# Properties

 Label 768.4.a.r.1.2 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.3134668844$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1436.1 Defining polynomial: $$x^{3} - 11x - 12$$ x^3 - 11*x - 12 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.76644$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +0.612661 q^{5} +22.7441 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +0.612661 q^{5} +22.7441 q^{7} +9.00000 q^{9} +60.2630 q^{11} +52.9062 q^{13} -1.83798 q^{15} +47.1643 q^{17} +29.1643 q^{19} -68.2324 q^{21} -109.488 q^{23} -124.625 q^{25} -27.0000 q^{27} -10.4250 q^{29} -220.881 q^{31} -180.789 q^{33} +13.9345 q^{35} +408.348 q^{37} -158.718 q^{39} +360.742 q^{41} -236.414 q^{43} +5.51395 q^{45} +129.113 q^{47} +174.296 q^{49} -141.493 q^{51} +117.819 q^{53} +36.9208 q^{55} -87.4928 q^{57} -262.854 q^{59} +273.465 q^{61} +204.697 q^{63} +32.4135 q^{65} +89.4077 q^{67} +328.465 q^{69} +350.521 q^{71} -532.610 q^{73} +373.874 q^{75} +1370.63 q^{77} -166.561 q^{79} +81.0000 q^{81} -361.934 q^{83} +28.8957 q^{85} +31.2750 q^{87} -40.3285 q^{89} +1203.31 q^{91} +662.642 q^{93} +17.8678 q^{95} -614.921 q^{97} +542.367 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 9 q^{3} + 10 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10})$$ 3 * q - 9 * q^3 + 10 * q^5 - 14 * q^7 + 27 * q^9 $$3 q - 9 q^{3} + 10 q^{5} - 14 q^{7} + 27 q^{9} + 52 q^{13} - 30 q^{15} + 26 q^{17} - 28 q^{19} + 42 q^{21} - 164 q^{23} + 53 q^{25} - 81 q^{27} + 174 q^{29} - 318 q^{31} + 92 q^{35} + 296 q^{37} - 156 q^{39} - 118 q^{41} + 260 q^{43} + 90 q^{45} - 204 q^{47} + 327 q^{49} - 78 q^{51} + 1086 q^{53} - 512 q^{55} + 84 q^{57} - 196 q^{59} + 1536 q^{61} - 126 q^{63} - 872 q^{65} + 660 q^{67} + 492 q^{69} + 852 q^{71} - 478 q^{73} - 159 q^{75} + 2304 q^{77} - 22 q^{79} + 243 q^{81} - 1136 q^{83} + 2732 q^{85} - 522 q^{87} + 110 q^{89} + 632 q^{91} + 954 q^{93} + 2552 q^{95} - 1222 q^{97}+O(q^{100})$$ 3 * q - 9 * q^3 + 10 * q^5 - 14 * q^7 + 27 * q^9 + 52 * q^13 - 30 * q^15 + 26 * q^17 - 28 * q^19 + 42 * q^21 - 164 * q^23 + 53 * q^25 - 81 * q^27 + 174 * q^29 - 318 * q^31 + 92 * q^35 + 296 * q^37 - 156 * q^39 - 118 * q^41 + 260 * q^43 + 90 * q^45 - 204 * q^47 + 327 * q^49 - 78 * q^51 + 1086 * q^53 - 512 * q^55 + 84 * q^57 - 196 * q^59 + 1536 * q^61 - 126 * q^63 - 872 * q^65 + 660 * q^67 + 492 * q^69 + 852 * q^71 - 478 * q^73 - 159 * q^75 + 2304 * q^77 - 22 * q^79 + 243 * q^81 - 1136 * q^83 + 2732 * q^85 - 522 * q^87 + 110 * q^89 + 632 * q^91 + 954 * q^93 + 2552 * q^95 - 1222 * q^97

## 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
$$5$$ 0.612661 0.0547981 0.0273990 0.999625i $$-0.491278\pi$$
0.0273990 + 0.999625i $$0.491278\pi$$
$$6$$ 0 0
$$7$$ 22.7441 1.22807 0.614034 0.789279i $$-0.289546\pi$$
0.614034 + 0.789279i $$0.289546\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 60.2630 1.65182 0.825908 0.563805i $$-0.190663\pi$$
0.825908 + 0.563805i $$0.190663\pi$$
$$12$$ 0 0
$$13$$ 52.9062 1.12873 0.564367 0.825524i $$-0.309120\pi$$
0.564367 + 0.825524i $$0.309120\pi$$
$$14$$ 0 0
$$15$$ −1.83798 −0.0316377
$$16$$ 0 0
$$17$$ 47.1643 0.672883 0.336442 0.941704i $$-0.390777\pi$$
0.336442 + 0.941704i $$0.390777\pi$$
$$18$$ 0 0
$$19$$ 29.1643 0.352144 0.176072 0.984377i $$-0.443661\pi$$
0.176072 + 0.984377i $$0.443661\pi$$
$$20$$ 0 0
$$21$$ −68.2324 −0.709026
$$22$$ 0 0
$$23$$ −109.488 −0.992604 −0.496302 0.868150i $$-0.665309\pi$$
−0.496302 + 0.868150i $$0.665309\pi$$
$$24$$ 0 0
$$25$$ −124.625 −0.996997
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −10.4250 −0.0667542 −0.0333771 0.999443i $$-0.510626\pi$$
−0.0333771 + 0.999443i $$0.510626\pi$$
$$30$$ 0 0
$$31$$ −220.881 −1.27972 −0.639860 0.768492i $$-0.721008\pi$$
−0.639860 + 0.768492i $$0.721008\pi$$
$$32$$ 0 0
$$33$$ −180.789 −0.953676
$$34$$ 0 0
$$35$$ 13.9345 0.0672958
$$36$$ 0 0
$$37$$ 408.348 1.81438 0.907188 0.420725i $$-0.138224\pi$$
0.907188 + 0.420725i $$0.138224\pi$$
$$38$$ 0 0
$$39$$ −158.718 −0.651674
$$40$$ 0 0
$$41$$ 360.742 1.37411 0.687054 0.726606i $$-0.258903\pi$$
0.687054 + 0.726606i $$0.258903\pi$$
$$42$$ 0 0
$$43$$ −236.414 −0.838436 −0.419218 0.907886i $$-0.637696\pi$$
−0.419218 + 0.907886i $$0.637696\pi$$
$$44$$ 0 0
$$45$$ 5.51395 0.0182660
$$46$$ 0 0
$$47$$ 129.113 0.400703 0.200352 0.979724i $$-0.435792\pi$$
0.200352 + 0.979724i $$0.435792\pi$$
$$48$$ 0 0
