# Properties

 Label 784.4.a.z.1.2 Level $784$ Weight $4$ Character 784.1 Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.2574974445$$ Analytic rank: $$0$$ 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 1.2 Root $$4.69042$$ of defining polynomial Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} +O(q^{10})$$ $$q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} -20.0000 q^{11} +65.6658 q^{13} +88.0000 q^{15} +56.2850 q^{17} -9.38083 q^{19} -48.0000 q^{23} -37.0000 q^{25} +318.948 q^{27} -166.000 q^{29} +206.378 q^{31} -187.617 q^{33} -78.0000 q^{37} +616.000 q^{39} +393.995 q^{41} -436.000 q^{43} +572.231 q^{45} -206.378 q^{47} +528.000 q^{51} +62.0000 q^{53} -187.617 q^{55} -88.0000 q^{57} +666.039 q^{59} +272.044 q^{61} +616.000 q^{65} -580.000 q^{67} -450.280 q^{69} +544.000 q^{71} -600.373 q^{73} -347.091 q^{75} +680.000 q^{79} +1345.00 q^{81} -196.997 q^{83} +528.000 q^{85} -1557.22 q^{87} -1500.93 q^{89} +1936.00 q^{93} -88.0000 q^{95} -656.658 q^{97} -1220.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 122q^{9} + O(q^{10})$$ $$2q + 122q^{9} - 40q^{11} + 176q^{15} - 96q^{23} - 74q^{25} - 332q^{29} - 156q^{37} + 1232q^{39} - 872q^{43} + 1056q^{51} + 124q^{53} - 176q^{57} + 1232q^{65} - 1160q^{67} + 1088q^{71} + 1360q^{79} + 2690q^{81} + 1056q^{85} + 3872q^{93} - 176q^{95} - 2440q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 9.38083 1.80534 0.902671 0.430331i $$-0.141603\pi$$
0.902671 + 0.430331i $$0.141603\pi$$
$$4$$ 0 0
$$5$$ 9.38083 0.839047 0.419524 0.907744i $$-0.362197\pi$$
0.419524 + 0.907744i $$0.362197\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 61.0000 2.25926
$$10$$ 0 0
$$11$$ −20.0000 −0.548202 −0.274101 0.961701i $$-0.588380\pi$$
−0.274101 + 0.961701i $$0.588380\pi$$
$$12$$ 0 0
$$13$$ 65.6658 1.40096 0.700478 0.713674i $$-0.252970\pi$$
0.700478 + 0.713674i $$0.252970\pi$$
$$14$$ 0 0
$$15$$ 88.0000 1.51477
$$16$$ 0 0
$$17$$ 56.2850 0.803007 0.401503 0.915858i $$-0.368488\pi$$
0.401503 + 0.915858i $$0.368488\pi$$
$$18$$ 0 0
$$19$$ −9.38083 −0.113269 −0.0566345 0.998395i $$-0.518037\pi$$
−0.0566345 + 0.998395i $$0.518037\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −48.0000 −0.435161 −0.217580 0.976042i $$-0.569816\pi$$
−0.217580 + 0.976042i $$0.569816\pi$$
$$24$$ 0 0
$$25$$ −37.0000 −0.296000
$$26$$ 0 0
$$27$$ 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 $$0.296022\pi$$
$$32$$ 0 0
$$33$$ −187.617 −0.989693
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −78.0000 −0.346571 −0.173285 0.984872i $$-0.555438\pi$$
−0.173285 + 0.984872i $$0.555438\pi$$
$$38$$ 0 0
$$39$$ 616.000 2.52920
$$40$$ 0 0
$$41$$ 393.995 1.50077 0.750386 0.661000i $$-0.229868\pi$$
0.750386 + 0.661000i $$0.229868\pi$$
$$42$$ 0 0
$$43$$ −436.000 −1.54626 −0.773132 0.634245i $$-0.781311\pi$$
−0.773132 + 0.634245i $$0.781311\pi$$
$$44$$ 0 0
$$45$$ 572.231 1.89562
$$46$$ 0 0
$$47$$ −206.378 −0.640497 −0.320249 0.947334i $$-0.603766\pi$$
−0.320249 + 0.947334i $$0.603766\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 528.000 1.44970
$$52$$ 0 0
$$53$$ 62.0000 0.160686 0.0803430 0.996767i $$-0.474398\pi$$
0.0803430 + 0.996767i $$0.474398\pi$$
$$54$$ 0 0
$$55$$ −187.617 −0.459968
$$56$$ 0 0
$$57$$ −88.0000 −0.204489
$$58$$ 0 0
$$59$$ 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 $$-0.407838\pi$$
0.285506 + 0.958377i $$0.407838\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 616.000 1.17547
$$66$$ 0 0
$$67$$ −580.000 −1.05759 −0.528793 0.848751i $$-0.677355\pi$$
−0.528793 + 0.848751i $$0.677355\pi$$
$$68$$ 0 0
$$69$$ −450.280 −0.785613
$$70$$ 0 0
$$71$$ 544.000 0.909309 0.454654 0.890668i $$-0.349763\pi$$
0.454654 + 0.890668i $$0.349763\pi$$
$$72$$ 0 0
$$73$$ −600.373 −0.962580 −0.481290 0.876561i $$-0.659832\pi$$
−0.481290 + 0.876561i $$0.659832\pi$$
$$74$$ 0 0
$$75$$ −347.091 −0.534381
