# Properties

 Label 1176.4.a.bb.1.2 Level $1176$ Weight $4$ Character 1176.1 Self dual yes Analytic conductor $69.386$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.3862461668$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.145408.2 Defining polynomial: $$x^{4} - 24 x^{2} + 142$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.66254$$ of defining polynomial Character $$\chi$$ $$=$$ 1176.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -4.11659 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -4.11659 q^{5} +9.00000 q^{9} +17.9344 q^{11} +23.4236 q^{13} +12.3498 q^{15} +76.2280 q^{17} +35.5964 q^{19} -40.7918 q^{23} -108.054 q^{25} -27.0000 q^{27} -178.805 q^{29} +31.6797 q^{31} -53.8032 q^{33} -54.8450 q^{37} -70.2708 q^{39} -190.547 q^{41} -131.964 q^{43} -37.0493 q^{45} +199.573 q^{47} -228.684 q^{51} +321.907 q^{53} -73.8287 q^{55} -106.789 q^{57} +163.392 q^{59} +265.230 q^{61} -96.4254 q^{65} +278.127 q^{67} +122.375 q^{69} -10.5794 q^{71} +584.292 q^{73} +324.161 q^{75} -183.078 q^{79} +81.0000 q^{81} -175.117 q^{83} -313.800 q^{85} +536.416 q^{87} +47.1437 q^{89} -95.0391 q^{93} -146.536 q^{95} -556.805 q^{97} +161.410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10})$$ $$4q - 12q^{3} + 8q^{5} + 36q^{9} - 40q^{11} + 48q^{13} - 24q^{15} - 152q^{17} + 224q^{19} - 8q^{23} - 28q^{25} - 108q^{27} - 144q^{29} + 400q^{31} + 120q^{33} - 304q^{37} - 144q^{39} - 152q^{41} + 160q^{43} + 72q^{45} + 544q^{47} + 456q^{51} - 1320q^{53} + 16q^{55} - 672q^{57} + 1040q^{59} + 896q^{61} - 648q^{65} - 416q^{67} + 24q^{69} + 248q^{71} - 752q^{73} + 84q^{75} + 864q^{79} + 324q^{81} + 1456q^{83} - 1608q^{85} + 432q^{87} + 2936q^{89} - 1200q^{93} - 80q^{95} + 144q^{97} - 360q^{99} + O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −4.11659 −0.368199 −0.184100 0.982908i $$-0.558937\pi$$
−0.184100 + 0.982908i $$0.558937\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 17.9344 0.491585 0.245792 0.969323i $$-0.420952\pi$$
0.245792 + 0.969323i $$0.420952\pi$$
$$12$$ 0 0
$$13$$ 23.4236 0.499733 0.249867 0.968280i $$-0.419613\pi$$
0.249867 + 0.968280i $$0.419613\pi$$
$$14$$ 0 0
$$15$$ 12.3498 0.212580
$$16$$ 0 0
$$17$$ 76.2280 1.08753 0.543765 0.839237i $$-0.316998\pi$$
0.543765 + 0.839237i $$0.316998\pi$$
$$18$$ 0 0
$$19$$ 35.5964 0.429809 0.214905 0.976635i $$-0.431056\pi$$
0.214905 + 0.976635i $$0.431056\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −40.7918 −0.369812 −0.184906 0.982756i $$-0.559198\pi$$
−0.184906 + 0.982756i $$0.559198\pi$$
$$24$$ 0 0
$$25$$ −108.054 −0.864429
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −178.805 −1.14494 −0.572471 0.819925i $$-0.694015\pi$$
−0.572471 + 0.819925i $$0.694015\pi$$
$$30$$ 0 0
$$31$$ 31.6797 0.183543 0.0917717 0.995780i $$-0.470747\pi$$
0.0917717 + 0.995780i $$0.470747\pi$$
$$32$$ 0 0
$$33$$ −53.8032 −0.283816
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −54.8450 −0.243688 −0.121844 0.992549i $$-0.538881\pi$$
−0.121844 + 0.992549i $$0.538881\pi$$
$$38$$ 0 0
$$39$$ −70.2708 −0.288521
$$40$$ 0 0
$$41$$ −190.547 −0.725815 −0.362907 0.931825i $$-0.618216\pi$$
−0.362907 + 0.931825i $$0.618216\pi$$
$$42$$ 0 0
$$43$$ −131.964 −0.468009 −0.234004 0.972236i $$-0.575183\pi$$
−0.234004 + 0.972236i $$0.575183\pi$$
$$44$$ 0 0
$$45$$ −37.0493 −0.122733
$$46$$ 0 0
$$47$$ 199.573 0.619377 0.309689 0.950838i $$-0.399775\pi$$
0.309689 + 0.950838i $$0.399775\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −228.684 −0.627886
$$52$$ 0 0
$$53$$ 321.907 0.834289 0.417145 0.908840i $$-0.363031\pi$$
0.417145 + 0.908840i $$0.363031\pi$$
$$54$$ 0 0
$$55$$ −73.8287 −0.181001
$$56$$ 0 0
$$57$$ −106.789 −0.248150
$$58$$ 0 0
$$59$$ 163.392 0.360540 0.180270 0.983617i $$-0.442303\pi$$
0.180270 + 0.983617i $$0.442303\pi$$
$$60$$ 0 0
$$61$$ 265.230 0.556710 0.278355 0.960478i $$-0.410211\pi$$
0.278355 + 0.960478i $$0.410211\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −96.4254 −0.184001
$$66$$ 0 0
$$67$$ 278.127 0.507144 0.253572 0.967316i $$-0.418394\pi$$
0.253572 + 0.967316i $$0.418394\pi$$
$$68$$ 0 0
$$69$$ 122.375 0.213511
