# Properties

 Label 1216.2.a.w.1.2 Level $1216$ Weight $2$ Character 1216.1 Self dual yes Analytic conductor $9.710$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.70980888579$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.69353$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.56155 q^{3} +4.20608 q^{5} +1.74252 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q-2.56155 q^{3} +4.20608 q^{5} +1.74252 q^{7} +3.56155 q^{9} -6.20608 q^{11} -4.82550 q^{13} -10.7741 q^{15} -5.12957 q^{17} -1.00000 q^{19} -4.46356 q^{21} +0.463560 q^{23} +12.6911 q^{25} -1.43845 q^{27} -4.82550 q^{29} -1.63806 q^{31} +15.8972 q^{33} +7.32919 q^{35} -3.63806 q^{37} +12.3608 q^{39} -5.28906 q^{41} -1.08298 q^{43} +14.9802 q^{45} -0.555087 q^{47} -3.96362 q^{49} +13.1397 q^{51} -2.65955 q^{53} -26.1033 q^{55} +2.56155 q^{57} +1.85061 q^{59} +9.32919 q^{61} +6.20608 q^{63} -20.2964 q^{65} -4.72751 q^{67} -1.18743 q^{69} -11.4850 q^{71} -0.00646614 q^{73} -32.5090 q^{75} -10.8142 q^{77} +6.76117 q^{79} -7.00000 q^{81} +11.8972 q^{83} -21.5754 q^{85} +12.3608 q^{87} -7.31909 q^{89} -8.40853 q^{91} +4.19598 q^{93} -4.20608 q^{95} -10.4122 q^{97} -22.1033 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 $$4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37} - 12 q^{39} - 2 q^{41} - 3 q^{43} + 7 q^{45} - 3 q^{47} + 7 q^{49} + 6 q^{51} - 4 q^{53} - 35 q^{55} + 2 q^{57} - 20 q^{59} + 3 q^{61} + 7 q^{63} - 12 q^{65} - 8 q^{67} + 4 q^{69} - 30 q^{71} + 9 q^{73} - 34 q^{75} + 7 q^{77} + 10 q^{79} - 28 q^{81} - 4 q^{83} - 19 q^{85} - 12 q^{87} - 16 q^{89} - 10 q^{91} + 20 q^{93} + q^{95} - 6 q^{97} - 19 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * q^99

## 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
$$3$$ −2.56155 −1.47891 −0.739457 0.673204i $$-0.764917\pi$$
−0.739457 + 0.673204i $$0.764917\pi$$
$$4$$ 0 0
$$5$$ 4.20608 1.88102 0.940508 0.339770i $$-0.110349\pi$$
0.940508 + 0.339770i $$0.110349\pi$$
$$6$$ 0 0
$$7$$ 1.74252 0.658611 0.329306 0.944223i $$-0.393185\pi$$
0.329306 + 0.944223i $$0.393185\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ −6.20608 −1.87120 −0.935602 0.353056i $$-0.885142\pi$$
−0.935602 + 0.353056i $$0.885142\pi$$
$$12$$ 0 0
$$13$$ −4.82550 −1.33835 −0.669176 0.743104i $$-0.733353\pi$$
−0.669176 + 0.743104i $$0.733353\pi$$
$$14$$ 0 0
$$15$$ −10.7741 −2.78186
$$16$$ 0 0
$$17$$ −5.12957 −1.24410 −0.622052 0.782976i $$-0.713701\pi$$
−0.622052 + 0.782976i $$0.713701\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −4.46356 −0.974029
$$22$$ 0 0
$$23$$ 0.463560 0.0966590 0.0483295 0.998831i $$-0.484610\pi$$
0.0483295 + 0.998831i $$0.484610\pi$$
$$24$$ 0 0
$$25$$ 12.6911 2.53822
$$26$$ 0 0
$$27$$ −1.43845 −0.276829
$$28$$ 0 0
$$29$$ −4.82550 −0.896072 −0.448036 0.894015i $$-0.647876\pi$$
−0.448036 + 0.894015i $$0.647876\pi$$
$$30$$ 0 0
$$31$$ −1.63806 −0.294205 −0.147102 0.989121i $$-0.546995\pi$$
−0.147102 + 0.989121i $$0.546995\pi$$
$$32$$ 0 0
$$33$$ 15.8972 2.76735
$$34$$ 0 0
$$35$$ 7.32919 1.23886
$$36$$ 0 0
$$37$$ −3.63806 −0.598094 −0.299047 0.954238i $$-0.596669\pi$$
−0.299047 + 0.954238i $$0.596669\pi$$
$$38$$ 0 0
$$39$$ 12.3608 1.97931
$$40$$ 0 0
$$41$$ −5.28906 −0.826012 −0.413006 0.910728i $$-0.635521\pi$$
−0.413006 + 0.910728i $$0.635521\pi$$
$$42$$ 0 0
$$43$$ −1.08298 −0.165152 −0.0825762 0.996585i $$-0.526315\pi$$
−0.0825762 + 0.996585i $$0.526315\pi$$
$$44$$ 0 0
$$45$$ 14.9802 2.23311
$$46$$ 0 0
$$47$$ −0.555087 −0.0809677 −0.0404839 0.999180i $$-0.512890\pi$$
−0.0404839 + 0.999180i $$0.512890\pi$$
$$48$$ 0 0
$$49$$ −3.96362 −0.566231
$$50$$ 0 0
$$51$$ 13.1397 1.83992
$$52$$ 0 0
$$53$$ −2.65955 −0.365317 −0.182658 0.983176i $$-0.558470\pi$$
−0.182658 + 0.983176i $$0.558470\pi$$
$$54$$ 0 0
$$55$$ −26.1033 −3.51977
$$56$$ 0 0
$$57$$ 2.56155 0.339286
$$58$$ 0 0
$$59$$ 1.85061 0.240929 0.120465 0.992718i $$-0.461562\pi$$
0.120465 + 0.992718i $$0.461562\pi$$
$$60$$ 0 0
$$61$$ 9.32919 1.19448 0.597240 0.802063i $$-0.296264\pi$$
0.597240 + 0.802063i $$0.296264\pi$$
$$62$$ 0 0
$$63$$ 6.20608 0.781893
$$64$$ 0 0
$$65$$ −20.2964 −2.51746
$$66$$ 0 0
$$67$$ −4.72751 −0.577557 −0.288778 0.957396i $$-0.593249\pi$$