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 680.000 0.968430 0.484215 0.874949i $$-0.339105\pi$$
0.484215 + 0.874949i $$0.339105\pi$$
$$80$$ 0 0
$$81$$ 1345.00 1.84499
$$82$$ 0 0
$$83$$ −196.997 −0.260521 −0.130261 0.991480i $$-0.541581\pi$$
−0.130261 + 0.991480i $$0.541581\pi$$
$$84$$ 0 0
$$85$$ 528.000 0.673760
$$86$$ 0 0
$$87$$ −1557.22 −1.91898
$$88$$ 0 0
$$89$$ −1500.93 −1.78762 −0.893812 0.448441i $$-0.851979\pi$$
−0.893812 + 0.448441i $$0.851979\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 1936.00 2.15864
$$94$$ 0 0
$$95$$ −88.0000 −0.0950380
$$96$$ 0 0
$$97$$ −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 $$-0.480867\pi$$
0.0600721 + 0.998194i $$0.480867\pi$$
$$102$$ 0 0
$$103$$ 1369.60 1.31020 0.655101 0.755541i $$-0.272626\pi$$
0.655101 + 0.755541i $$0.272626\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 260.000 0.234908 0.117454 0.993078i $$-0.462527\pi$$
0.117454 + 0.993078i $$0.462527\pi$$
$$108$$ 0 0
$$109$$ 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 $$-0.679800\pi$$
−0.535295 + 0.844665i $$0.679800\pi$$
$$114$$ 0 0
$$115$$ −450.280 −0.365120
$$116$$ 0 0
$$117$$ 4005.62 3.16512
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ 0 0
$$123$$ 3696.00 2.70941
$$124$$ 0 0
$$125$$ −1519.69 −1.08741
$$126$$ 0 0
$$127$$ −2312.00 −1.61541 −0.807704 0.589588i $$-0.799290\pi$$
−0.807704 + 0.589588i $$0.799290\pi$$
$$128$$ 0 0
$$129$$ −4090.04 −2.79154
$$130$$ 0 0
$$131$$ 253.282 0.168927 0.0844633 0.996427i $$-0.473082\pi$$
0.0844633 + 0.996427i $$0.473082\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 2992.00 1.90748
$$136$$ 0 0
$$137$$ −1114.00 −0.694711 −0.347356 0.937733i $$-0.612920\pi$$
−0.347356 + 0.937733i $$0.612920\pi$$
$$138$$ 0 0
$$139$$ −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
$$142$$ 0 0
$$143$$ −1313.32 −0.768007
$$144$$ 0 0
$$145$$ −1557.22 −0.891862
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −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 $$-0.571976\pi$$
−0.224196 + 0.974544i $$0.571976\pi$$
$$152$$ 0 0
$$153$$ 3433.38 1.81420
$$154$$ 0 0
$$155$$ 1936.00 1.00325
$$156$$ 0 0
$$157$$ 2879.92 1.46396 0.731982 0.681324i $$-0.238596\pi$$
0.731982 + 0.681324i $$0.238596\pi$$
$$158$$ 0 0
$$159$$ 581.612 0.290093
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −636.000 −0.305616 −0.152808 0.988256i $$-0.548832\pi$$
−0.152808 + 0.988256i $$0.548832\pi$$
$$164$$ 0 0
$$165$$ −1760.00 −0.830399
$$166$$ 0 0
$$167$$ 656.658 0.304274 0.152137 0.988359i $$-0.451385\pi$$
0.152137 + 0.988359i $$0.451385\pi$$
$$168$$ 0 0
$$169$$ 2115.00 0.962676
$$170$$ 0 0
$$171$$ −572.231 −0.255904
$$172$$ 0 0
$$173$$ 666.039 0.292705 0.146353 0.989232i $$-0.453247\pi$$
0.146353 + 0.989232i $$0.453247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6248.00 2.65327
$$178$$ 0 0
$$179$$ 3228.00 1.34789 0.673944 0.738782i $$-0.264599\pi$$
0.673944 + 0.738782i $$0.264599\pi$$
$$180$$ 0 0
$$181$$ −2823.63 −1.15955 −0.579776 0.814776i $$-0.696860\pi$$
−0.579776 + 0.814776i $$0.696860\pi$$
$$182$$ 0 0
$$183$$ 2552.00 1.03087
$$184$$ 0 0
$$185$$ −731.705 −0.290789
$$186$$ 0 0
$$187$$ −1125.70 −0.440210
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2136.00 0.809191 0.404596 0.914496i $$-0.367412\pi$$
0.404596 + 0.914496i $$0.367412\pi$$
$$192$$ 0 0
$$193$$ 1658.00 0.618370 0.309185 0.951002i $$-0.399944\pi$$
0.309185 + 0.951002i $$0.399944\pi$$
$$194$$ 0 0
$$195$$ 5778.59 2.12212
$$196$$ 0 0
$$197$$ −978.000 −0.353704 −0.176852 0.984237i $$-0.556591\pi$$
−0.176852 + 0.984237i $$0.556591\pi$$
$$198$$ 0 0
$$199$$ 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
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 3696.00 1.25922
$$206$$ 0 0
$$207$$ −2928.00 −0.983140
$$208$$ 0 0
$$209$$ 187.617 0.0620943
$$210$$ 0 0
$$211$$ −1556.00 −0.507675 −0.253838 0.967247i $$-0.581693\pi$$