$$70$$ 0 0
$$71$$ −10.5794 −0.0176836 −0.00884182 0.999961i $$-0.502814\pi$$
−0.00884182 + 0.999961i $$0.502814\pi$$
$$72$$ 0 0
$$73$$ 584.292 0.936797 0.468398 0.883517i $$-0.344831\pi$$
0.468398 + 0.883517i $$0.344831\pi$$
$$74$$ 0 0
$$75$$ 324.161 0.499079
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −183.078 −0.260733 −0.130366 0.991466i $$-0.541615\pi$$
−0.130366 + 0.991466i $$0.541615\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −175.117 −0.231585 −0.115792 0.993273i $$-0.536941\pi$$
−0.115792 + 0.993273i $$0.536941\pi$$
$$84$$ 0 0
$$85$$ −313.800 −0.400428
$$86$$ 0 0
$$87$$ 536.416 0.661033
$$88$$ 0 0
$$89$$ 47.1437 0.0561485 0.0280743 0.999606i $$-0.491063\pi$$
0.0280743 + 0.999606i $$0.491063\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −95.0391 −0.105969
$$94$$ 0 0
$$95$$ −146.536 −0.158255
$$96$$ 0 0
$$97$$ −556.805 −0.582834 −0.291417 0.956596i $$-0.594127\pi$$
−0.291417 + 0.956596i $$0.594127\pi$$
$$98$$ 0 0
$$99$$ 161.410 0.163862
$$100$$ 0 0
$$101$$ 128.804 0.126895 0.0634477 0.997985i $$-0.479790\pi$$
0.0634477 + 0.997985i $$0.479790\pi$$
$$102$$ 0 0
$$103$$ 2043.48 1.95485 0.977427 0.211271i $$-0.0677604\pi$$
0.977427 + 0.211271i $$0.0677604\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −268.860 −0.242913 −0.121456 0.992597i $$-0.538756\pi$$
−0.121456 + 0.992597i $$0.538756\pi$$
$$108$$ 0 0
$$109$$ 1335.64 1.17368 0.586841 0.809702i $$-0.300371\pi$$
0.586841 + 0.809702i $$0.300371\pi$$
$$110$$ 0 0
$$111$$ 164.535 0.140693
$$112$$ 0 0
$$113$$ −1675.40 −1.39476 −0.697382 0.716700i $$-0.745652\pi$$
−0.697382 + 0.716700i $$0.745652\pi$$
$$114$$ 0 0
$$115$$ 167.923 0.136164
$$116$$ 0 0
$$117$$ 210.812 0.166578
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1009.36 −0.758345
$$122$$ 0 0
$$123$$ 571.640 0.419049
$$124$$ 0 0
$$125$$ 959.387 0.686481
$$126$$ 0 0
$$127$$ 1626.28 1.13629 0.568144 0.822929i $$-0.307662\pi$$
0.568144 + 0.822929i $$0.307662\pi$$
$$128$$ 0 0
$$129$$ 395.893 0.270205
$$130$$ 0 0
$$131$$ 1534.98 1.02375 0.511877 0.859059i $$-0.328950\pi$$
0.511877 + 0.859059i $$0.328950\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 111.148 0.0708600
$$136$$ 0 0
$$137$$ 2466.49 1.53815 0.769076 0.639158i $$-0.220717\pi$$
0.769076 + 0.639158i $$0.220717\pi$$
$$138$$ 0 0
$$139$$ 1322.68 0.807107 0.403554 0.914956i $$-0.367775\pi$$
0.403554 + 0.914956i $$0.367775\pi$$
$$140$$ 0 0
$$141$$ −598.719 −0.357598
$$142$$ 0 0
$$143$$ 420.088 0.245661
$$144$$ 0 0
$$145$$ 736.069 0.421567
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −648.069 −0.356321 −0.178161 0.984001i $$-0.557015\pi$$
−0.178161 + 0.984001i $$0.557015\pi$$
$$150$$ 0 0
$$151$$ 2003.70 1.07986 0.539930 0.841710i $$-0.318451\pi$$
0.539930 + 0.841710i $$0.318451\pi$$
$$152$$ 0 0
$$153$$ 686.052 0.362510
$$154$$ 0 0
$$155$$ −130.412 −0.0675805
$$156$$ 0 0
$$157$$ −773.589 −0.393243 −0.196622 0.980479i $$-0.562997\pi$$
−0.196622 + 0.980479i $$0.562997\pi$$
$$158$$ 0 0
$$159$$ −965.721 −0.481677
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 66.4286 0.0319208 0.0159604 0.999873i $$-0.494919\pi$$
0.0159604 + 0.999873i $$0.494919\pi$$
$$164$$ 0 0
$$165$$ 221.486 0.104501
$$166$$ 0 0
$$167$$ 842.616 0.390441 0.195220 0.980759i $$-0.437458\pi$$
0.195220 + 0.980759i $$0.437458\pi$$
$$168$$ 0 0
$$169$$ −1648.34 −0.750266
$$170$$ 0 0
$$171$$ 320.368 0.143270
$$172$$ 0 0
$$173$$ 1570.69 0.690275 0.345138 0.938552i $$-0.387832\pi$$
0.345138 + 0.938552i $$0.387832\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −490.177 −0.208158
$$178$$ 0 0
$$179$$ 3670.74 1.53276 0.766380 0.642387i $$-0.222056\pi$$
0.766380 + 0.642387i $$0.222056\pi$$
$$180$$ 0 0
$$181$$ −802.417 −0.329520 −0.164760 0.986334i $$-0.552685\pi$$
−0.164760 + 0.986334i $$0.552685\pi$$
$$182$$ 0 0
$$183$$ −795.691 −0.321416
$$184$$ 0 0
$$185$$ 225.775 0.0897258
$$186$$ 0 0
$$187$$ 1367.11 0.534613
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4559.10 −1.72714 −0.863572 0.504225i $$-0.831778\pi$$
−0.863572 + 0.504225i $$0.831778\pi$$
$$192$$ 0 0
$$193$$ 3133.71 1.16875 0.584377 0.811482i $$-0.301339\pi$$