$$49$$ 174.296 0.508152
$$50$$ 0 0
$$51$$ −141.493 −0.388489
$$52$$ 0 0
$$53$$ 117.819 0.305353 0.152677 0.988276i $$-0.451211\pi$$
0.152677 + 0.988276i $$0.451211\pi$$
$$54$$ 0 0
$$55$$ 36.9208 0.0905163
$$56$$ 0 0
$$57$$ −87.4928 −0.203311
$$58$$ 0 0
$$59$$ −262.854 −0.580012 −0.290006 0.957025i $$-0.593657\pi$$
−0.290006 + 0.957025i $$0.593657\pi$$
$$60$$ 0 0
$$61$$ 273.465 0.573993 0.286996 0.957932i $$-0.407343\pi$$
0.286996 + 0.957932i $$0.407343\pi$$
$$62$$ 0 0
$$63$$ 204.697 0.409356
$$64$$ 0 0
$$65$$ 32.4135 0.0618524
$$66$$ 0 0
$$67$$ 89.4077 0.163028 0.0815141 0.996672i $$-0.474024\pi$$
0.0815141 + 0.996672i $$0.474024\pi$$
$$68$$ 0 0
$$69$$ 328.465 0.573080
$$70$$ 0 0
$$71$$ 350.521 0.585904 0.292952 0.956127i $$-0.405362\pi$$
0.292952 + 0.956127i $$0.405362\pi$$
$$72$$ 0 0
$$73$$ −532.610 −0.853936 −0.426968 0.904267i $$-0.640418\pi$$
−0.426968 + 0.904267i $$0.640418\pi$$
$$74$$ 0 0
$$75$$ 373.874 0.575617
$$76$$ 0 0
$$77$$ 1370.63 2.02854
$$78$$ 0 0
$$79$$ −166.561 −0.237210 −0.118605 0.992942i $$-0.537842\pi$$
−0.118605 + 0.992942i $$0.537842\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −361.934 −0.478644 −0.239322 0.970940i $$-0.576925\pi$$
−0.239322 + 0.970940i $$0.576925\pi$$
$$84$$ 0 0
$$85$$ 28.8957 0.0368727
$$86$$ 0 0
$$87$$ 31.2750 0.0385405
$$88$$ 0 0
$$89$$ −40.3285 −0.0480316 −0.0240158 0.999712i $$-0.507645\pi$$
−0.0240158 + 0.999712i $$0.507645\pi$$
$$90$$ 0 0
$$91$$ 1203.31 1.38616
$$92$$ 0 0
$$93$$ 662.642 0.738846
$$94$$ 0 0
$$95$$ 17.8678 0.0192968
$$96$$ 0 0
$$97$$ −614.921 −0.643667 −0.321834 0.946796i $$-0.604299\pi$$
−0.321834 + 0.946796i $$0.604299\pi$$
$$98$$ 0 0
$$99$$ 542.367 0.550605
$$100$$ 0 0
$$101$$ 1664.99 1.64033 0.820163 0.572130i $$-0.193883\pi$$
0.820163 + 0.572130i $$0.193883\pi$$
$$102$$ 0 0
$$103$$ −396.858 −0.379647 −0.189823 0.981818i $$-0.560792\pi$$
−0.189823 + 0.981818i $$0.560792\pi$$
$$104$$ 0 0
$$105$$ −41.8034 −0.0388532
$$106$$ 0 0
$$107$$ 350.630 0.316791 0.158396 0.987376i $$-0.449368\pi$$
0.158396 + 0.987376i $$0.449368\pi$$
$$108$$ 0 0
$$109$$ 597.009 0.524615 0.262308 0.964984i $$-0.415516\pi$$
0.262308 + 0.964984i $$0.415516\pi$$
$$110$$ 0 0
$$111$$ −1225.04 −1.04753
$$112$$ 0 0
$$113$$ 496.422 0.413270 0.206635 0.978418i $$-0.433749\pi$$
0.206635 + 0.978418i $$0.433749\pi$$
$$114$$ 0 0
$$115$$ −67.0792 −0.0543928
$$116$$ 0 0
$$117$$ 476.155 0.376244
$$118$$ 0 0
$$119$$ 1072.71 0.826346
$$120$$ 0 0
$$121$$ 2300.63 1.72849
$$122$$ 0 0
$$123$$ −1082.23 −0.793342
$$124$$ 0 0
$$125$$ −152.935 −0.109432
$$126$$ 0 0
$$127$$ 1799.85 1.25756 0.628782 0.777581i $$-0.283554\pi$$
0.628782 + 0.777581i $$0.283554\pi$$
$$128$$ 0 0
$$129$$ 709.241 0.484071
$$130$$ 0 0
$$131$$ 1121.45 0.747949 0.373974 0.927439i $$-0.377995\pi$$
0.373974 + 0.927439i $$0.377995\pi$$
$$132$$ 0 0
$$133$$ 663.316 0.432457
$$134$$ 0 0
$$135$$ −16.5419 −0.0105459
$$136$$ 0 0
$$137$$ −2449.55 −1.52759 −0.763793 0.645461i $$-0.776665\pi$$
−0.763793 + 0.645461i $$0.776665\pi$$
$$138$$ 0 0
$$139$$ −2457.56 −1.49962 −0.749811 0.661652i $$-0.769856\pi$$
−0.749811 + 0.661652i $$0.769856\pi$$
$$140$$ 0 0
$$141$$ −387.339 −0.231346
$$142$$ 0 0
$$143$$ 3188.28 1.86446
$$144$$ 0 0
$$145$$ −6.38698 −0.00365800
$$146$$ 0 0
$$147$$ −522.888 −0.293382
$$148$$ 0 0
$$149$$ 2084.96 1.14635 0.573177 0.819432i $$-0.305711\pi$$
0.573177 + 0.819432i $$0.305711\pi$$
$$150$$ 0 0
$$151$$ −1057.80 −0.570084 −0.285042 0.958515i $$-0.592008\pi$$
−0.285042 + 0.958515i $$0.592008\pi$$
$$152$$ 0 0
$$153$$ 424.478 0.224294
$$154$$ 0 0
$$155$$ −135.325 −0.0701262
$$156$$ 0 0
$$157$$ 3193.01 1.62312 0.811559 0.584270i $$-0.198619\pi$$
0.811559 + 0.584270i $$0.198619\pi$$
$$158$$ 0 0
$$159$$ −353.458 −0.176296
$$160$$ 0 0
$$161$$ −2490.22 −1.21899
$$162$$ 0 0
$$163$$ −846.854 −0.406937 −0.203469 0.979081i $$-0.565221\pi$$
−0.203469 + 0.979081i $$0.565221\pi$$
$$164$$ 0 0
$$165$$ −110.762 −0.0522596
$$166$$ 0 0
$$167$$ 2630.15 1.21873 0.609363 0.792892i $$-0.291425\pi$$
0.609363 + 0.792892i $$0.291425\pi$$
$$168$$ 0 0
$$169$$ 602.062 0.274038
$$170$$ 0 0
$$171$$ 262.478 0.117381
$$172$$ 0 0
$$173$$ 429.843 0.188904 0.0944519 0.995529i $$-0.469890\pi$$
0.0944519 + 0.995529i $$0.469890\pi$$
$$174$$ 0 0
$$175$$ −2834.48 −1.22438
$$176$$ 0 0
$$177$$ 788.563 0.334870
$$178$$ 0 0
$$179$$ −1516.30 −0.633149 −0.316574 0.948568i $$-0.602533\pi$$
