.4 even 2 98.4.a.h.1.2 yes 2
7.6 odd 2 inner 2450.4.a.bs.1.2 2
15.14 odd 2 882.4.a.w.1.2 2
20.19 odd 2 784.4.a.z.1.1 2
35.4 even 6 98.4.c.g.79.1 4
35.9 even 6 98.4.c.g.67.1 4
35.19 odd 6 98.4.c.g.67.2 4
35.24 odd 6 98.4.c.g.79.2 4
35.34 odd 2 98.4.a.h.1.1 2
105.44 odd 6 882.4.g.bi.361.1 4
105.59 even 6 882.4.g.bi.667.2 4
105.74 odd 6 882.4.g.bi.667.1 4
105.89 even 6 882.4.g.bi.361.2 4
105.104 even 2 882.4.a.w.1.1 2
140.139 even 2 784.4.a.z.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 35.34 odd 2
98.4.a.h.1.2 yes 2 5.4 even 2
98.4.c.g.67.1 4 35.9 even 6
98.4.c.g.67.2 4 35.19 odd 6
98.4.c.g.79.1 4 35.4 even 6
98.4.c.g.79.2 4 35.24 odd 6
784.4.a.z.1.1 2 20.19 odd 2
784.4.a.z.1.2 2 140.139 even 2
882.4.a.w.1.1 2 105.104 even 2
882.4.a.w.1.2 2 15.14 odd 2
882.4.g.bi.361.1 4 105.44 odd 6
882.4.g.bi.361.2 4 105.89 even 6
882.4.g.bi.667.1 4 105.74 odd 6
882.4.g.bi.667.2 4 105.59 even 6
2450.4.a.bs.1.1 2 1.1 even 1 trivial
2450.4.a.bs.1.2 2 7.6 odd 2 inner