−0.288778 + 0.957396i $$0.593249\pi$$
$$68$$ 0 0
$$69$$ −1.18743 −0.142950
$$70$$ 0 0
$$71$$ −11.4850 −1.36302 −0.681512 0.731807i $$-0.738677\pi$$
−0.681512 + 0.731807i $$0.738677\pi$$
$$72$$ 0 0
$$73$$ −0.00646614 −0.000756804 0 −0.000378402 1.00000i $$-0.500120\pi$$
−0.000378402 1.00000i $$0.500120\pi$$
$$74$$ 0 0
$$75$$ −32.5090 −3.75381
$$76$$ 0 0
$$77$$ −10.8142 −1.23240
$$78$$ 0 0
$$79$$ 6.76117 0.760691 0.380345 0.924844i $$-0.375805\pi$$
0.380345 + 0.924844i $$0.375805\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 11.8972 1.30589 0.652944 0.757406i $$-0.273534\pi$$
0.652944 + 0.757406i $$0.273534\pi$$
$$84$$ 0 0
$$85$$ −21.5754 −2.34018
$$86$$ 0 0
$$87$$ 12.3608 1.32521
$$88$$ 0 0
$$89$$ −7.31909 −0.775822 −0.387911 0.921697i $$-0.626803\pi$$
−0.387911 + 0.921697i $$0.626803\pi$$
$$90$$ 0 0
$$91$$ −8.40853 −0.881454
$$92$$ 0 0
$$93$$ 4.19598 0.435103
$$94$$ 0 0
$$95$$ −4.20608 −0.431535
$$96$$ 0 0
$$97$$ −10.4122 −1.05720 −0.528598 0.848873i $$-0.677282\pi$$
−0.528598 + 0.848873i $$0.677282\pi$$
$$98$$ 0 0
$$99$$ −22.1033 −2.22146
$$100$$ 0 0
$$101$$ −0.246211 −0.0244989 −0.0122495 0.999925i $$-0.503899\pi$$
−0.0122495 + 0.999925i $$0.503899\pi$$
$$102$$ 0 0
$$103$$ 0.710942 0.0700512 0.0350256 0.999386i $$-0.488849\pi$$
0.0350256 + 0.999386i $$0.488849\pi$$
$$104$$ 0 0
$$105$$ −18.7741 −1.83216
$$106$$ 0 0
$$107$$ 11.6976 1.13085 0.565424 0.824800i $$-0.308712\pi$$
0.565424 + 0.824800i $$0.308712\pi$$
$$108$$ 0 0
$$109$$ 2.99145 0.286529 0.143264 0.989684i $$-0.454240\pi$$
0.143264 + 0.989684i $$0.454240\pi$$
$$110$$ 0 0
$$111$$ 9.31909 0.884529
$$112$$ 0 0
$$113$$ 6.41216 0.603206 0.301603 0.953434i $$-0.402478\pi$$
0.301603 + 0.953434i $$0.402478\pi$$
$$114$$ 0 0
$$115$$ 1.94977 0.181817
$$116$$ 0 0
$$117$$ −17.1863 −1.58887
$$118$$ 0 0
$$119$$ −8.93839 −0.819381
$$120$$ 0 0
$$121$$ 27.5155 2.50140
$$122$$ 0 0
$$123$$ 13.5482 1.22160
$$124$$ 0 0
$$125$$ 32.3495 2.89343
$$126$$ 0 0
$$127$$ −18.1434 −1.60997 −0.804984 0.593297i $$-0.797826\pi$$
−0.804984 + 0.593297i $$0.797826\pi$$
$$128$$ 0 0
$$129$$ 2.77410 0.244246
$$130$$ 0 0
$$131$$ −12.4652 −1.08909 −0.544546 0.838731i $$-0.683298\pi$$
−0.544546 + 0.838731i $$0.683298\pi$$
$$132$$ 0 0
$$133$$ −1.74252 −0.151096
$$134$$ 0 0
$$135$$ −6.05023 −0.520721
$$136$$ 0 0
$$137$$ 8.41863 0.719252 0.359626 0.933097i $$-0.382904\pi$$
0.359626 + 0.933097i $$0.382904\pi$$
$$138$$ 0 0
$$139$$ −11.3292 −0.960929 −0.480465 0.877014i $$-0.659532\pi$$
−0.480465 + 0.877014i $$0.659532\pi$$
$$140$$ 0 0
$$141$$ 1.42188 0.119744
$$142$$ 0 0
$$143$$ 29.9474 2.50433
$$144$$ 0 0
$$145$$ −20.2964 −1.68553
$$146$$ 0 0
$$147$$ 10.1530 0.837407
$$148$$ 0 0
$$149$$ 0.321808 0.0263635 0.0131818 0.999913i $$-0.495804\pi$$
0.0131818 + 0.999913i $$0.495804\pi$$
$$150$$ 0 0
$$151$$ −2.95715 −0.240650 −0.120325 0.992735i $$-0.538394\pi$$
−0.120325 + 0.992735i $$0.538394\pi$$
$$152$$ 0 0
$$153$$ −18.2692 −1.47698
$$154$$ 0 0
$$155$$ −6.88983 −0.553404
$$156$$ 0 0
$$157$$ −6.52789 −0.520982 −0.260491 0.965476i $$-0.583884\pi$$
−0.260491 + 0.965476i $$0.583884\pi$$
$$158$$ 0 0
$$159$$ 6.81257 0.540272
$$160$$ 0 0
$$161$$ 0.807764 0.0636607
$$162$$ 0 0
$$163$$ −2.05023 −0.160586 −0.0802931 0.996771i $$-0.525586\pi$$
−0.0802931 + 0.996771i $$0.525586\pi$$
$$164$$ 0 0
$$165$$ 66.8650 5.20543
$$166$$ 0 0
$$167$$ −10.2462 −0.792876 −0.396438 0.918062i $$-0.629754\pi$$
−0.396438 + 0.918062i $$0.629754\pi$$
$$168$$ 0 0
$$169$$ 10.2854 0.791187
$$170$$ 0 0
$$171$$ −3.56155 −0.272359
$$172$$ 0 0
$$173$$ −8.05023 −0.612047 −0.306024 0.952024i $$-0.598999\pi$$
−0.306024 + 0.952024i $$0.598999\pi$$
$$174$$ 0 0
$$175$$ 22.1146 1.67170
$$176$$ 0 0
$$177$$ −4.74044 −0.356313
$$178$$ 0 0
$$179$$ −1.75379 −0.131084 −0.0655422 0.997850i $$-0.520878\pi$$
−0.0655422 + 0.997850i $$0.520878\pi$$
$$180$$ 0 0
$$181$$ 11.0203 0.819133 0.409567 0.912280i $$-0.365680\pi$$
0.409567 + 0.912280i $$0.365680\pi$$
$$182$$ 0 0
$$183$$ −23.8972 −1.76653
$$184$$ 0 0
$$185$$ −15.3020 −1.12502
$$186$$ 0 0
$$187$$ 31.8345 2.32797
$$188$$ 0 0
$$189$$ −2.50652 −0.182323
$$190$$ 0 0
$$191$$ −13.2276 −0.957113 −0.478556 0.878057i $$-0.658840\pi$$
−0.478556 + 0.878057i $$0.658840\pi$$
$$192$$ 0 0