0.584377 + 0.811482i $$0.301339\pi$$
$$194$$ 0 0
$$195$$ 289.276 0.106233
$$196$$ 0 0
$$197$$ −2805.70 −1.01471 −0.507354 0.861738i $$-0.669376\pi$$
−0.507354 + 0.861738i $$0.669376\pi$$
$$198$$ 0 0
$$199$$ 5257.19 1.87272 0.936362 0.351036i $$-0.114171\pi$$
0.936362 + 0.351036i $$0.114171\pi$$
$$200$$ 0 0
$$201$$ −834.382 −0.292800
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 784.403 0.267244
$$206$$ 0 0
$$207$$ −367.126 −0.123271
$$208$$ 0 0
$$209$$ 638.400 0.211287
$$210$$ 0 0
$$211$$ 3230.30 1.05395 0.526974 0.849882i $$-0.323327\pi$$
0.526974 + 0.849882i $$0.323327\pi$$
$$212$$ 0 0
$$213$$ 31.7381 0.0102097
$$214$$ 0 0
$$215$$ 543.243 0.172320
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1752.87 −0.540860
$$220$$ 0 0
$$221$$ 1785.53 0.543475
$$222$$ 0 0
$$223$$ 433.169 0.130077 0.0650385 0.997883i $$-0.479283\pi$$
0.0650385 + 0.997883i $$0.479283\pi$$
$$224$$ 0 0
$$225$$ −972.483 −0.288143
$$226$$ 0 0
$$227$$ 4108.60 1.20131 0.600655 0.799509i $$-0.294907\pi$$
0.600655 + 0.799509i $$0.294907\pi$$
$$228$$ 0 0
$$229$$ 4633.01 1.33693 0.668467 0.743742i $$-0.266951\pi$$
0.668467 + 0.743742i $$0.266951\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 643.222 0.180853 0.0904267 0.995903i $$-0.471177\pi$$
0.0904267 + 0.995903i $$0.471177\pi$$
$$234$$ 0 0
$$235$$ −821.561 −0.228054
$$236$$ 0 0
$$237$$ 549.235 0.150534
$$238$$ 0 0
$$239$$ 783.612 0.212082 0.106041 0.994362i $$-0.466182\pi$$
0.106041 + 0.994362i $$0.466182\pi$$
$$240$$ 0 0
$$241$$ −6338.58 −1.69421 −0.847103 0.531429i $$-0.821655\pi$$
−0.847103 + 0.531429i $$0.821655\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 833.795 0.214790
$$248$$ 0 0
$$249$$ 525.350 0.133706
$$250$$ 0 0
$$251$$ 5863.01 1.47438 0.737191 0.675685i $$-0.236152\pi$$
0.737191 + 0.675685i $$0.236152\pi$$
$$252$$ 0 0
$$253$$ −731.577 −0.181794
$$254$$ 0 0
$$255$$ 941.399 0.231187
$$256$$ 0 0
$$257$$ −4213.76 −1.02275 −0.511376 0.859357i $$-0.670864\pi$$
−0.511376 + 0.859357i $$0.670864\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1609.25 −0.381648
$$262$$ 0 0
$$263$$ −1955.85 −0.458566 −0.229283 0.973360i $$-0.573638\pi$$
−0.229283 + 0.973360i $$0.573638\pi$$
$$264$$ 0 0
$$265$$ −1325.16 −0.307185
$$266$$ 0 0
$$267$$ −141.431 −0.0324174
$$268$$ 0 0
$$269$$ 1527.99 0.346332 0.173166 0.984893i $$-0.444600\pi$$
0.173166 + 0.984893i $$0.444600\pi$$
$$270$$ 0 0
$$271$$ 7673.88 1.72013 0.860064 0.510185i $$-0.170423\pi$$
0.860064 + 0.510185i $$0.170423\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1937.88 −0.424940
$$276$$ 0 0
$$277$$ 6312.20 1.36918 0.684591 0.728927i $$-0.259981\pi$$
0.684591 + 0.728927i $$0.259981\pi$$
$$278$$ 0 0
$$279$$ 285.117 0.0611811
$$280$$ 0 0
$$281$$ −3334.66 −0.707932 −0.353966 0.935258i $$-0.615167\pi$$
−0.353966 + 0.935258i $$0.615167\pi$$
$$282$$ 0 0
$$283$$ −725.900 −0.152474 −0.0762372 0.997090i $$-0.524291\pi$$
−0.0762372 + 0.997090i $$0.524291\pi$$
$$284$$ 0 0
$$285$$ 439.607 0.0913688
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 897.713 0.182722
$$290$$ 0 0
$$291$$ 1670.41 0.336500
$$292$$ 0 0
$$293$$ −5681.47 −1.13282 −0.566408 0.824125i $$-0.691667\pi$$
−0.566408 + 0.824125i $$0.691667\pi$$
$$294$$ 0 0
$$295$$ −672.620 −0.132751
$$296$$ 0 0
$$297$$ −484.229 −0.0946055
$$298$$ 0 0
$$299$$ −955.490 −0.184807
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −386.411 −0.0732631
$$304$$ 0 0
$$305$$ −1091.85 −0.204980
$$306$$ 0 0
$$307$$ −2469.04 −0.459008 −0.229504 0.973308i $$-0.573710\pi$$
−0.229504 + 0.973308i $$0.573710\pi$$
$$308$$ 0 0
$$309$$ −6130.44 −1.12864
$$310$$ 0 0
$$311$$ 4068.00 0.741721 0.370860 0.928689i $$-0.379063\pi$$
0.370860 + 0.928689i $$0.379063\pi$$
$$312$$ 0 0
$$313$$ 4912.13 0.887062 0.443531 0.896259i $$-0.353726\pi$$
0.443531 + 0.896259i $$0.353726\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2761.17 0.489219 0.244610 0.969622i $$-0.421340\pi$$
0.244610 + 0.969622i $$0.421340\pi$$
$$318$$ 0 0
$$319$$ −3206.77 −0.562836
$$320$$ 0 0
$$321$$ 806.579 0.140246
$$322$$ 0 0
$$323$$ 2713.44 0.467430
$$324$$ 0 0
$$325$$ −2531.00 −0.431984
$$326$$ 0 0
$$327$$ −4006.93 −0.677626
