# Properties

 Label 4864.2.a.bt.1.1 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 2432) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.719482$$ of defining polynomial Character $$\chi$$ $$=$$ 4864.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.46871 q^{3} -3.22672 q^{5} -4.04213 q^{7} +3.09455 q^{9} +O(q^{10})$$ $$q-2.46871 q^{3} -3.22672 q^{5} -4.04213 q^{7} +3.09455 q^{9} +6.41171 q^{11} -0.815411 q^{13} +7.96585 q^{15} -2.62028 q^{17} +1.00000 q^{19} +9.97886 q^{21} -6.16717 q^{23} +5.41171 q^{25} -0.233424 q^{27} -3.85818 q^{29} -7.43670 q^{31} -15.8287 q^{33} +13.0428 q^{35} -5.13742 q^{37} +2.01302 q^{39} -11.5007 q^{41} -0.0837292 q^{43} -9.98525 q^{45} -3.99564 q^{47} +9.33880 q^{49} +6.46871 q^{51} -4.22608 q^{53} -20.6888 q^{55} -2.46871 q^{57} +8.53655 q^{59} -5.72239 q^{61} -12.5086 q^{63} +2.63110 q^{65} -11.4278 q^{67} +15.2250 q^{69} +1.10168 q^{71} -7.25458 q^{73} -13.3600 q^{75} -25.9169 q^{77} +13.9906 q^{79} -8.70740 q^{81} -6.51708 q^{83} +8.45489 q^{85} +9.52474 q^{87} -15.3867 q^{89} +3.29599 q^{91} +18.3591 q^{93} -3.22672 q^{95} -13.2654 q^{97} +19.8414 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 4q^{3} + 14q^{9} + O(q^{10})$$ $$10q + 4q^{3} + 14q^{9} + 20q^{11} + 4q^{17} + 10q^{19} + 10q^{25} + 28q^{27} - 8q^{33} + 36q^{35} - 12q^{41} - 4q^{43} + 26q^{49} + 36q^{51} + 4q^{57} + 52q^{59} - 24q^{65} + 12q^{67} + 12q^{73} - 12q^{75} + 34q^{81} + 16q^{83} - 20q^{89} + 60q^{91} - 28q^{97} + 60q^{99} + O(q^{100})$$

## 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
$$3$$ −2.46871 −1.42531 −0.712657 0.701513i $$-0.752508\pi$$
−0.712657 + 0.701513i $$0.752508\pi$$
$$4$$ 0 0
$$5$$ −3.22672 −1.44303 −0.721516 0.692398i $$-0.756554\pi$$
−0.721516 + 0.692398i $$0.756554\pi$$
$$6$$ 0 0
$$7$$ −4.04213 −1.52778 −0.763890 0.645346i $$-0.776713\pi$$
−0.763890 + 0.645346i $$0.776713\pi$$
$$8$$ 0 0
$$9$$ 3.09455 1.03152
$$10$$ 0 0
$$11$$ 6.41171 1.93320 0.966601 0.256286i $$-0.0824988\pi$$
0.966601 + 0.256286i $$0.0824988\pi$$
$$12$$ 0 0
$$13$$ −0.815411 −0.226154 −0.113077 0.993586i $$-0.536071\pi$$
−0.113077 + 0.993586i $$0.536071\pi$$
$$14$$ 0 0
$$15$$ 7.96585 2.05677
$$16$$ 0 0
$$17$$ −2.62028 −0.635510 −0.317755 0.948173i $$-0.602929\pi$$
−0.317755 + 0.948173i $$0.602929\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 9.97886 2.17757
$$22$$ 0 0
