# Properties

 Label 768.4.a.q.1.3 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ 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: $$1$$ 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 1.3 Root $$-1.28282$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +9.15486 q^{5} +27.4175 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +9.15486 q^{5} +27.4175 q^{7} +9.00000 q^{9} -20.5252 q^{11} -32.0471 q^{13} -27.4646 q^{15} -111.764 q^{17} -129.764 q^{19} -82.2524 q^{21} +9.16510 q^{23} -41.1885 q^{25} -27.0000 q^{27} -41.0606 q^{29} +187.606 q^{31} +61.5755 q^{33} +251.003 q^{35} +114.127 q^{37} +96.1414 q^{39} -282.915 q^{41} +89.3870 q^{43} +82.3937 q^{45} +54.6464 q^{47} +408.717 q^{49} +335.292 q^{51} -726.878 q^{53} -187.905 q^{55} +389.292 q^{57} +216.579 q^{59} -754.222 q^{61} +246.757 q^{63} -293.387 q^{65} -379.433 q^{67} -27.4953 q^{69} +302.080 q^{71} +504.396 q^{73} +123.566 q^{75} -562.748 q^{77} -301.780 q^{79} +81.0000 q^{81} -599.003 q^{83} -1023.18 q^{85} +123.182 q^{87} +277.528 q^{89} -878.651 q^{91} -562.818 q^{93} -1187.97 q^{95} -765.905 q^{97} -184.727 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10})$$ 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
$$5$$ 9.15486 0.818836 0.409418 0.912347i $$-0.365732\pi$$
0.409418 + 0.912347i $$0.365732\pi$$
$$6$$ 0 0
$$7$$ 27.4175 1.48040 0.740202 0.672385i $$-0.234730\pi$$
0.740202 + 0.672385i $$0.234730\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −20.5252 −0.562598 −0.281299 0.959620i $$-0.590765\pi$$
−0.281299 + 0.959620i $$0.590765\pi$$
$$12$$ 0 0
$$13$$ −32.0471 −0.683713 −0.341857 0.939752i $$-0.611056\pi$$
−0.341857 + 0.939752i $$0.611056\pi$$
$$14$$ 0 0
$$15$$ −27.4646 −0.472755
$$16$$ 0 0
$$17$$ −111.764 −1.59452 −0.797258 0.603639i $$-0.793717\pi$$
−0.797258 + 0.603639i $$0.793717\pi$$
$$18$$ 0 0
$$19$$ −129.764 −1.56684 −0.783419 0.621494i $$-0.786526\pi$$
−0.783419 + 0.621494i $$0.786526\pi$$
$$20$$ 0 0
$$21$$ −82.2524 −0.854711
$$22$$ 0 0
$$23$$ 9.16510 0.0830893 0.0415447 0.999137i $$-0.486772\pi$$
0.0415447 + 0.999137i $$0.486772\pi$$
$$24$$ 0 0
$$25$$ −41.1885 −0.329508
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −41.0606 −0.262923 −0.131461 0.991321i $$-0.541967\pi$$
−0.131461 + 0.991321i $$0.541967\pi$$
$$30$$ 0 0
$$31$$ 187.606 1.08694 0.543468 0.839430i $$-0.317111\pi$$
0.543468 + 0.839430i $$0.317111\pi$$
$$32$$ 0 0
$$33$$ 61.5755 0.324816
$$34$$ 0 0
$$35$$ 251.003 1.21221
$$36$$ 0 0
$$37$$ 114.127 0.507093 0.253546 0.967323i $$-0.418403\pi$$
0.253546 + 0.967323i $$0.418403\pi$$
$$38$$ 0 0
$$39$$ 96.1414 0.394742
$$40$$ 0 0
$$41$$ −282.915 −1.07766 −0.538828 0.842416i $$-0.681133\pi$$
−0.538828 + 0.842416i $$0.681133\pi$$
$$42$$ 0 0
$$43$$ 89.3870 0.317009 0.158505 0.987358i $$-0.449333\pi$$
0.158505 + 0.987358i $$0.449333\pi$$
$$44$$ 0 0
$$45$$ 82.3937 0.272945
$$46$$ 0 0
$$47$$ 54.6464 0.169596 0.0847978 0.996398i $$-0.472976\pi$$
0.0847978 + 0.996398i $$0.472976\pi$$
$$48$$ 0 0
$$49$$ 408.717 1.19159
$$50$$ 0 0
$$51$$ 335.292 0.920594
$$52$$ 0 0
$$53$$ −726.878 −1.88386 −0.941928 0.335815i $$-0.890988\pi$$
−0.941928 + 0.335815i $$0.890988\pi$$
$$54$$ 0 0
$$55$$ −187.905 −0.460675
$$56$$ 0 0
$$57$$ 389.292 0.904614
$$58$$ 0 0
$$59$$ 216.579 0.477900 0.238950 0.971032i $$-0.423197\pi$$
0.238950 + 0.971032i $$0.423197\pi$$
$$60$$ 0 0
$$61$$ −754.222 −1.58309 −0.791543 0.611114i $$-0.790722\pi$$
−0.791543 + 0.611114i $$0.790722\pi$$
$$62$$ 0 0
$$63$$ 246.757 0.493468
$$64$$ 0 0
$$65$$ −293.387 −0.559849
$$66$$ 0 0
$$67$$ −379.433 −0.691868 −0.345934 0.938259i $$-0.612438\pi$$
−0.345934 + 0.938259i $$0.612438\pi$$
$$68$$ 0 0
$$69$$ −27.4953 −0.0479717
$$70$$ 0 0
$$71$$ 302.080 0.504933 0.252467 0.967606i $$-0.418758\pi$$
0.252467 + 0.967606i $$0.418758\pi$$
$$72$$ 0 0
$$73$$ 504.396 0.808700 0.404350 0.914604i $$-0.367498\pi$$
0.404350 + 0.914604i $$0.367498\pi$$
$$74$$ 0 0
$$75$$ 123.566 0.190242
$$76$$ 0 0
$$77$$ −562.748 −0.832872
$$78$$ 0 0
$$79$$ −301.780 −0.429784 −0.214892 0.976638i $$-0.568940\pi$$
−0.214892 + 0.976638i $$0.568940\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −599.003 −0.792158 −0.396079 0.918216i $$-0.629629\pi$$