$$193$$ 1.47211 0.105965 0.0529824 0.998595i $$-0.483127\pi$$
0.0529824 + 0.998595i $$0.483127\pi$$
$$194$$ 0 0
$$195$$ 51.9904 3.72311
$$196$$ 0 0
$$197$$ −7.84698 −0.559074 −0.279537 0.960135i $$-0.590181\pi$$
−0.279537 + 0.960135i $$0.590181\pi$$
$$198$$ 0 0
$$199$$ −11.8057 −0.836882 −0.418441 0.908244i $$-0.637423\pi$$
−0.418441 + 0.908244i $$0.637423\pi$$
$$200$$ 0 0
$$201$$ 12.1098 0.854156
$$202$$ 0 0
$$203$$ −8.40853 −0.590163
$$204$$ 0 0
$$205$$ −22.2462 −1.55374
$$206$$ 0 0
$$207$$ 1.65100 0.114752
$$208$$ 0 0
$$209$$ 6.20608 0.429284
$$210$$ 0 0
$$211$$ 14.5745 1.00335 0.501674 0.865056i $$-0.332718\pi$$
0.501674 + 0.865056i $$0.332718\pi$$
$$212$$ 0 0
$$213$$ 29.4195 2.01579
$$214$$ 0 0
$$215$$ −4.55509 −0.310654
$$216$$ 0 0
$$217$$ −2.85436 −0.193767
$$218$$ 0 0
$$219$$ 0.0165634 0.00111925
$$220$$ 0 0
$$221$$ 24.7527 1.66505
$$222$$ 0 0
$$223$$ 26.6713 1.78604 0.893021 0.450014i $$-0.148581\pi$$
0.893021 + 0.450014i $$0.148581\pi$$
$$224$$ 0 0
$$225$$ 45.2001 3.01334
$$226$$ 0 0
$$227$$ 4.01656 0.266589 0.133294 0.991076i $$-0.457444\pi$$
0.133294 + 0.991076i $$0.457444\pi$$
$$228$$ 0 0
$$229$$ −23.1834 −1.53200 −0.766002 0.642838i $$-0.777757\pi$$
−0.766002 + 0.642838i $$0.777757\pi$$
$$230$$ 0 0
$$231$$ 27.7012 1.82261
$$232$$ 0 0
$$233$$ 22.2692 1.45891 0.729453 0.684031i $$-0.239775\pi$$
0.729453 + 0.684031i $$0.239775\pi$$
$$234$$ 0 0
$$235$$ −2.33474 −0.152302
$$236$$ 0 0
$$237$$ −17.3191 −1.12500
$$238$$ 0 0
$$239$$ −23.0818 −1.49304 −0.746519 0.665364i $$-0.768276\pi$$
−0.746519 + 0.665364i $$0.768276\pi$$
$$240$$ 0 0
$$241$$ −0.565184 −0.0364067 −0.0182033 0.999834i $$-0.505795\pi$$
−0.0182033 + 0.999834i $$0.505795\pi$$
$$242$$ 0 0
$$243$$ 22.2462 1.42710
$$244$$ 0 0
$$245$$ −16.6713 −1.06509
$$246$$ 0 0
$$247$$ 4.82550 0.307039
$$248$$ 0 0
$$249$$ −30.4753 −1.93130
$$250$$ 0 0
$$251$$ −15.8441 −1.00007 −0.500037 0.866004i $$-0.666680\pi$$
−0.500037 + 0.866004i $$0.666680\pi$$
$$252$$ 0 0
$$253$$ −2.87689 −0.180869
$$254$$ 0 0
$$255$$ 55.2665 3.46092
$$256$$ 0 0
$$257$$ 12.0632 0.752479 0.376240 0.926522i $$-0.377217\pi$$
0.376240 + 0.926522i $$0.377217\pi$$
$$258$$ 0 0
$$259$$ −6.33940 −0.393911
$$260$$ 0 0
$$261$$ −17.1863 −1.06380
$$262$$ 0 0
$$263$$ 23.2135 1.43140 0.715702 0.698406i $$-0.246107\pi$$
0.715702 + 0.698406i $$0.246107\pi$$
$$264$$ 0 0
$$265$$ −11.1863 −0.687167
$$266$$ 0 0
$$267$$ 18.7482 1.14737
$$268$$ 0 0
$$269$$ −26.2964 −1.60332 −0.801661 0.597779i $$-0.796050\pi$$
−0.801661 + 0.597779i $$0.796050\pi$$
$$270$$ 0 0
$$271$$ 12.9560 0.787020 0.393510 0.919320i $$-0.371261\pi$$
0.393510 + 0.919320i $$0.371261\pi$$
$$272$$ 0 0
$$273$$ 21.5389 1.30359
$$274$$ 0 0
$$275$$ −78.7622 −4.74954
$$276$$ 0 0
$$277$$ 10.8645 0.652782 0.326391 0.945235i $$-0.394167\pi$$
0.326391 + 0.945235i $$0.394167\pi$$
$$278$$ 0 0
$$279$$ −5.83405 −0.349275
$$280$$ 0 0
$$281$$ −15.8470 −0.945352 −0.472676 0.881236i $$-0.656712\pi$$
−0.472676 + 0.881236i $$0.656712\pi$$
$$282$$ 0 0
$$283$$ 27.6740 1.64505 0.822525 0.568729i $$-0.192565\pi$$
0.822525 + 0.568729i $$0.192565\pi$$
$$284$$ 0 0
$$285$$ 10.7741 0.638203
$$286$$ 0 0
$$287$$ −9.21630 −0.544021
$$288$$ 0 0
$$289$$ 9.31251 0.547795
$$290$$ 0 0
$$291$$ 26.6713 1.56350
$$292$$ 0 0
$$293$$ 21.3179 1.24541 0.622703 0.782458i $$-0.286034\pi$$
0.622703 + 0.782458i $$0.286034\pi$$
$$294$$ 0 0
$$295$$ 7.78382 0.453192
$$296$$ 0 0
$$297$$ 8.92712 0.518004
$$298$$ 0 0
$$299$$ −2.23691 −0.129364
$$300$$ 0 0
$$301$$ −1.88711 −0.108771
$$302$$ 0 0
$$303$$ 0.630683 0.0362318
$$304$$ 0 0
$$305$$ 39.2393 2.24684
$$306$$ 0 0
$$307$$ 26.6584 1.52147 0.760737 0.649060i $$-0.224838\pi$$
0.760737 + 0.649060i $$0.224838\pi$$
$$308$$ 0 0
$$309$$ −1.82112 −0.103600
$$310$$ 0 0
$$311$$ 20.0690 1.13801 0.569004 0.822335i $$-0.307329\pi$$
0.569004 + 0.822335i $$0.307329\pi$$
$$312$$ 0 0
$$313$$ −11.1397 −0.629651 −0.314826 0.949150i $$-0.601946\pi$$
−0.314826 + 0.949150i $$0.601946\pi$$
$$314$$ 0 0
$$315$$ 26.1033 1.47075
$$316$$ 0 0
$$317$$ −33.1349 −1.86104 −0.930520 0.366242i $$-0.880644\pi$$
−0.930520 + 0.366242i $$0.880644\pi$$
$$318$$ 0 0
$$319$$ 29.9474 1.67673
$$320$$ 0 0
$$321$$ −29.9640 −1.67243
$$322$$ 0 0