$$23$$ −6.16717 −1.28594 −0.642972 0.765890i $$-0.722299\pi$$
−0.642972 + 0.765890i $$0.722299\pi$$
$$24$$ 0 0
$$25$$ 5.41171 1.08234
$$26$$ 0 0
$$27$$ −0.233424 −0.0449225
$$28$$ 0 0
$$29$$ −3.85818 −0.716446 −0.358223 0.933636i $$-0.616617\pi$$
−0.358223 + 0.933636i $$0.616617\pi$$
$$30$$ 0 0
$$31$$ −7.43670 −1.33567 −0.667835 0.744309i $$-0.732779\pi$$
−0.667835 + 0.744309i $$0.732779\pi$$
$$32$$ 0 0
$$33$$ −15.8287 −2.75542
$$34$$ 0 0
$$35$$ 13.0428 2.20464
$$36$$ 0 0
$$37$$ −5.13742 −0.844586 −0.422293 0.906459i $$-0.638775\pi$$
−0.422293 + 0.906459i $$0.638775\pi$$
$$38$$ 0 0
$$39$$ 2.01302 0.322341
$$40$$ 0 0
$$41$$ −11.5007 −1.79611 −0.898054 0.439886i $$-0.855019\pi$$
−0.898054 + 0.439886i $$0.855019\pi$$
$$42$$ 0 0
$$43$$ −0.0837292 −0.0127686 −0.00638429 0.999980i $$-0.502032\pi$$
−0.00638429 + 0.999980i $$0.502032\pi$$
$$44$$ 0 0
$$45$$ −9.98525 −1.48851
$$46$$ 0 0
$$47$$ −3.99564 −0.582824 −0.291412 0.956598i $$-0.594125\pi$$
−0.291412 + 0.956598i $$0.594125\pi$$
$$48$$ 0 0
$$49$$ 9.33880 1.33411
$$50$$ 0 0
$$51$$ 6.46871 0.905801
$$52$$ 0 0
$$53$$ −4.22608 −0.580497 −0.290248 0.956951i $$-0.593738\pi$$
−0.290248 + 0.956951i $$0.593738\pi$$
$$54$$ 0 0
$$55$$ −20.6888 −2.78967
$$56$$ 0 0
$$57$$ −2.46871 −0.326989
$$58$$ 0 0
$$59$$ 8.53655 1.11136 0.555682 0.831395i $$-0.312457\pi$$
0.555682 + 0.831395i $$0.312457\pi$$
$$60$$ 0 0
$$61$$ −5.72239 −0.732677 −0.366339 0.930482i $$-0.619389\pi$$
−0.366339 + 0.930482i $$0.619389\pi$$
$$62$$ 0 0
$$63$$ −12.5086 −1.57593
$$64$$ 0 0
$$65$$ 2.63110 0.326348
$$66$$ 0 0
$$67$$ −11.4278 −1.39613 −0.698063 0.716036i $$-0.745955\pi$$
−0.698063 + 0.716036i $$0.745955\pi$$
$$68$$ 0 0
$$69$$ 15.2250 1.83287
$$70$$ 0 0
$$71$$ 1.10168 0.130745 0.0653725 0.997861i $$-0.479176\pi$$
0.0653725 + 0.997861i $$0.479176\pi$$
$$72$$ 0 0
$$73$$ −7.25458 −0.849085 −0.424542 0.905408i $$-0.639565\pi$$
−0.424542 + 0.905408i $$0.639565\pi$$
$$74$$ 0 0
$$75$$ −13.3600 −1.54268
$$76$$ 0 0
$$77$$ −25.9169 −2.95351
$$78$$ 0 0
$$79$$ 13.9906 1.57407 0.787033 0.616911i $$-0.211616\pi$$
0.787033 + 0.616911i $$0.211616\pi$$
$$80$$ 0 0
$$81$$ −8.70740 −0.967489
$$82$$ 0 0
$$83$$ −6.51708 −0.715343 −0.357671 0.933848i $$-0.616429\pi$$
−0.357671 + 0.933848i $$0.616429\pi$$
$$84$$ 0 0
$$85$$ 8.45489 0.917062
$$86$$ 0 0
$$87$$ 9.52474 1.02116
$$88$$ 0 0
$$89$$ −15.3867 −1.63098 −0.815492 0.578768i $$-0.803534\pi$$