−0.253838 + 0.967247i $$0.581693\pi$$
$$212$$ 0 0
$$213$$ 5103.17 1.64161
$$214$$ 0 0
$$215$$ −4090.04 −1.29739
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −5632.00 −1.73779
$$220$$ 0 0
$$221$$ 3696.00 1.12498
$$222$$ 0 0
$$223$$ −2889.30 −0.867630 −0.433815 0.901002i $$-0.642833\pi$$
−0.433815 + 0.901002i $$0.642833\pi$$
$$224$$ 0 0
$$225$$ −2257.00 −0.668741
$$226$$ 0 0
$$227$$ 1979.36 0.578742 0.289371 0.957217i $$-0.406554\pi$$
0.289371 + 0.957217i $$0.406554\pi$$
$$228$$ 0 0
$$229$$ 2767.35 0.798565 0.399282 0.916828i $$-0.369259\pi$$
0.399282 + 0.916828i $$0.369259\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6490.00 −1.82478 −0.912391 0.409321i $$-0.865766\pi$$
−0.912391 + 0.409321i $$0.865766\pi$$
$$234$$ 0 0
$$235$$ −1936.00 −0.537407
$$236$$ 0 0
$$237$$ 6378.97 1.74835
$$238$$ 0 0
$$239$$ 4296.00 1.16270 0.581350 0.813654i $$-0.302525\pi$$
0.581350 + 0.813654i $$0.302525\pi$$
$$240$$ 0 0
$$241$$ 4521.56 1.20854 0.604272 0.796778i $$-0.293464\pi$$
0.604272 + 0.796778i $$0.293464\pi$$
$$242$$ 0 0
$$243$$ 4005.62 1.05745
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −616.000 −0.158685
$$248$$ 0 0
$$249$$ −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
$$254$$ 0 0
$$255$$ 4953.08 1.21637
$$256$$ 0 0
$$257$$ −1500.93 −0.364302 −0.182151 0.983271i $$-0.558306\pi$$
−0.182151 + 0.983271i $$0.558306\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −10126.0 −2.40147
$$262$$ 0 0
$$263$$ 400.000 0.0937835 0.0468917 0.998900i $$-0.485068\pi$$
0.0468917 + 0.998900i $$0.485068\pi$$
$$264$$ 0 0
$$265$$ 581.612 0.134823
$$266$$ 0 0
$$267$$ −14080.0 −3.22727
$$268$$ 0 0
$$269$$ −272.044 −0.0616610 −0.0308305 0.999525i $$-0.509815\pi$$
−0.0308305 + 0.999525i $$0.509815\pi$$
$$270$$ 0 0
$$271$$ −6904.29 −1.54762 −0.773812 0.633416i $$-0.781652\pi$$
−0.773812 + 0.633416i $$0.781652\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 740.000 0.162268
$$276$$ 0 0
$$277$$ −6770.00 −1.46848 −0.734242 0.678888i $$-0.762462\pi$$
−0.734242 + 0.678888i $$0.762462\pi$$
$$278$$ 0 0
$$279$$ 12589.1 2.70139
$$280$$ 0 0
$$281$$ 1878.00 0.398691 0.199345 0.979929i $$-0.436118\pi$$
0.199345 + 0.979929i $$0.436118\pi$$
$$282$$ 0 0
$$283$$ −384.614 −0.0807878 −0.0403939 0.999184i $$-0.512861\pi$$
−0.0403939 + 0.999184i $$0.512861\pi$$
$$284$$ 0 0
$$285$$ −825.513 −0.171576
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1745.00 −0.355180
$$290$$ 0 0
$$291$$ −6160.00 −1.24091
$$292$$ 0 0
$$293$$ −3742.95 −0.746299 −0.373149 0.927771i $$-0.621722\pi$$
−0.373149 + 0.927771i $$0.621722\pi$$
$$294$$ 0 0
$$295$$ 6248.00 1.23313
$$296$$ 0 0
$$297$$ −6378.97 −1.24628
$$298$$ 0 0
$$299$$ −3151.96 −0.609641
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1144.00 0.216901
$$304$$ 0 0
$$305$$ 2552.00 0.479105
$$306$$ 0 0
$$307$$ −722.324 −0.134284 −0.0671420 0.997743i $$-0.521388\pi$$
−0.0671420 + 0.997743i $$0.521388\pi$$
$$308$$ 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 $$0.730989\pi$$
$$312$$ 0 0
$$313$$ 1519.69 0.274435 0.137218 0.990541i $$-0.456184\pi$$
0.137218 + 0.990541i $$0.456184\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2358.00 0.417787 0.208893 0.977938i $$-0.433014\pi$$
0.208893 + 0.977938i $$0.433014\pi$$
$$318$$ 0 0
$$319$$ 3320.00 0.582709
$$320$$ 0 0
$$321$$ 2439.02 0.424089
$$322$$ 0 0
$$323$$ −528.000 −0.0909557
$$324$$ 0 0
$$325$$ −2429.64 −0.414683
$$326$$ 0 0
$$327$$ 17654.7 2.98565
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2372.00 −0.393888 −0.196944 0.980415i $$-0.563102\pi$$
−0.196944 + 0.980415i $$0.563102\pi$$
$$332$$ 0 0
$$333$$ −4758.00 −0.782993
$$334$$ 0 0
$$335$$ −5440.88 −0.887365
$$336$$ 0 0
$$337$$ −250.000 −0.0404106 −0.0202053 0.999796i $$-0.506432\pi$$
−0.0202053 + 0.999796i $$0.506432\pi$$
$$338$$ 0 0
$$339$$ −12063.7 −1.93278
$$340$$ 0 0
$$341$$ −4127.57 −0.655485
