# Properties

 Label 1176.4.a.be.1.3 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: 4.4.391168.1 Defining polynomial: $$x^{4} - 40 x^{2} + 382$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-4.92368$$ of defining polynomial Character $$\chi$$ $$=$$ 1176.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +14.5121 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +14.5121 q^{5} +9.00000 q^{9} +10.1116 q^{11} -0.623659 q^{13} +43.5362 q^{15} -16.8120 q^{17} +75.7927 q^{19} +46.7115 q^{23} +85.5999 q^{25} +27.0000 q^{27} +73.5290 q^{29} -135.435 q^{31} +30.3348 q^{33} +133.220 q^{37} -1.87098 q^{39} -69.3010 q^{41} +279.261 q^{43} +130.609 q^{45} +372.176 q^{47} -50.4361 q^{51} +656.872 q^{53} +146.740 q^{55} +227.378 q^{57} -730.411 q^{59} +39.5121 q^{61} -9.05057 q^{65} +417.962 q^{67} +140.135 q^{69} -831.408 q^{71} +1192.72 q^{73} +256.800 q^{75} -1034.49 q^{79} +81.0000 q^{81} +117.618 q^{83} -243.977 q^{85} +220.587 q^{87} -678.390 q^{89} -406.305 q^{93} +1099.91 q^{95} +1659.31 q^{97} +91.0044 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 8q^{5} + 36q^{9} + 40q^{11} + 48q^{13} + 24q^{15} + 72q^{17} + 32q^{19} + 8q^{23} + 164q^{25} + 108q^{27} + 144q^{29} + 48q^{31} + 120q^{33} + 48q^{37} + 144q^{39} + 72q^{41} + 512q^{43} + 72q^{45} + 160q^{47} + 216q^{51} + 536q^{53} - 336q^{55} + 96q^{57} + 240q^{59} + 896q^{61} - 136q^{65} + 1088q^{67} + 24q^{69} + 1288q^{71} + 1488q^{73} + 492q^{75} + 416q^{79} + 324q^{81} - 112q^{83} - 1512q^{85} + 432q^{87} + 3160q^{89} + 144q^{93} - 240q^{95} + 2384q^{97} + 360q^{99} + O(q^{100})$$

## 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
$$5$$ 14.5121 1.29800 0.648999 0.760789i $$-0.275188\pi$$
0.648999 + 0.760789i $$0.275188\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 10.1116 0.277160 0.138580 0.990351i $$-0.455746\pi$$
0.138580 + 0.990351i $$0.455746\pi$$
$$12$$ 0 0
$$13$$ −0.623659 −0.0133055 −0.00665276 0.999978i $$-0.502118\pi$$
−0.00665276 + 0.999978i $$0.502118\pi$$
$$14$$ 0 0
$$15$$ 43.5362 0.749400
$$16$$ 0 0
$$17$$ −16.8120 −0.239854 −0.119927 0.992783i $$-0.538266\pi$$
−0.119927 + 0.992783i $$0.538266\pi$$
$$18$$ 0 0
$$19$$ 75.7927 0.915159 0.457580 0.889169i $$-0.348716\pi$$
0.457580 + 0.889169i $$0.348716\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 46.7115 0.423479 0.211740 0.977326i $$-0.432087\pi$$
0.211740 + 0.977326i $$0.432087\pi$$
$$24$$ 0 0
$$25$$ 85.5999 0.684799
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 73.5290 0.470827 0.235414 0.971895i $$-0.424356\pi$$
0.235414 + 0.971895i $$0.424356\pi$$
$$30$$ 0 0
$$31$$ −135.435 −0.784672 −0.392336 0.919822i $$-0.628333\pi$$
−0.392336 + 0.919822i $$0.628333\pi$$
$$32$$ 0 0
$$33$$ 30.3348 0.160018
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 133.220 0.591926 0.295963 0.955199i $$-0.404359\pi$$
0.295963 + 0.955199i $$0.404359\pi$$
$$38$$ 0 0
$$39$$ −1.87098 −0.00768195
$$40$$ 0 0
$$41$$ −69.3010 −0.263975 −0.131988 0.991251i $$-0.542136\pi$$
−0.131988 + 0.991251i $$0.542136\pi$$
$$42$$ 0 0
$$43$$ 279.261 0.990392 0.495196 0.868781i $$-0.335096\pi$$
0.495196 + 0.868781i $$0.335096\pi$$
$$44$$ 0 0
$$45$$ 130.609 0.432666
$$46$$ 0 0
$$47$$ 372.176 1.15505 0.577526 0.816372i $$-0.304018\pi$$
0.577526 + 0.816372i $$0.304018\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −50.4361 −0.138480
$$52$$ 0 0
$$53$$ 656.872 1.70242 0.851210 0.524825i $$-0.175869\pi$$
0.851210 + 0.524825i $$0.175869\pi$$
$$54$$ 0 0
$$55$$ 146.740 0.359753
$$56$$ 0 0
$$57$$ 227.378 0.528368
$$58$$ 0 0
$$59$$ −730.411 −1.61172 −0.805859 0.592107i $$-0.798296\pi$$
−0.805859 + 0.592107i $$0.798296\pi$$
$$60$$ 0 0
$$61$$ 39.5121 0.0829345 0.0414673 0.999140i $$-0.486797\pi$$
0.0414673 + 0.999140i $$0.486797\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −9.05057 −0.0172705
$$66$$ 0 0
$$67$$ 417.962 0.762122 0.381061 0.924550i $$-0.375559\pi$$
0.381061 + 0.924550i $$0.375559\pi$$
$$68$$ 0 0
$$69$$ 140.135 0.244496
$$70$$ 0 0
$$71$$ −831.408 −1.38972 −0.694859 0.719146i $$-0.744533\pi$$
−0.694859 + 0.719146i $$0.744533\pi$$
$$72$$ 0 0
