# Properties

 Label 7168.2.a.bb.1.3 Level $7168$ Weight $2$ Character 7168.1 Self dual yes Analytic conductor $57.237$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7168 = 2^{10} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7168.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.2367681689$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.9433055232.1 Defining polynomial: $$x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1792) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.21236$$ of defining polynomial Character $$\chi$$ $$=$$ 7168.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.78579 q^{3} -4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} +O(q^{10})$$ $$q-1.78579 q^{3} -4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} -4.50362 q^{11} -4.84427 q^{13} +7.46903 q^{15} +5.13834 q^{17} +2.12835 q^{19} -1.78579 q^{21} -7.11888 q^{23} +12.4932 q^{25} +5.01978 q^{27} -5.43932 q^{29} -0.831138 q^{31} +8.04251 q^{33} -4.18248 q^{35} +7.98492 q^{37} +8.65084 q^{39} -2.22639 q^{41} +2.28804 q^{43} -0.790669 q^{45} +7.83759 q^{47} +1.00000 q^{49} -9.17599 q^{51} +7.90235 q^{53} +18.8363 q^{55} -3.80079 q^{57} -2.62811 q^{59} -2.33370 q^{61} +0.189043 q^{63} +20.2611 q^{65} +8.16822 q^{67} +12.7128 q^{69} -6.04851 q^{71} -7.67177 q^{73} -22.3101 q^{75} -4.50362 q^{77} -1.90198 q^{79} -9.53139 q^{81} +11.2843 q^{83} -21.4910 q^{85} +9.71349 q^{87} -2.49938 q^{89} -4.84427 q^{91} +1.48424 q^{93} -8.90180 q^{95} -1.98784 q^{97} -0.851377 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10})$$ $$8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100})$$

## 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
$$3$$ −1.78579 −1.03103 −0.515513 0.856882i $$-0.672399\pi$$
−0.515513 + 0.856882i $$0.672399\pi$$
$$4$$ 0 0
$$5$$ −4.18248 −1.87046 −0.935231 0.354037i $$-0.884809\pi$$
−0.935231 + 0.354037i $$0.884809\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0.189043 0.0630143
$$10$$ 0 0
$$11$$ −4.50362 −1.35789 −0.678946 0.734188i $$-0.737563\pi$$
−0.678946 + 0.734188i $$0.737563\pi$$
$$12$$ 0 0
$$13$$ −4.84427 −1.34356 −0.671779 0.740752i $$-0.734469\pi$$
−0.671779 + 0.740752i $$0.734469\pi$$
$$14$$ 0 0
$$15$$ 7.46903 1.92850
$$16$$ 0 0
$$17$$ 5.13834 1.24623 0.623115 0.782130i $$-0.285867\pi$$
0.623115 + 0.782130i $$0.285867\pi$$
$$18$$ 0 0
$$19$$ 2.12835 0.488278 0.244139 0.969740i $$-0.421495\pi$$
0.244139 + 0.969740i $$0.421495\pi$$
$$20$$ 0 0
$$21$$ −1.78579 −0.389691
$$22$$ 0 0
$$23$$ −7.11888 −1.48439 −0.742195 0.670184i $$-0.766215\pi$$
−0.742195 + 0.670184i $$0.766215\pi$$
$$24$$ 0 0
$$25$$ 12.4932 2.49863
$$26$$ 0 0
$$27$$ 5.01978 0.966056
$$28$$ 0 0
$$29$$ −5.43932 −1.01006 −0.505029 0.863103i $$-0.668518\pi$$
−0.505029 + 0.863103i $$0.668518\pi$$
$$30$$ 0 0
$$31$$ −0.831138 −0.149277 −0.0746384 0.997211i $$-0.523780\pi$$
−0.0746384 + 0.997211i $$0.523780\pi$$
$$32$$ 0 0
$$33$$ 8.04251 1.40002
$$34$$ 0 0
$$35$$ −4.18248 −0.706969
$$36$$ 0 0
$$37$$ 7.98492 1.31271 0.656357 0.754451i $$-0.272097\pi$$
0.656357 + 0.754451i $$0.272097\pi$$
$$38$$ 0 0
$$39$$ 8.65084 1.38524
$$40$$ 0 0
$$41$$ −2.22639 −0.347704 −0.173852 0.984772i $$-0.555621\pi$$
−0.173852 + 0.984772i $$0.555621\pi$$
$$42$$ 0 0
$$43$$ 2.28804 0.348923 0.174462 0.984664i $$-0.444182\pi$$
0.174462 + 0.984664i $$0.444182\pi$$
$$44$$ 0 0
$$45$$ −0.790669 −0.117866
$$46$$ 0 0
$$47$$ 7.83759 1.14323 0.571615 0.820522i $$-0.306317\pi$$
0.571615 + 0.820522i $$0.306317\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −9.17599 −1.28490
$$52$$ 0 0
$$53$$ 7.90235 1.08547 0.542736 0.839903i $$-0.317388\pi$$
0.542736 + 0.839903i $$0.317388\pi$$
$$54$$ 0 0
$$55$$ 18.8363 2.53989
$$56$$ 0 0
$$57$$ −3.80079 −0.503427
$$58$$ 0 0
$$59$$ −2.62811 −0.342150 −0.171075 0.985258i $$-0.554724\pi$$
−0.171075 + 0.985258i $$0.554724\pi$$
$$60$$ 0 0
$$61$$ −2.33370 −0.298799 −0.149400 0.988777i $$-0.547734\pi$$
−0.149400 + 0.988777i $$0.547734\pi$$
$$62$$ 0 0
$$63$$ 0.189043 0.0238172
$$64$$ 0 0
$$65$$ 20.2611 2.51307
$$66$$ 0 0
$$67$$ 8.16822 0.997907 0.498954 0.866629i $$-0.333718\pi$$
0.498954 + 0.866629i $$0.333718\pi$$
$$68$$ 0 0
$$69$$ 12.7128 1.53044
$$70$$ 0 0
$$71$$ −6.04851 −0.717826 −0.358913 0.933371i $$-0.616853\pi$$