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4224.00 −0.659167
$$346$$ 0 0
$$347$$ −9540.00 −1.47589 −0.737945 0.674861i $$-0.764204\pi$$
−0.737945 + 0.674861i $$0.764204\pi$$
$$348$$ 0 0
$$349$$ −5712.93 −0.876235 −0.438117 0.898918i $$-0.644355\pi$$
−0.438117 + 0.898918i $$0.644355\pi$$
$$350$$ 0 0
$$351$$ 20944.0 3.18492
$$352$$ 0 0
$$353$$ 4390.23 0.661950 0.330975 0.943640i $$-0.392622\pi$$
0.330975 + 0.943640i $$0.392622\pi$$
$$354$$ 0 0
$$355$$ 5103.17 0.762953
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1840.00 −0.270506 −0.135253 0.990811i $$-0.543185\pi$$
−0.135253 + 0.990811i $$0.543185\pi$$
$$360$$ 0 0
$$361$$ −6771.00 −0.987170
$$362$$ 0 0
$$363$$ −8733.55 −1.26279
$$364$$ 0 0
$$365$$ −5632.00 −0.807650
$$366$$ 0 0
$$367$$ 2964.34 0.421628 0.210814 0.977526i $$-0.432389\pi$$
0.210814 + 0.977526i $$0.432389\pi$$
$$368$$ 0 0
$$369$$ 24033.7 3.39063
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3982.00 0.552762 0.276381 0.961048i $$-0.410865\pi$$
0.276381 + 0.961048i $$0.410865\pi$$
$$374$$ 0 0
$$375$$ −14256.0 −1.96314
$$376$$ 0 0
$$377$$ −10900.5 −1.48914
$$378$$ 0 0
$$379$$ −2676.00 −0.362683 −0.181342 0.983420i $$-0.558044\pi$$
−0.181342 + 0.983420i $$0.558044\pi$$
$$380$$ 0 0
$$381$$ −21688.5 −2.91636
$$382$$ 0 0
$$383$$ −7035.62 −0.938652 −0.469326 0.883025i $$-0.655503\pi$$
−0.469326 + 0.883025i $$0.655503\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −26596.0 −3.49341
$$388$$ 0 0
$$389$$ 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
$$394$$ 0 0
$$395$$ 6378.97 0.812558
$$396$$ 0 0
$$397$$ −9052.50 −1.14441 −0.572207 0.820109i $$-0.693912\pi$$
−0.572207 + 0.820109i $$0.693912\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5706.00 −0.710584 −0.355292 0.934755i $$-0.615619\pi$$
−0.355292 + 0.934755i $$0.615619\pi$$
$$402$$ 0 0
$$403$$ 13552.0 1.67512
$$404$$ 0 0
$$405$$ 12617.2 1.54804
$$406$$ 0 0
$$407$$ 1560.00 0.189991
$$408$$ 0 0
$$409$$ 2420.25 0.292601 0.146301 0.989240i $$-0.453263\pi$$
0.146301 + 0.989240i $$0.453263\pi$$
$$410$$ 0 0
$$411$$ −10450.2 −1.25419
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1848.00 −0.218590
$$416$$ 0 0
$$417$$ −12936.0 −1.51913
$$418$$ 0 0
$$419$$ 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 $$0.922741\pi$$
$$422$$ 0 0
$$423$$ −12589.1 −1.44705
$$424$$ 0 0
$$425$$ −2082.54 −0.237690
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −12320.0 −1.38652
$$430$$ 0 0
$$431$$ −1336.00 −0.149311 −0.0746553 0.997209i $$-0.523786\pi$$
−0.0746553 + 0.997209i $$0.523786\pi$$
$$432$$ 0 0
$$433$$ −11163.2 −1.23896 −0.619479 0.785013i $$-0.712656\pi$$
−0.619479 + 0.785013i $$0.712656\pi$$
$$434$$ 0 0
$$435$$ −14608.0 −1.61011
$$436$$ 0 0
$$437$$ 450.280 0.0492902
$$438$$ 0 0
$$439$$ 3602.24 0.391630 0.195815 0.980641i $$-0.437265\pi$$
0.195815 + 0.980641i $$0.437265\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −6348.00 −0.680818 −0.340409 0.940277i $$-0.610566\pi$$
−0.340409 + 0.940277i $$0.610566\pi$$
$$444$$ 0 0
$$445$$ −14080.0 −1.49990
$$446$$ 0 0
$$447$$ −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
$$452$$ 0 0
$$453$$ −7804.85 −0.809501
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6866.00 0.702796 0.351398 0.936226i $$-0.385706\pi$$
0.351398 + 0.936226i $$0.385706\pi$$
$$458$$ 0 0
$$459$$ 17952.0 1.82555
$$460$$ 0 0
$$461$$ 1378.98 0.139318 0.0696590 0.997571i $$-0.477809\pi$$
0.0696590 + 0.997571i $$0.477809\pi$$
$$462$$ 0 0
$$463$$ −2648.00 −0.265795 −0.132897 0.991130i $$-0.542428\pi$$
−0.132897 + 0.991130i $$0.542428\pi$$
$$464$$ 0 0
$$465$$ 18161.3 1.81120
$$466$$ 0 0
$$467$$ 12335.8 1.22234 0.611170 0.791500i $$-0.290699\pi$$
0.611170 + 0.791500i $$0.290699\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 27016.0 2.64295
$$472$$ 0 0
$$473$$ 8720.00 0.847666
$$474$$ 0 0
$$475$$ 347.091 0.0335276
$$476$$ 0 0
$$477$$ 3782.00 0.363031