−0.316574 + 0.948568i $$0.602533\pi$$
$$180$$ 0 0
$$181$$ 3380.20 1.38811 0.694056 0.719921i $$-0.255822\pi$$
0.694056 + 0.719921i $$0.255822\pi$$
$$182$$ 0 0
$$183$$ −820.394 −0.331395
$$184$$ 0 0
$$185$$ 250.179 0.0994244
$$186$$ 0 0
$$187$$ 2842.26 1.11148
$$188$$ 0 0
$$189$$ −614.092 −0.236342
$$190$$ 0 0
$$191$$ −2799.71 −1.06063 −0.530314 0.847801i $$-0.677926\pi$$
−0.530314 + 0.847801i $$0.677926\pi$$
$$192$$ 0 0
$$193$$ 624.106 0.232768 0.116384 0.993204i $$-0.462870\pi$$
0.116384 + 0.993204i $$0.462870\pi$$
$$194$$ 0 0
$$195$$ −97.2406 −0.0357105
$$196$$ 0 0
$$197$$ −4779.25 −1.72846 −0.864232 0.503094i $$-0.832195\pi$$
−0.864232 + 0.503094i $$0.832195\pi$$
$$198$$ 0 0
$$199$$ −2615.92 −0.931846 −0.465923 0.884825i $$-0.654278\pi$$
−0.465923 + 0.884825i $$0.654278\pi$$
$$200$$ 0 0
$$201$$ −268.223 −0.0941244
$$202$$ 0 0
$$203$$ −237.107 −0.0819787
$$204$$ 0 0
$$205$$ 221.013 0.0752985
$$206$$ 0 0
$$207$$ −985.395 −0.330868
$$208$$ 0 0
$$209$$ 1757.52 0.581677
$$210$$ 0 0
$$211$$ −1745.78 −0.569595 −0.284798 0.958588i $$-0.591926\pi$$
−0.284798 + 0.958588i $$0.591926\pi$$
$$212$$ 0 0
$$213$$ −1051.56 −0.338272
$$214$$ 0 0
$$215$$ −144.841 −0.0459447
$$216$$ 0 0
$$217$$ −5023.74 −1.57158
$$218$$ 0 0
$$219$$ 1597.83 0.493020
$$220$$ 0 0
$$221$$ 2495.28 0.759505
$$222$$ 0 0
$$223$$ −3385.60 −1.01667 −0.508333 0.861161i $$-0.669738\pi$$
−0.508333 + 0.861161i $$0.669738\pi$$
$$224$$ 0 0
$$225$$ −1121.62 −0.332332
$$226$$ 0 0
$$227$$ 3847.72 1.12503 0.562515 0.826787i $$-0.309834\pi$$
0.562515 + 0.826787i $$0.309834\pi$$
$$228$$ 0 0
$$229$$ −1335.15 −0.385279 −0.192640 0.981270i $$-0.561705\pi$$
−0.192640 + 0.981270i $$0.561705\pi$$
$$230$$ 0 0
$$231$$ −4111.89 −1.17118
$$232$$ 0 0
$$233$$ 5146.38 1.44700 0.723499 0.690325i $$-0.242532\pi$$
0.723499 + 0.690325i $$0.242532\pi$$
$$234$$ 0 0
$$235$$ 79.1025 0.0219578
$$236$$ 0 0
$$237$$ 499.683 0.136953
$$238$$ 0 0
$$239$$ 7085.07 1.91755 0.958777 0.284160i $$-0.0917146\pi$$
0.958777 + 0.284160i $$0.0917146\pi$$
$$240$$ 0 0
$$241$$ 2538.40 0.678476 0.339238 0.940701i $$-0.389831\pi$$
0.339238 + 0.940701i $$0.389831\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 106.784 0.0278458
$$246$$ 0 0
$$247$$ 1542.97 0.397477
$$248$$ 0 0
$$249$$ 1085.80 0.276345
$$250$$ 0 0
$$251$$ 1696.99 0.426746 0.213373 0.976971i $$-0.431555\pi$$
0.213373 + 0.976971i $$0.431555\pi$$
$$252$$ 0 0
$$253$$ −6598.09 −1.63960
$$254$$ 0 0
$$255$$ −86.6871 −0.0212885
$$256$$ 0 0
$$257$$ −382.902 −0.0929369 −0.0464685 0.998920i $$-0.514797\pi$$
−0.0464685 + 0.998920i $$0.514797\pi$$
$$258$$ 0 0
$$259$$ 9287.52 2.22818
$$260$$ 0 0
$$261$$ −93.8249 −0.0222514
$$262$$ 0 0
$$263$$ 5002.02 1.17277 0.586383 0.810034i $$-0.300551\pi$$
0.586383 + 0.810034i $$0.300551\pi$$
$$264$$ 0 0
$$265$$ 72.1833 0.0167328
$$266$$ 0 0
$$267$$ 120.986 0.0277311
$$268$$ 0 0
$$269$$ −6117.47 −1.38658 −0.693288 0.720661i $$-0.743839\pi$$
−0.693288 + 0.720661i $$0.743839\pi$$
$$270$$ 0 0
$$271$$ −3956.12 −0.886780 −0.443390 0.896329i $$-0.646224\pi$$
−0.443390 + 0.896329i $$0.646224\pi$$
$$272$$ 0 0
$$273$$ −3609.92 −0.800301
$$274$$ 0 0
$$275$$ −7510.25 −1.64686
$$276$$ 0 0
$$277$$ −4842.17 −1.05032 −0.525158 0.851004i $$-0.675994\pi$$
−0.525158 + 0.851004i $$0.675994\pi$$
$$278$$ 0 0
$$279$$ −1987.92 −0.426573
$$280$$ 0 0
$$281$$ −1878.68 −0.398835 −0.199417 0.979915i $$-0.563905\pi$$
−0.199417 + 0.979915i $$0.563905\pi$$
$$282$$ 0 0
$$283$$ −5724.87 −1.20250 −0.601251 0.799060i $$-0.705331\pi$$
−0.601251 + 0.799060i $$0.705331\pi$$
$$284$$ 0 0
$$285$$ −53.6034 −0.0111410
$$286$$ 0 0
$$287$$ 8204.77 1.68750
$$288$$ 0 0
$$289$$ −2688.53 −0.547228
$$290$$ 0 0
$$291$$ 1844.76 0.371622
$$292$$ 0 0
$$293$$ 5088.75 1.01464 0.507318 0.861759i $$-0.330637\pi$$
0.507318 + 0.861759i $$0.330637\pi$$
$$294$$ 0 0
$$295$$ −161.041 −0.0317836
$$296$$ 0 0
$$297$$ −1627.10 −0.317892
$$298$$ 0 0
$$299$$ −5792.61 −1.12038
$$300$$ 0 0
$$301$$ −5377.02 −1.02966
$$302$$ 0 0
$$303$$ −4994.98 −0.947043
$$304$$ 0 0
$$305$$ 167.541 0.0314537
$$306$$ 0 0
$$307$$ −7219.21 −1.34209 −0.671046 0.741416i $$-0.734155\pi$$
−0.671046 + 0.741416i $$0.734155\pi$$
$$308$$ 0 0
$$309$$ 1190.58 0.219189
$$310$$ 0 0
$$311$$ 1537.06 0.280252 0.140126 0.990134i $$-0.455249\pi$$
0.140126 + 0.990134i $$0.455249\pi$$
$$312$$ 0 0
$$313$$ 2200.93 0.397456 0.198728 0.980055i $$-0.436319\pi$$
0.198728 + 0.980055i $$0.436319\pi$$