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1191.72 0.197894 0.0989470 0.995093i $$-0.468453\pi$$
0.0989470 + 0.995093i $$0.468453\pi$$
$$332$$ 0 0
$$333$$ −493.605 −0.0812294
$$334$$ 0 0
$$335$$ −1144.94 −0.186730
$$336$$ 0 0
$$337$$ 4336.34 0.700937 0.350468 0.936575i $$-0.386022\pi$$
0.350468 + 0.936575i $$0.386022\pi$$
$$338$$ 0 0
$$339$$ 5026.20 0.805267
$$340$$ 0 0
$$341$$ 568.157 0.0902271
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −503.769 −0.0786146
$$346$$ 0 0
$$347$$ −9450.20 −1.46200 −0.730999 0.682379i $$-0.760946\pi$$
−0.730999 + 0.682379i $$0.760946\pi$$
$$348$$ 0 0
$$349$$ −10654.7 −1.63418 −0.817092 0.576507i $$-0.804415\pi$$
−0.817092 + 0.576507i $$0.804415\pi$$
$$350$$ 0 0
$$351$$ −632.437 −0.0961738
$$352$$ 0 0
$$353$$ −1702.81 −0.256746 −0.128373 0.991726i $$-0.540975\pi$$
−0.128373 + 0.991726i $$0.540975\pi$$
$$354$$ 0 0
$$355$$ 43.5509 0.00651110
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4108.61 −0.604023 −0.302011 0.953304i $$-0.597658\pi$$
−0.302011 + 0.953304i $$0.597658\pi$$
$$360$$ 0 0
$$361$$ −5591.90 −0.815264
$$362$$ 0 0
$$363$$ 3028.07 0.437830
$$364$$ 0 0
$$365$$ −2405.29 −0.344928
$$366$$ 0 0
$$367$$ 1179.35 0.167743 0.0838715 0.996477i $$-0.473271\pi$$
0.0838715 + 0.996477i $$0.473271\pi$$
$$368$$ 0 0
$$369$$ −1714.92 −0.241938
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7865.22 1.09181 0.545906 0.837846i $$-0.316186\pi$$
0.545906 + 0.837846i $$0.316186\pi$$
$$374$$ 0 0
$$375$$ −2878.16 −0.396340
$$376$$ 0 0
$$377$$ −4188.27 −0.572166
$$378$$ 0 0
$$379$$ 13076.8 1.77233 0.886164 0.463372i $$-0.153360\pi$$
0.886164 + 0.463372i $$0.153360\pi$$
$$380$$ 0 0
$$381$$ −4878.83 −0.656036
$$382$$ 0 0
$$383$$ −2410.38 −0.321578 −0.160789 0.986989i $$-0.551404\pi$$
−0.160789 + 0.986989i $$0.551404\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1187.68 −0.156003
$$388$$ 0 0
$$389$$ 6178.78 0.805339 0.402669 0.915346i $$-0.368082\pi$$
0.402669 + 0.915346i $$0.368082\pi$$
$$390$$ 0 0
$$391$$ −3109.48 −0.402182
$$392$$ 0 0
$$393$$ −4604.94 −0.591064
$$394$$ 0 0
$$395$$ 753.658 0.0960017
$$396$$ 0 0
$$397$$ 4809.52 0.608017 0.304009 0.952669i $$-0.401675\pi$$
0.304009 + 0.952669i $$0.401675\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6088.29 0.758191 0.379096 0.925357i $$-0.376235\pi$$
0.379096 + 0.925357i $$0.376235\pi$$
$$402$$ 0 0
$$403$$ 742.053 0.0917228
$$404$$ 0 0
$$405$$ −333.444 −0.0409110
$$406$$ 0 0
$$407$$ −983.613 −0.119793
$$408$$ 0 0
$$409$$ −7869.74 −0.951426 −0.475713 0.879600i $$-0.657810\pi$$
−0.475713 + 0.879600i $$0.657810\pi$$
$$410$$ 0 0
$$411$$ −7399.48 −0.888052
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 720.884 0.0852694
$$416$$ 0 0
$$417$$ −3968.03 −0.465984
$$418$$ 0 0
$$419$$ 15279.0 1.78145 0.890724 0.454544i $$-0.150198\pi$$
0.890724 + 0.454544i $$0.150198\pi$$
$$420$$ 0 0
$$421$$ 6909.70 0.799900 0.399950 0.916537i $$-0.369027\pi$$
0.399950 + 0.916537i $$0.369027\pi$$
$$422$$ 0 0
$$423$$ 1796.16 0.206459
$$424$$ 0 0
$$425$$ −8236.72 −0.940093
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1260.27 −0.141833
$$430$$ 0 0
$$431$$ −12627.4 −1.41124 −0.705618 0.708592i $$-0.749331\pi$$
−0.705618 + 0.708592i $$0.749331\pi$$
$$432$$ 0 0
$$433$$ −6738.74 −0.747906 −0.373953 0.927448i $$-0.621998\pi$$
−0.373953 + 0.927448i $$0.621998\pi$$
$$434$$ 0 0
$$435$$ −2208.21 −0.243392
$$436$$ 0 0
$$437$$ −1452.04 −0.158948
$$438$$ 0 0
$$439$$ −1752.54 −0.190533 −0.0952667 0.995452i $$-0.530370\pi$$
−0.0952667 + 0.995452i $$0.530370\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −14286.9 −1.53226 −0.766130 0.642686i $$-0.777820\pi$$
−0.766130 + 0.642686i $$0.777820\pi$$
$$444$$ 0 0
$$445$$ −194.071 −0.0206738
$$446$$ 0 0
$$447$$ 1944.21 0.205722
$$448$$ 0 0
$$449$$ −7524.11 −0.790835 −0.395417 0.918501i $$-0.629400\pi$$
−0.395417 + 0.918501i $$0.629400\pi$$
$$450$$ 0 0
$$451$$ −3417.35 −0.356799
$$452$$ 0 0
$$453$$ −6011.10 −0.623457
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10579.4 1.08290 0.541448 0.840734i $$-0.317876\pi$$
0.541448 + 0.840734i $$0.317876\pi$$
$$458$$ 0 0