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −6160.00 −0.576724
$$486$$ 0 0
$$487$$ −13936.0 −1.29672 −0.648358 0.761336i $$-0.724544\pi$$
−0.648358 + 0.761336i $$0.724544\pi$$
$$488$$ 0 0
$$489$$ −5966.21 −0.551741
$$490$$ 0 0
$$491$$ 12276.0 1.12833 0.564163 0.825663i $$-0.309199\pi$$
0.564163 + 0.825663i $$0.309199\pi$$
$$492$$ 0 0
$$493$$ −9343.31 −0.853553
$$494$$ 0 0
$$495$$ −11444.6 −1.03919
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2220.00 0.199160 0.0995800 0.995030i $$-0.468250\pi$$
0.0995800 + 0.995030i $$0.468250\pi$$
$$500$$ 0 0
$$501$$ 6160.00 0.549318
$$502$$ 0 0
$$503$$ 11294.5 1.00119 0.500594 0.865682i $$-0.333115\pi$$
0.500594 + 0.865682i $$0.333115\pi$$
$$504$$ 0 0
$$505$$ 1144.00 0.100807
$$506$$ 0 0
$$507$$ 19840.5 1.73796
$$508$$ 0 0
$$509$$ −15881.7 −1.38300 −0.691499 0.722377i $$-0.743049\pi$$
−0.691499 + 0.722377i $$0.743049\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2992.00 −0.257505
$$514$$ 0 0
$$515$$ 12848.0 1.09932
$$516$$ 0 0
$$517$$ 4127.57 0.351122
$$518$$ 0 0
$$519$$ 6248.00 0.528433
$$520$$ 0 0
$$521$$ 11613.5 0.976575 0.488287 0.872683i $$-0.337622\pi$$
0.488287 + 0.872683i $$0.337622\pi$$
$$522$$ 0 0
$$523$$ −12617.2 −1.05490 −0.527450 0.849586i $$-0.676852\pi$$
−0.527450 + 0.849586i $$0.676852\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 11616.0 0.960154
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ 0 0
$$531$$ 40628.4 3.32038
$$532$$ 0 0
$$533$$ 25872.0 2.10252
$$534$$ 0 0
$$535$$ 2439.02 0.197099
$$536$$ 0 0
$$537$$ 30281.3 2.43340
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1798.00 0.142887 0.0714437 0.997445i $$-0.477239\pi$$
0.0714437 + 0.997445i $$0.477239\pi$$
$$542$$ 0 0
$$543$$ −26488.0 −2.09339
$$544$$ 0 0
$$545$$ 17654.7 1.38761
$$546$$ 0 0
$$547$$ −1276.00 −0.0997401 −0.0498700 0.998756i $$-0.515881\pi$$
−0.0498700 + 0.998756i $$0.515881\pi$$
$$548$$ 0 0
$$549$$ 16594.7 1.29006
$$550$$ 0 0
$$551$$ 1557.22 0.120399
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −6864.00 −0.524974
$$556$$ 0 0
$$557$$ 2694.00 0.204934 0.102467 0.994736i $$-0.467326\pi$$
0.102467 + 0.994736i $$0.467326\pi$$
$$558$$ 0 0
$$559$$ −28630.3 −2.16625
$$560$$ 0 0
$$561$$ −10560.0 −0.794730
$$562$$ 0 0
$$563$$ 15769.2 1.18045 0.590223 0.807240i $$-0.299040\pi$$
0.590223 + 0.807240i $$0.299040\pi$$
$$564$$ 0 0
$$565$$ −12063.7 −0.898276
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 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 $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 20037.5 1.46087
$$574$$ 0 0
$$575$$ 1776.00 0.128808
$$576$$ 0 0
$$577$$ 14371.4 1.03690 0.518449 0.855108i $$-0.326509\pi$$
0.518449 + 0.855108i $$0.326509\pi$$
$$578$$ 0 0
$$579$$ 15553.4 1.11637
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1240.00 −0.0880884
$$584$$ 0 0
$$585$$ 37576.0 2.65569
$$586$$ 0 0
$$587$$ −18977.4 −1.33438 −0.667191 0.744887i $$-0.732503\pi$$
−0.667191 + 0.744887i $$0.732503\pi$$
$$588$$ 0 0
$$589$$ −1936.00 −0.135435
$$590$$ 0 0
$$591$$ −9174.45 −0.638556
$$592$$ 0 0
$$593$$ 8217.61 0.569067 0.284534 0.958666i $$-0.408161\pi$$
0.284534 + 0.958666i $$0.408161\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 46288.0 3.17327
$$598$$ 0 0
$$599$$ 19104.0 1.30312 0.651559 0.758598i $$-0.274115\pi$$
0.651559 + 0.758598i $$0.274115\pi$$
$$600$$ 0 0
$$601$$ 21538.4 1.46185 0.730923 0.682460i $$-0.239090\pi$$
0.730923 + 0.682460i $$0.239090\pi$$
$$602$$ 0 0
$$603$$ −35380.0 −2.38936
$$604$$ 0 0
$$605$$ −8733.55 −0.586892
$$606$$ 0 0
$$607$$ −13733.5 −0.918331 −0.459166 0.888351i $$-0.651852\pi$$
−0.459166 + 0.888351i $$0.651852\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13552.0 −0.897308
$$612$$ 0 0
$$613$$ 28034.0 1.84712 0.923558 0.383458i $$-0.125267\pi$$
0.923558 + 0.383458i $$0.125267\pi$$
$$614$$ 0 0
$$615$$ 34671.6 2.27332
$$616$$ 0 0