$$73$$ 1192.72 1.91229 0.956145 0.292894i $$-0.0946184\pi$$
0.956145 + 0.292894i $$0.0946184\pi$$
$$74$$ 0 0
$$75$$ 256.800 0.395369
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1034.49 −1.47327 −0.736637 0.676288i $$-0.763587\pi$$
−0.736637 + 0.676288i $$0.763587\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 117.618 0.155545 0.0777726 0.996971i $$-0.475219\pi$$
0.0777726 + 0.996971i $$0.475219\pi$$
$$84$$ 0 0
$$85$$ −243.977 −0.311330
$$86$$ 0 0
$$87$$ 220.587 0.271832
$$88$$ 0 0
$$89$$ −678.390 −0.807969 −0.403984 0.914766i $$-0.632375\pi$$
−0.403984 + 0.914766i $$0.632375\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −406.305 −0.453031
$$94$$ 0 0
$$95$$ 1099.91 1.18788
$$96$$ 0 0
$$97$$ 1659.31 1.73688 0.868439 0.495795i $$-0.165123\pi$$
0.868439 + 0.495795i $$0.165123\pi$$
$$98$$ 0 0
$$99$$ 91.0044 0.0923867
$$100$$ 0 0
$$101$$ 485.839 0.478641 0.239321 0.970941i $$-0.423075\pi$$
0.239321 + 0.970941i $$0.423075\pi$$
$$102$$ 0 0
$$103$$ 276.792 0.264787 0.132394 0.991197i $$-0.457734\pi$$
0.132394 + 0.991197i $$0.457734\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 103.639 0.0936373 0.0468187 0.998903i $$-0.485092\pi$$
0.0468187 + 0.998903i $$0.485092\pi$$
$$108$$ 0 0
$$109$$ −50.1571 −0.0440751 −0.0220375 0.999757i $$-0.507015\pi$$
−0.0220375 + 0.999757i $$0.507015\pi$$
$$110$$ 0 0
$$111$$ 399.661 0.341749
$$112$$ 0 0
$$113$$ −1954.66 −1.62725 −0.813623 0.581393i $$-0.802508\pi$$
−0.813623 + 0.581393i $$0.802508\pi$$
$$114$$ 0 0
$$115$$ 677.880 0.549675
$$116$$ 0 0
$$117$$ −5.61293 −0.00443517
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1228.76 −0.923182
$$122$$ 0 0
$$123$$ −207.903 −0.152406
$$124$$ 0 0
$$125$$ −571.776 −0.409130
$$126$$ 0 0
$$127$$ −1123.20 −0.784783 −0.392392 0.919798i $$-0.628352\pi$$
−0.392392 + 0.919798i $$0.628352\pi$$
$$128$$ 0 0
$$129$$ 837.782 0.571803
$$130$$ 0 0
$$131$$ 1562.85 1.04234 0.521171 0.853453i $$-0.325496\pi$$
0.521171 + 0.853453i $$0.325496\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 391.826 0.249800
$$136$$ 0 0
$$137$$ 1283.91 0.800673 0.400337 0.916368i $$-0.368893\pi$$
0.400337 + 0.916368i $$0.368893\pi$$
$$138$$ 0 0
$$139$$ −674.069 −0.411322 −0.205661 0.978623i $$-0.565934\pi$$
−0.205661 + 0.978623i $$0.565934\pi$$
$$140$$ 0 0
$$141$$ 1116.53 0.666869
$$142$$ 0 0
$$143$$ −6.30619 −0.00368776
$$144$$ 0 0
$$145$$ 1067.06 0.611133
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1293.72 −0.711314 −0.355657 0.934617i $$-0.615743\pi$$
−0.355657 + 0.934617i $$0.615743\pi$$
$$150$$ 0 0
$$151$$ 2266.65 1.22157 0.610786 0.791795i $$-0.290853\pi$$
0.610786 + 0.791795i $$0.290853\pi$$
$$152$$ 0 0
$$153$$ −151.308 −0.0799512
$$154$$ 0 0
$$155$$ −1965.44 −1.01850
$$156$$ 0 0
$$157$$ 1868.03 0.949585 0.474792 0.880098i $$-0.342523\pi$$
0.474792 + 0.880098i $$0.342523\pi$$
$$158$$ 0 0
$$159$$ 1970.61 0.982892
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1511.45 −0.726293 −0.363147 0.931732i $$-0.618298\pi$$
−0.363147 + 0.931732i $$0.618298\pi$$
$$164$$ 0 0
$$165$$ 440.220 0.207704
$$166$$ 0 0
$$167$$ −2371.40 −1.09883 −0.549415 0.835550i $$-0.685149\pi$$
−0.549415 + 0.835550i $$0.685149\pi$$
$$168$$ 0 0
$$169$$ −2196.61 −0.999823
$$170$$ 0 0
$$171$$ 682.134 0.305053
$$172$$ 0 0
$$173$$ 4124.67 1.81267 0.906337 0.422555i $$-0.138867\pi$$
0.906337 + 0.422555i $$0.138867\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2191.23 −0.930526
$$178$$ 0 0
$$179$$ 3194.53 1.33391 0.666955 0.745098i $$-0.267597\pi$$
0.666955 + 0.745098i $$0.267597\pi$$
$$180$$ 0 0
$$181$$ 1389.99 0.570815 0.285407 0.958406i $$-0.407871\pi$$
0.285407 + 0.958406i $$0.407871\pi$$
$$182$$ 0 0
$$183$$ 118.536 0.0478823
$$184$$ 0 0
$$185$$ 1933.30 0.768319
$$186$$ 0 0
$$187$$ −169.996 −0.0664779
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1915.99 0.725844 0.362922 0.931820i $$-0.381779\pi$$
0.362922 + 0.931820i $$0.381779\pi$$
$$192$$ 0 0
$$193$$ −1092.29 −0.407381 −0.203690 0.979035i $$-0.565294\pi$$
−0.203690 + 0.979035i $$0.565294\pi$$
$$194$$ 0 0
$$195$$ −27.1517 −0.00997115
$$196$$ 0 0