$$459$$ −2058.16 −0.209295
$$460$$ 0 0
$$461$$ 5889.66 0.595030 0.297515 0.954717i $$-0.403842\pi$$
0.297515 + 0.954717i $$0.403842\pi$$
$$462$$ 0 0
$$463$$ 12716.5 1.27643 0.638215 0.769858i $$-0.279673\pi$$
0.638215 + 0.769858i $$0.279673\pi$$
$$464$$ 0 0
$$465$$ 391.237 0.0390176
$$466$$ 0 0
$$467$$ −8362.46 −0.828626 −0.414313 0.910134i $$-0.635978\pi$$
−0.414313 + 0.910134i $$0.635978\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2320.77 0.227039
$$472$$ 0 0
$$473$$ −2366.70 −0.230066
$$474$$ 0 0
$$475$$ −3846.32 −0.371540
$$476$$ 0 0
$$477$$ 2897.16 0.278096
$$478$$ 0 0
$$479$$ −5404.57 −0.515535 −0.257768 0.966207i $$-0.582987\pi$$
−0.257768 + 0.966207i $$0.582987\pi$$
$$480$$ 0 0
$$481$$ −1284.67 −0.121779
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2292.14 0.214599
$$486$$ 0 0
$$487$$ −10659.5 −0.991842 −0.495921 0.868368i $$-0.665169\pi$$
−0.495921 + 0.868368i $$0.665169\pi$$
$$488$$ 0 0
$$489$$ −199.286 −0.0184295
$$490$$ 0 0
$$491$$ −277.130 −0.0254719 −0.0127360 0.999919i $$-0.504054\pi$$
−0.0127360 + 0.999919i $$0.504054\pi$$
$$492$$ 0 0
$$493$$ −13630.0 −1.24516
$$494$$ 0 0
$$495$$ −664.458 −0.0603337
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 18108.3 1.62453 0.812266 0.583288i $$-0.198234\pi$$
0.812266 + 0.583288i $$0.198234\pi$$
$$500$$ 0 0
$$501$$ −2527.85 −0.225421
$$502$$ 0 0
$$503$$ −3671.00 −0.325411 −0.162706 0.986675i $$-0.552022\pi$$
−0.162706 + 0.986675i $$0.552022\pi$$
$$504$$ 0 0
$$505$$ −530.232 −0.0467228
$$506$$ 0 0
$$507$$ 4945.01 0.433167
$$508$$ 0 0
$$509$$ −17364.1 −1.51208 −0.756040 0.654526i $$-0.772868\pi$$
−0.756040 + 0.654526i $$0.772868\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −961.103 −0.0827168
$$514$$ 0 0
$$515$$ −8412.17 −0.719776
$$516$$ 0 0
$$517$$ 3579.23 0.304476
$$518$$ 0 0
$$519$$ −4712.08 −0.398531
$$520$$ 0 0
$$521$$ 2870.45 0.241376 0.120688 0.992690i $$-0.461490\pi$$
0.120688 + 0.992690i $$0.461490\pi$$
$$522$$ 0 0
$$523$$ 5735.39 0.479524 0.239762 0.970832i $$-0.422931\pi$$
0.239762 + 0.970832i $$0.422931\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2414.88 0.199609
$$528$$ 0 0
$$529$$ −10503.0 −0.863239
$$530$$ 0 0
$$531$$ 1470.53 0.120180
$$532$$ 0 0
$$533$$ −4463.29 −0.362714
$$534$$ 0 0
$$535$$ 1106.79 0.0894402
$$536$$ 0 0
$$537$$ −11012.2 −0.884940
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2014.27 0.160074 0.0800372 0.996792i $$-0.474496\pi$$
0.0800372 + 0.996792i $$0.474496\pi$$
$$542$$ 0 0
$$543$$ 2407.25 0.190249
$$544$$ 0 0
$$545$$ −5498.30 −0.432149
$$546$$ 0 0
$$547$$ 18005.2 1.40740 0.703699 0.710498i $$-0.251530\pi$$
0.703699 + 0.710498i $$0.251530\pi$$
$$548$$ 0 0
$$549$$ 2387.07 0.185570
$$550$$ 0 0
$$551$$ −6364.83 −0.492107
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −677.324 −0.0518032
$$556$$ 0 0
$$557$$ 3093.70 0.235340 0.117670 0.993053i $$-0.462458\pi$$
0.117670 + 0.993053i $$0.462458\pi$$
$$558$$ 0 0
$$559$$ −3091.08 −0.233880
$$560$$ 0 0
$$561$$ −4101.32 −0.308659
$$562$$ 0 0
$$563$$ −18245.4 −1.36581 −0.682907 0.730505i $$-0.739285\pi$$
−0.682907 + 0.730505i $$0.739285\pi$$
$$564$$ 0 0
$$565$$ 6896.94 0.513551
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 17456.7 1.28616 0.643079 0.765800i $$-0.277657\pi$$
0.643079 + 0.765800i $$0.277657\pi$$
$$570$$ 0 0
$$571$$ 10013.7 0.733902 0.366951 0.930240i $$-0.380402\pi$$
0.366951 + 0.930240i $$0.380402\pi$$
$$572$$ 0 0
$$573$$ 13677.3 0.997167
$$574$$ 0 0
$$575$$ 4407.70 0.319676
$$576$$ 0 0
$$577$$ −26521.5 −1.91353 −0.956764 0.290867i $$-0.906056\pi$$
−0.956764 + 0.290867i $$0.906056\pi$$
$$578$$ 0 0
$$579$$ −9401.14 −0.674781
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 5773.21 0.410124
$$584$$ 0 0
$$585$$ −867.828 −0.0613338
$$586$$ 0 0
$$587$$ 6734.74 0.473547 0.236774 0.971565i $$-0.423910\pi$$
0.236774 + 0.971565i $$0.423910\pi$$
$$588$$ 0 0
$$589$$ 1127.68 0.0788886
$$590$$ 0 0
$$591$$ 8417.09 0.585842
$$592$$ 0 0
$$593$$ −26173.3 −1.81250 −0.906248 0.422746i $$-0.861066\pi$$
−0.906248 + 0.422746i $$0.861066\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −15771.6 −1.08122