$$323$$ 5.12957 0.285417
$$324$$ 0 0
$$325$$ −61.2410 −3.39704
$$326$$ 0 0
$$327$$ −7.66276 −0.423751
$$328$$ 0 0
$$329$$ −0.967250 −0.0533262
$$330$$ 0 0
$$331$$ −4.30241 −0.236482 −0.118241 0.992985i $$-0.537726\pi$$
−0.118241 + 0.992985i $$0.537726\pi$$
$$332$$ 0 0
$$333$$ −12.9572 −0.710048
$$334$$ 0 0
$$335$$ −19.8843 −1.08639
$$336$$ 0 0
$$337$$ 31.3393 1.70716 0.853580 0.520962i $$-0.174427\pi$$
0.853580 + 0.520962i $$0.174427\pi$$
$$338$$ 0 0
$$339$$ −16.4251 −0.892089
$$340$$ 0 0
$$341$$ 10.1660 0.550517
$$342$$ 0 0
$$343$$ −19.1043 −1.03154
$$344$$ 0 0
$$345$$ −4.99445 −0.268892
$$346$$ 0 0
$$347$$ 21.6782 1.16375 0.581873 0.813280i $$-0.302320\pi$$
0.581873 + 0.813280i $$0.302320\pi$$
$$348$$ 0 0
$$349$$ −4.50486 −0.241140 −0.120570 0.992705i $$-0.538472\pi$$
−0.120570 + 0.992705i $$0.538472\pi$$
$$350$$ 0 0
$$351$$ 6.94122 0.370495
$$352$$ 0 0
$$353$$ −22.4158 −1.19307 −0.596536 0.802586i $$-0.703457\pi$$
−0.596536 + 0.802586i $$0.703457\pi$$
$$354$$ 0 0
$$355$$ −48.3070 −2.56387
$$356$$ 0 0
$$357$$ 22.8962 1.21179
$$358$$ 0 0
$$359$$ 11.4940 0.606629 0.303314 0.952891i $$-0.401907\pi$$
0.303314 + 0.952891i $$0.401907\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −70.4823 −3.69936
$$364$$ 0 0
$$365$$ −0.0271971 −0.00142356
$$366$$ 0 0
$$367$$ −19.7944 −1.03326 −0.516630 0.856209i $$-0.672814\pi$$
−0.516630 + 0.856209i $$0.672814\pi$$
$$368$$ 0 0
$$369$$ −18.8373 −0.980629
$$370$$ 0 0
$$371$$ −4.63431 −0.240602
$$372$$ 0 0
$$373$$ 16.6199 0.860546 0.430273 0.902699i $$-0.358417\pi$$
0.430273 + 0.902699i $$0.358417\pi$$
$$374$$ 0 0
$$375$$ −82.8650 −4.27913
$$376$$ 0 0
$$377$$ 23.2854 1.19926
$$378$$ 0 0
$$379$$ −16.3956 −0.842185 −0.421093 0.907018i $$-0.638353\pi$$
−0.421093 + 0.907018i $$0.638353\pi$$
$$380$$ 0 0
$$381$$ 46.4753 2.38100
$$382$$ 0 0
$$383$$ −30.6915 −1.56826 −0.784131 0.620595i $$-0.786891\pi$$
−0.784131 + 0.620595i $$0.786891\pi$$
$$384$$ 0 0
$$385$$ −45.4855 −2.31816
$$386$$ 0 0
$$387$$ −3.85708 −0.196066
$$388$$ 0 0
$$389$$ 2.24905 0.114031 0.0570156 0.998373i $$-0.481842\pi$$
0.0570156 + 0.998373i $$0.481842\pi$$
$$390$$ 0 0
$$391$$ −2.37787 −0.120254
$$392$$ 0 0
$$393$$ 31.9303 1.61067
$$394$$ 0 0
$$395$$ 28.4380 1.43087
$$396$$ 0 0
$$397$$ 4.12582 0.207069 0.103535 0.994626i $$-0.466985\pi$$
0.103535 + 0.994626i $$0.466985\pi$$
$$398$$ 0 0
$$399$$ 4.46356 0.223458
$$400$$ 0 0
$$401$$ −22.3992 −1.11856 −0.559282 0.828977i $$-0.688923\pi$$
−0.559282 + 0.828977i $$0.688923\pi$$
$$402$$ 0 0
$$403$$ 7.90447 0.393750
$$404$$ 0 0
$$405$$ −29.4426 −1.46301
$$406$$ 0 0
$$407$$ 22.5781 1.11916
$$408$$ 0 0
$$409$$ 13.1992 0.652658 0.326329 0.945256i $$-0.394188\pi$$
0.326329 + 0.945256i $$0.394188\pi$$
$$410$$ 0 0
$$411$$ −21.5648 −1.06371
$$412$$ 0 0
$$413$$ 3.22473 0.158679
$$414$$ 0 0
$$415$$ 50.0406 2.45640
$$416$$ 0 0
$$417$$ 29.0203 1.42113
$$418$$ 0 0
$$419$$ 31.5984 1.54368 0.771842 0.635814i $$-0.219336\pi$$
0.771842 + 0.635814i $$0.219336\pi$$
$$420$$ 0 0
$$421$$ 25.8587 1.26028 0.630139 0.776482i $$-0.282998\pi$$
0.630139 + 0.776482i $$0.282998\pi$$
$$422$$ 0 0
$$423$$ −1.97697 −0.0961236
$$424$$ 0 0
$$425$$ −65.1000 −3.15782
$$426$$ 0 0
$$427$$ 16.2563 0.786698
$$428$$ 0 0
$$429$$ −76.7119 −3.70369
$$430$$ 0 0
$$431$$ −31.8972 −1.53643 −0.768217 0.640189i $$-0.778856\pi$$
−0.768217 + 0.640189i $$0.778856\pi$$
$$432$$ 0 0
$$433$$ 23.1006 1.11014 0.555071 0.831803i $$-0.312691\pi$$
0.555071 + 0.831803i $$0.312691\pi$$
$$434$$ 0 0
$$435$$ 51.9904 2.49275
$$436$$ 0 0
$$437$$ −0.463560 −0.0221751
$$438$$ 0 0
$$439$$ 15.7944 0.753826 0.376913 0.926249i $$-0.376986\pi$$
0.376913 + 0.926249i $$0.376986\pi$$
$$440$$ 0 0
$$441$$ −14.1166 −0.672221
$$442$$ 0 0
$$443$$ −40.5584 −1.92699 −0.963494 0.267729i $$-0.913727\pi$$
−0.963494 + 0.267729i $$0.913727\pi$$
$$444$$ 0 0
$$445$$ −30.7847 −1.45933
$$446$$ 0 0
$$447$$ −0.824327 −0.0389893
$$448$$ 0 0
$$449$$ −30.4026 −1.43479 −0.717393 0.696669i $$-0.754665\pi$$
−0.717393 + 0.696669i $$0.754665\pi$$
$$450$$ 0 0
$$451$$ 32.8243 1.54564
$$452$$ 0 0
$$453$$ 7.57490 0.355900
$$454$$ 0 0
$$455$$ −35.3670 −1.65803
$$456$$ 0 0
$$457$$ 1.33516 0.0624561 0.0312280 0.999512i $$-0.490058\pi$$
0.0312280 + 0.999512i $$0.490058\pi$$