$$197$$ 4727.17 1.70963 0.854815 0.518933i $$-0.173671\pi$$
0.854815 + 0.518933i $$0.173671\pi$$
$$198$$ 0 0
$$199$$ 634.583 0.226052 0.113026 0.993592i $$-0.463946\pi$$
0.113026 + 0.993592i $$0.463946\pi$$
$$200$$ 0 0
$$201$$ 1253.89 0.440011
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1005.70 −0.342640
$$206$$ 0 0
$$207$$ 420.404 0.141160
$$208$$ 0 0
$$209$$ 766.385 0.253646
$$210$$ 0 0
$$211$$ −1782.00 −0.581413 −0.290707 0.956812i $$-0.593890\pi$$
−0.290707 + 0.956812i $$0.593890\pi$$
$$212$$ 0 0
$$213$$ −2494.22 −0.802354
$$214$$ 0 0
$$215$$ 4052.65 1.28553
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3578.16 1.10406
$$220$$ 0 0
$$221$$ 10.4850 0.00319138
$$222$$ 0 0
$$223$$ −5362.96 −1.61045 −0.805225 0.592970i $$-0.797955\pi$$
−0.805225 + 0.592970i $$0.797955\pi$$
$$224$$ 0 0
$$225$$ 770.399 0.228266
$$226$$ 0 0
$$227$$ −734.718 −0.214824 −0.107412 0.994215i $$-0.534256\pi$$
−0.107412 + 0.994215i $$0.534256\pi$$
$$228$$ 0 0
$$229$$ 5160.26 1.48908 0.744541 0.667577i $$-0.232668\pi$$
0.744541 + 0.667577i $$0.232668\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4068.43 −1.14391 −0.571956 0.820284i $$-0.693815\pi$$
−0.571956 + 0.820284i $$0.693815\pi$$
$$234$$ 0 0
$$235$$ 5401.04 1.49925
$$236$$ 0 0
$$237$$ −3103.46 −0.850595
$$238$$ 0 0
$$239$$ −650.039 −0.175931 −0.0879655 0.996124i $$-0.528037\pi$$
−0.0879655 + 0.996124i $$0.528037\pi$$
$$240$$ 0 0
$$241$$ 3233.14 0.864170 0.432085 0.901833i $$-0.357778\pi$$
0.432085 + 0.901833i $$0.357778\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −47.2687 −0.0121767
$$248$$ 0 0
$$249$$ 352.854 0.0898041
$$250$$ 0 0
$$251$$ 1139.75 0.286615 0.143308 0.989678i $$-0.454226\pi$$
0.143308 + 0.989678i $$0.454226\pi$$
$$252$$ 0 0
$$253$$ 472.328 0.117372
$$254$$ 0 0
$$255$$ −731.931 −0.179746
$$256$$ 0 0
$$257$$ 2036.97 0.494408 0.247204 0.968963i $$-0.420488\pi$$
0.247204 + 0.968963i $$0.420488\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 661.761 0.156942
$$262$$ 0 0
$$263$$ −836.215 −0.196058 −0.0980289 0.995184i $$-0.531254\pi$$
−0.0980289 + 0.995184i $$0.531254\pi$$
$$264$$ 0 0
$$265$$ 9532.56 2.20974
$$266$$ 0 0
$$267$$ −2035.17 −0.466481
$$268$$ 0 0
$$269$$ 2682.99 0.608122 0.304061 0.952652i $$-0.401657\pi$$
0.304061 + 0.952652i $$0.401657\pi$$
$$270$$ 0 0
$$271$$ −6541.42 −1.46628 −0.733142 0.680076i $$-0.761947\pi$$
−0.733142 + 0.680076i $$0.761947\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 865.552 0.189799
$$276$$ 0 0
$$277$$ −1293.11 −0.280490 −0.140245 0.990117i $$-0.544789\pi$$
−0.140245 + 0.990117i $$0.544789\pi$$
$$278$$ 0 0
$$279$$ −1218.92 −0.261557
$$280$$ 0 0
$$281$$ 4684.69 0.994538 0.497269 0.867596i $$-0.334336\pi$$
0.497269 + 0.867596i $$0.334336\pi$$
$$282$$ 0 0
$$283$$ 565.240 0.118728 0.0593640 0.998236i $$-0.481093\pi$$
0.0593640 + 0.998236i $$0.481093\pi$$
$$284$$ 0 0
$$285$$ 3299.72 0.685820
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4630.36 −0.942470
$$290$$ 0 0
$$291$$ 4977.93 1.00279
$$292$$ 0 0
$$293$$ 809.451 0.161395 0.0806973 0.996739i $$-0.474285\pi$$
0.0806973 + 0.996739i $$0.474285\pi$$
$$294$$ 0 0
$$295$$ −10599.8 −2.09201
$$296$$ 0 0
$$297$$ 273.013 0.0533395
$$298$$ 0 0
$$299$$ −29.1320 −0.00563461
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1457.52 0.276344
$$304$$ 0 0
$$305$$ 573.402 0.107649
$$306$$ 0 0
$$307$$ 3667.95 0.681892 0.340946 0.940083i $$-0.389253\pi$$
0.340946 + 0.940083i $$0.389253\pi$$
$$308$$ 0 0
$$309$$ 830.375 0.152875
$$310$$ 0 0
$$311$$ −73.3047 −0.0133657 −0.00668284 0.999978i $$-0.502127\pi$$
−0.00668284 + 0.999978i $$0.502127\pi$$
$$312$$ 0 0
$$313$$ −8300.73 −1.49899 −0.749497 0.662008i $$-0.769705\pi$$
−0.749497 + 0.662008i $$0.769705\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1288.89 0.228364 0.114182 0.993460i $$-0.463575\pi$$
0.114182 + 0.993460i $$0.463575\pi$$
$$318$$ 0 0
$$319$$ 743.495 0.130494
$$320$$ 0 0
$$321$$ 310.918 0.0540615
$$322$$ 0 0
$$323$$ −1274.23 −0.219504
$$324$$ 0 0
$$325$$ −53.3851 −0.00911161
$$326$$ 0 0
$$327$$ −150.471 −0.0254468
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6517.90 −1.08234 −0.541172 0.840912i $$-0.682019\pi$$