−0.358913 + 0.933371i $$0.616853\pi$$
$$72$$ 0 0
$$73$$ −7.67177 −0.897913 −0.448957 0.893554i $$-0.648204\pi$$
−0.448957 + 0.893554i $$0.648204\pi$$
$$74$$ 0 0
$$75$$ −22.3101 −2.57615
$$76$$ 0 0
$$77$$ −4.50362 −0.513235
$$78$$ 0 0
$$79$$ −1.90198 −0.213989 −0.106995 0.994260i $$-0.534123\pi$$
−0.106995 + 0.994260i $$0.534123\pi$$
$$80$$ 0 0
$$81$$ −9.53139 −1.05904
$$82$$ 0 0
$$83$$ 11.2843 1.23861 0.619306 0.785149i $$-0.287414\pi$$
0.619306 + 0.785149i $$0.287414\pi$$
$$84$$ 0 0
$$85$$ −21.4910 −2.33103
$$86$$ 0 0
$$87$$ 9.71349 1.04139
$$88$$ 0 0
$$89$$ −2.49938 −0.264934 −0.132467 0.991187i $$-0.542290\pi$$
−0.132467 + 0.991187i $$0.542290\pi$$
$$90$$ 0 0
$$91$$ −4.84427 −0.507817
$$92$$ 0 0
$$93$$ 1.48424 0.153908
$$94$$ 0 0
$$95$$ −8.90180 −0.913306
$$96$$ 0 0
$$97$$ −1.98784 −0.201834 −0.100917 0.994895i $$-0.532178\pi$$
−0.100917 + 0.994895i $$0.532178\pi$$
$$98$$ 0 0
$$99$$ −0.851377 −0.0855666
$$100$$ 0 0
$$101$$ 15.7459 1.56678 0.783390 0.621531i $$-0.213489\pi$$
0.783390 + 0.621531i $$0.213489\pi$$
$$102$$ 0 0
$$103$$ 15.2106 1.49874 0.749371 0.662150i $$-0.230356\pi$$
0.749371 + 0.662150i $$0.230356\pi$$
$$104$$ 0 0
$$105$$ 7.46903 0.728903
$$106$$ 0 0
$$107$$ −1.26941 −0.122718 −0.0613592 0.998116i $$-0.519544\pi$$
−0.0613592 + 0.998116i $$0.519544\pi$$
$$108$$ 0 0
$$109$$ −4.02383 −0.385413 −0.192707 0.981256i $$-0.561727\pi$$
−0.192707 + 0.981256i $$0.561727\pi$$
$$110$$ 0 0
$$111$$ −14.2594 −1.35344
$$112$$ 0 0
$$113$$ 1.66487 0.156618 0.0783089 0.996929i $$-0.475048\pi$$
0.0783089 + 0.996929i $$0.475048\pi$$
$$114$$ 0 0
$$115$$ 29.7746 2.77650
$$116$$ 0 0
$$117$$ −0.915774 −0.0846634
$$118$$ 0 0
$$119$$ 5.13834 0.471031
$$120$$ 0 0
$$121$$ 9.28258 0.843871
$$122$$ 0 0
$$123$$ 3.97586 0.358492
$$124$$ 0 0
$$125$$ −31.3400 −2.80313
$$126$$ 0 0
$$127$$ 7.86069 0.697523 0.348762 0.937211i $$-0.386602\pi$$
0.348762 + 0.937211i $$0.386602\pi$$
$$128$$ 0 0
$$129$$ −4.08596 −0.359749
$$130$$ 0 0
$$131$$ 7.70009 0.672760 0.336380 0.941726i $$-0.390797\pi$$
0.336380 + 0.941726i $$0.390797\pi$$
$$132$$ 0 0
$$133$$ 2.12835 0.184552
$$134$$ 0 0
$$135$$ −20.9951 −1.80697
$$136$$ 0 0
$$137$$ −17.6977 −1.51201 −0.756007 0.654564i $$-0.772852\pi$$
−0.756007 + 0.654564i $$0.772852\pi$$
$$138$$ 0 0
$$139$$ 16.1930 1.37348 0.686738 0.726905i $$-0.259042\pi$$
0.686738 + 0.726905i $$0.259042\pi$$
$$140$$ 0 0
$$141$$ −13.9963 −1.17870
$$142$$ 0 0
$$143$$ 21.8167 1.82441
$$144$$ 0 0
$$145$$ 22.7499 1.88927
$$146$$ 0 0
$$147$$ −1.78579 −0.147289
$$148$$ 0 0
$$149$$ −12.1806 −0.997874 −0.498937 0.866638i $$-0.666276\pi$$
−0.498937 + 0.866638i $$0.666276\pi$$
$$150$$ 0 0
$$151$$ −17.7449 −1.44406 −0.722029 0.691863i $$-0.756790\pi$$
−0.722029 + 0.691863i $$0.756790\pi$$
$$152$$ 0 0
$$153$$ 0.971366 0.0785303
$$154$$ 0 0
$$155$$ 3.47622 0.279217
$$156$$ 0 0
$$157$$ 13.9789 1.11564 0.557818 0.829963i $$-0.311639\pi$$
0.557818 + 0.829963i $$0.311639\pi$$
$$158$$ 0 0
$$159$$ −14.1119 −1.11915
$$160$$ 0 0
$$161$$ −7.11888 −0.561047
$$162$$ 0 0
$$163$$ 13.5962 1.06494 0.532468 0.846450i $$-0.321265\pi$$
0.532468 + 0.846450i $$0.321265\pi$$
$$164$$ 0 0
$$165$$ −33.6377 −2.61869
$$166$$ 0 0
$$167$$ −6.70735 −0.519030 −0.259515 0.965739i $$-0.583563\pi$$
−0.259515 + 0.965739i $$0.583563\pi$$
$$168$$ 0 0
$$169$$ 10.4669 0.805147
$$170$$ 0 0
$$171$$ 0.402350 0.0307685
$$172$$ 0 0
$$173$$ −20.8358 −1.58411 −0.792057 0.610448i $$-0.790990\pi$$
−0.792057 + 0.610448i $$0.790990\pi$$
$$174$$ 0 0
$$175$$ 12.4932 0.944394
$$176$$ 0 0
$$177$$ 4.69325 0.352766
$$178$$ 0 0
$$179$$ 22.4086 1.67490 0.837448 0.546517i $$-0.184047\pi$$
0.837448 + 0.546517i $$0.184047\pi$$
$$180$$ 0 0
$$181$$ −4.42942 −0.329236 −0.164618 0.986357i $$-0.552639\pi$$
−0.164618 + 0.986357i $$0.552639\pi$$
$$182$$ 0 0
$$183$$ 4.16749 0.308070
$$184$$ 0 0
$$185$$ −33.3968 −2.45538
$$186$$ 0 0
$$187$$ −23.1411 −1.69225
$$188$$ 0 0
$$189$$ 5.01978 0.365135
$$190$$ 0 0
$$191$$ −7.38976 −0.534704 −0.267352 0.963599i $$-0.586149\pi$$
−0.267352 + 0.963599i $$0.586149\pi$$
$$192$$ 0 0
$$193$$ 0.139138 0.0100154 0.00500769 0.999987i $$-0.498406\pi$$
0.00500769 + 0.999987i $$0.498406\pi$$
$$194$$ 0 0
$$195$$ −36.1820 −2.59104
$$196$$ 0 0
$$197$$ −10.4057 −0.741377 −0.370688 0.928757i $$-0.620878\pi$$