$$458$$ 0 0
$$459$$ 7.37862 0.344404
$$460$$ 0 0
$$461$$ 23.0604 1.07403 0.537016 0.843572i $$-0.319552\pi$$
0.537016 + 0.843572i $$0.319552\pi$$
$$462$$ 0 0
$$463$$ 9.29916 0.432168 0.216084 0.976375i $$-0.430671\pi$$
0.216084 + 0.976375i $$0.430671\pi$$
$$464$$ 0 0
$$465$$ 17.6487 0.818437
$$466$$ 0 0
$$467$$ 18.8344 0.871553 0.435777 0.900055i $$-0.356474\pi$$
0.435777 + 0.900055i $$0.356474\pi$$
$$468$$ 0 0
$$469$$ −8.23778 −0.380385
$$470$$ 0 0
$$471$$ 16.7215 0.770488
$$472$$ 0 0
$$473$$ 6.72104 0.309034
$$474$$ 0 0
$$475$$ −12.6911 −0.582309
$$476$$ 0 0
$$477$$ −9.47211 −0.433698
$$478$$ 0 0
$$479$$ 10.9701 0.501236 0.250618 0.968086i $$-0.419366\pi$$
0.250618 + 0.968086i $$0.419366\pi$$
$$480$$ 0 0
$$481$$ 17.5555 0.800460
$$482$$ 0 0
$$483$$ −2.06913 −0.0941487
$$484$$ 0 0
$$485$$ −43.7944 −1.98860
$$486$$ 0 0
$$487$$ 19.3895 0.878623 0.439311 0.898335i $$-0.355223\pi$$
0.439311 + 0.898335i $$0.355223\pi$$
$$488$$ 0 0
$$489$$ 5.25176 0.237493
$$490$$ 0 0
$$491$$ −33.1562 −1.49632 −0.748160 0.663518i $$-0.769062\pi$$
−0.748160 + 0.663518i $$0.769062\pi$$
$$492$$ 0 0
$$493$$ 24.7527 1.11481
$$494$$ 0 0
$$495$$ −92.9682 −4.17861
$$496$$ 0 0
$$497$$ −20.0129 −0.897703
$$498$$ 0 0
$$499$$ 25.8814 1.15861 0.579306 0.815110i $$-0.303324\pi$$
0.579306 + 0.815110i $$0.303324\pi$$
$$500$$ 0 0
$$501$$ 26.2462 1.17259
$$502$$ 0 0
$$503$$ 10.3178 0.460048 0.230024 0.973185i $$-0.426120\pi$$
0.230024 + 0.973185i $$0.426120\pi$$
$$504$$ 0 0
$$505$$ −1.03558 −0.0460829
$$506$$ 0 0
$$507$$ −26.3467 −1.17010
$$508$$ 0 0
$$509$$ 6.21618 0.275527 0.137764 0.990465i $$-0.456009\pi$$
0.137764 + 0.990465i $$0.456009\pi$$
$$510$$ 0 0
$$511$$ −0.0112674 −0.000498440 0
$$512$$ 0 0
$$513$$ 1.43845 0.0635090
$$514$$ 0 0
$$515$$ 2.99028 0.131767
$$516$$ 0 0
$$517$$ 3.44491 0.151507
$$518$$ 0 0
$$519$$ 20.6211 0.905165
$$520$$ 0 0
$$521$$ 22.8917 1.00290 0.501451 0.865186i $$-0.332800\pi$$
0.501451 + 0.865186i $$0.332800\pi$$
$$522$$ 0 0
$$523$$ −9.02628 −0.394692 −0.197346 0.980334i $$-0.563232\pi$$
−0.197346 + 0.980334i $$0.563232\pi$$
$$524$$ 0 0
$$525$$ −56.6476 −2.47230
$$526$$ 0 0
$$527$$ 8.40256 0.366021
$$528$$ 0 0
$$529$$ −22.7851 −0.990657
$$530$$ 0 0
$$531$$ 6.59105 0.286027
$$532$$ 0 0
$$533$$ 25.5223 1.10550
$$534$$ 0 0
$$535$$ 49.2010 2.12715
$$536$$ 0 0
$$537$$ 4.49242 0.193862
$$538$$ 0 0
$$539$$ 24.5985 1.05953
$$540$$ 0 0
$$541$$ 21.2937 0.915489 0.457744 0.889084i $$-0.348658\pi$$
0.457744 + 0.889084i $$0.348658\pi$$
$$542$$ 0 0
$$543$$ −28.2291 −1.21143
$$544$$ 0 0
$$545$$ 12.5823 0.538966
$$546$$ 0 0
$$547$$ −3.60803 −0.154268 −0.0771341 0.997021i $$-0.524577\pi$$
−0.0771341 + 0.997021i $$0.524577\pi$$
$$548$$ 0 0
$$549$$ 33.2264 1.41807
$$550$$ 0 0
$$551$$ 4.82550 0.205573
$$552$$ 0 0
$$553$$ 11.7815 0.501000
$$554$$ 0 0
$$555$$ 39.1969 1.66381
$$556$$ 0 0
$$557$$ 34.8645 1.47725 0.738627 0.674114i $$-0.235474\pi$$
0.738627 + 0.674114i $$0.235474\pi$$
$$558$$ 0 0
$$559$$ 5.22590 0.221032
$$560$$ 0 0
$$561$$ −81.5459 −3.44287
$$562$$ 0 0
$$563$$ −41.3167 −1.74129 −0.870647 0.491909i $$-0.836299\pi$$
−0.870647 + 0.491909i $$0.836299\pi$$
$$564$$ 0 0
$$565$$ 26.9701 1.13464
$$566$$ 0 0
$$567$$ −12.1976 −0.512253
$$568$$ 0 0
$$569$$ 41.0835 1.72231 0.861154 0.508344i $$-0.169742\pi$$
0.861154 + 0.508344i $$0.169742\pi$$
$$570$$ 0 0
$$571$$ 36.8243 1.54105 0.770525 0.637410i $$-0.219994\pi$$
0.770525 + 0.637410i $$0.219994\pi$$
$$572$$ 0 0
$$573$$ 33.8831 1.41549
$$574$$ 0 0
$$575$$ 5.88310 0.245342
$$576$$ 0 0
$$577$$ 15.7004 0.653617 0.326809 0.945091i $$-0.394027\pi$$
0.326809 + 0.945091i $$0.394027\pi$$
$$578$$ 0 0
$$579$$ −3.77089 −0.156713
$$580$$ 0 0
$$581$$ 20.7311 0.860072
$$582$$ 0 0
$$583$$ 16.5054 0.683582
$$584$$ 0 0
$$585$$ −72.2868 −2.98869
$$586$$ 0 0
$$587$$ −34.3099 −1.41612 −0.708060 0.706152i $$-0.750429\pi$$
−0.708060 + 0.706152i $$0.750429\pi$$
$$588$$ 0 0
$$589$$ 1.63806 0.0674952
$$590$$ 0 0
$$591$$ 20.1005 0.826822
$$592$$ 0 0
$$593$$ 7.11584 0.292213 0.146106 0.989269i $$-0.453326\pi$$
0.146106 + 0.989269i $$0.453326\pi$$
$$594$$ 0 0
$$595$$ −37.5956 −1.54127
$$596$$ 0 0
$$597$$ 30.2409 1.23768
$$598$$ 0 0