−0.541172 + 0.840912i $$0.682019\pi$$
$$332$$ 0 0
$$333$$ 1198.98 0.197309
$$334$$ 0 0
$$335$$ 6065.49 0.989233
$$336$$ 0 0
$$337$$ −7477.45 −1.20867 −0.604337 0.796729i $$-0.706562\pi$$
−0.604337 + 0.796729i $$0.706562\pi$$
$$338$$ 0 0
$$339$$ −5863.97 −0.939491
$$340$$ 0 0
$$341$$ −1369.46 −0.217480
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2033.64 0.317355
$$346$$ 0 0
$$347$$ −1228.91 −0.190119 −0.0950596 0.995472i $$-0.530304\pi$$
−0.0950596 + 0.995472i $$0.530304\pi$$
$$348$$ 0 0
$$349$$ 1660.55 0.254691 0.127345 0.991858i $$-0.459354\pi$$
0.127345 + 0.991858i $$0.459354\pi$$
$$350$$ 0 0
$$351$$ −16.8388 −0.00256065
$$352$$ 0 0
$$353$$ 10483.4 1.58066 0.790330 0.612681i $$-0.209909\pi$$
0.790330 + 0.612681i $$0.209909\pi$$
$$354$$ 0 0
$$355$$ −12065.4 −1.80385
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −5329.76 −0.783549 −0.391775 0.920061i $$-0.628139\pi$$
−0.391775 + 0.920061i $$0.628139\pi$$
$$360$$ 0 0
$$361$$ −1114.47 −0.162483
$$362$$ 0 0
$$363$$ −3686.27 −0.533000
$$364$$ 0 0
$$365$$ 17308.8 2.48215
$$366$$ 0 0
$$367$$ −6608.43 −0.939937 −0.469969 0.882683i $$-0.655735\pi$$
−0.469969 + 0.882683i $$0.655735\pi$$
$$368$$ 0 0
$$369$$ −623.709 −0.0879918
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6969.99 0.967540 0.483770 0.875195i $$-0.339267\pi$$
0.483770 + 0.875195i $$0.339267\pi$$
$$374$$ 0 0
$$375$$ −1715.33 −0.236211
$$376$$ 0 0
$$377$$ −45.8570 −0.00626460
$$378$$ 0 0
$$379$$ −7383.56 −1.00071 −0.500353 0.865821i $$-0.666797\pi$$
−0.500353 + 0.865821i $$0.666797\pi$$
$$380$$ 0 0
$$381$$ −3369.59 −0.453095
$$382$$ 0 0
$$383$$ 845.491 0.112800 0.0564002 0.998408i $$-0.482038\pi$$
0.0564002 + 0.998408i $$0.482038\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2513.35 0.330131
$$388$$ 0 0
$$389$$ −5897.87 −0.768725 −0.384362 0.923182i $$-0.625579\pi$$
−0.384362 + 0.923182i $$0.625579\pi$$
$$390$$ 0 0
$$391$$ −785.315 −0.101573
$$392$$ 0 0
$$393$$ 4688.55 0.601796
$$394$$ 0 0
$$395$$ −15012.5 −1.91231
$$396$$ 0 0
$$397$$ −10274.5 −1.29890 −0.649448 0.760406i $$-0.725000\pi$$
−0.649448 + 0.760406i $$0.725000\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4261.85 0.530740 0.265370 0.964147i $$-0.414506\pi$$
0.265370 + 0.964147i $$0.414506\pi$$
$$402$$ 0 0
$$403$$ 84.4652 0.0104405
$$404$$ 0 0
$$405$$ 1175.48 0.144222
$$406$$ 0 0
$$407$$ 1347.07 0.164058
$$408$$ 0 0
$$409$$ 6389.21 0.772436 0.386218 0.922408i $$-0.373781\pi$$
0.386218 + 0.922408i $$0.373781\pi$$
$$410$$ 0 0
$$411$$ 3851.74 0.462269
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1706.88 0.201898
$$416$$ 0 0
$$417$$ −2022.21 −0.237477
$$418$$ 0 0
$$419$$ −7.77360 −0.000906361 0 −0.000453180 1.00000i $$-0.500144\pi$$
−0.000453180 1.00000i $$0.500144\pi$$
$$420$$ 0 0
$$421$$ 8500.33 0.984040 0.492020 0.870584i $$-0.336259\pi$$
0.492020 + 0.870584i $$0.336259\pi$$
$$422$$ 0 0
$$423$$ 3349.58 0.385017
$$424$$ 0 0
$$425$$ −1439.11 −0.164252
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −18.9186 −0.00212913
$$430$$ 0 0
$$431$$ −7264.32 −0.811856 −0.405928 0.913905i $$-0.633052\pi$$
−0.405928 + 0.913905i $$0.633052\pi$$
$$432$$ 0 0
$$433$$ 1023.35 0.113578 0.0567890 0.998386i $$-0.481914\pi$$
0.0567890 + 0.998386i $$0.481914\pi$$
$$434$$ 0 0
$$435$$ 3201.17 0.352838
$$436$$ 0 0
$$437$$ 3540.39 0.387551
$$438$$ 0 0
$$439$$ −8081.20 −0.878576 −0.439288 0.898346i $$-0.644769\pi$$
−0.439288 + 0.898346i $$0.644769\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4900.66 0.525592 0.262796 0.964851i $$-0.415355\pi$$
0.262796 + 0.964851i $$0.415355\pi$$
$$444$$ 0 0
$$445$$ −9844.84 −1.04874
$$446$$ 0 0
$$447$$ −3881.16 −0.410677
$$448$$ 0 0
$$449$$ 8493.30 0.892703 0.446351 0.894858i $$-0.352723\pi$$
0.446351 + 0.894858i $$0.352723\pi$$
$$450$$ 0 0
$$451$$ −700.744 −0.0731635
$$452$$ 0 0
$$453$$ 6799.96 0.705275
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13042.6 −1.33503 −0.667514 0.744597i $$-0.732642\pi$$
−0.667514 + 0.744597i $$0.732642\pi$$
$$458$$ 0 0
$$459$$ −453.925 −0.0461599
$$460$$ 0 0