−0.370688 + 0.928757i $$0.620878\pi$$
$$198$$ 0 0
$$199$$ 5.09550 0.361211 0.180605 0.983556i $$-0.442194\pi$$
0.180605 + 0.983556i $$0.442194\pi$$
$$200$$ 0 0
$$201$$ −14.5867 −1.02887
$$202$$ 0 0
$$203$$ −5.43932 −0.381766
$$204$$ 0 0
$$205$$ 9.31184 0.650367
$$206$$ 0 0
$$207$$ −1.34577 −0.0935378
$$208$$ 0 0
$$209$$ −9.58529 −0.663029
$$210$$ 0 0
$$211$$ −12.5589 −0.864592 −0.432296 0.901732i $$-0.642296\pi$$
−0.432296 + 0.901732i $$0.642296\pi$$
$$212$$ 0 0
$$213$$ 10.8014 0.740097
$$214$$ 0 0
$$215$$ −9.56970 −0.652648
$$216$$ 0 0
$$217$$ −0.831138 −0.0564213
$$218$$ 0 0
$$219$$ 13.7002 0.925772
$$220$$ 0 0
$$221$$ −24.8915 −1.67438
$$222$$ 0 0
$$223$$ 9.66949 0.647517 0.323758 0.946140i $$-0.395054\pi$$
0.323758 + 0.946140i $$0.395054\pi$$
$$224$$ 0 0
$$225$$ 2.36174 0.157450
$$226$$ 0 0
$$227$$ 2.99594 0.198847 0.0994237 0.995045i $$-0.468300\pi$$
0.0994237 + 0.995045i $$0.468300\pi$$
$$228$$ 0 0
$$229$$ −8.99569 −0.594452 −0.297226 0.954807i $$-0.596061\pi$$
−0.297226 + 0.954807i $$0.596061\pi$$
$$230$$ 0 0
$$231$$ 8.04251 0.529158
$$232$$ 0 0
$$233$$ 7.53066 0.493350 0.246675 0.969098i $$-0.420662\pi$$
0.246675 + 0.969098i $$0.420662\pi$$
$$234$$ 0 0
$$235$$ −32.7806 −2.13837
$$236$$ 0 0
$$237$$ 3.39653 0.220629
$$238$$ 0 0
$$239$$ −1.87072 −0.121007 −0.0605034 0.998168i $$-0.519271\pi$$
−0.0605034 + 0.998168i $$0.519271\pi$$
$$240$$ 0 0
$$241$$ 14.4911 0.933454 0.466727 0.884401i $$-0.345433\pi$$
0.466727 + 0.884401i $$0.345433\pi$$
$$242$$ 0 0
$$243$$ 1.96173 0.125845
$$244$$ 0 0
$$245$$ −4.18248 −0.267209
$$246$$ 0 0
$$247$$ −10.3103 −0.656029
$$248$$ 0 0
$$249$$ −20.1514 −1.27704
$$250$$ 0 0
$$251$$ −6.33974 −0.400161 −0.200080 0.979779i $$-0.564120\pi$$
−0.200080 + 0.979779i $$0.564120\pi$$
$$252$$ 0 0
$$253$$ 32.0607 2.01564
$$254$$ 0 0
$$255$$ 38.3784 2.40335
$$256$$ 0 0
$$257$$ 12.1594 0.758483 0.379241 0.925298i $$-0.376185\pi$$
0.379241 + 0.925298i $$0.376185\pi$$
$$258$$ 0 0
$$259$$ 7.98492 0.496159
$$260$$ 0 0
$$261$$ −1.02827 −0.0636480
$$262$$ 0 0
$$263$$ 0.0299529 0.00184698 0.000923488 1.00000i $$-0.499706\pi$$
0.000923488 1.00000i $$0.499706\pi$$
$$264$$ 0 0
$$265$$ −33.0514 −2.03033
$$266$$ 0 0
$$267$$ 4.46336 0.273153
$$268$$ 0 0
$$269$$ 18.0622 1.10127 0.550637 0.834745i $$-0.314385\pi$$
0.550637 + 0.834745i $$0.314385\pi$$
$$270$$ 0 0
$$271$$ 10.0906 0.612958 0.306479 0.951877i $$-0.400849\pi$$
0.306479 + 0.951877i $$0.400849\pi$$
$$272$$ 0 0
$$273$$ 8.65084 0.523573
$$274$$ 0 0
$$275$$ −56.2644 −3.39287
$$276$$ 0 0
$$277$$ −8.86651 −0.532737 −0.266368 0.963871i $$-0.585824\pi$$
−0.266368 + 0.963871i $$0.585824\pi$$
$$278$$ 0 0
$$279$$ −0.157121 −0.00940657
$$280$$ 0 0
$$281$$ −11.6731 −0.696356 −0.348178 0.937428i $$-0.613200\pi$$
−0.348178 + 0.937428i $$0.613200\pi$$
$$282$$ 0 0
$$283$$ 12.5547 0.746297 0.373149 0.927772i $$-0.378278\pi$$
0.373149 + 0.927772i $$0.378278\pi$$
$$284$$ 0 0
$$285$$ 15.8967 0.941642
$$286$$ 0 0
$$287$$ −2.22639 −0.131420
$$288$$ 0 0
$$289$$ 9.40252 0.553090
$$290$$ 0 0
$$291$$ 3.54986 0.208096
$$292$$ 0 0
$$293$$ −24.3152 −1.42051 −0.710256 0.703944i $$-0.751421\pi$$
−0.710256 + 0.703944i $$0.751421\pi$$
$$294$$ 0 0
$$295$$ 10.9920 0.639980
$$296$$ 0 0
$$297$$ −22.6072 −1.31180
$$298$$ 0 0
$$299$$ 34.4858 1.99436
$$300$$ 0 0
$$301$$ 2.28804 0.131881
$$302$$ 0 0
$$303$$ −28.1189 −1.61539
$$304$$ 0 0
$$305$$ 9.76065 0.558893
$$306$$ 0 0
$$307$$ −27.2536 −1.55544 −0.777722 0.628608i $$-0.783625\pi$$
−0.777722 + 0.628608i $$0.783625\pi$$
$$308$$ 0 0
$$309$$ −27.1629 −1.54524
$$310$$ 0 0
$$311$$ −5.78650 −0.328122 −0.164061 0.986450i $$-0.552459\pi$$
−0.164061 + 0.986450i $$0.552459\pi$$
$$312$$ 0 0
$$313$$ 3.03673 0.171646 0.0858231 0.996310i $$-0.472648\pi$$
0.0858231 + 0.996310i $$0.472648\pi$$
$$314$$ 0 0
$$315$$ −0.790669 −0.0445491
$$316$$ 0 0
$$317$$ −4.69828 −0.263882 −0.131941 0.991258i $$-0.542121\pi$$
−0.131941 + 0.991258i $$0.542121\pi$$
$$318$$ 0 0
$$319$$ 24.4966 1.37155
$$320$$ 0 0
$$321$$ 2.26690 0.126526
$$322$$ 0 0
$$323$$ 10.9362 0.608507
$$324$$ 0 0
$$325$$ −60.5202 −3.35706
$$326$$ 0 0
$$327$$ 7.18572 0.397371
$$328$$ 0 0
$$329$$ 7.83759 0.432101
$$330$$ 0 0
$$331$$ 36.0648 1.98230 0.991151 0.132738i $$-0.0423768\pi$$
0.991151 + 0.132738i $$0.0423768\pi$$