$$599$$ −13.8340 −0.565244 −0.282622 0.959231i $$-0.591204\pi$$
−0.282622 + 0.959231i $$0.591204\pi$$
$$600$$ 0 0
$$601$$ 39.2140 1.59957 0.799785 0.600286i $$-0.204947\pi$$
0.799785 + 0.600286i $$0.204947\pi$$
$$602$$ 0 0
$$603$$ −16.8373 −0.685666
$$604$$ 0 0
$$605$$ 115.732 4.70518
$$606$$ 0 0
$$607$$ 35.1733 1.42764 0.713821 0.700328i $$-0.246963\pi$$
0.713821 + 0.700328i $$0.246963\pi$$
$$608$$ 0 0
$$609$$ 21.5389 0.872800
$$610$$ 0 0
$$611$$ 2.67857 0.108363
$$612$$ 0 0
$$613$$ −6.24905 −0.252397 −0.126198 0.992005i $$-0.540278\pi$$
−0.126198 + 0.992005i $$0.540278\pi$$
$$614$$ 0 0
$$615$$ 56.9848 2.29785
$$616$$ 0 0
$$617$$ −42.0378 −1.69238 −0.846189 0.532883i $$-0.821109\pi$$
−0.846189 + 0.532883i $$0.821109\pi$$
$$618$$ 0 0
$$619$$ −31.0633 −1.24854 −0.624269 0.781209i $$-0.714603\pi$$
−0.624269 + 0.781209i $$0.714603\pi$$
$$620$$ 0 0
$$621$$ −0.666807 −0.0267581
$$622$$ 0 0
$$623$$ −12.7537 −0.510965
$$624$$ 0 0
$$625$$ 72.6090 2.90436
$$626$$ 0 0
$$627$$ −15.8972 −0.634873
$$628$$ 0 0
$$629$$ 18.6617 0.744091
$$630$$ 0 0
$$631$$ −40.0378 −1.59388 −0.796940 0.604059i $$-0.793549\pi$$
−0.796940 + 0.604059i $$0.793549\pi$$
$$632$$ 0 0
$$633$$ −37.3333 −1.48387
$$634$$ 0 0
$$635$$ −76.3127 −3.02838
$$636$$ 0 0
$$637$$ 19.1264 0.757817
$$638$$ 0 0
$$639$$ −40.9046 −1.61816
$$640$$ 0 0
$$641$$ −20.3619 −0.804248 −0.402124 0.915585i $$-0.631728\pi$$
−0.402124 + 0.915585i $$0.631728\pi$$
$$642$$ 0 0
$$643$$ −13.7041 −0.540435 −0.270218 0.962799i $$-0.587096\pi$$
−0.270218 + 0.962799i $$0.587096\pi$$
$$644$$ 0 0
$$645$$ 11.6681 0.459431
$$646$$ 0 0
$$647$$ 26.9588 1.05986 0.529930 0.848041i $$-0.322218\pi$$
0.529930 + 0.848041i $$0.322218\pi$$
$$648$$ 0 0
$$649$$ −11.4850 −0.450827
$$650$$ 0 0
$$651$$ 7.31159 0.286564
$$652$$ 0 0
$$653$$ −30.7842 −1.20468 −0.602339 0.798240i $$-0.705765\pi$$
−0.602339 + 0.798240i $$0.705765\pi$$
$$654$$ 0 0
$$655$$ −52.4298 −2.04860
$$656$$ 0 0
$$657$$ −0.0230295 −0.000898466 0
$$658$$ 0 0
$$659$$ −7.57854 −0.295218 −0.147609 0.989046i $$-0.547158\pi$$
−0.147609 + 0.989046i $$0.547158\pi$$
$$660$$ 0 0
$$661$$ −47.8191 −1.85995 −0.929974 0.367626i $$-0.880171\pi$$
−0.929974 + 0.367626i $$0.880171\pi$$
$$662$$ 0 0
$$663$$ −63.4054 −2.46246
$$664$$ 0 0
$$665$$ −7.32919 −0.284214
$$666$$ 0 0
$$667$$ −2.23691 −0.0866135
$$668$$ 0 0
$$669$$ −68.3200 −2.64140
$$670$$ 0 0
$$671$$ −57.8977 −2.23512
$$672$$ 0 0
$$673$$ 12.2389 0.471777 0.235888 0.971780i $$-0.424200\pi$$
0.235888 + 0.971780i $$0.424200\pi$$
$$674$$ 0 0
$$675$$ −18.2555 −0.702655
$$676$$ 0 0
$$677$$ −23.3609 −0.897832 −0.448916 0.893574i $$-0.648190\pi$$
−0.448916 + 0.893574i $$0.648190\pi$$
$$678$$ 0 0
$$679$$ −18.1434 −0.696280
$$680$$ 0 0
$$681$$ −10.2886 −0.394262
$$682$$ 0 0
$$683$$ −24.7239 −0.946033 −0.473016 0.881054i $$-0.656835\pi$$
−0.473016 + 0.881054i $$0.656835\pi$$
$$684$$ 0 0
$$685$$ 35.4094 1.35293
$$686$$ 0 0
$$687$$ 59.3856 2.26570
$$688$$ 0 0
$$689$$ 12.8336 0.488922
$$690$$ 0 0
$$691$$ −6.78837 −0.258242 −0.129121 0.991629i $$-0.541215\pi$$
−0.129121 + 0.991629i $$0.541215\pi$$
$$692$$ 0 0
$$693$$ −38.5155 −1.46308
$$694$$ 0 0
$$695$$ −47.6515 −1.80752
$$696$$ 0 0
$$697$$ 27.1306 1.02764
$$698$$ 0 0
$$699$$ −57.0438 −2.15760
$$700$$ 0 0
$$701$$ −26.9175 −1.01666 −0.508330 0.861162i $$-0.669737\pi$$
−0.508330 + 0.861162i $$0.669737\pi$$
$$702$$ 0 0
$$703$$ 3.63806 0.137212
$$704$$ 0 0
$$705$$ 5.98056 0.225241
$$706$$ 0 0
$$707$$ −0.429028 −0.0161353
$$708$$ 0 0
$$709$$ 21.5984 0.811146 0.405573 0.914063i $$-0.367072\pi$$
0.405573 + 0.914063i $$0.367072\pi$$
$$710$$ 0 0
$$711$$ 24.0803 0.903080
$$712$$ 0 0
$$713$$ −0.759341 −0.0284376
$$714$$ 0 0
$$715$$ 125.961 4.71069
$$716$$ 0 0
$$717$$ 59.1253 2.20807
$$718$$ 0 0
$$719$$ −22.2575 −0.830064 −0.415032 0.909807i $$-0.636230\pi$$
−0.415032 + 0.909807i $$0.636230\pi$$
$$720$$ 0 0
$$721$$ 1.23883 0.0461365
$$722$$ 0 0
$$723$$ 1.44775 0.0538423
$$724$$ 0 0
$$725$$ −61.2410 −2.27443
$$726$$ 0 0
$$727$$ −51.9118 −1.92530 −0.962651 0.270745i $$-0.912730\pi$$
−0.962651 + 0.270745i $$0.912730\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 5.55520 0.205467
$$732$$ 0 0
$$733$$ −20.0575 −0.740840 −0.370420 0.928864i $$-0.620786\pi$$