$$461$$ −8900.01 −0.899164 −0.449582 0.893239i $$-0.648427\pi$$
−0.449582 + 0.893239i $$0.648427\pi$$
$$462$$ 0 0
$$463$$ −17336.4 −1.74015 −0.870076 0.492917i $$-0.835930\pi$$
−0.870076 + 0.492917i $$0.835930\pi$$
$$464$$ 0 0
$$465$$ −5896.32 −0.588033
$$466$$ 0 0
$$467$$ 3071.51 0.304352 0.152176 0.988353i $$-0.451372\pi$$
0.152176 + 0.988353i $$0.451372\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5604.08 0.548243
$$472$$ 0 0
$$473$$ 2823.77 0.274497
$$474$$ 0 0
$$475$$ 6487.85 0.626701
$$476$$ 0 0
$$477$$ 5911.84 0.567473
$$478$$ 0 0
$$479$$ −16272.9 −1.55225 −0.776125 0.630580i $$-0.782817\pi$$
−0.776125 + 0.630580i $$0.782817\pi$$
$$480$$ 0 0
$$481$$ −83.0839 −0.00787589
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 24080.0 2.25447
$$486$$ 0 0
$$487$$ −10851.9 −1.00975 −0.504874 0.863193i $$-0.668461\pi$$
−0.504874 + 0.863193i $$0.668461\pi$$
$$488$$ 0 0
$$489$$ −4534.35 −0.419326
$$490$$ 0 0
$$491$$ 744.455 0.0684252 0.0342126 0.999415i $$-0.489108\pi$$
0.0342126 + 0.999415i $$0.489108\pi$$
$$492$$ 0 0
$$493$$ −1236.17 −0.112930
$$494$$ 0 0
$$495$$ 1320.66 0.119918
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 13272.2 1.19067 0.595334 0.803478i $$-0.297020\pi$$
0.595334 + 0.803478i $$0.297020\pi$$
$$500$$ 0 0
$$501$$ −7114.21 −0.634410
$$502$$ 0 0
$$503$$ −19485.3 −1.72725 −0.863626 0.504133i $$-0.831812\pi$$
−0.863626 + 0.504133i $$0.831812\pi$$
$$504$$ 0 0
$$505$$ 7050.52 0.621275
$$506$$ 0 0
$$507$$ −6589.83 −0.577248
$$508$$ 0 0
$$509$$ 1348.71 0.117447 0.0587235 0.998274i $$-0.481297\pi$$
0.0587235 + 0.998274i $$0.481297\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 2046.40 0.176123
$$514$$ 0 0
$$515$$ 4016.82 0.343693
$$516$$ 0 0
$$517$$ 3763.29 0.320134
$$518$$ 0 0
$$519$$ 12374.0 1.04655
$$520$$ 0 0
$$521$$ 3349.98 0.281699 0.140849 0.990031i $$-0.455017\pi$$
0.140849 + 0.990031i $$0.455017\pi$$
$$522$$ 0 0
$$523$$ −7588.42 −0.634452 −0.317226 0.948350i $$-0.602751\pi$$
−0.317226 + 0.948350i $$0.602751\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2276.94 0.188207
$$528$$ 0 0
$$529$$ −9985.03 −0.820665
$$530$$ 0 0
$$531$$ −6573.70 −0.537239
$$532$$ 0 0
$$533$$ 43.2202 0.00351233
$$534$$ 0 0
$$535$$ 1504.02 0.121541
$$536$$ 0 0
$$537$$ 9583.58 0.770134
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −13690.4 −1.08797 −0.543987 0.839093i $$-0.683086\pi$$
−0.543987 + 0.839093i $$0.683086\pi$$
$$542$$ 0 0
$$543$$ 4169.98 0.329560
$$544$$ 0 0
$$545$$ −727.884 −0.0572094
$$546$$ 0 0
$$547$$ 15250.2 1.19205 0.596024 0.802967i $$-0.296746\pi$$
0.596024 + 0.802967i $$0.296746\pi$$
$$548$$ 0 0
$$549$$ 355.609 0.0276448
$$550$$ 0 0
$$551$$ 5572.96 0.430882
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 5799.90 0.443589
$$556$$ 0 0
$$557$$ −14231.6 −1.08261 −0.541305 0.840826i $$-0.682070\pi$$
−0.541305 + 0.840826i $$0.682070\pi$$
$$558$$ 0 0
$$559$$ −174.163 −0.0131777
$$560$$ 0 0
$$561$$ −509.989 −0.0383810
$$562$$ 0 0
$$563$$ −23001.7 −1.72186 −0.860928 0.508727i $$-0.830116\pi$$
−0.860928 + 0.508727i $$0.830116\pi$$
$$564$$ 0 0
$$565$$ −28366.1 −2.11216
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 11996.5 0.883866 0.441933 0.897048i $$-0.354293\pi$$
0.441933 + 0.897048i $$0.354293\pi$$
$$570$$ 0 0
$$571$$ −25436.0 −1.86421 −0.932106 0.362186i $$-0.882030\pi$$
−0.932106 + 0.362186i $$0.882030\pi$$
$$572$$ 0 0
$$573$$ 5747.97 0.419066
$$574$$ 0 0
$$575$$ 3998.50 0.289998
$$576$$ 0 0
$$577$$ 1605.24 0.115818 0.0579090 0.998322i $$-0.481557\pi$$
0.0579090 + 0.998322i $$0.481557\pi$$
$$578$$ 0 0
$$579$$ −3276.86 −0.235201
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6642.02 0.471843
$$584$$ 0 0
$$585$$ −81.4552 −0.00575685
$$586$$ 0 0
$$587$$ −20510.8 −1.44220 −0.721101 0.692829i $$-0.756364\pi$$
−0.721101 + 0.692829i $$0.756364\pi$$
$$588$$ 0 0
$$589$$ −10265.0 −0.718100
$$590$$ 0 0
$$591$$ 14181.5 0.987055
$$592$$ 0 0
$$593$$ 899.970 0.0623227 0.0311613 0.999514i $$-0.490079\pi$$
0.0311613 + 0.999514i $$0.490079\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1903.75 0.130511
$$598$$ 0 0