$$598$$ 0 0
$$599$$ −5944.71 −0.405499 −0.202750 0.979231i $$-0.564988\pi$$
−0.202750 + 0.979231i $$0.564988\pi$$
$$600$$ 0 0
$$601$$ −1988.96 −0.134994 −0.0674970 0.997719i $$-0.521501\pi$$
−0.0674970 + 0.997719i $$0.521501\pi$$
$$602$$ 0 0
$$603$$ 2503.15 0.169048
$$604$$ 0 0
$$605$$ 4155.11 0.279222
$$606$$ 0 0
$$607$$ 1383.89 0.0925376 0.0462688 0.998929i $$-0.485267\pi$$
0.0462688 + 0.998929i $$0.485267\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4674.72 0.309524
$$612$$ 0 0
$$613$$ 1500.35 0.0988558 0.0494279 0.998778i $$-0.484260\pi$$
0.0494279 + 0.998778i $$0.484260\pi$$
$$614$$ 0 0
$$615$$ −2353.21 −0.154294
$$616$$ 0 0
$$617$$ 15497.7 1.01121 0.505604 0.862766i $$-0.331270\pi$$
0.505604 + 0.862766i $$0.331270\pi$$
$$618$$ 0 0
$$619$$ 515.002 0.0334405 0.0167203 0.999860i $$-0.494678\pi$$
0.0167203 + 0.999860i $$0.494678\pi$$
$$620$$ 0 0
$$621$$ 1101.38 0.0711703
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 9557.30 0.611667
$$626$$ 0 0
$$627$$ −1915.20 −0.121987
$$628$$ 0 0
$$629$$ −4180.73 −0.265018
$$630$$ 0 0
$$631$$ 22060.1 1.39176 0.695880 0.718158i $$-0.255015\pi$$
0.695880 + 0.718158i $$0.255015\pi$$
$$632$$ 0 0
$$633$$ −9690.89 −0.608497
$$634$$ 0 0
$$635$$ −6694.71 −0.418380
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −95.2142 −0.00589455
$$640$$ 0 0
$$641$$ −17094.1 −1.05332 −0.526660 0.850076i $$-0.676556\pi$$
−0.526660 + 0.850076i $$0.676556\pi$$
$$642$$ 0 0
$$643$$ 2999.98 0.183993 0.0919965 0.995759i $$-0.470675\pi$$
0.0919965 + 0.995759i $$0.470675\pi$$
$$644$$ 0 0
$$645$$ −1629.73 −0.0994892
$$646$$ 0 0
$$647$$ −8255.72 −0.501648 −0.250824 0.968033i $$-0.580701\pi$$
−0.250824 + 0.968033i $$0.580701\pi$$
$$648$$ 0 0
$$649$$ 2930.35 0.177236
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −7953.02 −0.476609 −0.238304 0.971191i $$-0.576592\pi$$
−0.238304 + 0.971191i $$0.576592\pi$$
$$654$$ 0 0
$$655$$ −6318.88 −0.376945
$$656$$ 0 0
$$657$$ 5258.62 0.312266
$$658$$ 0 0
$$659$$ −4420.98 −0.261331 −0.130665 0.991427i $$-0.541711\pi$$
−0.130665 + 0.991427i $$0.541711\pi$$
$$660$$ 0 0
$$661$$ 5830.81 0.343105 0.171552 0.985175i $$-0.445122\pi$$
0.171552 + 0.985175i $$0.445122\pi$$
$$662$$ 0 0
$$663$$ −5356.60 −0.313776
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7293.79 0.423413
$$668$$ 0 0
$$669$$ −1299.51 −0.0751000
$$670$$ 0 0
$$671$$ 4756.75 0.273670
$$672$$ 0 0
$$673$$ 24614.2 1.40982 0.704910 0.709297i $$-0.250987\pi$$
0.704910 + 0.709297i $$0.250987\pi$$
$$674$$ 0 0
$$675$$ 2917.45 0.166360
$$676$$ 0 0
$$677$$ 130.989 0.00743619 0.00371810 0.999993i $$-0.498816\pi$$
0.00371810 + 0.999993i $$0.498816\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12325.8 −0.693576
$$682$$ 0 0
$$683$$ −4912.86 −0.275235 −0.137617 0.990485i $$-0.543944\pi$$
−0.137617 + 0.990485i $$0.543944\pi$$
$$684$$ 0 0
$$685$$ −10153.5 −0.566346
$$686$$ 0 0
$$687$$ −13899.0 −0.771879
$$688$$ 0 0
$$689$$ 7540.22 0.416922
$$690$$ 0 0
$$691$$ 9639.37 0.530678 0.265339 0.964155i $$-0.414516\pi$$
0.265339 + 0.964155i $$0.414516\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −5444.92 −0.297176
$$696$$ 0 0
$$697$$ −14525.0 −0.789346
$$698$$ 0 0
$$699$$ −1929.66 −0.104416
$$700$$ 0 0
$$701$$ −18377.9 −0.990188 −0.495094 0.868839i $$-0.664866\pi$$
−0.495094 + 0.868839i $$0.664866\pi$$
$$702$$ 0 0
$$703$$ −1952.28 −0.104739
$$704$$ 0 0
$$705$$ 2464.68 0.131667
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −8667.23 −0.459104 −0.229552 0.973296i $$-0.573726\pi$$
−0.229552 + 0.973296i $$0.573726\pi$$
$$710$$ 0 0
$$711$$ −1647.70 −0.0869110
$$712$$ 0 0
$$713$$ −1292.27 −0.0678765
$$714$$ 0 0
$$715$$ −1729.33 −0.0904523
$$716$$ 0 0
$$717$$ −2350.84 −0.122446
$$718$$ 0 0
$$719$$ 13439.1 0.697070 0.348535 0.937296i $$-0.386679\pi$$
0.348535 + 0.937296i $$0.386679\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 19015.7 0.978150
$$724$$ 0 0
$$725$$ 19320.6 0.989722
$$726$$ 0 0
$$727$$ −25085.9 −1.27976 −0.639879 0.768476i $$-0.721015\pi$$
−0.639879 + 0.768476i $$0.721015\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −10059.4 −0.508973
$$732$$ 0 0