−0.370420 + 0.928864i $$0.620786\pi$$
$$734$$ 0 0
$$735$$ 42.7044 1.57518
$$736$$ 0 0
$$737$$ 29.3393 1.08073
$$738$$ 0 0
$$739$$ −7.36465 −0.270913 −0.135457 0.990783i $$-0.543250\pi$$
−0.135457 + 0.990783i $$0.543250\pi$$
$$740$$ 0 0
$$741$$ −12.3608 −0.454084
$$742$$ 0 0
$$743$$ −2.85436 −0.104716 −0.0523581 0.998628i $$-0.516674\pi$$
−0.0523581 + 0.998628i $$0.516674\pi$$
$$744$$ 0 0
$$745$$ 1.35355 0.0495902
$$746$$ 0 0
$$747$$ 42.3725 1.55033
$$748$$ 0 0
$$749$$ 20.3833 0.744790
$$750$$ 0 0
$$751$$ −22.3725 −0.816385 −0.408193 0.912896i $$-0.633841\pi$$
−0.408193 + 0.912896i $$0.633841\pi$$
$$752$$ 0 0
$$753$$ 40.5856 1.47902
$$754$$ 0 0
$$755$$ −12.4380 −0.452666
$$756$$ 0 0
$$757$$ −6.07326 −0.220736 −0.110368 0.993891i $$-0.535203\pi$$
−0.110368 + 0.993891i $$0.535203\pi$$
$$758$$ 0 0
$$759$$ 7.36932 0.267489
$$760$$ 0 0
$$761$$ 13.1166 0.475478 0.237739 0.971329i $$-0.423594\pi$$
0.237739 + 0.971329i $$0.423594\pi$$
$$762$$ 0 0
$$763$$ 5.21267 0.188711
$$764$$ 0 0
$$765$$ −76.8419 −2.77823
$$766$$ 0 0
$$767$$ −8.93012 −0.322448
$$768$$ 0 0
$$769$$ −44.5791 −1.60757 −0.803783 0.594923i $$-0.797182\pi$$
−0.803783 + 0.594923i $$0.797182\pi$$
$$770$$ 0 0
$$771$$ −30.9004 −1.11285
$$772$$ 0 0
$$773$$ 4.43920 0.159667 0.0798334 0.996808i $$-0.474561\pi$$
0.0798334 + 0.996808i $$0.474561\pi$$
$$774$$ 0 0
$$775$$ −20.7889 −0.746758
$$776$$ 0 0
$$777$$ 16.2387 0.582561
$$778$$ 0 0
$$779$$ 5.28906 0.189500
$$780$$ 0 0
$$781$$ 71.2771 2.55050
$$782$$ 0 0
$$783$$ 6.94122 0.248059
$$784$$ 0 0
$$785$$ −27.4568 −0.979977
$$786$$ 0 0
$$787$$ −0.315342 −0.0112407 −0.00562036 0.999984i $$-0.501789\pi$$
−0.00562036 + 0.999984i $$0.501789\pi$$
$$788$$ 0 0
$$789$$ −59.4625 −2.11692
$$790$$ 0 0
$$791$$ 11.1733 0.397278
$$792$$ 0 0
$$793$$ −45.0180 −1.59864
$$794$$ 0 0
$$795$$ 28.6542 1.01626
$$796$$ 0 0
$$797$$ −6.16712 −0.218451 −0.109225 0.994017i $$-0.534837\pi$$
−0.109225 + 0.994017i $$0.534837\pi$$
$$798$$ 0 0
$$799$$ 2.84736 0.100732
$$800$$ 0 0
$$801$$ −26.0673 −0.921044
$$802$$ 0 0
$$803$$ 0.0401294 0.00141614
$$804$$ 0 0
$$805$$ 3.39752 0.119747
$$806$$ 0 0
$$807$$ 67.3597 2.37117
$$808$$ 0 0
$$809$$ 20.4057 0.717426 0.358713 0.933448i $$-0.383216\pi$$
0.358713 + 0.933448i $$0.383216\pi$$
$$810$$ 0 0
$$811$$ 36.5763 1.28437 0.642184 0.766551i $$-0.278029\pi$$
0.642184 + 0.766551i $$0.278029\pi$$
$$812$$ 0 0
$$813$$ −33.1874 −1.16393
$$814$$ 0 0
$$815$$ −8.62342 −0.302065
$$816$$ 0 0
$$817$$ 1.08298 0.0378885
$$818$$ 0 0
$$819$$ −29.9474 −1.04645
$$820$$ 0 0
$$821$$ −19.6686 −0.686439 −0.343219 0.939255i $$-0.611517\pi$$
−0.343219 + 0.939255i $$0.611517\pi$$
$$822$$ 0 0
$$823$$ −36.7328 −1.28042 −0.640212 0.768198i $$-0.721154\pi$$
−0.640212 + 0.768198i $$0.721154\pi$$
$$824$$ 0 0
$$825$$ 201.753 7.02415
$$826$$ 0 0
$$827$$ −27.4920 −0.955991 −0.477995 0.878362i $$-0.658636\pi$$
−0.477995 + 0.878362i $$0.658636\pi$$
$$828$$ 0 0
$$829$$ 12.1016 0.420307 0.210153 0.977668i $$-0.432604\pi$$
0.210153 + 0.977668i $$0.432604\pi$$
$$830$$ 0 0
$$831$$ −27.8299 −0.965408
$$832$$ 0 0
$$833$$ 20.3317 0.704451
$$834$$ 0 0
$$835$$ −43.0964 −1.49141
$$836$$ 0 0
$$837$$ 2.35627 0.0814445
$$838$$ 0 0
$$839$$ −51.4731 −1.77705 −0.888524 0.458829i $$-0.848269\pi$$
−0.888524 + 0.458829i $$0.848269\pi$$
$$840$$ 0 0
$$841$$ −5.71457 −0.197054
$$842$$ 0 0
$$843$$ 40.5929 1.39809
$$844$$ 0 0
$$845$$ 43.2613 1.48824
$$846$$ 0 0
$$847$$ 47.9463 1.64745
$$848$$ 0 0
$$849$$ −70.8885 −2.43289
$$850$$ 0 0
$$851$$ −1.68646 −0.0578112
$$852$$ 0 0
$$853$$ 42.6187 1.45924 0.729619 0.683854i $$-0.239697\pi$$
0.729619 + 0.683854i $$0.239697\pi$$
$$854$$ 0 0
$$855$$ −14.9802 −0.512311
$$856$$ 0 0
$$857$$ 3.45501 0.118021 0.0590105 0.998257i $$-0.481205\pi$$
0.0590105 + 0.998257i $$0.481205\pi$$
$$858$$ 0 0
$$859$$ −17.1161 −0.583994 −0.291997 0.956419i $$-0.594320\pi$$
−0.291997 + 0.956419i $$0.594320\pi$$
$$860$$ 0 0
$$861$$ 23.6080 0.804560
$$862$$ 0 0
$$863$$ −13.0130 −0.442969 −0.221485 0.975164i $$-0.571090\pi$$
−0.221485 + 0.975164i $$0.571090\pi$$
$$864$$ 0 0
$$865$$ −33.8599 −1.15127
$$866$$ 0 0
$$867$$ −23.8545 −0.810141
$$868$$ 0 0
$$869$$ −41.9604 −1.42341
$$870$$ 0 0
$$871$$ 22.8126 0.772974
$$872$$ 0 0