$$599$$ −17326.2 −1.18185 −0.590926 0.806726i $$-0.701238\pi$$
−0.590926 + 0.806726i $$0.701238\pi$$
$$600$$ 0 0
$$601$$ −10193.7 −0.691860 −0.345930 0.938260i $$-0.612437\pi$$
−0.345930 + 0.938260i $$0.612437\pi$$
$$602$$ 0 0
$$603$$ 3761.66 0.254041
$$604$$ 0 0
$$605$$ −17831.8 −1.19829
$$606$$ 0 0
$$607$$ −10674.8 −0.713803 −0.356901 0.934142i $$-0.616167\pi$$
−0.356901 + 0.934142i $$0.616167\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −232.111 −0.0153686
$$612$$ 0 0
$$613$$ −20442.8 −1.34694 −0.673471 0.739214i $$-0.735197\pi$$
−0.673471 + 0.739214i $$0.735197\pi$$
$$614$$ 0 0
$$615$$ −3017.10 −0.197823
$$616$$ 0 0
$$617$$ −12084.9 −0.788523 −0.394261 0.918998i $$-0.629000\pi$$
−0.394261 + 0.918998i $$0.629000\pi$$
$$618$$ 0 0
$$619$$ 23057.3 1.49718 0.748589 0.663035i $$-0.230732\pi$$
0.748589 + 0.663035i $$0.230732\pi$$
$$620$$ 0 0
$$621$$ 1261.21 0.0814986
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −18997.6 −1.21585
$$626$$ 0 0
$$627$$ 2299.15 0.146442
$$628$$ 0 0
$$629$$ −2239.70 −0.141976
$$630$$ 0 0
$$631$$ −19201.2 −1.21139 −0.605694 0.795697i $$-0.707105\pi$$
−0.605694 + 0.795697i $$0.707105\pi$$
$$632$$ 0 0
$$633$$ −5346.01 −0.335679
$$634$$ 0 0
$$635$$ −16299.9 −1.01865
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −7482.67 −0.463239
$$640$$ 0 0
$$641$$ 10581.2 0.651998 0.325999 0.945370i $$-0.394299\pi$$
0.325999 + 0.945370i $$0.394299\pi$$
$$642$$ 0 0
$$643$$ 29823.2 1.82910 0.914550 0.404474i $$-0.132545\pi$$
0.914550 + 0.404474i $$0.132545\pi$$
$$644$$ 0 0
$$645$$ 12157.9 0.742199
$$646$$ 0 0
$$647$$ 143.283 0.00870639 0.00435320 0.999991i $$-0.498614\pi$$
0.00435320 + 0.999991i $$0.498614\pi$$
$$648$$ 0 0
$$649$$ −7385.62 −0.446704
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 7516.26 0.450435 0.225217 0.974309i $$-0.427691\pi$$
0.225217 + 0.974309i $$0.427691\pi$$
$$654$$ 0 0
$$655$$ 22680.2 1.35296
$$656$$ 0 0
$$657$$ 10734.5 0.637430
$$658$$ 0 0
$$659$$ −4791.03 −0.283205 −0.141603 0.989924i $$-0.545226\pi$$
−0.141603 + 0.989924i $$0.545226\pi$$
$$660$$ 0 0
$$661$$ 4711.63 0.277249 0.138624 0.990345i $$-0.455732\pi$$
0.138624 + 0.990345i $$0.455732\pi$$
$$662$$ 0 0
$$663$$ 31.4549 0.00184254
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3434.65 0.199386
$$668$$ 0 0
$$669$$ −16088.9 −0.929793
$$670$$ 0 0
$$671$$ 399.530 0.0229861
$$672$$ 0 0
$$673$$ −10953.3 −0.627367 −0.313683 0.949528i $$-0.601563\pi$$
−0.313683 + 0.949528i $$0.601563\pi$$
$$674$$ 0 0
$$675$$ 2311.20 0.131790
$$676$$ 0 0
$$677$$ −27537.8 −1.56331 −0.781656 0.623710i $$-0.785625\pi$$
−0.781656 + 0.623710i $$0.785625\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −2204.15 −0.124028
$$682$$ 0 0
$$683$$ 3280.93 0.183808 0.0919042 0.995768i $$-0.470705\pi$$
0.0919042 + 0.995768i $$0.470705\pi$$
$$684$$ 0 0
$$685$$ 18632.2 1.03927
$$686$$ 0 0
$$687$$ 15480.8 0.859722
$$688$$ 0 0
$$689$$ −409.664 −0.0226516
$$690$$ 0 0
$$691$$ −20420.4 −1.12421 −0.562106 0.827065i $$-0.690009\pi$$
−0.562106 + 0.827065i $$0.690009\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −9782.13 −0.533895
$$696$$ 0 0
$$697$$ 1165.09 0.0633155
$$698$$ 0 0
$$699$$ −12205.3 −0.660438
$$700$$ 0 0
$$701$$ −19238.0 −1.03653 −0.518267 0.855219i $$-0.673423\pi$$
−0.518267 + 0.855219i $$0.673423\pi$$
$$702$$ 0 0
$$703$$ 10097.1 0.541707
$$704$$ 0 0
$$705$$ 16203.1 0.865595
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −34685.9 −1.83731 −0.918657 0.395057i $$-0.870725\pi$$
−0.918657 + 0.395057i $$0.870725\pi$$
$$710$$ 0 0
$$711$$ −9310.37 −0.491091
$$712$$ 0 0
$$713$$ −6326.37 −0.332293
$$714$$ 0 0
$$715$$ −91.5157 −0.00478671
$$716$$ 0 0
$$717$$ −1950.12 −0.101574
$$718$$ 0 0
$$719$$ 21896.6 1.13575 0.567877 0.823114i $$-0.307765\pi$$
0.567877 + 0.823114i $$0.307765\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 9699.42 0.498929
$$724$$ 0 0
$$725$$ 6294.07 0.322422
$$726$$ 0 0
$$727$$ −22962.5 −1.17143 −0.585717 0.810516i $$-0.699187\pi$$
−0.585717 + 0.810516i $$0.699187\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −4694.94 −0.237549
$$732$$ 0 0
$$733$$ −22990.5 −1.15849 −0.579245 0.815153i $$-0.696653\pi$$