−0.396079 + 0.918216i $$0.629629\pi$$
$$84$$ 0 0
$$85$$ −1023.18 −1.30565
$$86$$ 0 0
$$87$$ 123.182 0.151799
$$88$$ 0 0
$$89$$ 277.528 0.330538 0.165269 0.986248i $$-0.447151\pi$$
0.165269 + 0.986248i $$0.447151\pi$$
$$90$$ 0 0
$$91$$ −878.651 −1.01217
$$92$$ 0 0
$$93$$ −562.818 −0.627543
$$94$$ 0 0
$$95$$ −1187.97 −1.28298
$$96$$ 0 0
$$97$$ −765.905 −0.801710 −0.400855 0.916141i $$-0.631287\pi$$
−0.400855 + 0.916141i $$0.631287\pi$$
$$98$$ 0 0
$$99$$ −184.727 −0.187533
$$100$$ 0 0
$$101$$ 201.253 0.198272 0.0991360 0.995074i $$-0.468392\pi$$
0.0991360 + 0.995074i $$0.468392\pi$$
$$102$$ 0 0
$$103$$ 682.440 0.652843 0.326421 0.945224i $$-0.394157\pi$$
0.326421 + 0.945224i $$0.394157\pi$$
$$104$$ 0 0
$$105$$ −753.009 −0.699868
$$106$$ 0 0
$$107$$ −457.252 −0.413123 −0.206562 0.978434i $$-0.566227\pi$$
−0.206562 + 0.978434i $$0.566227\pi$$
$$108$$ 0 0
$$109$$ −625.812 −0.549926 −0.274963 0.961455i $$-0.588666\pi$$
−0.274963 + 0.961455i $$0.588666\pi$$
$$110$$ 0 0
$$111$$ −342.382 −0.292770
$$112$$ 0 0
$$113$$ 981.151 0.816805 0.408402 0.912802i $$-0.366086\pi$$
0.408402 + 0.912802i $$0.366086\pi$$
$$114$$ 0 0
$$115$$ 83.9052 0.0680365
$$116$$ 0 0
$$117$$ −288.424 −0.227904
$$118$$ 0 0
$$119$$ −3064.29 −2.36053
$$120$$ 0 0
$$121$$ −909.717 −0.683484
$$122$$ 0 0
$$123$$ 848.745 0.622185
$$124$$ 0 0
$$125$$ −1521.43 −1.08865
$$126$$ 0 0
$$127$$ 808.055 0.564593 0.282296 0.959327i $$-0.408904\pi$$
0.282296 + 0.959327i $$0.408904\pi$$
$$128$$ 0 0
$$129$$ −268.161 −0.183025
$$130$$ 0 0
$$131$$ 1110.85 0.740884 0.370442 0.928856i $$-0.379206\pi$$
0.370442 + 0.928856i $$0.379206\pi$$
$$132$$ 0 0
$$133$$ −3557.80 −2.31955
$$134$$ 0 0
$$135$$ −247.181 −0.157585
$$136$$ 0 0
$$137$$ 466.765 0.291084 0.145542 0.989352i $$-0.453507\pi$$
0.145542 + 0.989352i $$0.453507\pi$$
$$138$$ 0 0
$$139$$ −351.773 −0.214654 −0.107327 0.994224i $$-0.534229\pi$$
−0.107327 + 0.994224i $$0.534229\pi$$
$$140$$ 0 0
$$141$$ −163.939 −0.0979161
$$142$$ 0 0
$$143$$ 657.773 0.384656
$$144$$ 0 0
$$145$$ −375.904 −0.215291
$$146$$ 0 0
$$147$$ −1226.15 −0.687967
$$148$$ 0 0
$$149$$ −1290.49 −0.709540 −0.354770 0.934954i $$-0.615441\pi$$
−0.354770 + 0.934954i $$0.615441\pi$$
$$150$$ 0 0
$$151$$ 1175.51 0.633521 0.316761 0.948505i $$-0.397405\pi$$
0.316761 + 0.948505i $$0.397405\pi$$
$$152$$ 0 0
$$153$$ −1005.88 −0.531505
$$154$$ 0 0
$$155$$ 1717.51 0.890022
$$156$$ 0 0
$$157$$ 1092.09 0.555148 0.277574 0.960704i $$-0.410470\pi$$
0.277574 + 0.960704i $$0.410470\pi$$
$$158$$ 0 0
$$159$$ 2180.63 1.08764
$$160$$ 0 0
$$161$$ 251.284 0.123006
$$162$$ 0 0
$$163$$ 3626.97 1.74286 0.871430 0.490519i $$-0.163193\pi$$
0.871430 + 0.490519i $$0.163193\pi$$
$$164$$ 0 0
$$165$$ 563.716 0.265971
$$166$$ 0 0
$$167$$ 45.8012 0.0212228 0.0106114 0.999944i $$-0.496622\pi$$
0.0106114 + 0.999944i $$0.496622\pi$$
$$168$$ 0 0
$$169$$ −1169.98 −0.532536
$$170$$ 0 0
$$171$$ −1167.88 −0.522279
$$172$$ 0 0
$$173$$ −2455.02 −1.07891 −0.539455 0.842014i $$-0.681370\pi$$
−0.539455 + 0.842014i $$0.681370\pi$$
$$174$$ 0 0
$$175$$ −1129.28 −0.487805
$$176$$ 0 0
$$177$$ −649.736 −0.275916
$$178$$ 0 0
$$179$$ −1026.28 −0.428533 −0.214267 0.976775i $$-0.568736\pi$$
−0.214267 + 0.976775i $$0.568736\pi$$
$$180$$ 0 0
$$181$$ 3699.05 1.51905 0.759526 0.650477i $$-0.225431\pi$$
0.759526 + 0.650477i $$0.225431\pi$$
$$182$$ 0 0
$$183$$ 2262.66 0.913995
$$184$$ 0 0
$$185$$ 1044.82 0.415226
$$186$$ 0 0
$$187$$ 2293.98 0.897071
$$188$$ 0 0
$$189$$ −740.271 −0.284904
$$190$$ 0 0
$$191$$ −5108.93 −1.93544 −0.967721 0.252023i $$-0.918904\pi$$
−0.967721 + 0.252023i $$0.918904\pi$$
$$192$$ 0 0
$$193$$ −1414.13 −0.527417 −0.263709 0.964602i $$-0.584946\pi$$
−0.263709 + 0.964602i $$0.584946\pi$$
$$194$$ 0 0
$$195$$ 880.161 0.323229
$$196$$ 0 0
$$197$$ −2816.66 −1.01867 −0.509337 0.860567i $$-0.670109\pi$$
−0.509337 + 0.860567i $$0.670109\pi$$
$$198$$ 0 0
$$199$$ −948.556 −0.337896 −0.168948 0.985625i $$-0.554037\pi$$
−0.168948 + 0.985625i $$0.554037\pi$$
$$200$$ 0 0
$$201$$ 1138.30 0.399450
$$202$$ 0 0
$$203$$ −1125.78 −0.389232
$$204$$ 0 0
$$205$$ −2590.05 −0.882424
$$206$$ 0 0
$$207$$ 82.4859 0.0276964
$$208$$ 0 0
$$209$$ 2663.43 0.881499
$$210$$ 0 0
$$211$$ −4487.28 −1.46406 −0.732032 0.681271i $$-0.761428\pi$$