$$873$$ −37.0835 −1.25509
$$874$$ 0 0
$$875$$ 56.3697 1.90564
$$876$$ 0 0
$$877$$ −0.908097 −0.0306642 −0.0153321 0.999882i $$-0.504881\pi$$
−0.0153321 + 0.999882i $$0.504881\pi$$
$$878$$ 0 0
$$879$$ −54.6070 −1.84185
$$880$$ 0 0
$$881$$ 29.1332 0.981523 0.490761 0.871294i $$-0.336719\pi$$
0.490761 + 0.871294i $$0.336719\pi$$
$$882$$ 0 0
$$883$$ 31.4780 1.05932 0.529660 0.848210i $$-0.322319\pi$$
0.529660 + 0.848210i $$0.322319\pi$$
$$884$$ 0 0
$$885$$ −19.9387 −0.670231
$$886$$ 0 0
$$887$$ 37.0933 1.24547 0.622736 0.782432i $$-0.286021\pi$$
0.622736 + 0.782432i $$0.286021\pi$$
$$888$$ 0 0
$$889$$ −31.6153 −1.06034
$$890$$ 0 0
$$891$$ 43.4426 1.45538
$$892$$ 0 0
$$893$$ 0.555087 0.0185753
$$894$$ 0 0
$$895$$ −7.37658 −0.246572
$$896$$ 0 0
$$897$$ 5.72996 0.191318
$$898$$ 0 0
$$899$$ 7.90447 0.263629
$$900$$ 0 0
$$901$$ 13.6423 0.454492
$$902$$ 0 0
$$903$$ 4.83393 0.160863
$$904$$ 0 0
$$905$$ 46.3523 1.54080
$$906$$ 0 0
$$907$$ 24.0295 0.797886 0.398943 0.916976i $$-0.369377\pi$$
0.398943 + 0.916976i $$0.369377\pi$$
$$908$$ 0 0
$$909$$ −0.876894 −0.0290848
$$910$$ 0 0
$$911$$ −45.7644 −1.51624 −0.758121 0.652114i $$-0.773882\pi$$
−0.758121 + 0.652114i $$0.773882\pi$$
$$912$$ 0 0
$$913$$ −73.8350 −2.44358
$$914$$ 0 0
$$915$$ −100.514 −3.32288
$$916$$ 0 0
$$917$$ −21.7209 −0.717288
$$918$$ 0 0
$$919$$ −2.91536 −0.0961688 −0.0480844 0.998843i $$-0.515312\pi$$
−0.0480844 + 0.998843i $$0.515312\pi$$
$$920$$ 0 0
$$921$$ −68.2868 −2.25013
$$922$$ 0 0
$$923$$ 55.4210 1.82421
$$924$$ 0 0
$$925$$ −46.1711 −1.51810
$$926$$ 0 0
$$927$$ 2.53206 0.0831637
$$928$$ 0 0
$$929$$ −14.4417 −0.473815 −0.236908 0.971532i $$-0.576134\pi$$
−0.236908 + 0.971532i $$0.576134\pi$$
$$930$$ 0 0
$$931$$ 3.96362 0.129902
$$932$$ 0 0
$$933$$ −51.4078 −1.68302
$$934$$ 0 0
$$935$$ 133.899 4.37896
$$936$$ 0 0
$$937$$ 39.3960 1.28701 0.643505 0.765442i $$-0.277479\pi$$
0.643505 + 0.765442i $$0.277479\pi$$
$$938$$ 0 0
$$939$$ 28.5349 0.931200
$$940$$ 0 0
$$941$$ −37.5954 −1.22558 −0.612788 0.790247i $$-0.709952\pi$$
−0.612788 + 0.790247i $$0.709952\pi$$
$$942$$ 0 0
$$943$$ −2.45180 −0.0798415
$$944$$ 0 0
$$945$$ −10.5426 −0.342952
$$946$$ 0 0
$$947$$ 21.3522 0.693854 0.346927 0.937892i $$-0.387225\pi$$
0.346927 + 0.937892i $$0.387225\pi$$
$$948$$ 0 0
$$949$$ 0.0312023 0.00101287
$$950$$ 0 0
$$951$$ 84.8767 2.75232
$$952$$ 0 0
$$953$$ 37.5726 1.21709 0.608547 0.793518i $$-0.291753\pi$$
0.608547 + 0.793518i $$0.291753\pi$$
$$954$$ 0 0
$$955$$ −55.6362 −1.80035
$$956$$ 0 0
$$957$$ −76.7119 −2.47974
$$958$$ 0 0
$$959$$ 14.6696 0.473707
$$960$$ 0 0
$$961$$ −28.3167 −0.913444
$$962$$ 0 0
$$963$$ 41.6616 1.34253
$$964$$ 0 0
$$965$$ 6.19182 0.199322
$$966$$ 0 0
$$967$$ 11.5676 0.371990 0.185995 0.982551i $$-0.440449\pi$$
0.185995 + 0.982551i $$0.440449\pi$$
$$968$$ 0 0
$$969$$ −13.1397 −0.422107
$$970$$ 0 0
$$971$$ 60.2009 1.93194 0.965970 0.258656i $$-0.0832796\pi$$
0.965970 + 0.258656i $$0.0832796\pi$$
$$972$$ 0 0
$$973$$ −19.7414 −0.632879
$$974$$ 0 0
$$975$$ 156.872 5.02393
$$976$$ 0 0
$$977$$ 41.2738 1.32047 0.660233 0.751061i $$-0.270458\pi$$
0.660233 + 0.751061i $$0.270458\pi$$
$$978$$ 0 0
$$979$$ 45.4229 1.45172
$$980$$ 0 0
$$981$$ 10.6542 0.340163
$$982$$ 0 0
$$983$$ 16.5610 0.528214 0.264107 0.964493i $$-0.414923\pi$$
0.264107 + 0.964493i $$0.414923\pi$$
$$984$$ 0 0
$$985$$ −33.0050 −1.05163
$$986$$ 0 0
$$987$$ 2.47766 0.0788649
$$988$$ 0 0
$$989$$ −0.502025 −0.0159635
$$990$$ 0 0
$$991$$ 35.6243 1.13164 0.565821 0.824528i $$-0.308559\pi$$
0.565821 + 0.824528i $$0.308559\pi$$
$$992$$ 0 0
$$993$$ 11.0208 0.349736
$$994$$ 0 0
$$995$$ −49.6557 −1.57419
$$996$$ 0 0
$$997$$ −41.0563 −1.30027 −0.650133 0.759821i $$-0.725287\pi$$
−0.650133 + 0.759821i $$0.725287\pi$$
$$998$$ 0 0
$$999$$ 5.23316 0.165570
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 1216.2.a.w.1.2 4
4.3 odd 2 1216.2.a.x.1.4 4
8.3 odd 2 608.2.a.i.1.1 4
8.5 even 2 608.2.a.j.1.3 yes 4
24.5 odd 2 5472.2.a.bs.1.4 4
24.11 even 2 5472.2.a.bt.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.1 4 8.3 odd 2
608.2.a.j.1.3 yes 4 8.5 even 2
1216.2.a.w.1.2 4 1.1 even 1 trivial
1216.2.a.x.1.4 4 4.3 odd 2
5472.2.a.bs.1.4 4 24.5 odd 2
5472.2.a.bt.1.4 4 24.11 even 2