$$314$$ 0 0
$$315$$ 125.410 0.0224319
$$316$$ 0 0
$$317$$ −2840.41 −0.503260 −0.251630 0.967824i $$-0.580967\pi$$
−0.251630 + 0.967824i $$0.580967\pi$$
$$318$$ 0 0
$$319$$ −628.240 −0.110266
$$320$$ 0 0
$$321$$ −1051.89 −0.182899
$$322$$ 0 0
$$323$$ 1375.51 0.236952
$$324$$ 0 0
$$325$$ −6593.41 −1.12534
$$326$$ 0 0
$$327$$ −1791.03 −0.302887
$$328$$ 0 0
$$329$$ 2936.56 0.492091
$$330$$ 0 0
$$331$$ 2118.52 0.351795 0.175898 0.984408i $$-0.443717\pi$$
0.175898 + 0.984408i $$0.443717\pi$$
$$332$$ 0 0
$$333$$ 3675.13 0.604792
$$334$$ 0 0
$$335$$ 54.7766 0.00893364
$$336$$ 0 0
$$337$$ −659.599 −0.106619 −0.0533096 0.998578i $$-0.516977\pi$$
−0.0533096 + 0.998578i $$0.516977\pi$$
$$338$$ 0 0
$$339$$ −1489.27 −0.238601
$$340$$ 0 0
$$341$$ −13310.9 −2.11386
$$342$$ 0 0
$$343$$ −3837.03 −0.604023
$$344$$ 0 0
$$345$$ 201.238 0.0314037
$$346$$ 0 0
$$347$$ 8377.42 1.29603 0.648017 0.761626i $$-0.275599\pi$$
0.648017 + 0.761626i $$0.275599\pi$$
$$348$$ 0 0
$$349$$ −3254.18 −0.499119 −0.249559 0.968360i $$-0.580286\pi$$
−0.249559 + 0.968360i $$0.580286\pi$$
$$350$$ 0 0
$$351$$ −1428.47 −0.217225
$$352$$ 0 0
$$353$$ 11117.5 1.67627 0.838137 0.545459i $$-0.183645\pi$$
0.838137 + 0.545459i $$0.183645\pi$$
$$354$$ 0 0
$$355$$ 214.750 0.0321064
$$356$$ 0 0
$$357$$ −3218.13 −0.477091
$$358$$ 0 0
$$359$$ 4756.56 0.699281 0.349640 0.936884i $$-0.386304\pi$$
0.349640 + 0.936884i $$0.386304\pi$$
$$360$$ 0 0
$$361$$ −6008.45 −0.875994
$$362$$ 0 0
$$363$$ −6901.88 −0.997946
$$364$$ 0 0
$$365$$ −326.310 −0.0467940
$$366$$ 0 0
$$367$$ 1837.40 0.261339 0.130670 0.991426i $$-0.458287\pi$$
0.130670 + 0.991426i $$0.458287\pi$$
$$368$$ 0 0
$$369$$ 3246.68 0.458036
$$370$$ 0 0
$$371$$ 2679.70 0.374995
$$372$$ 0 0
$$373$$ −5598.07 −0.777097 −0.388549 0.921428i $$-0.627023\pi$$
−0.388549 + 0.921428i $$0.627023\pi$$
$$374$$ 0 0
$$375$$ 458.806 0.0631804
$$376$$ 0 0
$$377$$ −551.546 −0.0753476
$$378$$ 0 0
$$379$$ 3460.18 0.468965 0.234482 0.972120i $$-0.424661\pi$$
0.234482 + 0.972120i $$0.424661\pi$$
$$380$$ 0 0
$$381$$ −5399.55 −0.726055
$$382$$ 0 0
$$383$$ 5059.63 0.675027 0.337513 0.941321i $$-0.390414\pi$$
0.337513 + 0.941321i $$0.390414\pi$$
$$384$$ 0 0
$$385$$ 839.732 0.111160
$$386$$ 0 0
$$387$$ −2127.72 −0.279479
$$388$$ 0 0
$$389$$ −2192.22 −0.285732 −0.142866 0.989742i $$-0.545632\pi$$
−0.142866 + 0.989742i $$0.545632\pi$$
$$390$$ 0 0
$$391$$ −5163.93 −0.667906
$$392$$ 0 0
$$393$$ −3364.34 −0.431828
$$394$$ 0 0
$$395$$ −102.045 −0.0129986
$$396$$ 0 0
$$397$$ 5519.94 0.697828 0.348914 0.937155i $$-0.386551\pi$$
0.348914 + 0.937155i $$0.386551\pi$$
$$398$$ 0 0
$$399$$ −1989.95 −0.249679
$$400$$ 0 0
$$401$$ −7352.64 −0.915645 −0.457822 0.889044i $$-0.651370\pi$$
−0.457822 + 0.889044i $$0.651370\pi$$
$$402$$ 0 0
$$403$$ −11685.9 −1.44446
$$404$$ 0 0
$$405$$ 49.6256 0.00608868
$$406$$ 0 0
$$407$$ 24608.2 2.99702
$$408$$ 0 0
$$409$$ 11311.2 1.36749 0.683745 0.729721i $$-0.260350\pi$$
0.683745 + 0.729721i $$0.260350\pi$$
$$410$$ 0 0
$$411$$ 7348.66 0.881952
$$412$$ 0 0
$$413$$ −5978.40 −0.712295
$$414$$ 0 0
$$415$$ −221.743 −0.0262288
$$416$$ 0 0
$$417$$ 7372.68 0.865808
$$418$$ 0 0
$$419$$ 13042.2 1.52065 0.760325 0.649543i $$-0.225040\pi$$
0.760325 + 0.649543i $$0.225040\pi$$
$$420$$ 0 0
$$421$$ 4544.38 0.526080 0.263040 0.964785i $$-0.415275\pi$$
0.263040 + 0.964785i $$0.415275\pi$$
$$422$$ 0 0
$$423$$ 1162.02 0.133568
$$424$$ 0 0
$$425$$ −5877.83 −0.670863
$$426$$ 0 0
$$427$$ 6219.72 0.704903
$$428$$ 0 0
$$429$$ −9564.85 −1.07645
$$430$$ 0 0
$$431$$ 7713.83 0.862093 0.431047 0.902330i $$-0.358144\pi$$
0.431047 + 0.902330i $$0.358144\pi$$
$$432$$ 0 0
$$433$$ −15068.3 −1.67237 −0.836183 0.548451i $$-0.815218\pi$$
−0.836183 + 0.548451i $$0.815218\pi$$
$$434$$ 0 0
$$435$$ 19.1609 0.00211195
$$436$$ 0 0
$$437$$ −3193.14 −0.349540
$$438$$ 0 0
$$439$$ −11004.7 −1.19642 −0.598208 0.801341i $$-0.704120\pi$$
−0.598208 + 0.801341i $$0.704120\pi$$
$$440$$ 0 0
$$441$$ 1568.67 0.169384
$$442$$ 0 0
$$443$$ −2513.04 −0.269522 −0.134761 0.990878i $$-0.543027\pi$$
−0.134761 + 0.990878i $$0.543027\pi$$
$$444$$ 0 0
$$445$$ −24.7077 −0.00263204
$$446$$ 0 0
$$447$$ −6254.89 −0.661848
$$448$$ 0 0
$$449$$ −15752.7 −1.65571 −0.827855 0.560942i $$-0.810439\pi$$
−0.827855 + 0.560942i $$0.810439\pi$$
$$450$$ 0 0
$$451$$ 21739.4 2.26977
$$452$$ 0 0
$$453$$ 3173.40 0.329138
$$454$$ 0 0
$$455$$ 737.218 0.0759590