−0.732032 + 0.681271i $$0.761428\pi$$
$$212$$ 0 0
$$213$$ −906.239 −0.291523
$$214$$ 0 0
$$215$$ 818.326 0.259578
$$216$$ 0 0
$$217$$ 5143.68 1.60910
$$218$$ 0 0
$$219$$ −1513.19 −0.466903
$$220$$ 0 0
$$221$$ 3581.72 1.09019
$$222$$ 0 0
$$223$$ 4590.98 1.37863 0.689315 0.724462i $$-0.257912\pi$$
0.689315 + 0.724462i $$0.257912\pi$$
$$224$$ 0 0
$$225$$ −370.697 −0.109836
$$226$$ 0 0
$$227$$ −2897.47 −0.847189 −0.423594 0.905852i $$-0.639232\pi$$
−0.423594 + 0.905852i $$0.639232\pi$$
$$228$$ 0 0
$$229$$ −34.6293 −0.00999288 −0.00499644 0.999988i $$-0.501590\pi$$
−0.00499644 + 0.999988i $$0.501590\pi$$
$$230$$ 0 0
$$231$$ 1688.24 0.480859
$$232$$ 0 0
$$233$$ 1054.02 0.296355 0.148178 0.988961i $$-0.452659\pi$$
0.148178 + 0.988961i $$0.452659\pi$$
$$234$$ 0 0
$$235$$ 500.280 0.138871
$$236$$ 0 0
$$237$$ 905.341 0.248136
$$238$$ 0 0
$$239$$ 654.700 0.177192 0.0885962 0.996068i $$-0.471762\pi$$
0.0885962 + 0.996068i $$0.471762\pi$$
$$240$$ 0 0
$$241$$ 3194.00 0.853707 0.426854 0.904321i $$-0.359622\pi$$
0.426854 + 0.904321i $$0.359622\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 3741.74 0.975719
$$246$$ 0 0
$$247$$ 4158.57 1.07127
$$248$$ 0 0
$$249$$ 1797.01 0.457353
$$250$$ 0 0
$$251$$ −5042.90 −1.26815 −0.634074 0.773273i $$-0.718618\pi$$
−0.634074 + 0.773273i $$0.718618\pi$$
$$252$$ 0 0
$$253$$ −188.115 −0.0467459
$$254$$ 0 0
$$255$$ 3069.55 0.753815
$$256$$ 0 0
$$257$$ −5166.64 −1.25403 −0.627016 0.779007i $$-0.715724\pi$$
−0.627016 + 0.779007i $$0.715724\pi$$
$$258$$ 0 0
$$259$$ 3129.08 0.750702
$$260$$ 0 0
$$261$$ −369.545 −0.0876409
$$262$$ 0 0
$$263$$ −7366.11 −1.72705 −0.863524 0.504308i $$-0.831748\pi$$
−0.863524 + 0.504308i $$0.831748\pi$$
$$264$$ 0 0
$$265$$ −6654.47 −1.54257
$$266$$ 0 0
$$267$$ −832.584 −0.190836
$$268$$ 0 0
$$269$$ 7877.80 1.78557 0.892784 0.450484i $$-0.148749\pi$$
0.892784 + 0.450484i $$0.148749\pi$$
$$270$$ 0 0
$$271$$ 5399.92 1.21041 0.605206 0.796069i $$-0.293091\pi$$
0.605206 + 0.796069i $$0.293091\pi$$
$$272$$ 0 0
$$273$$ 2635.95 0.584378
$$274$$ 0 0
$$275$$ 845.402 0.185381
$$276$$ 0 0
$$277$$ 4416.07 0.957892 0.478946 0.877844i $$-0.341019\pi$$
0.478946 + 0.877844i $$0.341019\pi$$
$$278$$ 0 0
$$279$$ 1688.45 0.362312
$$280$$ 0 0
$$281$$ −8068.94 −1.71300 −0.856499 0.516148i $$-0.827365\pi$$
−0.856499 + 0.516148i $$0.827365\pi$$
$$282$$ 0 0
$$283$$ 5241.13 1.10089 0.550447 0.834870i $$-0.314457\pi$$
0.550447 + 0.834870i $$0.314457\pi$$
$$284$$ 0 0
$$285$$ 3563.92 0.740730
$$286$$ 0 0
$$287$$ −7756.81 −1.59537
$$288$$ 0 0
$$289$$ 7578.21 1.54248
$$290$$ 0 0
$$291$$ 2297.72 0.462868
$$292$$ 0 0
$$293$$ 6372.75 1.27065 0.635324 0.772246i $$-0.280867\pi$$
0.635324 + 0.772246i $$0.280867\pi$$
$$294$$ 0 0
$$295$$ 1982.75 0.391322
$$296$$ 0 0
$$297$$ 554.180 0.108272
$$298$$ 0 0
$$299$$ −293.715 −0.0568093
$$300$$ 0 0
$$301$$ 2450.76 0.469301
$$302$$ 0 0
$$303$$ −603.760 −0.114472
$$304$$ 0 0
$$305$$ −6904.79 −1.29629
$$306$$ 0 0
$$307$$ −3810.22 −0.708342 −0.354171 0.935181i $$-0.615237\pi$$
−0.354171 + 0.935181i $$0.615237\pi$$
$$308$$ 0 0
$$309$$ −2047.32 −0.376919
$$310$$ 0 0
$$311$$ −8106.73 −1.47810 −0.739052 0.673648i $$-0.764726\pi$$
−0.739052 + 0.673648i $$0.764726\pi$$
$$312$$ 0 0
$$313$$ 559.983 0.101125 0.0505625 0.998721i $$-0.483899\pi$$
0.0505625 + 0.998721i $$0.483899\pi$$
$$314$$ 0 0
$$315$$ 2259.03 0.404069
$$316$$ 0 0
$$317$$ −5828.98 −1.03277 −0.516385 0.856357i $$-0.672723\pi$$
−0.516385 + 0.856357i $$0.672723\pi$$
$$318$$ 0 0
$$319$$ 842.776 0.147920
$$320$$ 0 0
$$321$$ 1371.76 0.238517
$$322$$ 0 0
$$323$$ 14503.0 2.49835
$$324$$ 0 0
$$325$$ 1319.97 0.225289
$$326$$ 0 0
$$327$$ 1877.44 0.317500
$$328$$ 0 0
$$329$$ 1498.26 0.251070
$$330$$ 0 0
$$331$$ −2847.98 −0.472928 −0.236464 0.971640i $$-0.575989\pi$$
−0.236464 + 0.971640i $$0.575989\pi$$
$$332$$ 0 0
$$333$$ 1027.15 0.169031
$$334$$ 0 0
$$335$$ −3473.66 −0.566526
$$336$$ 0 0
$$337$$ −10127.8 −1.63707 −0.818537 0.574454i $$-0.805214\pi$$
−0.818537 + 0.574454i $$0.805214\pi$$
$$338$$ 0 0
$$339$$ −2943.45 −0.471582
$$340$$ 0 0
$$341$$ −3850.65 −0.611508
$$342$$ 0 0
$$343$$ 1801.78 0.283636
$$344$$ 0 0
$$345$$ −251.716 −0.0392809
$$346$$ 0 0