$$332$$ 0 0
$$333$$ 1.50949 0.0827197
$$334$$ 0 0
$$335$$ −34.1634 −1.86655
$$336$$ 0 0
$$337$$ −31.3282 −1.70656 −0.853279 0.521454i $$-0.825390\pi$$
−0.853279 + 0.521454i $$0.825390\pi$$
$$338$$ 0 0
$$339$$ −2.97311 −0.161477
$$340$$ 0 0
$$341$$ 3.74313 0.202702
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −53.1712 −2.86264
$$346$$ 0 0
$$347$$ −34.1006 −1.83062 −0.915308 0.402755i $$-0.868053\pi$$
−0.915308 + 0.402755i $$0.868053\pi$$
$$348$$ 0 0
$$349$$ −29.6604 −1.58769 −0.793843 0.608123i $$-0.791923\pi$$
−0.793843 + 0.608123i $$0.791923\pi$$
$$350$$ 0 0
$$351$$ −24.3171 −1.29795
$$352$$ 0 0
$$353$$ −0.605671 −0.0322366 −0.0161183 0.999870i $$-0.505131\pi$$
−0.0161183 + 0.999870i $$0.505131\pi$$
$$354$$ 0 0
$$355$$ 25.2978 1.34267
$$356$$ 0 0
$$357$$ −9.17599 −0.485645
$$358$$ 0 0
$$359$$ 11.6214 0.613354 0.306677 0.951814i $$-0.400783\pi$$
0.306677 + 0.951814i $$0.400783\pi$$
$$360$$ 0 0
$$361$$ −14.4701 −0.761585
$$362$$ 0 0
$$363$$ −16.5767 −0.870052
$$364$$ 0 0
$$365$$ 32.0871 1.67951
$$366$$ 0 0
$$367$$ 13.1299 0.685376 0.342688 0.939449i $$-0.388663\pi$$
0.342688 + 0.939449i $$0.388663\pi$$
$$368$$ 0 0
$$369$$ −0.420883 −0.0219103
$$370$$ 0 0
$$371$$ 7.90235 0.410270
$$372$$ 0 0
$$373$$ −18.4801 −0.956863 −0.478431 0.878125i $$-0.658794\pi$$
−0.478431 + 0.878125i $$0.658794\pi$$
$$374$$ 0 0
$$375$$ 55.9666 2.89010
$$376$$ 0 0
$$377$$ 26.3495 1.35707
$$378$$ 0 0
$$379$$ −4.13943 −0.212628 −0.106314 0.994333i $$-0.533905\pi$$
−0.106314 + 0.994333i $$0.533905\pi$$
$$380$$ 0 0
$$381$$ −14.0375 −0.719164
$$382$$ 0 0
$$383$$ 34.3667 1.75606 0.878029 0.478608i $$-0.158858\pi$$
0.878029 + 0.478608i $$0.158858\pi$$
$$384$$ 0 0
$$385$$ 18.8363 0.959987
$$386$$ 0 0
$$387$$ 0.432538 0.0219872
$$388$$ 0 0
$$389$$ 10.5339 0.534089 0.267044 0.963684i $$-0.413953\pi$$
0.267044 + 0.963684i $$0.413953\pi$$
$$390$$ 0 0
$$391$$ −36.5792 −1.84989
$$392$$ 0 0
$$393$$ −13.7507 −0.693633
$$394$$ 0 0
$$395$$ 7.95499 0.400259
$$396$$ 0 0
$$397$$ 11.2088 0.562554 0.281277 0.959627i $$-0.409242\pi$$
0.281277 + 0.959627i $$0.409242\pi$$
$$398$$ 0 0
$$399$$ −3.80079 −0.190278
$$400$$ 0 0
$$401$$ −15.4031 −0.769192 −0.384596 0.923085i $$-0.625659\pi$$
−0.384596 + 0.923085i $$0.625659\pi$$
$$402$$ 0 0
$$403$$ 4.02625 0.200562
$$404$$ 0 0
$$405$$ 39.8649 1.98090
$$406$$ 0 0
$$407$$ −35.9610 −1.78252
$$408$$ 0 0
$$409$$ 22.6029 1.11764 0.558820 0.829289i $$-0.311254\pi$$
0.558820 + 0.829289i $$0.311254\pi$$
$$410$$ 0 0
$$411$$ 31.6043 1.55892
$$412$$ 0 0
$$413$$ −2.62811 −0.129321
$$414$$ 0 0
$$415$$ −47.1964 −2.31678
$$416$$ 0 0
$$417$$ −28.9174 −1.41609
$$418$$ 0 0
$$419$$ −12.7468 −0.622720 −0.311360 0.950292i $$-0.600785\pi$$
−0.311360 + 0.950292i $$0.600785\pi$$
$$420$$ 0 0
$$421$$ −30.9413 −1.50799 −0.753994 0.656882i $$-0.771875\pi$$
−0.753994 + 0.656882i $$0.771875\pi$$
$$422$$ 0 0
$$423$$ 1.48164 0.0720399
$$424$$ 0 0
$$425$$ 64.1941 3.11387
$$426$$ 0 0
$$427$$ −2.33370 −0.112936
$$428$$ 0 0
$$429$$ −38.9601 −1.88101
$$430$$ 0 0
$$431$$ −13.1089 −0.631434 −0.315717 0.948853i $$-0.602245\pi$$
−0.315717 + 0.948853i $$0.602245\pi$$
$$432$$ 0 0
$$433$$ 35.1859 1.69093 0.845463 0.534033i $$-0.179324\pi$$
0.845463 + 0.534033i $$0.179324\pi$$
$$434$$ 0 0
$$435$$ −40.6265 −1.94789
$$436$$ 0 0
$$437$$ −15.1515 −0.724795
$$438$$ 0 0
$$439$$ 4.82251 0.230166 0.115083 0.993356i $$-0.463287\pi$$
0.115083 + 0.993356i $$0.463287\pi$$
$$440$$ 0 0
$$441$$ 0.189043 0.00900204
$$442$$ 0 0
$$443$$ 17.7618 0.843888 0.421944 0.906622i $$-0.361348\pi$$
0.421944 + 0.906622i $$0.361348\pi$$
$$444$$ 0 0
$$445$$ 10.4536 0.495548
$$446$$ 0 0
$$447$$ 21.7520 1.02883
$$448$$ 0 0
$$449$$ −29.9204 −1.41203 −0.706015 0.708197i $$-0.749509\pi$$
−0.706015 + 0.708197i $$0.749509\pi$$
$$450$$ 0 0
$$451$$ 10.0268 0.472144
$$452$$ 0 0
$$453$$ 31.6886 1.48886
$$454$$ 0 0
$$455$$ 20.2611 0.949853
$$456$$ 0 0
$$457$$ 29.1293 1.36261 0.681305 0.732000i $$-0.261413\pi$$
0.681305 + 0.732000i $$0.261413\pi$$
$$458$$ 0 0
$$459$$ 25.7933 1.20393
$$460$$ 0 0
$$461$$ −10.1181 −0.471247 −0.235624 0.971844i $$-0.575713\pi$$
−0.235624 + 0.971844i $$0.575713\pi$$
$$462$$ 0 0
$$463$$ −40.1547 −1.86615 −0.933074 0.359686i $$-0.882884\pi$$
−0.933074 + 0.359686i $$0.882884\pi$$