−0.579245 + 0.815153i $$0.696653\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4226.26 0.211230
$$738$$ 0 0
$$739$$ 5010.14 0.249392 0.124696 0.992195i $$-0.460204\pi$$
0.124696 + 0.992195i $$0.460204\pi$$
$$740$$ 0 0
$$741$$ −141.806 −0.00703021
$$742$$ 0 0
$$743$$ 14389.0 0.710472 0.355236 0.934777i $$-0.384401\pi$$
0.355236 + 0.934777i $$0.384401\pi$$
$$744$$ 0 0
$$745$$ −18774.6 −0.923284
$$746$$ 0 0
$$747$$ 1058.56 0.0518484
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 35586.5 1.72912 0.864560 0.502529i $$-0.167597\pi$$
0.864560 + 0.502529i $$0.167597\pi$$
$$752$$ 0 0
$$753$$ 3419.25 0.165477
$$754$$ 0 0
$$755$$ 32893.8 1.58560
$$756$$ 0 0
$$757$$ −12140.6 −0.582905 −0.291453 0.956585i $$-0.594139\pi$$
−0.291453 + 0.956585i $$0.594139\pi$$
$$758$$ 0 0
$$759$$ 1416.98 0.0677645
$$760$$ 0 0
$$761$$ 22998.0 1.09550 0.547750 0.836642i $$-0.315484\pi$$
0.547750 + 0.836642i $$0.315484\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2195.79 −0.103777
$$766$$ 0 0
$$767$$ 455.527 0.0214448
$$768$$ 0 0
$$769$$ 24004.6 1.12566 0.562828 0.826574i $$-0.309713\pi$$
0.562828 + 0.826574i $$0.309713\pi$$
$$770$$ 0 0
$$771$$ 6110.92 0.285447
$$772$$ 0 0
$$773$$ 2777.97 0.129258 0.0646292 0.997909i $$-0.479414\pi$$
0.0646292 + 0.997909i $$0.479414\pi$$
$$774$$ 0 0
$$775$$ −11593.2 −0.537343
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −5252.50 −0.241580
$$780$$ 0 0
$$781$$ −8406.87 −0.385175
$$782$$ 0 0
$$783$$ 1985.28 0.0906107
$$784$$ 0 0
$$785$$ 27108.9 1.23256
$$786$$ 0 0
$$787$$ 25824.5 1.16969 0.584843 0.811147i $$-0.301156\pi$$
0.584843 + 0.811147i $$0.301156\pi$$
$$788$$ 0 0
$$789$$ −2508.64 −0.113194
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −24.6421 −0.00110349
$$794$$ 0 0
$$795$$ 28597.7 1.27579
$$796$$ 0 0
$$797$$ −5363.77 −0.238387 −0.119193 0.992871i $$-0.538031\pi$$
−0.119193 + 0.992871i $$0.538031\pi$$
$$798$$ 0 0
$$799$$ −6257.03 −0.277043
$$800$$ 0 0
$$801$$ −6105.51 −0.269323
$$802$$ 0 0
$$803$$ 12060.3 0.530010
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 8048.98 0.351100
$$808$$ 0 0
$$809$$ 15270.5 0.663638 0.331819 0.943343i $$-0.392338\pi$$
0.331819 + 0.943343i $$0.392338\pi$$
$$810$$ 0 0
$$811$$ 2993.47 0.129612 0.0648058 0.997898i $$-0.479357\pi$$
0.0648058 + 0.997898i $$0.479357\pi$$
$$812$$ 0 0
$$813$$ −19624.3 −0.846559
$$814$$ 0 0
$$815$$ −21934.2 −0.942728
$$816$$ 0 0
$$817$$ 21165.9 0.906367
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16002.9 0.680272 0.340136 0.940376i $$-0.389527\pi$$
0.340136 + 0.940376i $$0.389527\pi$$
$$822$$ 0 0
$$823$$ −11836.7 −0.501340 −0.250670 0.968073i $$-0.580651\pi$$
−0.250670 + 0.968073i $$0.580651\pi$$
$$824$$ 0 0
$$825$$ 2596.66 0.109581
$$826$$ 0 0
$$827$$ −5675.18 −0.238628 −0.119314 0.992857i $$-0.538069\pi$$
−0.119314 + 0.992857i $$0.538069\pi$$
$$828$$ 0 0
$$829$$ −28989.5 −1.21453 −0.607265 0.794499i $$-0.707734\pi$$
−0.607265 + 0.794499i $$0.707734\pi$$
$$830$$ 0 0
$$831$$ −3879.34 −0.161941
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −34413.9 −1.42628
$$836$$ 0 0
$$837$$ −3656.75 −0.151010
$$838$$ 0 0
$$839$$ 10021.9 0.412390 0.206195 0.978511i $$-0.433892\pi$$
0.206195 + 0.978511i $$0.433892\pi$$
$$840$$ 0 0
$$841$$ −18982.5 −0.778322
$$842$$ 0 0
$$843$$ 14054.1 0.574197
$$844$$ 0 0
$$845$$ −31877.4 −1.29777
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 1695.72 0.0685477
$$850$$ 0 0
$$851$$ 6222.92 0.250668
$$852$$ 0 0
$$853$$ −19446.0 −0.780562 −0.390281 0.920696i $$-0.627622\pi$$
−0.390281 + 0.920696i $$0.627622\pi$$
$$854$$ 0 0
$$855$$ 9899.17 0.395958
$$856$$ 0 0
$$857$$ 39813.3 1.58693 0.793464 0.608617i $$-0.208275\pi$$
0.793464 + 0.608617i $$0.208275\pi$$
$$858$$ 0 0
$$859$$ 46968.9 1.86561 0.932804 0.360384i $$-0.117354\pi$$
0.932804 + 0.360384i $$0.117354\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −13859.8 −0.546689 −0.273344 0.961916i $$-0.588130\pi$$
−0.273344 + 0.961916i $$0.588130\pi$$
$$864$$ 0 0
$$865$$ 59857.4 2.35285
$$866$$ 0 0
$$867$$ −13891.1 −0.544135
$$868$$ 0 0
$$869$$ −10460.3 −0.408333