$$347$$ 10148.2 1.56999 0.784993 0.619505i $$-0.212667\pi$$
0.784993 + 0.619505i $$0.212667\pi$$
$$348$$ 0 0
$$349$$ 9515.96 1.45954 0.729768 0.683695i $$-0.239628\pi$$
0.729768 + 0.683695i $$0.239628\pi$$
$$350$$ 0 0
$$351$$ 865.273 0.131581
$$352$$ 0 0
$$353$$ 2813.56 0.424223 0.212111 0.977245i $$-0.431966\pi$$
0.212111 + 0.977245i $$0.431966\pi$$
$$354$$ 0 0
$$355$$ 2765.50 0.413457
$$356$$ 0 0
$$357$$ 9192.86 1.36285
$$358$$ 0 0
$$359$$ −2427.25 −0.356839 −0.178419 0.983955i $$-0.557098\pi$$
−0.178419 + 0.983955i $$0.557098\pi$$
$$360$$ 0 0
$$361$$ 9979.71 1.45498
$$362$$ 0 0
$$363$$ 2729.15 0.394610
$$364$$ 0 0
$$365$$ 4617.67 0.662192
$$366$$ 0 0
$$367$$ −5021.46 −0.714219 −0.357109 0.934063i $$-0.616238\pi$$
−0.357109 + 0.934063i $$0.616238\pi$$
$$368$$ 0 0
$$369$$ −2546.24 −0.359219
$$370$$ 0 0
$$371$$ −19929.1 −2.78887
$$372$$ 0 0
$$373$$ 3182.40 0.441765 0.220882 0.975300i $$-0.429106\pi$$
0.220882 + 0.975300i $$0.429106\pi$$
$$374$$ 0 0
$$375$$ 4564.30 0.628532
$$376$$ 0 0
$$377$$ 1315.87 0.179764
$$378$$ 0 0
$$379$$ −5868.93 −0.795426 −0.397713 0.917510i $$-0.630196\pi$$
−0.397713 + 0.917510i $$0.630196\pi$$
$$380$$ 0 0
$$381$$ −2424.16 −0.325968
$$382$$ 0 0
$$383$$ −7350.18 −0.980618 −0.490309 0.871549i $$-0.663116\pi$$
−0.490309 + 0.871549i $$0.663116\pi$$
$$384$$ 0 0
$$385$$ −5151.88 −0.681985
$$386$$ 0 0
$$387$$ 804.483 0.105670
$$388$$ 0 0
$$389$$ 13009.1 1.69560 0.847800 0.530317i $$-0.177927\pi$$
0.847800 + 0.530317i $$0.177927\pi$$
$$390$$ 0 0
$$391$$ −1024.33 −0.132487
$$392$$ 0 0
$$393$$ −3332.56 −0.427750
$$394$$ 0 0
$$395$$ −2762.76 −0.351923
$$396$$ 0 0
$$397$$ 4877.88 0.616659 0.308330 0.951280i $$-0.400230\pi$$
0.308330 + 0.951280i $$0.400230\pi$$
$$398$$ 0 0
$$399$$ 10673.4 1.33919
$$400$$ 0 0
$$401$$ 5552.33 0.691446 0.345723 0.938337i $$-0.387634\pi$$
0.345723 + 0.938337i $$0.387634\pi$$
$$402$$ 0 0
$$403$$ −6012.23 −0.743153
$$404$$ 0 0
$$405$$ 741.544 0.0909817
$$406$$ 0 0
$$407$$ −2342.49 −0.285289
$$408$$ 0 0
$$409$$ −6989.27 −0.844981 −0.422491 0.906367i $$-0.638844\pi$$
−0.422491 + 0.906367i $$0.638844\pi$$
$$410$$ 0 0
$$411$$ −1400.30 −0.168057
$$412$$ 0 0
$$413$$ 5938.03 0.707485
$$414$$ 0 0
$$415$$ −5483.79 −0.648647
$$416$$ 0 0
$$417$$ 1055.32 0.123931
$$418$$ 0 0
$$419$$ 10461.0 1.21970 0.609849 0.792518i $$-0.291230\pi$$
0.609849 + 0.792518i $$0.291230\pi$$
$$420$$ 0 0
$$421$$ 4648.55 0.538139 0.269070 0.963121i $$-0.413284\pi$$
0.269070 + 0.963121i $$0.413284\pi$$
$$422$$ 0 0
$$423$$ 491.817 0.0565319
$$424$$ 0 0
$$425$$ 4603.40 0.525406
$$426$$ 0 0
$$427$$ −20678.8 −2.34360
$$428$$ 0 0
$$429$$ −1973.32 −0.222081
$$430$$ 0 0
$$431$$ −12490.7 −1.39595 −0.697975 0.716122i $$-0.745915\pi$$
−0.697975 + 0.716122i $$0.745915\pi$$
$$432$$ 0 0
$$433$$ 9446.37 1.04842 0.524208 0.851590i $$-0.324362\pi$$
0.524208 + 0.851590i $$0.324362\pi$$
$$434$$ 0 0
$$435$$ 1127.71 0.124298
$$436$$ 0 0
$$437$$ −1189.30 −0.130188
$$438$$ 0 0
$$439$$ 2793.60 0.303716 0.151858 0.988402i $$-0.451474\pi$$
0.151858 + 0.988402i $$0.451474\pi$$
$$440$$ 0 0
$$441$$ 3678.45 0.397198
$$442$$ 0 0
$$443$$ 7601.37 0.815241 0.407621 0.913151i $$-0.366359\pi$$
0.407621 + 0.913151i $$0.366359\pi$$
$$444$$ 0 0
$$445$$ 2540.73 0.270657
$$446$$ 0 0
$$447$$ 3871.48 0.409653
$$448$$ 0 0
$$449$$ 10708.8 1.12557 0.562785 0.826603i $$-0.309730\pi$$
0.562785 + 0.826603i $$0.309730\pi$$
$$450$$ 0 0
$$451$$ 5806.89 0.606287
$$452$$ 0 0
$$453$$ −3526.53 −0.365764
$$454$$ 0 0
$$455$$ −8043.92 −0.828802
$$456$$ 0 0
$$457$$ −233.840 −0.0239356 −0.0119678 0.999928i $$-0.503810\pi$$
−0.0119678 + 0.999928i $$0.503810\pi$$
$$458$$ 0 0
$$459$$ 3017.63 0.306865
$$460$$ 0 0
$$461$$ 981.307 0.0991410 0.0495705 0.998771i $$-0.484215\pi$$
0.0495705 + 0.998771i $$0.484215\pi$$
$$462$$ 0 0
$$463$$ 14082.7 1.41356 0.706782 0.707431i $$-0.250146\pi$$
0.706782 + 0.707431i $$0.250146\pi$$
$$464$$ 0 0
$$465$$ −5152.52 −0.513855
$$466$$ 0 0
$$467$$ −9286.49 −0.920188 −0.460094 0.887870i $$-0.652184\pi$$
−0.460094 + 0.887870i $$0.652184\pi$$
$$468$$ 0 0
$$469$$ −10403.1 −1.02424
$$470$$ 0 0
$$471$$ −3276.27 −0.320515
$$472$$ 0 0
$$473$$ −1834.68 −0.178349
$$474$$ 0 0
$$475$$ 5344.79 0.516286
$$476$$ 0 0
$$477$$ −6541.90 −0.627952