$$456$$ 0 0
$$457$$ 5257.06 0.538107 0.269053 0.963125i $$-0.413289\pi$$
0.269053 + 0.963125i $$0.413289\pi$$
$$458$$ 0 0
$$459$$ −1273.43 −0.129496
$$460$$ 0 0
$$461$$ −8066.31 −0.814936 −0.407468 0.913220i $$-0.633588\pi$$
−0.407468 + 0.913220i $$0.633588\pi$$
$$462$$ 0 0
$$463$$ 5683.43 0.570478 0.285239 0.958456i $$-0.407927\pi$$
0.285239 + 0.958456i $$0.407927\pi$$
$$464$$ 0 0
$$465$$ 405.975 0.0404874
$$466$$ 0 0
$$467$$ −11139.3 −1.10378 −0.551891 0.833916i $$-0.686094\pi$$
−0.551891 + 0.833916i $$0.686094\pi$$
$$468$$ 0 0
$$469$$ 2033.50 0.200210
$$470$$ 0 0
$$471$$ −9579.02 −0.937108
$$472$$ 0 0
$$473$$ −14247.0 −1.38494
$$474$$ 0 0
$$475$$ −3634.59 −0.351087
$$476$$ 0 0
$$477$$ 1060.37 0.101784
$$478$$ 0 0
$$479$$ 3477.35 0.331699 0.165850 0.986151i $$-0.446963\pi$$
0.165850 + 0.986151i $$0.446963\pi$$
$$480$$ 0 0
$$481$$ 21604.1 2.04795
$$482$$ 0 0
$$483$$ 7470.65 0.703782
$$484$$ 0 0
$$485$$ −376.738 −0.0352717
$$486$$ 0 0
$$487$$ 478.797 0.0445510 0.0222755 0.999752i $$-0.492909\pi$$
0.0222755 + 0.999752i $$0.492909\pi$$
$$488$$ 0 0
$$489$$ 2540.56 0.234945
$$490$$ 0 0
$$491$$ −16601.8 −1.52592 −0.762961 0.646444i $$-0.776255\pi$$
−0.762961 + 0.646444i $$0.776255\pi$$
$$492$$ 0 0
$$493$$ −491.687 −0.0449178
$$494$$ 0 0
$$495$$ 332.287 0.0301721
$$496$$ 0 0
$$497$$ 7972.29 0.719530
$$498$$ 0 0
$$499$$ −9482.20 −0.850664 −0.425332 0.905037i $$-0.639843\pi$$
−0.425332 + 0.905037i $$0.639843\pi$$
$$500$$ 0 0
$$501$$ −7890.45 −0.703631
$$502$$ 0 0
$$503$$ −16561.2 −1.46805 −0.734023 0.679124i $$-0.762360\pi$$
−0.734023 + 0.679124i $$0.762360\pi$$
$$504$$ 0 0
$$505$$ 1020.08 0.0898867
$$506$$ 0 0
$$507$$ −1806.19 −0.158216
$$508$$ 0 0
$$509$$ 4197.35 0.365509 0.182755 0.983159i $$-0.441499\pi$$
0.182755 + 0.983159i $$0.441499\pi$$
$$510$$ 0 0
$$511$$ −12113.8 −1.04869
$$512$$ 0 0
$$513$$ −787.435 −0.0677702
$$514$$ 0 0
$$515$$ −243.140 −0.0208039
$$516$$ 0 0
$$517$$ 7780.73 0.661888
$$518$$ 0 0
$$519$$ −1289.53 −0.109064
$$520$$ 0 0
$$521$$ 15755.5 1.32488 0.662440 0.749115i $$-0.269521\pi$$
0.662440 + 0.749115i $$0.269521\pi$$
$$522$$ 0 0
$$523$$ −11555.1 −0.966098 −0.483049 0.875593i $$-0.660471\pi$$
−0.483049 + 0.875593i $$0.660471\pi$$
$$524$$ 0 0
$$525$$ 8503.44 0.706897
$$526$$ 0 0
$$527$$ −10417.7 −0.861102
$$528$$ 0 0
$$529$$ −179.314 −0.0147378
$$530$$ 0 0
$$531$$ −2365.69 −0.193337
$$532$$ 0 0
$$533$$ 19085.5 1.55100
$$534$$ 0 0
$$535$$ 214.817 0.0173595
$$536$$ 0 0
$$537$$ 4548.90 0.365549
$$538$$ 0 0
$$539$$ 10503.6 0.839373
$$540$$ 0 0
$$541$$ 7475.65 0.594091 0.297045 0.954863i $$-0.403999\pi$$
0.297045 + 0.954863i $$0.403999\pi$$
$$542$$ 0 0
$$543$$ −10140.6 −0.801427
$$544$$ 0 0
$$545$$ 365.764 0.0287479
$$546$$ 0 0
$$547$$ −6028.08 −0.471192 −0.235596 0.971851i $$-0.575704\pi$$
−0.235596 + 0.971851i $$0.575704\pi$$
$$548$$ 0 0
$$549$$ 2461.18 0.191331
$$550$$ 0 0
$$551$$ −304.037 −0.0235071
$$552$$ 0 0
$$553$$ −3788.29 −0.291310
$$554$$ 0 0
$$555$$ −750.536 −0.0574027
$$556$$ 0 0
$$557$$ 19381.5 1.47436 0.737181 0.675696i $$-0.236157\pi$$
0.737181 + 0.675696i $$0.236157\pi$$
$$558$$ 0 0
$$559$$ −12507.7 −0.946370
$$560$$ 0 0
$$561$$ −8526.77 −0.641712
$$562$$ 0 0
$$563$$ −20565.0 −1.53946 −0.769728 0.638372i $$-0.779608\pi$$
−0.769728 + 0.638372i $$0.779608\pi$$
$$564$$ 0 0
$$565$$ 304.139 0.0226464
$$566$$ 0 0
$$567$$ 1842.28 0.136452
$$568$$ 0 0
$$569$$ −15252.9 −1.12379 −0.561895 0.827209i $$-0.689927\pi$$
−0.561895 + 0.827209i $$0.689927\pi$$
$$570$$ 0 0
$$571$$ 16492.8 1.20876 0.604379 0.796697i $$-0.293421\pi$$
0.604379 + 0.796697i $$0.293421\pi$$
$$572$$ 0 0
$$573$$ 8399.13 0.612354
$$574$$ 0 0
$$575$$ 13644.9 0.989623
$$576$$ 0 0
$$577$$ −10298.2 −0.743016 −0.371508 0.928430i $$-0.621159\pi$$
−0.371508 + 0.928430i $$0.621159\pi$$
$$578$$ 0 0
$$579$$ −1872.32 −0.134388
$$580$$ 0 0
$$581$$ −8231.89 −0.587808
$$582$$ 0 0
$$583$$ 7100.14 0.504387
$$584$$ 0 0
$$585$$ 291.722 0.0206175
$$586$$ 0 0
$$587$$ 13104.8 0.921453 0.460727 0.887542i $$-0.347589\pi$$
0.460727 + 0.887542i $$0.347589\pi$$
$$588$$ 0 0
$$589$$ −6441.82 −0.450646
$$590$$ 0 0
$$591$$ 14337.7 0.997929
$$592$$ 0 0
$$593$$ 4163.34 0.288310 0.144155 0.989555i $$-0.453954\pi$$
0.144155 + 0.989555i $$0.453954\pi$$
$$594$$ 0 0
$$595$$ 657.208 0.0452822
$$596$$ 0 0
$$597$$ 7847.75 0.538001
$$598$$ 0 0
$$599$$ 5718.60 0.390076 0.195038 0.980796i $$-0.437517\pi$$