$$464$$ 0 0
$$465$$ −6.20779 −0.287880
$$466$$ 0 0
$$467$$ 40.3127 1.86545 0.932726 0.360587i $$-0.117423\pi$$
0.932726 + 0.360587i $$0.117423\pi$$
$$468$$ 0 0
$$469$$ 8.16822 0.377174
$$470$$ 0 0
$$471$$ −24.9633 −1.15025
$$472$$ 0 0
$$473$$ −10.3045 −0.473800
$$474$$ 0 0
$$475$$ 26.5899 1.22003
$$476$$ 0 0
$$477$$ 1.49388 0.0684002
$$478$$ 0 0
$$479$$ 12.6994 0.580249 0.290124 0.956989i $$-0.406303\pi$$
0.290124 + 0.956989i $$0.406303\pi$$
$$480$$ 0 0
$$481$$ −38.6811 −1.76371
$$482$$ 0 0
$$483$$ 12.7128 0.578454
$$484$$ 0 0
$$485$$ 8.31409 0.377523
$$486$$ 0 0
$$487$$ −34.2373 −1.55144 −0.775721 0.631076i $$-0.782614\pi$$
−0.775721 + 0.631076i $$0.782614\pi$$
$$488$$ 0 0
$$489$$ −24.2799 −1.09798
$$490$$ 0 0
$$491$$ 30.2843 1.36671 0.683356 0.730085i $$-0.260520\pi$$
0.683356 + 0.730085i $$0.260520\pi$$
$$492$$ 0 0
$$493$$ −27.9491 −1.25876
$$494$$ 0 0
$$495$$ 3.56087 0.160049
$$496$$ 0 0
$$497$$ −6.04851 −0.271313
$$498$$ 0 0
$$499$$ 22.2171 0.994574 0.497287 0.867586i $$-0.334330\pi$$
0.497287 + 0.867586i $$0.334330\pi$$
$$500$$ 0 0
$$501$$ 11.9779 0.535134
$$502$$ 0 0
$$503$$ −28.6238 −1.27627 −0.638137 0.769923i $$-0.720294\pi$$
−0.638137 + 0.769923i $$0.720294\pi$$
$$504$$ 0 0
$$505$$ −65.8571 −2.93060
$$506$$ 0 0
$$507$$ −18.6917 −0.830128
$$508$$ 0 0
$$509$$ −21.2622 −0.942431 −0.471215 0.882018i $$-0.656185\pi$$
−0.471215 + 0.882018i $$0.656185\pi$$
$$510$$ 0 0
$$511$$ −7.67177 −0.339379
$$512$$ 0 0
$$513$$ 10.6839 0.471704
$$514$$ 0 0
$$515$$ −63.6180 −2.80334
$$516$$ 0 0
$$517$$ −35.2975 −1.55238
$$518$$ 0 0
$$519$$ 37.2083 1.63326
$$520$$ 0 0
$$521$$ −23.7033 −1.03846 −0.519230 0.854635i $$-0.673781\pi$$
−0.519230 + 0.854635i $$0.673781\pi$$
$$522$$ 0 0
$$523$$ −19.6640 −0.859847 −0.429923 0.902865i $$-0.641459\pi$$
−0.429923 + 0.902865i $$0.641459\pi$$
$$524$$ 0 0
$$525$$ −22.3101 −0.973694
$$526$$ 0 0
$$527$$ −4.27067 −0.186033
$$528$$ 0 0
$$529$$ 27.6785 1.20341
$$530$$ 0 0
$$531$$ −0.496825 −0.0215604
$$532$$ 0 0
$$533$$ 10.7852 0.467160
$$534$$ 0 0
$$535$$ 5.30928 0.229540
$$536$$ 0 0
$$537$$ −40.0170 −1.72686
$$538$$ 0 0
$$539$$ −4.50362 −0.193985
$$540$$ 0 0
$$541$$ 12.2602 0.527106 0.263553 0.964645i $$-0.415106\pi$$
0.263553 + 0.964645i $$0.415106\pi$$
$$542$$ 0 0
$$543$$ 7.91002 0.339451
$$544$$ 0 0
$$545$$ 16.8296 0.720901
$$546$$ 0 0
$$547$$ 5.75986 0.246274 0.123137 0.992390i $$-0.460705\pi$$
0.123137 + 0.992390i $$0.460705\pi$$
$$548$$ 0 0
$$549$$ −0.441169 −0.0188286
$$550$$ 0 0
$$551$$ −11.5768 −0.493188
$$552$$ 0 0
$$553$$ −1.90198 −0.0808804
$$554$$ 0 0
$$555$$ 59.6396 2.53156
$$556$$ 0 0
$$557$$ −2.57936 −0.109291 −0.0546455 0.998506i $$-0.517403\pi$$
−0.0546455 + 0.998506i $$0.517403\pi$$
$$558$$ 0 0
$$559$$ −11.0839 −0.468798
$$560$$ 0 0
$$561$$ 41.3252 1.74475
$$562$$ 0 0
$$563$$ 15.1195 0.637212 0.318606 0.947887i $$-0.396785\pi$$
0.318606 + 0.947887i $$0.396785\pi$$
$$564$$ 0 0
$$565$$ −6.96329 −0.292948
$$566$$ 0 0
$$567$$ −9.53139 −0.400281
$$568$$ 0 0
$$569$$ 11.0034 0.461288 0.230644 0.973038i $$-0.425917\pi$$
0.230644 + 0.973038i $$0.425917\pi$$
$$570$$ 0 0
$$571$$ −40.9982 −1.71572 −0.857861 0.513881i $$-0.828207\pi$$
−0.857861 + 0.513881i $$0.828207\pi$$
$$572$$ 0 0
$$573$$ 13.1966 0.551294
$$574$$ 0 0
$$575$$ −88.9373 −3.70894
$$576$$ 0 0
$$577$$ 2.85596 0.118895 0.0594476 0.998231i $$-0.481066\pi$$
0.0594476 + 0.998231i $$0.481066\pi$$
$$578$$ 0 0
$$579$$ −0.248472 −0.0103261
$$580$$ 0 0
$$581$$ 11.2843 0.468152
$$582$$ 0 0
$$583$$ −35.5892 −1.47395
$$584$$ 0 0
$$585$$ 3.83021 0.158360
$$586$$ 0 0
$$587$$ −24.8280 −1.02476 −0.512381 0.858758i $$-0.671237\pi$$
−0.512381 + 0.858758i $$0.671237\pi$$
$$588$$ 0 0
$$589$$ −1.76895 −0.0728885
$$590$$ 0 0
$$591$$ 18.5824 0.764379
$$592$$ 0 0
$$593$$ −7.72713 −0.317315 −0.158658 0.987334i $$-0.550717\pi$$
−0.158658 + 0.987334i $$0.550717\pi$$
$$594$$ 0 0
$$595$$ −21.4910 −0.881045
$$596$$ 0 0
$$597$$ −9.09950 −0.372418
$$598$$ 0 0
$$599$$ 7.28771 0.297768 0.148884 0.988855i $$-0.452432\pi$$
0.148884 + 0.988855i $$0.452432\pi$$
$$600$$ 0 0
$$601$$ 20.0313 0.817095 0.408547 0.912737i $$-0.366035\pi$$
0.408547 + 0.912737i $$0.366035\pi$$
$$602$$ 0 0
$$603$$ 1.54414 0.0628824
$$604$$ 0 0
$$605$$ −38.8242 −1.57843
$$606$$ 0 0