$$478$$ 0 0
$$479$$ 19409.3 1.85143 0.925715 0.378222i $$-0.123464\pi$$
0.925715 + 0.378222i $$0.123464\pi$$
$$480$$ 0 0
$$481$$ −3657.46 −0.346706
$$482$$ 0 0
$$483$$ −753.851 −0.0710174
$$484$$ 0 0
$$485$$ −7011.76 −0.656469
$$486$$ 0 0
$$487$$ 12124.8 1.12818 0.564091 0.825712i $$-0.309227\pi$$
0.564091 + 0.825712i $$0.309227\pi$$
$$488$$ 0 0
$$489$$ −10880.9 −1.00624
$$490$$ 0 0
$$491$$ −5100.69 −0.468820 −0.234410 0.972138i $$-0.575316\pi$$
−0.234410 + 0.972138i $$0.575316\pi$$
$$492$$ 0 0
$$493$$ 4589.10 0.419235
$$494$$ 0 0
$$495$$ −1691.15 −0.153558
$$496$$ 0 0
$$497$$ 8282.26 0.747505
$$498$$ 0 0
$$499$$ 85.2797 0.00765058 0.00382529 0.999993i $$-0.498782\pi$$
0.00382529 + 0.999993i $$0.498782\pi$$
$$500$$ 0 0
$$501$$ −137.404 −0.0122530
$$502$$ 0 0
$$503$$ −12287.2 −1.08918 −0.544592 0.838701i $$-0.683316\pi$$
−0.544592 + 0.838701i $$0.683316\pi$$
$$504$$ 0 0
$$505$$ 1842.45 0.162352
$$506$$ 0 0
$$507$$ 3509.94 0.307460
$$508$$ 0 0
$$509$$ −450.441 −0.0392248 −0.0196124 0.999808i $$-0.506243\pi$$
−0.0196124 + 0.999808i $$0.506243\pi$$
$$510$$ 0 0
$$511$$ 13829.2 1.19720
$$512$$ 0 0
$$513$$ 3503.63 0.301538
$$514$$ 0 0
$$515$$ 6247.64 0.534571
$$516$$ 0 0
$$517$$ −1121.63 −0.0954141
$$518$$ 0 0
$$519$$ 7365.05 0.622909
$$520$$ 0 0
$$521$$ 15088.1 1.26876 0.634378 0.773023i $$-0.281256\pi$$
0.634378 + 0.773023i $$0.281256\pi$$
$$522$$ 0 0
$$523$$ 17719.4 1.48149 0.740743 0.671789i $$-0.234474\pi$$
0.740743 + 0.671789i $$0.234474\pi$$
$$524$$ 0 0
$$525$$ 3387.85 0.281634
$$526$$ 0 0
$$527$$ −20967.6 −1.73314
$$528$$ 0 0
$$529$$ −12083.0 −0.993096
$$530$$ 0 0
$$531$$ 1949.21 0.159300
$$532$$ 0 0
$$533$$ 9066.62 0.736808
$$534$$ 0 0
$$535$$ −4186.08 −0.338280
$$536$$ 0 0
$$537$$ 3078.83 0.247414
$$538$$ 0 0
$$539$$ −8388.98 −0.670388
$$540$$ 0 0
$$541$$ −12244.5 −0.973074 −0.486537 0.873660i $$-0.661740\pi$$
−0.486537 + 0.873660i $$0.661740\pi$$
$$542$$ 0 0
$$543$$ −11097.2 −0.877025
$$544$$ 0 0
$$545$$ −5729.22 −0.450299
$$546$$ 0 0
$$547$$ 7822.46 0.611452 0.305726 0.952119i $$-0.401101\pi$$
0.305726 + 0.952119i $$0.401101\pi$$
$$548$$ 0 0
$$549$$ −6787.99 −0.527695
$$550$$ 0 0
$$551$$ 5328.19 0.411957
$$552$$ 0 0
$$553$$ −8274.05 −0.636254
$$554$$ 0 0
$$555$$ −3134.46 −0.239731
$$556$$ 0 0
$$557$$ −16555.5 −1.25938 −0.629692 0.776845i $$-0.716819\pi$$
−0.629692 + 0.776845i $$0.716819\pi$$
$$558$$ 0 0
$$559$$ −2864.60 −0.216743
$$560$$ 0 0
$$561$$ −6881.93 −0.517924
$$562$$ 0 0
$$563$$ 12580.7 0.941766 0.470883 0.882196i $$-0.343935\pi$$
0.470883 + 0.882196i $$0.343935\pi$$
$$564$$ 0 0
$$565$$ 8982.30 0.668829
$$566$$ 0 0
$$567$$ 2220.81 0.164489
$$568$$ 0 0
$$569$$ −2657.93 −0.195828 −0.0979141 0.995195i $$-0.531217\pi$$
−0.0979141 + 0.995195i $$0.531217\pi$$
$$570$$ 0 0
$$571$$ 17669.0 1.29496 0.647481 0.762081i $$-0.275822\pi$$
0.647481 + 0.762081i $$0.275822\pi$$
$$572$$ 0 0
$$573$$ 15326.8 1.11743
$$574$$ 0 0
$$575$$ −377.497 −0.0273786
$$576$$ 0 0
$$577$$ 14617.7 1.05467 0.527334 0.849658i $$-0.323192\pi$$
0.527334 + 0.849658i $$0.323192\pi$$
$$578$$ 0 0
$$579$$ 4242.40 0.304505
$$580$$ 0 0
$$581$$ −16423.1 −1.17271
$$582$$ 0 0
$$583$$ 14919.3 1.05985
$$584$$ 0 0
$$585$$ −2640.48 −0.186616
$$586$$ 0 0
$$587$$ −4096.53 −0.288044 −0.144022 0.989574i $$-0.546004\pi$$
−0.144022 + 0.989574i $$0.546004\pi$$
$$588$$ 0 0
$$589$$ −24344.5 −1.70305
$$590$$ 0 0
$$591$$ 8449.99 0.588132
$$592$$ 0 0
$$593$$ −21988.3 −1.52269 −0.761343 0.648349i $$-0.775460\pi$$
−0.761343 + 0.648349i $$0.775460\pi$$
$$594$$ 0 0
$$595$$ −28053.1 −1.93288
$$596$$ 0 0
$$597$$ 2845.67 0.195084
$$598$$ 0 0
$$599$$ 20767.7 1.41660 0.708302 0.705909i $$-0.249461\pi$$
0.708302 + 0.705909i $$0.249461\pi$$
$$600$$ 0 0
$$601$$ −5382.61 −0.365326 −0.182663 0.983176i $$-0.558472\pi$$
−0.182663 + 0.983176i $$0.558472\pi$$
$$602$$ 0 0
$$603$$ −3414.90 −0.230623
$$604$$ 0 0
$$605$$ −8328.33 −0.559661
$$606$$ 0 0
$$607$$ −11165.4 −0.746607 −0.373304 0.927709i $$-0.621775\pi$$
−0.373304 + 0.927709i $$0.621775\pi$$
$$608$$ 0 0
$$609$$ 3377.33 0.224723
$$610$$ 0 0
$$611$$ −1751.26 −0.115955
$$612$$ 0 0
$$613$$ −16413.5 −1.08146 −0.540731 0.841195i $$-0.681852\pi$$
−0.540731 + 0.841195i $$0.681852\pi$$
$$614$$ 0 0