−0.815492 + 0.578768i $$0.803534\pi$$
$$90$$ 0 0
$$91$$ 3.29599 0.345514
$$92$$ 0 0
$$93$$ 18.3591 1.90375
$$94$$ 0 0
$$95$$ −3.22672 −0.331054
$$96$$ 0 0
$$97$$ −13.2654 −1.34690 −0.673449 0.739234i $$-0.735188\pi$$
−0.673449 + 0.739234i $$0.735188\pi$$
$$98$$ 0 0
$$99$$ 19.8414 1.99413
$$100$$ 0 0
$$101$$ −2.76311 −0.274940 −0.137470 0.990506i $$-0.543897\pi$$
−0.137470 + 0.990506i $$0.543897\pi$$
$$102$$ 0 0
$$103$$ −0.529145 −0.0521382 −0.0260691 0.999660i $$-0.508299\pi$$
−0.0260691 + 0.999660i $$0.508299\pi$$
$$104$$ 0 0
$$105$$ −32.1990 −3.14230
$$106$$ 0 0
$$107$$ 0.211777 0.0204732 0.0102366 0.999948i $$-0.496742\pi$$
0.0102366 + 0.999948i $$0.496742\pi$$
$$108$$ 0 0
$$109$$ −9.73908 −0.932835 −0.466417 0.884565i $$-0.654456\pi$$
−0.466417 + 0.884565i $$0.654456\pi$$
$$110$$ 0 0
$$111$$ 12.6828 1.20380
$$112$$ 0 0
$$113$$ −6.09269 −0.573152 −0.286576 0.958058i $$-0.592517\pi$$
−0.286576 + 0.958058i $$0.592517\pi$$
$$114$$ 0 0
$$115$$ 19.8997 1.85566
$$116$$ 0 0
$$117$$ −2.52333 −0.233282
$$118$$ 0 0
$$119$$ 10.5915 0.970921
$$120$$ 0 0
$$121$$ 30.1100 2.73727
$$122$$ 0 0
$$123$$ 28.3919 2.56002
$$124$$ 0 0
$$125$$ −1.32846 −0.118821
$$126$$ 0 0
$$127$$ 6.64012 0.589215 0.294608 0.955618i $$-0.404811\pi$$
0.294608 + 0.955618i $$0.404811\pi$$
$$128$$ 0 0
$$129$$ 0.206704 0.0181992
$$130$$ 0 0
$$131$$ 3.61315 0.315682 0.157841 0.987465i $$-0.449547\pi$$
0.157841 + 0.987465i $$0.449547\pi$$
$$132$$ 0 0
$$133$$ −4.04213 −0.350497
$$134$$ 0 0
$$135$$ 0.753194 0.0648246
$$136$$ 0 0
$$137$$ −11.7468 −1.00360 −0.501799 0.864984i $$-0.667328\pi$$
−0.501799 + 0.864984i $$0.667328\pi$$
$$138$$ 0 0
$$139$$ −20.4757 −1.73672 −0.868362 0.495931i $$-0.834827\pi$$
−0.868362 + 0.495931i $$0.834827\pi$$
$$140$$ 0 0
$$141$$ 9.86410 0.830707
$$142$$ 0 0
$$143$$ −5.22817 −0.437202
$$144$$ 0 0
$$145$$ 12.4493 1.03385
$$146$$ 0 0
$$147$$ −23.0548 −1.90153
$$148$$ 0 0
$$149$$ 20.4213 1.67298 0.836490 0.547982i $$-0.184604\pi$$
0.836490 + 0.547982i $$0.184604\pi$$
$$150$$ 0 0
$$151$$ 13.1686 1.07164 0.535822 0.844331i $$-0.320002\pi$$
0.535822 + 0.844331i $$0.320002\pi$$
$$152$$ 0 0
$$153$$ −8.10858 −0.655540
$$154$$ 0 0
$$155$$ 23.9961 1.92742
$$156$$ 0 0
$$157$$ 15.0668 1.20246 0.601232 0.799074i $$-0.294677\pi$$
0.601232 + 0.799074i $$0.294677\pi$$
$$158$$ 0 0
$$159$$ 10.4330 0.827389
$$160$$ 0 0
$$161$$ 24.9285 1.96464