0.195038 + 0.980796i $$0.437517\pi$$
$$600$$ 0 0
$$601$$ −17473.0 −1.18592 −0.592959 0.805233i $$-0.702040\pi$$
−0.592959 + 0.805233i $$0.702040\pi$$
$$602$$ 0 0
$$603$$ 804.670 0.0543428
$$604$$ 0 0
$$605$$ 1409.50 0.0947181
$$606$$ 0 0
$$607$$ −5647.60 −0.377643 −0.188821 0.982011i $$-0.560467\pi$$
−0.188821 + 0.982011i $$0.560467\pi$$
$$608$$ 0 0
$$609$$ 711.322 0.0473304
$$610$$ 0 0
$$611$$ 6830.87 0.452287
$$612$$ 0 0
$$613$$ −16023.6 −1.05577 −0.527884 0.849317i $$-0.677014\pi$$
−0.527884 + 0.849317i $$0.677014\pi$$
$$614$$ 0 0
$$615$$ −663.038 −0.0434736
$$616$$ 0 0
$$617$$ 21022.1 1.37167 0.685834 0.727758i $$-0.259438\pi$$
0.685834 + 0.727758i $$0.259438\pi$$
$$618$$ 0 0
$$619$$ −17824.2 −1.15737 −0.578686 0.815550i $$-0.696434\pi$$
−0.578686 + 0.815550i $$0.696434\pi$$
$$620$$ 0 0
$$621$$ 2956.18 0.191027
$$622$$ 0 0
$$623$$ −917.238 −0.0589861
$$624$$ 0 0
$$625$$ 15484.4 0.991001
$$626$$ 0 0
$$627$$ −5272.57 −0.335831
$$628$$ 0 0
$$629$$ 19259.4 1.22086
$$630$$ 0 0
$$631$$ 22339.0 1.40935 0.704677 0.709528i $$-0.251092\pi$$
0.704677 + 0.709528i $$0.251092\pi$$
$$632$$ 0 0
$$633$$ 5237.35 0.328856
$$634$$ 0 0
$$635$$ 1102.70 0.0689121
$$636$$ 0 0
$$637$$ 9221.34 0.573568
$$638$$ 0 0
$$639$$ 3154.69 0.195301
$$640$$ 0 0
$$641$$ −5268.43 −0.324634 −0.162317 0.986739i $$-0.551897\pi$$
−0.162317 + 0.986739i $$0.551897\pi$$
$$642$$ 0 0
$$643$$ 21965.8 1.34719 0.673597 0.739099i $$-0.264748\pi$$
0.673597 + 0.739099i $$0.264748\pi$$
$$644$$ 0 0
$$645$$ 434.524 0.0265262
$$646$$ 0 0
$$647$$ −3165.40 −0.192341 −0.0961706 0.995365i $$-0.530659\pi$$
−0.0961706 + 0.995365i $$0.530659\pi$$
$$648$$ 0 0
$$649$$ −15840.4 −0.958073
$$650$$ 0 0
$$651$$ 15071.2 0.907354
$$652$$ 0 0
$$653$$ 12094.7 0.724814 0.362407 0.932020i $$-0.381955\pi$$
0.362407 + 0.932020i $$0.381955\pi$$
$$654$$ 0 0
$$655$$ 687.067 0.0409861
$$656$$ 0 0
$$657$$ −4793.49 −0.284645
$$658$$ 0 0
$$659$$ 7523.17 0.444706 0.222353 0.974966i $$-0.428626\pi$$
0.222353 + 0.974966i $$0.428626\pi$$
$$660$$ 0 0
$$661$$ 24141.1 1.42054 0.710271 0.703928i $$-0.248572\pi$$
0.710271 + 0.703928i $$0.248572\pi$$
$$662$$ 0 0
$$663$$ −7485.84 −0.438501
$$664$$ 0 0
$$665$$ 406.388 0.0236978
$$666$$ 0 0
$$667$$ 1141.41 0.0662604
$$668$$ 0 0
$$669$$ 10156.8 0.586972
$$670$$ 0 0
$$671$$ 16479.8 0.948130
$$672$$ 0 0
$$673$$ 944.143 0.0540773 0.0270387 0.999634i $$-0.491392\pi$$
0.0270387 + 0.999634i $$0.491392\pi$$
$$674$$ 0 0
$$675$$ 3364.87 0.191872
$$676$$ 0 0
$$677$$ 4450.51 0.252654 0.126327 0.991989i $$-0.459681\pi$$
0.126327 + 0.991989i $$0.459681\pi$$
$$678$$ 0 0
$$679$$ −13985.8 −0.790468
$$680$$ 0 0
$$681$$ −11543.1 −0.649536
$$682$$ 0 0
$$683$$ −1726.40 −0.0967189 −0.0483594 0.998830i $$-0.515399\pi$$
−0.0483594 + 0.998830i $$0.515399\pi$$
$$684$$ 0 0
$$685$$ −1500.75 −0.0837088
$$686$$ 0 0
$$687$$ 4005.44 0.222441
$$688$$ 0 0
$$689$$ 6233.37 0.344662
$$690$$ 0 0
$$691$$ 683.143 0.0376092 0.0188046 0.999823i $$-0.494014\pi$$
0.0188046 + 0.999823i $$0.494014\pi$$
$$692$$ 0 0
$$693$$ 12335.7 0.676181
$$694$$ 0 0
$$695$$ −1505.65 −0.0821764
$$696$$ 0 0
$$697$$ 17014.1 0.924614
$$698$$ 0 0
$$699$$ −15439.1 −0.835425
$$700$$ 0 0
$$701$$ 19533.2 1.05244 0.526220 0.850349i $$-0.323609\pi$$
0.526220 + 0.850349i $$0.323609\pi$$
$$702$$ 0 0
$$703$$ 11909.2 0.638922
$$704$$ 0 0
$$705$$ −237.307 −0.0126773
$$706$$ 0 0
$$707$$ 37868.8 2.01443
$$708$$ 0 0
$$709$$ 26081.9 1.38156 0.690779 0.723066i $$-0.257268\pi$$
0.690779 + 0.723066i $$0.257268\pi$$
$$710$$ 0 0
$$711$$ −1499.05 −0.0790699
$$712$$ 0 0
$$713$$ 24183.8 1.27025
$$714$$ 0 0
$$715$$ 1953.34 0.102169
$$716$$ 0 0
$$717$$ −21255.2 −1.10710
$$718$$ 0 0
$$719$$ −30077.3 −1.56007 −0.780036 0.625734i $$-0.784800\pi$$
−0.780036 + 0.625734i $$0.784800\pi$$
$$720$$ 0 0
$$721$$ −9026.21 −0.466232
$$722$$ 0 0
$$723$$ −7615.20 −0.391718
$$724$$ 0 0
$$725$$ 1299.21 0.0665537
$$726$$ 0 0
$$727$$ −23049.9 −1.17589 −0.587946 0.808900i $$-0.700063\pi$$
−0.587946 + 0.808900i $$0.700063\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −11150.3 −0.564169
$$732$$ 0 0
$$733$$ 4444.57 0.223962 0.111981 0.993710i $$-0.464280\pi$$
0.111981 + 0.993710i $$0.464280\pi$$
$$734$$ 0 0
$$735$$ −320.353 −0.0160768
$$736$$ 0 0
$$737$$ 5387.98 0.269293
$$738$$ 0 0
$$739$$ −28465.1 −1.41692 −0.708462 0.705749i $$-0.750610\pi$$
−0.708462 + 0.705749i $$0.750610\pi$$
$$740$$ 0 0