$$615$$ 7770.15 0.509468
$$616$$ 0 0
$$617$$ 51.5882 0.00336607 0.00168303 0.999999i $$-0.499464\pi$$
0.00168303 + 0.999999i $$0.499464\pi$$
$$618$$ 0 0
$$619$$ −6349.55 −0.412294 −0.206147 0.978521i $$-0.566093\pi$$
−0.206147 + 0.978521i $$0.566093\pi$$
$$620$$ 0 0
$$621$$ −247.458 −0.0159906
$$622$$ 0 0
$$623$$ 7609.11 0.489330
$$624$$ 0 0
$$625$$ −8779.94 −0.561916
$$626$$ 0 0
$$627$$ −7990.29 −0.508934
$$628$$ 0 0
$$629$$ −12755.3 −0.808567
$$630$$ 0 0
$$631$$ −13379.1 −0.844078 −0.422039 0.906578i $$-0.638685\pi$$
−0.422039 + 0.906578i $$0.638685\pi$$
$$632$$ 0 0
$$633$$ 13461.9 0.845277
$$634$$ 0 0
$$635$$ 7397.63 0.462309
$$636$$ 0 0
$$637$$ −13098.2 −0.814709
$$638$$ 0 0
$$639$$ 2718.72 0.168311
$$640$$ 0 0
$$641$$ −20406.3 −1.25741 −0.628705 0.777644i $$-0.716415\pi$$
−0.628705 + 0.777644i $$0.716415\pi$$
$$642$$ 0 0
$$643$$ 19415.1 1.19076 0.595378 0.803446i $$-0.297002\pi$$
0.595378 + 0.803446i $$0.297002\pi$$
$$644$$ 0 0
$$645$$ −2454.98 −0.149868
$$646$$ 0 0
$$647$$ −8167.12 −0.496264 −0.248132 0.968726i $$-0.579817\pi$$
−0.248132 + 0.968726i $$0.579817\pi$$
$$648$$ 0 0
$$649$$ −4445.31 −0.268866
$$650$$ 0 0
$$651$$ −15431.0 −0.929017
$$652$$ 0 0
$$653$$ −7444.93 −0.446160 −0.223080 0.974800i $$-0.571611\pi$$
−0.223080 + 0.974800i $$0.571611\pi$$
$$654$$ 0 0
$$655$$ 10169.7 0.606662
$$656$$ 0 0
$$657$$ 4539.56 0.269567
$$658$$ 0 0
$$659$$ −23780.4 −1.40569 −0.702846 0.711342i $$-0.748088\pi$$
−0.702846 + 0.711342i $$0.748088\pi$$
$$660$$ 0 0
$$661$$ 2528.90 0.148809 0.0744046 0.997228i $$-0.476294\pi$$
0.0744046 + 0.997228i $$0.476294\pi$$
$$662$$ 0 0
$$663$$ −10745.2 −0.629423
$$664$$ 0 0
$$665$$ −32571.2 −1.89933
$$666$$ 0 0
$$667$$ −376.324 −0.0218461
$$668$$ 0 0
$$669$$ −13772.9 −0.795953
$$670$$ 0 0
$$671$$ 15480.5 0.890640
$$672$$ 0 0
$$673$$ 16733.7 0.958447 0.479224 0.877693i $$-0.340918\pi$$
0.479224 + 0.877693i $$0.340918\pi$$
$$674$$ 0 0
$$675$$ 1112.09 0.0634139
$$676$$ 0 0
$$677$$ −24191.5 −1.37335 −0.686673 0.726966i $$-0.740930\pi$$
−0.686673 + 0.726966i $$0.740930\pi$$
$$678$$ 0 0
$$679$$ −20999.2 −1.18685
$$680$$ 0 0
$$681$$ 8692.41 0.489125
$$682$$ 0 0
$$683$$ −13965.2 −0.782376 −0.391188 0.920311i $$-0.627936\pi$$
−0.391188 + 0.920311i $$0.627936\pi$$
$$684$$ 0 0
$$685$$ 4273.17 0.238350
$$686$$ 0 0
$$687$$ 103.888 0.00576939
$$688$$ 0 0
$$689$$ 23294.4 1.28802
$$690$$ 0 0
$$691$$ −8685.63 −0.478172 −0.239086 0.970998i $$-0.576848\pi$$
−0.239086 + 0.970998i $$0.576848\pi$$
$$692$$ 0 0
$$693$$ −5064.73 −0.277624
$$694$$ 0 0
$$695$$ −3220.43 −0.175767
$$696$$ 0 0
$$697$$ 31619.7 1.71834
$$698$$ 0 0
$$699$$ −3162.05 −0.171101
$$700$$ 0 0
$$701$$ 25942.2 1.39775 0.698876 0.715243i $$-0.253684\pi$$
0.698876 + 0.715243i $$0.253684\pi$$
$$702$$ 0 0
$$703$$ −14809.6 −0.794532
$$704$$ 0 0
$$705$$ −1500.84 −0.0801772
$$706$$ 0 0
$$707$$ 5517.86 0.293522
$$708$$ 0 0
$$709$$ −5487.75 −0.290687 −0.145343 0.989381i $$-0.546429\pi$$
−0.145343 + 0.989381i $$0.546429\pi$$
$$710$$ 0 0
$$711$$ −2716.02 −0.143261
$$712$$ 0 0
$$713$$ 1719.43 0.0903128
$$714$$ 0 0
$$715$$ 6021.82 0.314970
$$716$$ 0 0
$$717$$ −1964.10 −0.102302
$$718$$ 0 0
$$719$$ −17141.2 −0.889094 −0.444547 0.895756i $$-0.646635\pi$$
−0.444547 + 0.895756i $$0.646635\pi$$
$$720$$ 0 0
$$721$$ 18710.8 0.966470
$$722$$ 0 0
$$723$$ −9581.99 −0.492888
$$724$$ 0 0
$$725$$ 1691.23 0.0866352
$$726$$ 0 0
$$727$$ −15946.4 −0.813508 −0.406754 0.913538i $$-0.633339\pi$$
−0.406754 + 0.913538i $$0.633339\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −9990.26 −0.505476
$$732$$ 0 0
$$733$$ −15914.2 −0.801917 −0.400958 0.916096i $$-0.631323\pi$$
−0.400958 + 0.916096i $$0.631323\pi$$
$$734$$ 0 0
$$735$$ −11225.2 −0.563332
$$736$$ 0 0
$$737$$ 7787.94 0.389243
$$738$$ 0 0
$$739$$ 13555.4 0.674755 0.337377 0.941369i $$-0.390460\pi$$
0.337377 + 0.941369i $$0.390460\pi$$
$$740$$ 0 0
$$741$$ −12475.7 −0.618497
$$742$$ 0 0
$$743$$ −1772.73 −0.0875303 −0.0437652 0.999042i $$-0.513935\pi$$
−0.0437652 + 0.999042i $$0.513935\pi$$
$$744$$ 0 0
$$745$$ −11814.3 −0.580997
$$746$$ 0 0
$$747$$ −5391.03 −0.264053
$$748$$ 0 0
$$749$$ −12536.7 −0.611589
$$750$$ 0 0
$$751$$ 1006.65 0.0489124 0.0244562 0.999701i $$-0.492215\pi$$