$$162$$ 0 0
$$163$$ 11.8860 0.930982 0.465491 0.885053i $$-0.345878\pi$$
0.465491 + 0.885053i $$0.345878\pi$$
$$164$$ 0 0
$$165$$ 51.0747 3.97616
$$166$$ 0 0
$$167$$ 19.2419 1.48898 0.744491 0.667632i $$-0.232692\pi$$
0.744491 + 0.667632i $$0.232692\pi$$
$$168$$ 0 0
$$169$$ −12.3351 −0.948854
$$170$$ 0 0
$$171$$ 3.09455 0.236646
$$172$$ 0 0
$$173$$ −1.53489 −0.116696 −0.0583478 0.998296i $$-0.518583\pi$$
−0.0583478 + 0.998296i $$0.518583\pi$$
$$174$$ 0 0
$$175$$ −21.8748 −1.65358
$$176$$ 0 0
$$177$$ −21.0743 −1.58404
$$178$$ 0 0
$$179$$ −3.82867 −0.286169 −0.143084 0.989711i $$-0.545702\pi$$
−0.143084 + 0.989711i $$0.545702\pi$$
$$180$$ 0 0
$$181$$ −5.89663 −0.438293 −0.219147 0.975692i $$-0.570327\pi$$
−0.219147 + 0.975692i $$0.570327\pi$$
$$182$$ 0 0
$$183$$ 14.1270 1.04429
$$184$$ 0 0
$$185$$ 16.5770 1.21877
$$186$$ 0 0
$$187$$ −16.8004 −1.22857
$$188$$ 0 0
$$189$$ 0.943531 0.0686318
$$190$$ 0 0
$$191$$ −19.8427 −1.43577 −0.717885 0.696162i $$-0.754890\pi$$
−0.717885 + 0.696162i $$0.754890\pi$$
$$192$$ 0 0
$$193$$ −3.62263 −0.260763 −0.130381 0.991464i $$-0.541620\pi$$
−0.130381 + 0.991464i $$0.541620\pi$$
$$194$$ 0 0
$$195$$ −6.49544 −0.465148
$$196$$ 0 0
$$197$$ 13.1190 0.934690 0.467345 0.884075i $$-0.345211\pi$$
0.467345 + 0.884075i $$0.345211\pi$$
$$198$$ 0 0
$$199$$ −15.9156 −1.12823 −0.564113 0.825697i $$-0.690782\pi$$
−0.564113 + 0.825697i $$0.690782\pi$$
$$200$$ 0 0
$$201$$ 28.2120 1.98992
$$202$$ 0 0
$$203$$ 15.5953 1.09457
$$204$$ 0 0
$$205$$ 37.1095 2.59184
$$206$$ 0 0
$$207$$ −19.0846 −1.32647
$$208$$ 0 0
$$209$$ 6.41171 0.443507
$$210$$ 0 0
$$211$$ 5.00713 0.344705 0.172352 0.985035i $$-0.444863\pi$$
0.172352 + 0.985035i $$0.444863\pi$$
$$212$$ 0 0
$$213$$ −2.71973 −0.186352
$$214$$ 0 0
$$215$$ 0.270170 0.0184255
$$216$$ 0 0
$$217$$ 30.0601 2.04061
$$218$$ 0 0
$$219$$ 17.9095 1.21021
$$220$$ 0 0
$$221$$ 2.13660 0.143723
$$222$$ 0 0
$$223$$ −8.30156 −0.555913 −0.277957 0.960594i $$-0.589657\pi$$
−0.277957 + 0.960594i $$0.589657\pi$$
$$224$$ 0 0
$$225$$ 16.7468 1.11645
$$226$$ 0 0
$$227$$ −26.8177 −1.77995 −0.889976 0.456007i $$-0.849279\pi$$
−0.889976 + 0.456007i $$0.849279\pi$$
$$228$$ 0 0
$$229$$ −6.15107 −0.406474 −0.203237 0.979130i $$-0.565146\pi$$
−0.203237 + 0.979130i $$0.565146\pi$$
$$230$$ 0 0
$$231$$ 63.9815 4.20968
$$232$$ 0 0
$$233$$ 7.79905 0.510933 0.255466 0.966818i $$-0.417771\pi$$