$$870$$ 0 0
$$871$$ −260.665 −0.0101404
$$872$$ 0 0
$$873$$ 14933.8 0.578960
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 27382.5 1.05432 0.527162 0.849765i $$-0.323256\pi$$
0.527162 + 0.849765i $$0.323256\pi$$
$$878$$ 0 0
$$879$$ 2428.35 0.0931812
$$880$$ 0 0
$$881$$ −17720.1 −0.677647 −0.338823 0.940850i $$-0.610029\pi$$
−0.338823 + 0.940850i $$0.610029\pi$$
$$882$$ 0 0
$$883$$ 45478.3 1.73326 0.866629 0.498953i $$-0.166282\pi$$
0.866629 + 0.498953i $$0.166282\pi$$
$$884$$ 0 0
$$885$$ −31799.3 −1.20782
$$886$$ 0 0
$$887$$ 35483.6 1.34321 0.671603 0.740911i $$-0.265606\pi$$
0.671603 + 0.740911i $$0.265606\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 819.039 0.0307956
$$892$$ 0 0
$$893$$ 28208.2 1.05706
$$894$$ 0 0
$$895$$ 46359.1 1.73141
$$896$$ 0 0
$$897$$ −87.3961 −0.00325315
$$898$$ 0 0
$$899$$ −9958.40 −0.369445
$$900$$ 0 0
$$901$$ −11043.3 −0.408332
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 20171.7 0.740917
$$906$$ 0 0
$$907$$ −4269.33 −0.156296 −0.0781481 0.996942i $$-0.524901\pi$$
−0.0781481 + 0.996942i $$0.524901\pi$$
$$908$$ 0 0
$$909$$ 4372.55 0.159547
$$910$$ 0 0
$$911$$ 45428.0 1.65214 0.826069 0.563569i $$-0.190572\pi$$
0.826069 + 0.563569i $$0.190572\pi$$
$$912$$ 0 0
$$913$$ 1189.31 0.0431110
$$914$$ 0 0
$$915$$ 1720.21 0.0621511
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −28035.2 −1.00631 −0.503153 0.864198i $$-0.667827\pi$$
−0.503153 + 0.864198i $$0.667827\pi$$
$$920$$ 0 0
$$921$$ 11003.8 0.393690
$$922$$ 0 0
$$923$$ 518.515 0.0184909
$$924$$ 0 0
$$925$$ 11403.6 0.405351
$$926$$ 0 0
$$927$$ 2491.12 0.0882624
$$928$$ 0 0
$$929$$ 45234.1 1.59751 0.798754 0.601658i $$-0.205493\pi$$
0.798754 + 0.601658i $$0.205493\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −219.914 −0.00771668
$$934$$ 0 0
$$935$$ −2467.00 −0.0862882
$$936$$ 0 0
$$937$$ −26573.2 −0.926476 −0.463238 0.886234i $$-0.653312\pi$$
−0.463238 + 0.886234i $$0.653312\pi$$
$$938$$ 0 0
$$939$$ −24902.2 −0.865445
$$940$$ 0 0
$$941$$ 37470.6 1.29809 0.649047 0.760748i $$-0.275168\pi$$
0.649047 + 0.760748i $$0.275168\pi$$
$$942$$ 0 0
$$943$$ −3237.15 −0.111788
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −46525.9 −1.59650 −0.798252 0.602323i $$-0.794242\pi$$
−0.798252 + 0.602323i $$0.794242\pi$$
$$948$$ 0 0
$$949$$ −743.849 −0.0254440
$$950$$ 0 0
$$951$$ 3866.68 0.131846
$$952$$ 0 0
$$953$$ 7066.68 0.240202 0.120101 0.992762i $$-0.461678\pi$$
0.120101 + 0.992762i $$0.461678\pi$$
$$954$$ 0 0
$$955$$ 27805.0 0.942144
$$956$$ 0 0
$$957$$ 2230.49 0.0753410
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −11448.4 −0.384289
$$962$$ 0 0
$$963$$ 932.754 0.0312124
$$964$$ 0 0
$$965$$ −15851.3 −0.528779
$$966$$ 0 0
$$967$$ −31681.8 −1.05359 −0.526793 0.849994i $$-0.676606\pi$$
−0.526793 + 0.849994i $$0.676606\pi$$
$$968$$ 0 0
$$969$$ −3822.68 −0.126731
$$970$$ 0 0
$$971$$ 53411.5 1.76525 0.882625 0.470078i $$-0.155774\pi$$
0.882625 + 0.470078i $$0.155774\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −160.155 −0.00526059
$$976$$ 0 0
$$977$$ 56743.2 1.85811 0.929056 0.369939i $$-0.120622\pi$$
0.929056 + 0.369939i $$0.120622\pi$$
$$978$$ 0 0
$$979$$ −6859.61 −0.223937
$$980$$ 0 0
$$981$$ −451.414 −0.0146917
$$982$$ 0 0
$$983$$ 40797.4 1.32374 0.661869 0.749620i $$-0.269764\pi$$
0.661869 + 0.749620i $$0.269764\pi$$
$$984$$ 0 0
$$985$$ 68601.0 2.21910
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 13044.7 0.419411
$$990$$ 0 0
$$991$$ 37456.6 1.20066 0.600328 0.799754i $$-0.295037\pi$$
0.600328 + 0.799754i $$0.295037\pi$$
$$992$$ 0 0
$$993$$ −19553.7 −0.624892
$$994$$ 0 0
$$995$$ 9209.10 0.293415
$$996$$ 0 0
$$997$$ −9679.23 −0.307467 −0.153733 0.988112i $$-0.549130\pi$$
−0.153733 + 0.988112i $$0.549130\pi$$
$$998$$ 0 0
$$999$$ 3596.95 0.113916
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.be.1.3 yes 4
4.3 odd 2 2352.4.a.cn.1.3 4
7.6 odd 2 1176.4.a.z.1.2 4
28.27 even 2 2352.4.a.co.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.z.1.2 4 7.6 odd 2
1176.4.a.be.1.3 yes 4 1.1 even 1 trivial
2352.4.a.cn.1.3 4 4.3 odd 2
2352.4.a.co.1.2 4 28.27 even 2