0.0244562 + 0.999701i $$0.492215\pi$$
$$752$$ 0 0
$$753$$ 15128.7 0.732165
$$754$$ 0 0
$$755$$ 10761.6 0.518750
$$756$$ 0 0
$$757$$ −28774.0 −1.38152 −0.690758 0.723086i $$-0.742723\pi$$
−0.690758 + 0.723086i $$0.742723\pi$$
$$758$$ 0 0
$$759$$ 564.346 0.0269887
$$760$$ 0 0
$$761$$ −18393.5 −0.876166 −0.438083 0.898934i $$-0.644342\pi$$
−0.438083 + 0.898934i $$0.644342\pi$$
$$762$$ 0 0
$$763$$ −17158.2 −0.814112
$$764$$ 0 0
$$765$$ −9208.66 −0.435215
$$766$$ 0 0
$$767$$ −6940.72 −0.326747
$$768$$ 0 0
$$769$$ −14672.2 −0.688027 −0.344014 0.938965i $$-0.611787\pi$$
−0.344014 + 0.938965i $$0.611787\pi$$
$$770$$ 0 0
$$771$$ 15499.9 0.724016
$$772$$ 0 0
$$773$$ 16256.4 0.756405 0.378202 0.925723i $$-0.376542\pi$$
0.378202 + 0.925723i $$0.376542\pi$$
$$774$$ 0 0
$$775$$ −7727.21 −0.358154
$$776$$ 0 0
$$777$$ −9387.25 −0.433418
$$778$$ 0 0
$$779$$ 36712.2 1.68851
$$780$$ 0 0
$$781$$ −6200.24 −0.284074
$$782$$ 0 0
$$783$$ 1108.64 0.0505995
$$784$$ 0 0
$$785$$ 9997.92 0.454575
$$786$$ 0 0
$$787$$ 23988.3 1.08652 0.543259 0.839565i $$-0.317190\pi$$
0.543259 + 0.839565i $$0.317190\pi$$
$$788$$ 0 0
$$789$$ 22098.3 0.997112
$$790$$ 0 0
$$791$$ 26900.7 1.20920
$$792$$ 0 0
$$793$$ 24170.6 1.08238
$$794$$ 0 0
$$795$$ 19963.4 0.890602
$$796$$ 0 0
$$797$$ −32966.9 −1.46518 −0.732589 0.680672i $$-0.761688\pi$$
−0.732589 + 0.680672i $$0.761688\pi$$
$$798$$ 0 0
$$799$$ −6107.50 −0.270423
$$800$$ 0 0
$$801$$ 2497.75 0.110179
$$802$$ 0 0
$$803$$ −10352.8 −0.454973
$$804$$ 0 0
$$805$$ 2300.47 0.100721
$$806$$ 0 0
$$807$$ −23633.4 −1.03090
$$808$$ 0 0
$$809$$ 41700.3 1.81224 0.906122 0.423017i $$-0.139029\pi$$
0.906122 + 0.423017i $$0.139029\pi$$
$$810$$ 0 0
$$811$$ 5981.80 0.259000 0.129500 0.991579i $$-0.458663\pi$$
0.129500 + 0.991579i $$0.458663\pi$$
$$812$$ 0 0
$$813$$ −16199.7 −0.698832
$$814$$ 0 0
$$815$$ 33204.4 1.42712
$$816$$ 0 0
$$817$$ −11599.2 −0.496702
$$818$$ 0 0
$$819$$ −7907.86 −0.337391
$$820$$ 0 0
$$821$$ −9846.06 −0.418550 −0.209275 0.977857i $$-0.567110\pi$$
−0.209275 + 0.977857i $$0.567110\pi$$
$$822$$ 0 0
$$823$$ 47001.9 1.99074 0.995372 0.0960935i $$-0.0306348\pi$$
0.995372 + 0.0960935i $$0.0306348\pi$$
$$824$$ 0 0
$$825$$ −2536.21 −0.107030
$$826$$ 0 0
$$827$$ 21727.4 0.913587 0.456794 0.889573i $$-0.348998\pi$$
0.456794 + 0.889573i $$0.348998\pi$$
$$828$$ 0 0
$$829$$ 22772.3 0.954058 0.477029 0.878888i $$-0.341714\pi$$
0.477029 + 0.878888i $$0.341714\pi$$
$$830$$ 0 0
$$831$$ −13248.2 −0.553039
$$832$$ 0 0
$$833$$ −45679.8 −1.90002
$$834$$ 0 0
$$835$$ 419.304 0.0173780
$$836$$ 0 0
$$837$$ −5065.36 −0.209181
$$838$$ 0 0
$$839$$ 11010.4 0.453064 0.226532 0.974004i $$-0.427261\pi$$
0.226532 + 0.974004i $$0.427261\pi$$
$$840$$ 0 0
$$841$$ −22703.0 −0.930872
$$842$$ 0 0
$$843$$ 24206.8 0.989000
$$844$$ 0 0
$$845$$ −10711.0 −0.436059
$$846$$ 0 0
$$847$$ −24942.1 −1.01183
$$848$$ 0 0
$$849$$ −15723.4 −0.635602
$$850$$ 0 0
$$851$$ 1045.99 0.0421340
$$852$$ 0 0
$$853$$ −38177.4 −1.53244 −0.766219 0.642579i $$-0.777864\pi$$
−0.766219 + 0.642579i $$0.777864\pi$$
$$854$$ 0 0
$$855$$ −10691.7 −0.427661
$$856$$ 0 0
$$857$$ −8848.01 −0.352675 −0.176337 0.984330i $$-0.556425\pi$$
−0.176337 + 0.984330i $$0.556425\pi$$
$$858$$ 0 0
$$859$$ 4347.66 0.172690 0.0863448 0.996265i $$-0.472481\pi$$
0.0863448 + 0.996265i $$0.472481\pi$$
$$860$$ 0 0
$$861$$ 23270.4 0.921085
$$862$$ 0 0
$$863$$ 33669.9 1.32808 0.664042 0.747695i $$-0.268839\pi$$
0.664042 + 0.747695i $$0.268839\pi$$
$$864$$ 0 0
$$865$$ −22475.3 −0.883450
$$866$$ 0 0
$$867$$ −22734.6 −0.890552
$$868$$ 0 0
$$869$$ 6194.10 0.241796
$$870$$ 0 0
$$871$$ 12159.7 0.473039
$$872$$ 0 0
$$873$$ −6893.15 −0.267237
$$874$$ 0 0
$$875$$ −41713.8 −1.61164
$$876$$ 0 0
$$877$$ 50102.0 1.92910 0.964552 0.263892i $$-0.0850062\pi$$
0.964552 + 0.263892i $$0.0850062\pi$$
$$878$$ 0 0
$$879$$ −19118.2 −0.733609
$$880$$ 0 0
$$881$$ 18716.9 0.715766 0.357883 0.933766i $$-0.383499\pi$$
0.357883 + 0.933766i $$0.383499\pi$$
$$882$$ 0 0
$$883$$ 7514.19 0.286379 0.143189 0.989695i $$-0.454264\pi$$
0.143189 + 0.989695i $$0.454264\pi$$
$$884$$ 0 0
$$885$$ −5948.24 −0.225930
$$886$$ 0 0
$$887$$ −15544.6 −0.588429 −0.294215 0.955739i $$-0.595058\pi$$
−0.294215 + 0.955739i $$0.595058\pi$$
$$888$$ 0 0