0.255466 + 0.966818i $$0.417771\pi$$
$$234$$ 0 0
$$235$$ 12.8928 0.841034
$$236$$ 0 0
$$237$$ −34.5388 −2.24354
$$238$$ 0 0
$$239$$ 27.7964 1.79800 0.898999 0.437951i $$-0.144296\pi$$
0.898999 + 0.437951i $$0.144296\pi$$
$$240$$ 0 0
$$241$$ −5.53347 −0.356442 −0.178221 0.983990i $$-0.557034\pi$$
−0.178221 + 0.983990i $$0.557034\pi$$
$$242$$ 0 0
$$243$$ 22.1964 1.42390
$$244$$ 0 0
$$245$$ −30.1337 −1.92517
$$246$$ 0 0
$$247$$ −0.815411 −0.0518833
$$248$$ 0 0
$$249$$ 16.0888 1.01959
$$250$$ 0 0
$$251$$ −14.9928 −0.946334 −0.473167 0.880973i $$-0.656889\pi$$
−0.473167 + 0.880973i $$0.656889\pi$$
$$252$$ 0 0
$$253$$ −39.5421 −2.48599
$$254$$ 0 0
$$255$$ −20.8727 −1.30710
$$256$$ 0 0
$$257$$ 13.6003 0.848365 0.424182 0.905577i $$-0.360562\pi$$
0.424182 + 0.905577i $$0.360562\pi$$
$$258$$ 0 0
$$259$$ 20.7661 1.29034
$$260$$ 0 0
$$261$$ −11.9393 −0.739026
$$262$$ 0 0
$$263$$ −26.4415 −1.63046 −0.815228 0.579140i $$-0.803388\pi$$
−0.815228 + 0.579140i $$0.803388\pi$$
$$264$$ 0 0
$$265$$ 13.6364 0.837675
$$266$$ 0 0
$$267$$ 37.9853 2.32466
$$268$$ 0 0
$$269$$ −19.6429 −1.19765 −0.598825 0.800880i $$-0.704365\pi$$
−0.598825 + 0.800880i $$0.704365\pi$$
$$270$$ 0 0
$$271$$ 14.0146 0.851327 0.425663 0.904882i $$-0.360041\pi$$
0.425663 + 0.904882i $$0.360041\pi$$
$$272$$ 0 0
$$273$$ −8.13687 −0.492466
$$274$$ 0 0
$$275$$ 34.6983 2.09238
$$276$$ 0 0
$$277$$ −4.92581 −0.295963 −0.147982 0.988990i $$-0.547278\pi$$
−0.147982 + 0.988990i $$0.547278\pi$$
$$278$$ 0 0
$$279$$ −23.0133 −1.37777
$$280$$ 0 0
$$281$$ 4.90466 0.292587 0.146294 0.989241i $$-0.453266\pi$$
0.146294 + 0.989241i $$0.453266\pi$$
$$282$$ 0 0
$$283$$ −2.07261 −0.123204 −0.0616018 0.998101i $$-0.519621\pi$$
−0.0616018 + 0.998101i $$0.519621\pi$$
$$284$$ 0 0
$$285$$ 7.96585 0.471856
$$286$$ 0 0
$$287$$ 46.4873 2.74406
$$288$$ 0 0
$$289$$ −10.1342 −0.596127
$$290$$ 0 0
$$291$$ 32.7485 1.91975
$$292$$ 0 0
$$293$$ 10.4606 0.611117 0.305559 0.952173i $$-0.401157\pi$$
0.305559 + 0.952173i $$0.401157\pi$$
$$294$$ 0 0
$$295$$ −27.5450 −1.60373
$$296$$ 0 0
$$297$$ −1.49665 −0.0868443
$$298$$ 0 0
$$299$$ 5.02878 0.290822
$$300$$ 0 0
$$301$$ 0.338444 0.0195076
$$302$$ 0 0
$$303$$ 6.82134 0.391875
$$304$$ 0 0
$$305$$ 18.4645 1.05728
$$306$$ 0 0
$$307$$ 5.87873 0.335517 0.167758 0.985828i $$-0.446347\pi$$
0.167758 + 0.985828i $$0.446347\pi$$
$$308$$ 0 0
$$309$$ 1.30631 0.0743132