$$733$$ 31020.6 1.56313 0.781563 0.623826i $$-0.214423\pi$$
0.781563 + 0.623826i $$0.214423\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4988.05 0.249304
$$738$$ 0 0
$$739$$ 12992.4 0.646730 0.323365 0.946274i $$-0.395186\pi$$
0.323365 + 0.946274i $$0.395186\pi$$
$$740$$ 0 0
$$741$$ −2501.39 −0.124009
$$742$$ 0 0
$$743$$ −23950.7 −1.18259 −0.591297 0.806454i $$-0.701384\pi$$
−0.591297 + 0.806454i $$0.701384\pi$$
$$744$$ 0 0
$$745$$ 2667.83 0.131197
$$746$$ 0 0
$$747$$ −1576.05 −0.0771950
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 13677.5 0.664580 0.332290 0.943177i $$-0.392179\pi$$
0.332290 + 0.943177i $$0.392179\pi$$
$$752$$ 0 0
$$753$$ −17589.0 −0.851235
$$754$$ 0 0
$$755$$ −8248.41 −0.397603
$$756$$ 0 0
$$757$$ −9295.64 −0.446309 −0.223154 0.974783i $$-0.571635\pi$$
−0.223154 + 0.974783i $$0.571635\pi$$
$$758$$ 0 0
$$759$$ 2194.73 0.104959
$$760$$ 0 0
$$761$$ −19586.0 −0.932970 −0.466485 0.884529i $$-0.654480\pi$$
−0.466485 + 0.884529i $$0.654480\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2824.20 −0.133476
$$766$$ 0 0
$$767$$ 3827.24 0.180174
$$768$$ 0 0
$$769$$ 5952.29 0.279122 0.139561 0.990213i $$-0.455431\pi$$
0.139561 + 0.990213i $$0.455431\pi$$
$$770$$ 0 0
$$771$$ 12641.3 0.590486
$$772$$ 0 0
$$773$$ 34379.6 1.59968 0.799838 0.600216i $$-0.204919\pi$$
0.799838 + 0.600216i $$0.204919\pi$$
$$774$$ 0 0
$$775$$ −3423.11 −0.158660
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6782.78 −0.311962
$$780$$ 0 0
$$781$$ −189.735 −0.00869300
$$782$$ 0 0
$$783$$ 4827.75 0.220344
$$784$$ 0 0
$$785$$ 3184.55 0.144792
$$786$$ 0 0
$$787$$ −19167.6 −0.868172 −0.434086 0.900872i $$-0.642929\pi$$
−0.434086 + 0.900872i $$0.642929\pi$$
$$788$$ 0 0
$$789$$ 5867.55 0.264753
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6212.65 0.278206
$$794$$ 0 0
$$795$$ 3975.48 0.177353
$$796$$ 0 0
$$797$$ 2679.24 0.119076 0.0595380 0.998226i $$-0.481037\pi$$
0.0595380 + 0.998226i $$0.481037\pi$$
$$798$$ 0 0
$$799$$ 15213.1 0.673592
$$800$$ 0 0
$$801$$ 424.293 0.0187162
$$802$$ 0 0
$$803$$ 10478.9 0.460515
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −4583.98 −0.199955
$$808$$ 0 0
$$809$$ 8075.98 0.350972 0.175486 0.984482i $$-0.443850\pi$$
0.175486 + 0.984482i $$0.443850\pi$$
$$810$$ 0 0
$$811$$ −31378.3 −1.35862 −0.679311 0.733851i $$-0.737721\pi$$
−0.679311 + 0.733851i $$0.737721\pi$$
$$812$$ 0 0
$$813$$ −23021.6 −0.993117
$$814$$ 0 0
$$815$$ −273.459 −0.0117532
$$816$$ 0 0
$$817$$ −4697.45 −0.201154
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3099.20 0.131745 0.0658726 0.997828i $$-0.479017\pi$$
0.0658726 + 0.997828i $$0.479017\pi$$
$$822$$ 0 0
$$823$$ −965.740 −0.0409035 −0.0204517 0.999791i $$-0.506510\pi$$
−0.0204517 + 0.999791i $$0.506510\pi$$
$$824$$ 0 0
$$825$$ 5813.64 0.245339
$$826$$ 0 0
$$827$$ −7538.11 −0.316960 −0.158480 0.987362i $$-0.550659\pi$$
−0.158480 + 0.987362i $$0.550659\pi$$
$$828$$ 0 0
$$829$$ 12841.2 0.537991 0.268995 0.963141i $$-0.413308\pi$$
0.268995 + 0.963141i $$0.413308\pi$$
$$830$$ 0 0
$$831$$ −18936.6 −0.790498
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3468.71 −0.143760
$$836$$ 0 0
$$837$$ −855.352 −0.0353229
$$838$$ 0 0
$$839$$ 13909.3 0.572351 0.286176 0.958177i $$-0.407616\pi$$
0.286176 + 0.958177i $$0.407616\pi$$
$$840$$ 0 0
$$841$$ 7582.39 0.310894
$$842$$ 0 0
$$843$$ 10004.0 0.408725
$$844$$ 0 0
$$845$$ 6785.52 0.276248
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2177.70 0.0880311
$$850$$ 0 0
$$851$$ 2237.22 0.0901188
$$852$$ 0 0
$$853$$ 33036.0 1.32606 0.663030 0.748593i $$-0.269270\pi$$
0.663030 + 0.748593i $$0.269270\pi$$
$$854$$ 0 0
$$855$$ −1318.82 −0.0527518
$$856$$ 0 0
$$857$$ 15827.5 0.630873 0.315437 0.948947i $$-0.397849\pi$$
0.315437 + 0.948947i $$0.397849\pi$$
$$858$$ 0 0
$$859$$ 6933.95 0.275417 0.137709 0.990473i $$-0.456026\pi$$
0.137709 + 0.990473i $$0.456026\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −25595.4 −1.00959 −0.504795 0.863239i $$-0.668432\pi$$
−0.504795 + 0.863239i $$0.668432\pi$$
$$864$$ 0 0
$$865$$ −6465.91 −0.254159
$$866$$ 0 0
$$867$$ −2693.14 −0.105495
$$868$$ 0 0
$$869$$ −3283.40 −0.128172