$$889$$ 22154.8 0.835825
$$890$$ 0 0
$$891$$ −1662.54 −0.0625109
$$892$$ 0 0
$$893$$ −7091.14 −0.265729
$$894$$ 0 0
$$895$$ −9395.42 −0.350898
$$896$$ 0 0
$$897$$ 881.145 0.0327989
$$898$$ 0 0
$$899$$ −7703.21 −0.285780
$$900$$ 0 0
$$901$$ 81238.8 3.00384
$$902$$ 0 0
$$903$$ −7352.29 −0.270951
$$904$$ 0 0
$$905$$ 33864.3 1.24385
$$906$$ 0 0
$$907$$ −8713.10 −0.318979 −0.159489 0.987200i $$-0.550985\pi$$
−0.159489 + 0.987200i $$0.550985\pi$$
$$908$$ 0 0
$$909$$ 1811.28 0.0660906
$$910$$ 0 0
$$911$$ −1975.97 −0.0718627 −0.0359313 0.999354i $$-0.511440\pi$$
−0.0359313 + 0.999354i $$0.511440\pi$$
$$912$$ 0 0
$$913$$ 12294.6 0.445666
$$914$$ 0 0
$$915$$ 20714.4 0.748411
$$916$$ 0 0
$$917$$ 30456.8 1.09681
$$918$$ 0 0
$$919$$ −18430.5 −0.661552 −0.330776 0.943709i $$-0.607311\pi$$
−0.330776 + 0.943709i $$0.607311\pi$$
$$920$$ 0 0
$$921$$ 11430.7 0.408961
$$922$$ 0 0
$$923$$ −9680.79 −0.345230
$$924$$ 0 0
$$925$$ −4700.74 −0.167091
$$926$$ 0 0
$$927$$ 6141.96 0.217614
$$928$$ 0 0
$$929$$ 12506.8 0.441697 0.220848 0.975308i $$-0.429117\pi$$
0.220848 + 0.975308i $$0.429117\pi$$
$$930$$ 0 0
$$931$$ −53036.7 −1.86703
$$932$$ 0 0
$$933$$ 24320.2 0.853384
$$934$$ 0 0
$$935$$ 21001.0 0.734554
$$936$$ 0 0
$$937$$ −39267.5 −1.36906 −0.684532 0.728982i $$-0.739994\pi$$
−0.684532 + 0.728982i $$0.739994\pi$$
$$938$$ 0 0
$$939$$ −1679.95 −0.0583846
$$940$$ 0 0
$$941$$ 23727.2 0.821981 0.410991 0.911640i $$-0.365183\pi$$
0.410991 + 0.911640i $$0.365183\pi$$
$$942$$ 0 0
$$943$$ −2592.94 −0.0895418
$$944$$ 0 0
$$945$$ −6777.08 −0.233289
$$946$$ 0 0
$$947$$ 23399.8 0.802948 0.401474 0.915870i $$-0.368498\pi$$
0.401474 + 0.915870i $$0.368498\pi$$
$$948$$ 0 0
$$949$$ −16164.4 −0.552919
$$950$$ 0 0
$$951$$ 17486.9 0.596270
$$952$$ 0 0
$$953$$ −41497.1 −1.41052 −0.705258 0.708950i $$-0.749169\pi$$
−0.705258 + 0.708950i $$0.749169\pi$$
$$954$$ 0 0
$$955$$ −46771.6 −1.58481
$$956$$ 0 0
$$957$$ −2528.33 −0.0854015
$$958$$ 0 0
$$959$$ 12797.5 0.430921
$$960$$ 0 0
$$961$$ 5405.00 0.181431
$$962$$ 0 0
$$963$$ −4115.27 −0.137708
$$964$$ 0 0
$$965$$ −12946.2 −0.431868
$$966$$ 0 0
$$967$$ −49123.6 −1.63362 −0.816808 0.576909i $$-0.804259\pi$$
−0.816808 + 0.576909i $$0.804259\pi$$
$$968$$ 0 0
$$969$$ −43508.9 −1.44242
$$970$$ 0 0
$$971$$ 20346.8 0.672462 0.336231 0.941780i $$-0.390848\pi$$
0.336231 + 0.941780i $$0.390848\pi$$
$$972$$ 0 0
$$973$$ −9644.71 −0.317775
$$974$$ 0 0
$$975$$ −3959.92 −0.130071
$$976$$ 0 0
$$977$$ 40602.1 1.32955 0.664777 0.747042i $$-0.268526\pi$$
0.664777 + 0.747042i $$0.268526\pi$$
$$978$$ 0 0
$$979$$ −5696.32 −0.185960
$$980$$ 0 0
$$981$$ −5632.31 −0.183309
$$982$$ 0 0
$$983$$ 50425.9 1.63615 0.818075 0.575112i $$-0.195041\pi$$
0.818075 + 0.575112i $$0.195041\pi$$
$$984$$ 0 0
$$985$$ −25786.2 −0.834127
$$986$$ 0 0
$$987$$ −4494.79 −0.144955
$$988$$ 0 0
$$989$$ 819.241 0.0263401
$$990$$ 0 0
$$991$$ 8511.62 0.272836 0.136418 0.990651i $$-0.456441\pi$$
0.136418 + 0.990651i $$0.456441\pi$$
$$992$$ 0 0
$$993$$ 8543.94 0.273045
$$994$$ 0 0
$$995$$ −8683.90 −0.276681
$$996$$ 0 0
$$997$$ −25302.1 −0.803738 −0.401869 0.915697i $$-0.631639\pi$$
−0.401869 + 0.915697i $$0.631639\pi$$
$$998$$ 0 0
$$999$$ −3081.44 −0.0975900
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.q.1.3 3
3.2 odd 2 2304.4.a.bw.1.1 3
4.3 odd 2 768.4.a.s.1.3 3
8.3 odd 2 768.4.a.r.1.1 3
8.5 even 2 768.4.a.t.1.1 3
12.11 even 2 2304.4.a.bv.1.1 3
16.3 odd 4 24.4.d.a.13.4 yes 6
16.5 even 4 96.4.d.a.49.6 6
16.11 odd 4 24.4.d.a.13.3 6
16.13 even 4 96.4.d.a.49.1 6
24.5 odd 2 2304.4.a.bu.1.3 3
24.11 even 2 2304.4.a.bt.1.3 3
48.5 odd 4 288.4.d.d.145.2 6
48.11 even 4 72.4.d.d.37.4 6
48.29 odd 4 288.4.d.d.145.5 6
48.35 even 4 72.4.d.d.37.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.3 6 16.11 odd 4
24.4.d.a.13.4 yes 6 16.3 odd 4
72.4.d.d.37.3 6 48.35 even 4
72.4.d.d.37.4 6 48.11 even 4
96.4.d.a.49.1 6 16.13 even 4
96.4.d.a.49.6 6 16.5 even 4
288.4.d.d.145.2 6 48.5 odd 4
288.4.d.d.145.5 6 48.29 odd 4
768.4.a.q.1.3 3 1.1 even 1 trivial
768.4.a.r.1.1 3 8.3 odd 2
768.4.a.s.1.3 3 4.3 odd 2
768.4.a.t.1.1 3 8.5 even 2
2304.4.a.bt.1.3 3 24.11 even 2
2304.4.a.bu.1.3 3 24.5 odd 2
2304.4.a.bv.1.1 3 12.11 even 2
2304.4.a.bw.1.1 3 3.2 odd 2