$$607$$ −17.2219 −0.699014 −0.349507 0.936934i $$-0.613651\pi$$
−0.349507 + 0.936934i $$0.613651\pi$$
$$608$$ 0 0
$$609$$ 9.71349 0.393610
$$610$$ 0 0
$$611$$ −37.9674 −1.53600
$$612$$ 0 0
$$613$$ 37.3993 1.51054 0.755271 0.655412i $$-0.227505\pi$$
0.755271 + 0.655412i $$0.227505\pi$$
$$614$$ 0 0
$$615$$ −16.6290 −0.670545
$$616$$ 0 0
$$617$$ −22.1036 −0.889856 −0.444928 0.895566i $$-0.646771\pi$$
−0.444928 + 0.895566i $$0.646771\pi$$
$$618$$ 0 0
$$619$$ −30.4908 −1.22553 −0.612764 0.790266i $$-0.709942\pi$$
−0.612764 + 0.790266i $$0.709942\pi$$
$$620$$ 0 0
$$621$$ −35.7352 −1.43400
$$622$$ 0 0
$$623$$ −2.49938 −0.100135
$$624$$ 0 0
$$625$$ 68.6132 2.74453
$$626$$ 0 0
$$627$$ 17.1173 0.683600
$$628$$ 0 0
$$629$$ 41.0292 1.63594
$$630$$ 0 0
$$631$$ 40.3151 1.60492 0.802458 0.596708i $$-0.203525\pi$$
0.802458 + 0.596708i $$0.203525\pi$$
$$632$$ 0 0
$$633$$ 22.4276 0.891417
$$634$$ 0 0
$$635$$ −32.8772 −1.30469
$$636$$ 0 0
$$637$$ −4.84427 −0.191937
$$638$$ 0 0
$$639$$ −1.14343 −0.0452333
$$640$$ 0 0
$$641$$ −0.875535 −0.0345815 −0.0172908 0.999851i $$-0.505504\pi$$
−0.0172908 + 0.999851i $$0.505504\pi$$
$$642$$ 0 0
$$643$$ −40.4431 −1.59492 −0.797459 0.603373i $$-0.793823\pi$$
−0.797459 + 0.603373i $$0.793823\pi$$
$$644$$ 0 0
$$645$$ 17.0895 0.672897
$$646$$ 0 0
$$647$$ 3.32973 0.130905 0.0654526 0.997856i $$-0.479151\pi$$
0.0654526 + 0.997856i $$0.479151\pi$$
$$648$$ 0 0
$$649$$ 11.8360 0.464603
$$650$$ 0 0
$$651$$ 1.48424 0.0581718
$$652$$ 0 0
$$653$$ −29.3589 −1.14890 −0.574452 0.818539i $$-0.694785\pi$$
−0.574452 + 0.818539i $$0.694785\pi$$
$$654$$ 0 0
$$655$$ −32.2055 −1.25837
$$656$$ 0 0
$$657$$ −1.45029 −0.0565814
$$658$$ 0 0
$$659$$ −29.2836 −1.14073 −0.570364 0.821392i $$-0.693198\pi$$
−0.570364 + 0.821392i $$0.693198\pi$$
$$660$$ 0 0
$$661$$ −13.7308 −0.534066 −0.267033 0.963687i $$-0.586043\pi$$
−0.267033 + 0.963687i $$0.586043\pi$$
$$662$$ 0 0
$$663$$ 44.4509 1.72633
$$664$$ 0 0
$$665$$ −8.90180 −0.345197
$$666$$ 0 0
$$667$$ 38.7219 1.49932
$$668$$ 0 0
$$669$$ −17.2677 −0.667606
$$670$$ 0 0
$$671$$ 10.5101 0.405737
$$672$$ 0 0
$$673$$ 49.3202 1.90115 0.950577 0.310490i $$-0.100493\pi$$
0.950577 + 0.310490i $$0.100493\pi$$
$$674$$ 0 0
$$675$$ 62.7129 2.41382
$$676$$ 0 0
$$677$$ 17.3000 0.664892 0.332446 0.943122i $$-0.392126\pi$$
0.332446 + 0.943122i $$0.392126\pi$$
$$678$$ 0 0
$$679$$ −1.98784 −0.0762861
$$680$$ 0 0
$$681$$ −5.35011 −0.205017
$$682$$ 0 0
$$683$$ 5.37135 0.205529 0.102764 0.994706i $$-0.467231\pi$$
0.102764 + 0.994706i $$0.467231\pi$$
$$684$$ 0 0
$$685$$ 74.0202 2.82816
$$686$$ 0 0
$$687$$ 16.0644 0.612895
$$688$$ 0 0
$$689$$ −38.2811 −1.45839
$$690$$ 0 0
$$691$$ −44.8853 −1.70752 −0.853759 0.520668i $$-0.825683\pi$$
−0.853759 + 0.520668i $$0.825683\pi$$
$$692$$ 0 0
$$693$$ −0.851377 −0.0323411
$$694$$ 0 0
$$695$$ −67.7271 −2.56904
$$696$$ 0 0
$$697$$ −11.4399 −0.433319
$$698$$ 0 0
$$699$$ −13.4482 −0.508657
$$700$$ 0 0
$$701$$ 8.61013 0.325200 0.162600 0.986692i $$-0.448012\pi$$
0.162600 + 0.986692i $$0.448012\pi$$
$$702$$ 0 0
$$703$$ 16.9947 0.640969
$$704$$ 0 0
$$705$$ 58.5392 2.20472
$$706$$ 0 0
$$707$$ 15.7459 0.592187
$$708$$ 0 0
$$709$$ −3.07942 −0.115650 −0.0578250 0.998327i $$-0.518417\pi$$
−0.0578250 + 0.998327i $$0.518417\pi$$
$$710$$ 0 0
$$711$$ −0.359556 −0.0134844
$$712$$ 0 0
$$713$$ 5.91677 0.221585
$$714$$ 0 0
$$715$$ −91.2481 −3.41248
$$716$$ 0 0
$$717$$ 3.34071 0.124761
$$718$$ 0 0
$$719$$ 42.5677 1.58751 0.793754 0.608239i $$-0.208124\pi$$
0.793754 + 0.608239i $$0.208124\pi$$
$$720$$ 0 0
$$721$$ 15.2106 0.566471
$$722$$ 0 0
$$723$$ −25.8781 −0.962416
$$724$$ 0 0
$$725$$ −67.9543 −2.52376
$$726$$ 0 0
$$727$$ −3.44163 −0.127643 −0.0638214 0.997961i $$-0.520329\pi$$
−0.0638214 + 0.997961i $$0.520329\pi$$
$$728$$ 0 0
$$729$$ 25.0909 0.929294
$$730$$ 0 0
$$731$$ 11.7567 0.434839
$$732$$ 0 0
$$733$$ −12.1501 −0.448775 −0.224388 0.974500i $$-0.572038\pi$$
−0.224388 + 0.974500i $$0.572038\pi$$
$$734$$ 0 0
$$735$$ 7.46903 0.275499
$$736$$ 0 0
$$737$$ −36.7866 −1.35505
$$738$$ 0 0
$$739$$ −27.9556 −1.02836 −0.514181 0.857682i $$-0.671904\pi$$
−0.514181 + 0.857682i $$0.671904\pi$$
$$740$$ 0 0
$$741$$ 18.4120 0.676383
$$742$$ 0 0
$$743$$ −14.0786 −0.516495 −0.258248 0.966079i $$-0.583145\pi$$