$$870$$ 0 0
$$871$$ 6514.74 0.253437
$$872$$ 0 0
$$873$$ −5011.24 −0.194278
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −44274.9 −1.70474 −0.852369 0.522940i $$-0.824835\pi$$
−0.852369 + 0.522940i $$0.824835\pi$$
$$878$$ 0 0
$$879$$ 17044.4 0.654032
$$880$$ 0 0
$$881$$ 5773.97 0.220806 0.110403 0.993887i $$-0.464786\pi$$
0.110403 + 0.993887i $$0.464786\pi$$
$$882$$ 0 0
$$883$$ −48171.4 −1.83590 −0.917948 0.396701i $$-0.870155\pi$$
−0.917948 + 0.396701i $$0.870155\pi$$
$$884$$ 0 0
$$885$$ 2017.86 0.0766437
$$886$$ 0 0
$$887$$ 48168.5 1.82338 0.911692 0.410875i $$-0.134777\pi$$
0.911692 + 0.410875i $$0.134777\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1452.69 0.0546205
$$892$$ 0 0
$$893$$ 7104.08 0.266214
$$894$$ 0 0
$$895$$ −15110.9 −0.564361
$$896$$ 0 0
$$897$$ 2866.47 0.106699
$$898$$ 0 0
$$899$$ −5664.51 −0.210147
$$900$$ 0 0
$$901$$ 24538.3 0.907315
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3303.22 0.121329
$$906$$ 0 0
$$907$$ 31992.5 1.17122 0.585608 0.810595i $$-0.300856\pi$$
0.585608 + 0.810595i $$0.300856\pi$$
$$908$$ 0 0
$$909$$ 1159.23 0.0422985
$$910$$ 0 0
$$911$$ −21184.7 −0.770450 −0.385225 0.922823i $$-0.625876\pi$$
−0.385225 + 0.922823i $$0.625876\pi$$
$$912$$ 0 0
$$913$$ −3140.61 −0.113844
$$914$$ 0 0
$$915$$ 3275.54 0.118345
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 21647.9 0.777040 0.388520 0.921440i $$-0.372986\pi$$
0.388520 + 0.921440i $$0.372986\pi$$
$$920$$ 0 0
$$921$$ 7407.11 0.265008
$$922$$ 0 0
$$923$$ −247.807 −0.00883711
$$924$$ 0 0
$$925$$ 5926.20 0.210651
$$926$$ 0 0
$$927$$ 18391.3 0.651618
$$928$$ 0 0
$$929$$ 9480.99 0.334834 0.167417 0.985886i $$-0.446457\pi$$
0.167417 + 0.985886i $$0.446457\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −12204.0 −0.428233
$$934$$ 0 0
$$935$$ −5627.82 −0.196844
$$936$$ 0 0
$$937$$ −2941.86 −0.102568 −0.0512841 0.998684i $$-0.516331\pi$$
−0.0512841 + 0.998684i $$0.516331\pi$$
$$938$$ 0 0
$$939$$ −14736.4 −0.512145
$$940$$ 0 0
$$941$$ 52683.8 1.82513 0.912563 0.408937i $$-0.134100\pi$$
0.912563 + 0.408937i $$0.134100\pi$$
$$942$$ 0 0
$$943$$ 7772.74 0.268415
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −45003.1 −1.54425 −0.772125 0.635471i $$-0.780806\pi$$
−0.772125 + 0.635471i $$0.780806\pi$$
$$948$$ 0 0
$$949$$ 13686.2 0.468149
$$950$$ 0 0
$$951$$ −8283.50 −0.282451
$$952$$ 0 0
$$953$$ 40137.2 1.36429 0.682146 0.731216i $$-0.261047\pi$$
0.682146 + 0.731216i $$0.261047\pi$$
$$954$$ 0 0
$$955$$ 18767.9 0.635933
$$956$$ 0 0
$$957$$ 9620.31 0.324954
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −28787.4 −0.966312
$$962$$ 0 0
$$963$$ −2419.74 −0.0809709
$$964$$ 0 0
$$965$$ −12900.2 −0.430335
$$966$$ 0 0
$$967$$ −42055.0 −1.39855 −0.699276 0.714852i $$-0.746494\pi$$
−0.699276 + 0.714852i $$0.746494\pi$$
$$968$$ 0 0
$$969$$ −8140.33 −0.269871
$$970$$ 0 0
$$971$$ −9772.69 −0.322987 −0.161494 0.986874i $$-0.551631\pi$$
−0.161494 + 0.986874i $$0.551631\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 7593.01 0.249406
$$976$$ 0 0
$$977$$ −11978.1 −0.392236 −0.196118 0.980580i $$-0.562834\pi$$
−0.196118 + 0.980580i $$0.562834\pi$$
$$978$$ 0 0
$$979$$ 845.494 0.0276018
$$980$$ 0 0
$$981$$ 12020.8 0.391228
$$982$$ 0 0
$$983$$ −56474.0 −1.83239 −0.916197 0.400729i $$-0.868757\pi$$
−0.916197 + 0.400729i $$0.868757\pi$$
$$984$$ 0 0
$$985$$ 11549.9 0.373615
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 5383.06 0.173075
$$990$$ 0 0
$$991$$ 48427.6 1.55233 0.776163 0.630533i $$-0.217163\pi$$
0.776163 + 0.630533i $$0.217163\pi$$
$$992$$ 0 0
$$993$$ −3575.16 −0.114254
$$994$$ 0 0
$$995$$ −21641.7 −0.689535
$$996$$ 0 0
$$997$$ 29268.6 0.929734 0.464867 0.885381i $$-0.346102\pi$$
0.464867 + 0.885381i $$0.346102\pi$$
$$998$$ 0 0
$$999$$ 1480.81 0.0468978
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 1176.4.a.bb.1.2 4
4.3 odd 2 2352.4.a.cr.1.2 4
7.6 odd 2 1176.4.a.bc.1.3 yes 4
28.27 even 2 2352.4.a.ck.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.2 4 1.1 even 1 trivial
1176.4.a.bc.1.3 yes 4 7.6 odd 2
2352.4.a.ck.1.3 4 28.27 even 2
2352.4.a.cr.1.2 4 4.3 odd 2