$$617$$ −8258.00 −0.538824 −0.269412 0.963025i $$-0.586829\pi$$
−0.269412 + 0.963025i $$0.586829\pi$$
$$618$$ 0 0
$$619$$ 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
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9631.00 −0.616384
$$626$$ 0 0
$$627$$ 1760.00 0.112101
$$628$$ 0 0
$$629$$ −4390.23 −0.278299
$$630$$ 0 0
$$631$$ −912.000 −0.0575375 −0.0287687 0.999586i $$-0.509159\pi$$
−0.0287687 + 0.999586i $$0.509159\pi$$
$$632$$ 0 0
$$633$$ −14596.6 −0.916527
$$634$$ 0 0
$$635$$ −21688.5 −1.35540
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 33184.0 2.05436
$$640$$ 0 0
$$641$$ −890.000 −0.0548407 −0.0274203 0.999624i $$-0.508729\pi$$
−0.0274203 + 0.999624i $$0.508729\pi$$
$$642$$ 0 0
$$643$$ 29352.6 1.80024 0.900120 0.435642i $$-0.143479\pi$$
0.900120 + 0.435642i $$0.143479\pi$$
$$644$$ 0 0
$$645$$ −38368.0 −2.34223
$$646$$ 0 0
$$647$$ −11876.1 −0.721637 −0.360818 0.932636i $$-0.617503\pi$$
−0.360818 + 0.932636i $$0.617503\pi$$
$$648$$ 0 0
$$649$$ −13320.8 −0.805680
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −21526.0 −1.29001 −0.645006 0.764178i $$-0.723145\pi$$
−0.645006 + 0.764178i $$0.723145\pi$$
$$654$$ 0 0
$$655$$ 2376.00 0.141737
$$656$$ 0 0
$$657$$ −36622.8 −2.17472
$$658$$ 0 0
$$659$$ −23452.0 −1.38628 −0.693141 0.720802i $$-0.743774\pi$$
−0.693141 + 0.720802i $$0.743774\pi$$
$$660$$ 0 0
$$661$$ −26669.7 −1.56934 −0.784668 0.619916i $$-0.787167\pi$$
−0.784668 + 0.619916i $$0.787167\pi$$
$$662$$ 0 0
$$663$$ 34671.6 2.03097
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7968.00 0.462552
$$668$$ 0 0
$$669$$ −27104.0 −1.56637
$$670$$ 0 0
$$671$$ −5440.88 −0.313030
$$672$$ 0 0
$$673$$ −13858.0 −0.793739 −0.396870 0.917875i $$-0.629904\pi$$
−0.396870 + 0.917875i $$0.629904\pi$$
$$674$$ 0 0
$$675$$ −11801.1 −0.672924
$$676$$ 0 0
$$677$$ 32448.3 1.84208 0.921041 0.389466i $$-0.127340\pi$$
0.921041 + 0.389466i $$0.127340\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18568.0 1.04483
$$682$$ 0 0
$$683$$ 27812.0 1.55812 0.779060 0.626949i $$-0.215696\pi$$
0.779060 + 0.626949i $$0.215696\pi$$
$$684$$ 0 0
$$685$$ −10450.2 −0.582895
$$686$$ 0 0
$$687$$ 25960.0 1.44168
$$688$$ 0 0
$$689$$ 4071.28 0.225114
$$690$$ 0 0
$$691$$ 1303.94 0.0717859 0.0358929 0.999356i $$-0.488572\pi$$
0.0358929 + 0.999356i $$0.488572\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −12936.0 −0.706029
$$696$$ 0 0
$$697$$ 22176.0 1.20513
$$698$$ 0 0
$$699$$ −60881.6 −3.29435
$$700$$ 0 0
$$701$$ 22906.0 1.23416 0.617081 0.786900i $$-0.288315\pi$$
0.617081 + 0.786900i $$0.288315\pi$$
$$702$$ 0 0
$$703$$ 731.705 0.0392557
$$704$$ 0 0
$$705$$ −18161.3 −0.970204
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −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
$$712$$ 0 0
$$713$$ −9906.16 −0.520321
$$714$$ 0 0
$$715$$ −12320.0 −0.644394
$$716$$ 0 0
$$717$$ 40300.1 2.09907
$$718$$ 0 0
$$719$$ 20544.0 1.06559 0.532797 0.846243i $$-0.321141\pi$$
0.532797 + 0.846243i $$0.321141\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 42416.0 2.18184
$$724$$ 0 0
$$725$$ 6142.00 0.314632
$$726$$ 0 0
$$727$$ 7223.24 0.368494 0.184247 0.982880i $$-0.441015\pi$$
0.184247 + 0.982880i $$0.441015\pi$$
$$728$$ 0 0
$$729$$ 1261.00 0.0640654
$$730$$ 0 0
$$731$$ −24540.3 −1.24166
$$732$$ 0 0
$$733$$ 29427.7 1.48286 0.741430 0.671031i $$-0.234148\pi$$
0.741430 + 0.671031i $$0.234148\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 11600.0 0.579771
$$738$$ 0 0
$$739$$ −32668.0 −1.62613 −0.813066 0.582171i $$-0.802203\pi$$
−0.813066 + 0.582171i $$0.802203\pi$$
$$740$$ 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 $$0.132316\pi$$
$$744$$ 0 0
$$745$$ −8874.27 −0.436413
$$746$$ 0 0
$$747$$ −12016.8 −0.588586
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 19608.0 0.952738 0.476369 0.879246i $$-0.341953\pi$$
0.476369 + 0.879246i $$0.341953\pi$$
$$752$$ 0 0