$$741$$ −4628.91 −0.229483
$$742$$ 0 0
$$743$$ −4389.76 −0.216749 −0.108374 0.994110i $$-0.534565\pi$$
−0.108374 + 0.994110i $$0.534565\pi$$
$$744$$ 0 0
$$745$$ 1277.38 0.0628180
$$746$$ 0 0
$$747$$ −3257.41 −0.159548
$$748$$ 0 0
$$749$$ 7974.77 0.389041
$$750$$ 0 0
$$751$$ 14121.9 0.686171 0.343085 0.939304i $$-0.388528\pi$$
0.343085 + 0.939304i $$0.388528\pi$$
$$752$$ 0 0
$$753$$ −5090.98 −0.246382
$$754$$ 0 0
$$755$$ −648.074 −0.0312395
$$756$$ 0 0
$$757$$ 17006.3 0.816516 0.408258 0.912866i $$-0.366136\pi$$
0.408258 + 0.912866i $$0.366136\pi$$
$$758$$ 0 0
$$759$$ 19794.3 0.946622
$$760$$ 0 0
$$761$$ 4603.38 0.219280 0.109640 0.993971i $$-0.465030\pi$$
0.109640 + 0.993971i $$0.465030\pi$$
$$762$$ 0 0
$$763$$ 13578.5 0.644264
$$764$$ 0 0
$$765$$ 260.061 0.0122909
$$766$$ 0 0
$$767$$ −13906.6 −0.654679
$$768$$ 0 0
$$769$$ −12459.1 −0.584248 −0.292124 0.956380i $$-0.594362\pi$$
−0.292124 + 0.956380i $$0.594362\pi$$
$$770$$ 0 0
$$771$$ 1148.71 0.0536571
$$772$$ 0 0
$$773$$ 27068.1 1.25947 0.629737 0.776808i $$-0.283163\pi$$
0.629737 + 0.776808i $$0.283163\pi$$
$$774$$ 0 0
$$775$$ 27527.2 1.27588
$$776$$ 0 0
$$777$$ −27862.6 −1.28644
$$778$$ 0 0
$$779$$ 10520.8 0.483884
$$780$$ 0 0
$$781$$ 21123.4 0.967804
$$782$$ 0 0
$$783$$ 281.475 0.0128468
$$784$$ 0 0
$$785$$ 1956.23 0.0889438
$$786$$ 0 0
$$787$$ −6986.86 −0.316461 −0.158230 0.987402i $$-0.550579\pi$$
−0.158230 + 0.987402i $$0.550579\pi$$
$$788$$ 0 0
$$789$$ −15006.1 −0.677097
$$790$$ 0 0
$$791$$ 11290.7 0.507523
$$792$$ 0 0
$$793$$ 14468.0 0.647885
$$794$$ 0 0
$$795$$ −216.550 −0.00966067
$$796$$ 0 0
$$797$$ −27271.5 −1.21205 −0.606027 0.795444i $$-0.707238\pi$$
−0.606027 + 0.795444i $$0.707238\pi$$
$$798$$ 0 0
$$799$$ 6089.52 0.269627
$$800$$ 0 0
$$801$$ −362.957 −0.0160105
$$802$$ 0 0
$$803$$ −32096.7 −1.41054
$$804$$ 0 0
$$805$$ −1525.66 −0.0667981
$$806$$ 0 0
$$807$$ 18352.4 0.800540
$$808$$ 0 0
$$809$$ −23785.9 −1.03371 −0.516853 0.856074i $$-0.672897\pi$$
−0.516853 + 0.856074i $$0.672897\pi$$
$$810$$ 0 0
$$811$$ −21703.5 −0.939718 −0.469859 0.882741i $$-0.655695\pi$$
−0.469859 + 0.882741i $$0.655695\pi$$
$$812$$ 0 0
$$813$$ 11868.4 0.511982
$$814$$ 0 0
$$815$$ −518.835 −0.0222994
$$816$$ 0 0
$$817$$ −6894.83 −0.295250
$$818$$ 0 0
$$819$$ 10829.7 0.462054
$$820$$ 0 0
$$821$$ −33240.4 −1.41303 −0.706515 0.707698i $$-0.749734\pi$$
−0.706515 + 0.707698i $$0.749734\pi$$
$$822$$ 0 0
$$823$$ 17227.5 0.729665 0.364832 0.931073i $$-0.381126\pi$$
0.364832 + 0.931073i $$0.381126\pi$$
$$824$$ 0 0
$$825$$ 22530.8 0.950812
$$826$$ 0 0
$$827$$ −21678.4 −0.911528 −0.455764 0.890101i $$-0.650634\pi$$
−0.455764 + 0.890101i $$0.650634\pi$$
$$828$$ 0 0
$$829$$ 34269.8 1.43575 0.717877 0.696170i $$-0.245114\pi$$
0.717877 + 0.696170i $$0.245114\pi$$
$$830$$ 0 0
$$831$$ 14526.5 0.606401
$$832$$ 0 0
$$833$$ 8220.55 0.341927
$$834$$ 0 0
$$835$$ 1611.39 0.0667838
$$836$$ 0 0
$$837$$ 5963.77 0.246282
$$838$$ 0 0
$$839$$ −33444.1 −1.37618 −0.688092 0.725623i $$-0.741552\pi$$
−0.688092 + 0.725623i $$0.741552\pi$$
$$840$$ 0 0
$$841$$ −24280.3 −0.995544
$$842$$ 0 0
$$843$$ 5636.04 0.230267
$$844$$ 0 0
$$845$$ 368.860 0.0150168
$$846$$ 0 0
$$847$$ 52325.8 2.12271
$$848$$ 0 0
$$849$$ 17174.6 0.694265
$$850$$ 0 0
$$851$$ −44709.3 −1.80096
$$852$$ 0 0
$$853$$ −37037.3 −1.48667 −0.743337 0.668917i $$-0.766758\pi$$
−0.743337 + 0.668917i $$0.766758\pi$$
$$854$$ 0 0
$$855$$ 160.810 0.00643227
$$856$$ 0 0
$$857$$ −35794.2 −1.42673 −0.713365 0.700793i $$-0.752830\pi$$
−0.713365 + 0.700793i $$0.752830\pi$$
$$858$$ 0 0
$$859$$ 20582.1 0.817522 0.408761 0.912641i $$-0.365961\pi$$
0.408761 + 0.912641i $$0.365961\pi$$
$$860$$ 0 0
$$861$$ −24614.3 −0.974278
$$862$$ 0 0
$$863$$ −25677.7 −1.01284 −0.506420 0.862287i $$-0.669031\pi$$
−0.506420 + 0.862287i $$0.669031\pi$$
$$864$$ 0 0
$$865$$ 263.348 0.0103516
$$866$$ 0 0
$$867$$ 8065.60 0.315942
$$868$$ 0 0
$$869$$ −10037.5 −0.391827
$$870$$ 0 0
$$871$$ 4730.22 0.184015
$$872$$ 0 0
$$873$$ −5534.29 −0.214556
$$874$$ 0 0
$$875$$ −3478.38 −0.134389
$$876$$ 0 0
$$877$$ 32783.0 1.26226 0.631131 0.775676i $$-0.282591\pi$$
0.631131 + 0.775676i $$0.282591\pi$$
$$878$$ 0 0
$$879$$ −15266.3 −0.585800
$$880$$ 0 0
$$881$$ 26615.7 1.01783 0.508914 0.860818i $$-0.330047\pi$$
0.508914 + 0.860818i $$0.330047\pi$$
$$882$$ 0 0
$$883$$ 19203.4 0.731875 0.365937 0.930639i $$-0.380748\pi$$
0.365937 + 0.930639i $$0.380748\pi$$