−0.258248 + 0.966079i $$0.583145\pi$$
$$744$$ 0 0
$$745$$ 50.9452 1.86649
$$746$$ 0 0
$$747$$ 2.13322 0.0780503
$$748$$ 0 0
$$749$$ −1.26941 −0.0463832
$$750$$ 0 0
$$751$$ 27.6318 1.00830 0.504148 0.863617i $$-0.331806\pi$$
0.504148 + 0.863617i $$0.331806\pi$$
$$752$$ 0 0
$$753$$ 11.3214 0.412576
$$754$$ 0 0
$$755$$ 74.2176 2.70106
$$756$$ 0 0
$$757$$ 2.76084 0.100344 0.0501722 0.998741i $$-0.484023\pi$$
0.0501722 + 0.998741i $$0.484023\pi$$
$$758$$ 0 0
$$759$$ −57.2537 −2.07818
$$760$$ 0 0
$$761$$ 34.5598 1.25279 0.626396 0.779505i $$-0.284529\pi$$
0.626396 + 0.779505i $$0.284529\pi$$
$$762$$ 0 0
$$763$$ −4.02383 −0.145673
$$764$$ 0 0
$$765$$ −4.06272 −0.146888
$$766$$ 0 0
$$767$$ 12.7313 0.459699
$$768$$ 0 0
$$769$$ 49.7370 1.79356 0.896781 0.442474i $$-0.145899\pi$$
0.896781 + 0.442474i $$0.145899\pi$$
$$770$$ 0 0
$$771$$ −21.7141 −0.782015
$$772$$ 0 0
$$773$$ −14.1791 −0.509986 −0.254993 0.966943i $$-0.582073\pi$$
−0.254993 + 0.966943i $$0.582073\pi$$
$$774$$ 0 0
$$775$$ −10.3835 −0.372988
$$776$$ 0 0
$$777$$ −14.2594 −0.511553
$$778$$ 0 0
$$779$$ −4.73855 −0.169776
$$780$$ 0 0
$$781$$ 27.2402 0.974730
$$782$$ 0 0
$$783$$ −27.3042 −0.975772
$$784$$ 0 0
$$785$$ −58.4664 −2.08675
$$786$$ 0 0
$$787$$ 13.1039 0.467104 0.233552 0.972344i $$-0.424965\pi$$
0.233552 + 0.972344i $$0.424965\pi$$
$$788$$ 0 0
$$789$$ −0.0534896 −0.00190428
$$790$$ 0 0
$$791$$ 1.66487 0.0591960
$$792$$ 0 0
$$793$$ 11.3050 0.401454
$$794$$ 0 0
$$795$$ 59.0229 2.09333
$$796$$ 0 0
$$797$$ −30.2441 −1.07130 −0.535651 0.844440i $$-0.679934\pi$$
−0.535651 + 0.844440i $$0.679934\pi$$
$$798$$ 0 0
$$799$$ 40.2722 1.42473
$$800$$ 0 0
$$801$$ −0.472490 −0.0166946
$$802$$ 0 0
$$803$$ 34.5507 1.21927
$$804$$ 0 0
$$805$$ 29.7746 1.04942
$$806$$ 0 0
$$807$$ −32.2553 −1.13544
$$808$$ 0 0
$$809$$ −1.36939 −0.0481450 −0.0240725 0.999710i $$-0.507663\pi$$
−0.0240725 + 0.999710i $$0.507663\pi$$
$$810$$ 0 0
$$811$$ 29.7374 1.04422 0.522110 0.852878i $$-0.325145\pi$$
0.522110 + 0.852878i $$0.325145\pi$$
$$812$$ 0 0
$$813$$ −18.0196 −0.631976
$$814$$ 0 0
$$815$$ −56.8659 −1.99192
$$816$$ 0 0
$$817$$ 4.86976 0.170371
$$818$$ 0 0
$$819$$ −0.915774 −0.0319997
$$820$$ 0 0
$$821$$ 18.1869 0.634727 0.317363 0.948304i $$-0.397203\pi$$
0.317363 + 0.948304i $$0.397203\pi$$
$$822$$ 0 0
$$823$$ 41.5715 1.44909 0.724546 0.689226i $$-0.242049\pi$$
0.724546 + 0.689226i $$0.242049\pi$$
$$824$$ 0 0
$$825$$ 100.476 3.49814
$$826$$ 0 0
$$827$$ −46.0708 −1.60204 −0.801019 0.598639i $$-0.795708\pi$$
−0.801019 + 0.598639i $$0.795708\pi$$
$$828$$ 0 0
$$829$$ 0.574936 0.0199683 0.00998417 0.999950i $$-0.496822\pi$$
0.00998417 + 0.999950i $$0.496822\pi$$
$$830$$ 0 0
$$831$$ 15.8337 0.549265
$$832$$ 0 0
$$833$$ 5.13834 0.178033
$$834$$ 0 0
$$835$$ 28.0534 0.970827
$$836$$ 0 0
$$837$$ −4.17213 −0.144210
$$838$$ 0 0
$$839$$ 28.6700 0.989799 0.494899 0.868950i $$-0.335205\pi$$
0.494899 + 0.868950i $$0.335205\pi$$
$$840$$ 0 0
$$841$$ 0.586242 0.0202152
$$842$$ 0 0
$$843$$ 20.8456 0.717961
$$844$$ 0 0
$$845$$ −43.7777 −1.50600
$$846$$ 0 0
$$847$$ 9.28258 0.318953
$$848$$ 0 0
$$849$$ −22.4200 −0.769452
$$850$$ 0 0
$$851$$ −56.8437 −1.94858
$$852$$ 0 0
$$853$$ 11.8044 0.404175 0.202087 0.979367i $$-0.435227\pi$$
0.202087 + 0.979367i $$0.435227\pi$$
$$854$$ 0 0
$$855$$ −1.68282 −0.0575513
$$856$$ 0 0
$$857$$ 48.6998 1.66355 0.831777 0.555110i $$-0.187324\pi$$
0.831777 + 0.555110i $$0.187324\pi$$
$$858$$ 0 0
$$859$$ −0.0381681 −0.00130228 −0.000651140 1.00000i $$-0.500207\pi$$
−0.000651140 1.00000i $$0.500207\pi$$
$$860$$ 0 0
$$861$$ 3.97586 0.135497
$$862$$ 0 0
$$863$$ −1.64855 −0.0561173 −0.0280587 0.999606i $$-0.508933\pi$$
−0.0280587 + 0.999606i $$0.508933\pi$$
$$864$$ 0 0
$$865$$ 87.1452 2.96303
$$866$$ 0 0
$$867$$ −16.7909 −0.570250
$$868$$ 0 0
$$869$$ 8.56579 0.290574
$$870$$ 0 0
$$871$$ −39.5690 −1.34075
$$872$$ 0 0
$$873$$ −0.375786 −0.0127184
$$874$$ 0 0
$$875$$ −31.3400 −1.05949
$$876$$ 0 0
$$877$$ −44.2259 −1.49340 −0.746701 0.665160i $$-0.768363\pi$$
−0.746701 + 0.665160i $$0.768363\pi$$
$$878$$ 0 0
$$879$$ 43.4219 1.46458
$$880$$ 0 0
$$881$$ 1.02249 0.0344486 0.0172243 0.999852i $$-0.494517\pi$$
0.0172243 + 0.999852i $$0.494517\pi$$
$$882$$ 0 0
$$883$$ 10.6948 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$884$$ 0 0