$$753$$ −52360.0 −2.53400
$$754$$ 0 0
$$755$$ −7804.85 −0.376222
$$756$$ 0 0
$$757$$ 19378.0 0.930390 0.465195 0.885208i $$-0.345984\pi$$
0.465195 + 0.885208i $$0.345984\pi$$
$$758$$ 0 0
$$759$$ 9005.60 0.430675
$$760$$ 0 0
$$761$$ −13977.4 −0.665810 −0.332905 0.942960i $$-0.608029\pi$$
−0.332905 + 0.942960i $$0.608029\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 32208.0 1.52220
$$766$$ 0 0
$$767$$ 43736.0 2.05895
$$768$$ 0 0
$$769$$ 8536.56 0.400307 0.200154 0.979765i $$-0.435856\pi$$
0.200154 + 0.979765i $$0.435856\pi$$
$$770$$ 0 0
$$771$$ −14080.0 −0.657690
$$772$$ 0 0
$$773$$ −29296.3 −1.36315 −0.681576 0.731748i $$-0.738705\pi$$
−0.681576 + 0.731748i $$0.738705\pi$$
$$774$$ 0 0
$$775$$ −7636.00 −0.353927
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3696.00 −0.169991
$$780$$ 0 0
$$781$$ −10880.0 −0.498485
$$782$$ 0 0
$$783$$ −52945.4 −2.41649
$$784$$ 0 0
$$785$$ 27016.0 1.22833
$$786$$ 0 0
$$787$$ −13780.4 −0.624167 −0.312084 0.950055i $$-0.601027\pi$$
−0.312084 + 0.950055i $$0.601027\pi$$
$$788$$ 0 0
$$789$$ 3752.33 0.169311
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 17864.0 0.799961
$$794$$ 0 0
$$795$$ 5456.00 0.243402
$$796$$ 0 0
$$797$$ −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
$$802$$ 0 0
$$803$$ 12007.5 0.527689
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −2552.00 −0.111319
$$808$$ 0 0
$$809$$ 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
$$814$$ 0 0
$$815$$ −5966.21 −0.256426
$$816$$ 0 0
$$817$$ 4090.04 0.175144
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 28622.0 1.21670 0.608352 0.793667i $$-0.291831\pi$$
0.608352 + 0.793667i $$0.291831\pi$$
$$822$$ 0 0
$$823$$ −24688.0 −1.04565 −0.522825 0.852440i $$-0.675122\pi$$
−0.522825 + 0.852440i $$0.675122\pi$$
$$824$$ 0 0
$$825$$ 6941.82 0.292949
$$826$$ 0 0
$$827$$ 30756.0 1.29322 0.646609 0.762822i $$-0.276187\pi$$
0.646609 + 0.762822i $$0.276187\pi$$
$$828$$ 0 0
$$829$$ 23236.3 0.973499 0.486750 0.873542i $$-0.338182\pi$$
0.486750 + 0.873542i $$0.338182\pi$$
$$830$$ 0 0
$$831$$ −63508.2 −2.65111
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 6160.00 0.255300
$$836$$ 0 0
$$837$$ 65824.0 2.71829
$$838$$ 0 0
$$839$$ 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
$$842$$ 0 0
$$843$$ 17617.2 0.719773
$$844$$ 0 0
$$845$$ 19840.5 0.807731
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −3608.00 −0.145850
$$850$$ 0 0
$$851$$ 3744.00 0.150814
$$852$$ 0 0
$$853$$ −23574.0 −0.946260 −0.473130 0.880993i $$-0.656876\pi$$
−0.473130 + 0.880993i $$0.656876\pi$$
$$854$$ 0 0
$$855$$ −5368.00 −0.214715
$$856$$ 0 0
$$857$$ 24484.0 0.975912 0.487956 0.872868i $$-0.337743\pi$$
0.487956 + 0.872868i $$0.337743\pi$$
$$858$$ 0 0
$$859$$ 32954.9 1.30897 0.654485 0.756075i $$-0.272885\pi$$
0.654485 + 0.756075i $$0.272885\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −40872.0 −1.61217 −0.806083 0.591803i $$-0.798416\pi$$
−0.806083 + 0.591803i $$0.798416\pi$$
$$864$$ 0 0
$$865$$ 6248.00 0.245593
$$866$$ 0 0
$$867$$ −16369.6 −0.641222
$$868$$ 0 0
$$869$$ −13600.0 −0.530896
$$870$$ 0 0
$$871$$ −38086.2 −1.48163
$$872$$ 0 0
$$873$$ −40056.2 −1.55292
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12006.0 −0.462273 −0.231137 0.972921i $$-0.574244\pi$$
−0.231137 + 0.972921i $$0.574244\pi$$
$$878$$ 0 0
$$879$$ −35112.0 −1.34732
$$880$$ 0 0
$$881$$ 35722.2 1.36607 0.683037 0.730383i $$-0.260659\pi$$
0.683037 + 0.730383i $$0.260659\pi$$
$$882$$ 0 0
$$883$$ −19588.0 −0.746533 −0.373267 0.927724i $$-0.621762\pi$$
−0.373267 + 0.927724i $$0.621762\pi$$
$$884$$ 0 0
$$885$$ 58611.4 2.22622
$$886$$ 0 0
$$887$$ 40243.8 1.52340 0.761699 0.647931i $$-0.224366\pi$$
0.761699 + 0.647931i $$0.224366\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −26900.0 −1.01143
$$892$$ 0 0
$$893$$ 1936.00 0.0725485
$$894$$ 0 0
$$895$$ 30281.3 1.13094
$$896$$ 0 0