$$884$$ 0 0
$$885$$ 483.122 0.0183503
$$886$$ 0 0
$$887$$ −23198.0 −0.878144 −0.439072 0.898452i $$-0.644693\pi$$
−0.439072 + 0.898452i $$0.644693\pi$$
$$888$$ 0 0
$$889$$ 40936.0 1.54438
$$890$$ 0 0
$$891$$ 4881.30 0.183535
$$892$$ 0 0
$$893$$ 3765.48 0.141105
$$894$$ 0 0
$$895$$ −928.979 −0.0346953
$$896$$ 0 0
$$897$$ 17377.8 0.646854
$$898$$ 0 0
$$899$$ 2302.68 0.0854266
$$900$$ 0 0
$$901$$ 5556.86 0.205467
$$902$$ 0 0
$$903$$ 16131.1 0.594472
$$904$$ 0 0
$$905$$ 2070.92 0.0760659
$$906$$ 0 0
$$907$$ 16151.3 0.591285 0.295643 0.955299i $$-0.404466\pi$$
0.295643 + 0.955299i $$0.404466\pi$$
$$908$$ 0 0
$$909$$ 14984.9 0.546776
$$910$$ 0 0
$$911$$ −3487.34 −0.126829 −0.0634143 0.997987i $$-0.520199\pi$$
−0.0634143 + 0.997987i $$0.520199\pi$$
$$912$$ 0 0
$$913$$ −21811.2 −0.790632
$$914$$ 0 0
$$915$$ −502.624 −0.0181598
$$916$$ 0 0
$$917$$ 25506.3 0.918532
$$918$$ 0 0
$$919$$ −17055.5 −0.612198 −0.306099 0.952000i $$-0.599024\pi$$
−0.306099 + 0.952000i $$0.599024\pi$$
$$920$$ 0 0
$$921$$ 21657.6 0.774857
$$922$$ 0 0
$$923$$ 18544.7 0.661329
$$924$$ 0 0
$$925$$ −50890.2 −1.80893
$$926$$ 0 0
$$927$$ −3571.73 −0.126549
$$928$$ 0 0
$$929$$ −55339.3 −1.95438 −0.977192 0.212359i $$-0.931885\pi$$
−0.977192 + 0.212359i $$0.931885\pi$$
$$930$$ 0 0
$$931$$ 5083.22 0.178943
$$932$$ 0 0
$$933$$ −4611.17 −0.161804
$$934$$ 0 0
$$935$$ 1741.34 0.0609069
$$936$$ 0 0
$$937$$ 20457.6 0.713255 0.356627 0.934247i $$-0.383927\pi$$
0.356627 + 0.934247i $$0.383927\pi$$
$$938$$ 0 0
$$939$$ −6602.78 −0.229471
$$940$$ 0 0
$$941$$ −55891.0 −1.93623 −0.968116 0.250502i $$-0.919404\pi$$
−0.968116 + 0.250502i $$0.919404\pi$$
$$942$$ 0 0
$$943$$ −39497.0 −1.36395
$$944$$ 0 0
$$945$$ −376.230 −0.0129511
$$946$$ 0 0
$$947$$ 54727.0 1.87792 0.938960 0.344027i $$-0.111791\pi$$
0.938960 + 0.344027i $$0.111791\pi$$
$$948$$ 0 0
$$949$$ −28178.4 −0.963865
$$950$$ 0 0
$$951$$ 8521.23 0.290557
$$952$$ 0 0
$$953$$ −29958.8 −1.01832 −0.509160 0.860672i $$-0.670044\pi$$
−0.509160 + 0.860672i $$0.670044\pi$$
$$954$$ 0 0
$$955$$ −1715.27 −0.0581204
$$956$$ 0 0
$$957$$ 1884.72 0.0636619
$$958$$ 0 0
$$959$$ −55713.0 −1.87598
$$960$$ 0 0
$$961$$ 18997.2 0.637682
$$962$$ 0 0
$$963$$ 3155.67 0.105597
$$964$$ 0 0
$$965$$ 382.365 0.0127552
$$966$$ 0 0
$$967$$ −13498.5 −0.448896 −0.224448 0.974486i $$-0.572058\pi$$
−0.224448 + 0.974486i $$0.572058\pi$$
$$968$$ 0 0
$$969$$ −4126.53 −0.136804
$$970$$ 0 0
$$971$$ 9652.36 0.319010 0.159505 0.987197i $$-0.449010\pi$$
0.159505 + 0.987197i $$0.449010\pi$$
$$972$$ 0 0
$$973$$ −55895.1 −1.84164
$$974$$ 0 0
$$975$$ 19780.2 0.649717
$$976$$ 0 0
$$977$$ 12169.8 0.398511 0.199256 0.979948i $$-0.436148\pi$$
0.199256 + 0.979948i $$0.436148\pi$$
$$978$$ 0 0
$$979$$ −2430.32 −0.0793394
$$980$$ 0 0
$$981$$ 5373.08 0.174872
$$982$$ 0 0
$$983$$ −47545.2 −1.54268 −0.771341 0.636423i $$-0.780413\pi$$
−0.771341 + 0.636423i $$0.780413\pi$$
$$984$$ 0 0
$$985$$ −2928.06 −0.0947165
$$986$$ 0 0
$$987$$ −8809.69 −0.284109
$$988$$ 0 0
$$989$$ 25884.5 0.832234
$$990$$ 0 0
$$991$$ −892.350 −0.0286039 −0.0143019 0.999898i $$-0.504553\pi$$
−0.0143019 + 0.999898i $$0.504553\pi$$
$$992$$ 0 0
$$993$$ −6355.55 −0.203109
$$994$$ 0 0
$$995$$ −1602.67 −0.0510634
$$996$$ 0 0
$$997$$ 4458.57 0.141629 0.0708146 0.997489i $$-0.477440\pi$$
0.0708146 + 0.997489i $$0.477440\pi$$
$$998$$ 0 0
$$999$$ −11025.4 −0.349177
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 768.4.a.r.1.2 3
3.2 odd 2 2304.4.a.bt.1.2 3
4.3 odd 2 768.4.a.t.1.2 3
8.3 odd 2 768.4.a.q.1.2 3
8.5 even 2 768.4.a.s.1.2 3
12.11 even 2 2304.4.a.bu.1.2 3
16.3 odd 4 96.4.d.a.49.5 6
16.5 even 4 24.4.d.a.13.5 6
16.11 odd 4 96.4.d.a.49.2 6
16.13 even 4 24.4.d.a.13.6 yes 6
24.5 odd 2 2304.4.a.bv.1.2 3
24.11 even 2 2304.4.a.bw.1.2 3
48.5 odd 4 72.4.d.d.37.2 6
48.11 even 4 288.4.d.d.145.3 6
48.29 odd 4 72.4.d.d.37.1 6
48.35 even 4 288.4.d.d.145.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.5 6 16.5 even 4
24.4.d.a.13.6 yes 6 16.13 even 4
72.4.d.d.37.1 6 48.29 odd 4
72.4.d.d.37.2 6 48.5 odd 4
96.4.d.a.49.2 6 16.11 odd 4
96.4.d.a.49.5 6 16.3 odd 4
288.4.d.d.145.3 6 48.11 even 4
288.4.d.d.145.4 6 48.35 even 4
768.4.a.q.1.2 3 8.3 odd 2
768.4.a.r.1.2 3 1.1 even 1 trivial
768.4.a.s.1.2 3 8.5 even 2
768.4.a.t.1.2 3 4.3 odd 2
2304.4.a.bt.1.2 3 3.2 odd 2
2304.4.a.bu.1.2 3 12.11 even 2
2304.4.a.bv.1.2 3 24.5 odd 2
2304.4.a.bw.1.2 3 24.11 even 2