$$885$$ −19.6294 −0.659836
$$886$$ 0 0
$$887$$ −4.15374 −0.139469 −0.0697344 0.997566i $$-0.522215\pi$$
−0.0697344 + 0.997566i $$0.522215\pi$$
$$888$$ 0 0
$$889$$ 7.86069 0.263639
$$890$$ 0 0
$$891$$ 42.9257 1.43807
$$892$$ 0 0
$$893$$ 16.6812 0.558214
$$894$$ 0 0
$$895$$ −93.7235 −3.13283
$$896$$ 0 0
$$897$$ −61.5843 −2.05624
$$898$$ 0 0
$$899$$ 4.52083 0.150778
$$900$$ 0 0
$$901$$ 40.6050 1.35275
$$902$$ 0 0
$$903$$ −4.08596 −0.135972
$$904$$ 0 0
$$905$$ 18.5260 0.615825
$$906$$ 0 0
$$907$$ −4.85902 −0.161341 −0.0806706 0.996741i $$-0.525706\pi$$
−0.0806706 + 0.996741i $$0.525706\pi$$
$$908$$ 0 0
$$909$$ 2.97666 0.0987295
$$910$$ 0 0
$$911$$ 21.2511 0.704082 0.352041 0.935985i $$-0.385488\pi$$
0.352041 + 0.935985i $$0.385488\pi$$
$$912$$ 0 0
$$913$$ −50.8202 −1.68190
$$914$$ 0 0
$$915$$ −17.4305 −0.576233
$$916$$ 0 0
$$917$$ 7.70009 0.254279
$$918$$ 0 0
$$919$$ −14.2571 −0.470297 −0.235149 0.971959i $$-0.575558\pi$$
−0.235149 + 0.971959i $$0.575558\pi$$
$$920$$ 0 0
$$921$$ 48.6692 1.60370
$$922$$ 0 0
$$923$$ 29.3006 0.964441
$$924$$ 0 0
$$925$$ 99.7569 3.27999
$$926$$ 0 0
$$927$$ 2.87545 0.0944422
$$928$$ 0 0
$$929$$ 31.9394 1.04790 0.523948 0.851750i $$-0.324458\pi$$
0.523948 + 0.851750i $$0.324458\pi$$
$$930$$ 0 0
$$931$$ 2.12835 0.0697540
$$932$$ 0 0
$$933$$ 10.3335 0.338303
$$934$$ 0 0
$$935$$ 96.7873 3.16528
$$936$$ 0 0
$$937$$ 4.91109 0.160438 0.0802191 0.996777i $$-0.474438\pi$$
0.0802191 + 0.996777i $$0.474438\pi$$
$$938$$ 0 0
$$939$$ −5.42296 −0.176972
$$940$$ 0 0
$$941$$ 55.0908 1.79591 0.897954 0.440090i $$-0.145053\pi$$
0.897954 + 0.440090i $$0.145053\pi$$
$$942$$ 0 0
$$943$$ 15.8494 0.516128
$$944$$ 0 0
$$945$$ −20.9951 −0.682972
$$946$$ 0 0
$$947$$ 49.0597 1.59423 0.797113 0.603830i $$-0.206359\pi$$
0.797113 + 0.603830i $$0.206359\pi$$
$$948$$ 0 0
$$949$$ 37.1641 1.20640
$$950$$ 0 0
$$951$$ 8.39014 0.272069
$$952$$ 0 0
$$953$$ −56.6312 −1.83446 −0.917232 0.398354i $$-0.869582\pi$$
−0.917232 + 0.398354i $$0.869582\pi$$
$$954$$ 0 0
$$955$$ 30.9075 1.00014
$$956$$ 0 0
$$957$$ −43.7458 −1.41410
$$958$$ 0 0
$$959$$ −17.6977 −0.571487
$$960$$ 0 0
$$961$$ −30.3092 −0.977716
$$962$$ 0 0
$$963$$ −0.239973 −0.00773302
$$964$$ 0 0
$$965$$ −0.581943 −0.0187334
$$966$$ 0 0
$$967$$ 47.2953 1.52092 0.760458 0.649388i $$-0.224975\pi$$
0.760458 + 0.649388i $$0.224975\pi$$
$$968$$ 0 0
$$969$$ −19.5298 −0.627386
$$970$$ 0 0
$$971$$ 2.80126 0.0898966 0.0449483 0.998989i $$-0.485688\pi$$
0.0449483 + 0.998989i $$0.485688\pi$$
$$972$$ 0 0
$$973$$ 16.1930 0.519125
$$974$$ 0 0
$$975$$ 108.076 3.46121
$$976$$ 0 0
$$977$$ −22.0570 −0.705665 −0.352832 0.935687i $$-0.614781\pi$$
−0.352832 + 0.935687i $$0.614781\pi$$
$$978$$ 0 0
$$979$$ 11.2562 0.359751
$$980$$ 0 0
$$981$$ −0.760677 −0.0242865
$$982$$ 0 0
$$983$$ 55.5608 1.77212 0.886058 0.463575i $$-0.153434\pi$$
0.886058 + 0.463575i $$0.153434\pi$$
$$984$$ 0 0
$$985$$ 43.5217 1.38672
$$986$$ 0 0
$$987$$ −13.9963 −0.445507
$$988$$ 0 0
$$989$$ −16.2883 −0.517938
$$990$$ 0 0
$$991$$ −13.9880 −0.444345 −0.222173 0.975007i $$-0.571315\pi$$
−0.222173 + 0.975007i $$0.571315\pi$$
$$992$$ 0 0
$$993$$ −64.4042 −2.04380
$$994$$ 0 0
$$995$$ −21.3119 −0.675631
$$996$$ 0 0
$$997$$ 13.9439 0.441608 0.220804 0.975318i $$-0.429132\pi$$
0.220804 + 0.975318i $$0.429132\pi$$
$$998$$ 0 0
$$999$$ 40.0825 1.26816
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 7168.2.a.bb.1.3 8
4.3 odd 2 7168.2.a.ba.1.6 8
8.3 odd 2 7168.2.a.be.1.3 8
8.5 even 2 7168.2.a.bf.1.6 8
32.3 odd 8 1792.2.m.f.1345.7 yes 16
32.5 even 8 1792.2.m.e.449.7 16
32.11 odd 8 1792.2.m.f.449.7 yes 16
32.13 even 8 1792.2.m.e.1345.7 yes 16
32.19 odd 8 1792.2.m.g.1345.2 yes 16
32.21 even 8 1792.2.m.h.449.2 yes 16
32.27 odd 8 1792.2.m.g.449.2 yes 16
32.29 even 8 1792.2.m.h.1345.2 yes 16

By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.7 16 32.5 even 8
1792.2.m.e.1345.7 yes 16 32.13 even 8
1792.2.m.f.449.7 yes 16 32.11 odd 8
1792.2.m.f.1345.7 yes 16 32.3 odd 8
1792.2.m.g.449.2 yes 16 32.27 odd 8
1792.2.m.g.1345.2 yes 16 32.19 odd 8
1792.2.m.h.449.2 yes 16 32.21 even 8
1792.2.m.h.1345.2 yes 16 32.29 even 8
7168.2.a.ba.1.6 8 4.3 odd 2
7168.2.a.bb.1.3 8 1.1 even 1 trivial
7168.2.a.be.1.3 8 8.3 odd 2
7168.2.a.bf.1.6 8 8.5 even 2