$$897$$ −29568.0 −1.10061
$$898$$ 0 0
$$899$$ −34258.8 −1.27096
$$900$$ 0 0
$$901$$ 3489.67 0.129032
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −26488.0 −0.972918
$$906$$ 0 0
$$907$$ −15868.0 −0.580913 −0.290457 0.956888i $$-0.593807\pi$$
−0.290457 + 0.956888i $$0.593807\pi$$
$$908$$ 0 0
$$909$$ 7439.00 0.271437
$$910$$ 0 0
$$911$$ −39832.0 −1.44862 −0.724310 0.689474i $$-0.757842\pi$$
−0.724310 + 0.689474i $$0.757842\pi$$
$$912$$ 0 0
$$913$$ 3939.95 0.142818
$$914$$ 0 0
$$915$$ 23939.9 0.864949
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 30528.0 1.09578 0.547892 0.836549i $$-0.315430\pi$$
0.547892 + 0.836549i $$0.315430\pi$$
$$920$$ 0 0
$$921$$ −6776.00 −0.242429
$$922$$ 0 0
$$923$$ 35722.2 1.27390
$$924$$ 0 0
$$925$$ 2886.00 0.102585
$$926$$ 0 0
$$927$$ 83545.7 2.96009
$$928$$ 0 0
$$929$$ −16604.1 −0.586396 −0.293198 0.956052i $$-0.594720\pi$$
−0.293198 + 0.956052i $$0.594720\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −68288.0 −2.39619
$$934$$ 0 0
$$935$$ −10560.0 −0.369357
$$936$$ 0 0
$$937$$ 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 $$-0.470320\pi$$
0.0931068 + 0.995656i $$0.470320\pi$$
$$942$$ 0 0
$$943$$ −18911.8 −0.653077
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −45212.0 −1.55142 −0.775709 0.631091i $$-0.782607\pi$$
−0.775709 + 0.631091i $$0.782607\pi$$
$$948$$ 0 0
$$949$$ −39424.0 −1.34853
$$950$$ 0 0
$$951$$ 22120.0 0.754248
$$952$$ 0 0
$$953$$ 34218.0 1.16310 0.581548 0.813512i $$-0.302447\pi$$
0.581548 + 0.813512i $$0.302447\pi$$
$$954$$ 0 0
$$955$$ 20037.5 0.678950
$$956$$ 0 0
$$957$$ 31144.4 1.05199
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 12801.0 0.429694
$$962$$ 0 0
$$963$$ 15860.0 0.530718
$$964$$ 0 0
$$965$$ 15553.4 0.518842
$$966$$ 0 0
$$967$$ −14464.0 −0.481004 −0.240502 0.970649i $$-0.577312\pi$$
−0.240502 + 0.970649i $$0.577312\pi$$
$$968$$ 0 0
$$969$$ −4953.08 −0.164206
$$970$$ 0 0
$$971$$ −37832.9 −1.25038 −0.625188 0.780474i $$-0.714978\pi$$
−0.625188 + 0.780474i $$0.714978\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −22792.0 −0.748644
$$976$$ 0 0
$$977$$ 42062.0 1.37736 0.688681 0.725065i $$-0.258190\pi$$
0.688681 + 0.725065i $$0.258190\pi$$
$$978$$ 0 0
$$979$$ 30018.7 0.979980
$$980$$ 0 0
$$981$$ 114802. 3.73634
$$982$$ 0 0
$$983$$ 43020.5 1.39587 0.697935 0.716161i $$-0.254102\pi$$
0.697935 + 0.716161i $$0.254102\pi$$
$$984$$ 0 0
$$985$$ −9174.45 −0.296774
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 20928.0 0.672873
$$990$$ 0 0
$$991$$ −21272.0 −0.681864 −0.340932 0.940088i $$-0.610743\pi$$
−0.340932 + 0.940088i $$0.610743\pi$$
$$992$$ 0 0
$$993$$ −22251.3 −0.711102
$$994$$ 0 0
$$995$$ 46288.0 1.47480
$$996$$ 0 0
$$997$$ 121.951 0.00387384 0.00193692 0.999998i $$-0.499383\pi$$
0.00193692 + 0.999998i $$0.499383\pi$$
$$998$$ 0 0
$$999$$ −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 784.4.a.z.1.2 2
4.3 odd 2 98.4.a.h.1.1 2
7.6 odd 2 inner 784.4.a.z.1.1 2
12.11 even 2 882.4.a.w.1.1 2
20.19 odd 2 2450.4.a.bs.1.2 2
28.3 even 6 98.4.c.g.79.1 4
28.11 odd 6 98.4.c.g.79.2 4
28.19 even 6 98.4.c.g.67.1 4
28.23 odd 6 98.4.c.g.67.2 4
28.27 even 2 98.4.a.h.1.2 yes 2
84.11 even 6 882.4.g.bi.667.2 4
84.23 even 6 882.4.g.bi.361.2 4
84.47 odd 6 882.4.g.bi.361.1 4
84.59 odd 6 882.4.g.bi.667.1 4
84.83 odd 2 882.4.a.w.1.2 2
140.139 even 2 2450.4.a.bs.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 4.3 odd 2
98.4.a.h.1.2 yes 2 28.27 even 2
98.4.c.g.67.1 4 28.19 even 6
98.4.c.g.67.2 4 28.23 odd 6
98.4.c.g.79.1 4 28.3 even 6
98.4.c.g.79.2 4 28.11 odd 6
784.4.a.z.1.1 2 7.6 odd 2 inner
784.4.a.z.1.2 2 1.1 even 1 trivial
882.4.a.w.1.1 2 12.11 even 2
882.4.a.w.1.2 2 84.83 odd 2
882.4.g.bi.361.1 4 84.47 odd 6
882.4.g.bi.361.2 4 84.23 even 6
882.4.g.bi.667.1 4 84.59 odd 6
882.4.g.bi.667.2 4 84.11 even 6
2450.4.a.bs.1.1 2 140.139 even 2
2450.4.a.bs.1.2 2 20.19 odd 2