$$310$$ 0 0
$$311$$ 12.6861 0.719365 0.359683 0.933075i $$-0.382885\pi$$
0.359683 + 0.933075i $$0.382885\pi$$
$$312$$ 0 0
$$313$$ 6.25089 0.353321 0.176661 0.984272i $$-0.443471\pi$$
0.176661 + 0.984272i $$0.443471\pi$$
$$314$$ 0 0
$$315$$ 40.3617 2.27412
$$316$$ 0 0
$$317$$ −4.72978 −0.265651 −0.132826 0.991139i $$-0.542405\pi$$
−0.132826 + 0.991139i $$0.542405\pi$$
$$318$$ 0 0
$$319$$ −24.7375 −1.38503
$$320$$ 0 0
$$321$$ −0.522816 −0.0291808
$$322$$ 0 0
$$323$$ −2.62028 −0.145796
$$324$$ 0 0
$$325$$ −4.41276 −0.244776
$$326$$ 0 0
$$327$$ 24.0430 1.32958
$$328$$ 0 0
$$329$$ 16.1509 0.890428
$$330$$ 0 0
$$331$$ 19.8060 1.08864 0.544318 0.838879i $$-0.316788\pi$$
0.544318 + 0.838879i $$0.316788\pi$$
$$332$$ 0 0
$$333$$ −15.8980 −0.871206
$$334$$ 0 0
$$335$$ 36.8743 2.01466
$$336$$ 0 0
$$337$$ 9.40107 0.512109 0.256054 0.966662i $$-0.417577\pi$$
0.256054 + 0.966662i $$0.417577\pi$$
$$338$$ 0 0
$$339$$ 15.0411 0.816921
$$340$$ 0 0
$$341$$ −47.6819 −2.58212
$$342$$ 0 0
$$343$$ −9.45373 −0.510454
$$344$$ 0 0
$$345$$ −49.1267 −2.64489
$$346$$ 0 0
$$347$$ −25.2991 −1.35813 −0.679063 0.734080i $$-0.737614\pi$$
−0.679063 + 0.734080i $$0.737614\pi$$
$$348$$ 0 0
$$349$$ 28.9343 1.54882 0.774408 0.632687i $$-0.218048\pi$$
0.774408 + 0.632687i $$0.218048\pi$$
$$350$$ 0 0
$$351$$ 0.190337 0.0101594
$$352$$ 0 0
$$353$$ −22.4897 −1.19701 −0.598503 0.801121i $$-0.704238\pi$$
−0.598503 + 0.801121i $$0.704238\pi$$
$$354$$ 0 0
$$355$$ −3.55480 −0.188669
$$356$$ 0 0
$$357$$ −26.1474 −1.38387
$$358$$ 0 0
$$359$$ −25.7745 −1.36033 −0.680164 0.733060i $$-0.738091\pi$$
−0.680164 + 0.733060i $$0.738091\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −74.3330 −3.90147
$$364$$ 0 0
$$365$$ 23.4085 1.22526
$$366$$ 0 0
$$367$$ 3.91733 0.204483 0.102242 0.994760i $$-0.467399\pi$$
0.102242 + 0.994760i $$0.467399\pi$$
$$368$$ 0 0
$$369$$ −35.5895 −1.85272
$$370$$ 0 0
$$371$$ 17.0824 0.886871
$$372$$ 0 0
$$373$$ −7.17543 −0.371530 −0.185765 0.982594i $$-0.559476\pi$$
−0.185765 + 0.982594i $$0.559476\pi$$
$$374$$ 0 0
$$375$$ 3.27959 0.169357
$$376$$ 0 0
$$377$$ 3.14600 0.162027
$$378$$ 0 0
$$379$$ −9.11028 −0.467964 −0.233982 0.972241i $$-0.575176\pi$$
−0.233982 + 0.972241i $$0.575176\pi$$
$$380$$ 0 0
$$381$$ −16.3926 −0.839816
$$382$$ 0 0
$$383$$ 28.4774 1.45513 0.727564 0.686040i $$-0.240652\pi$$
0.727564 + 0.686040i $$0.240652\pi$$
$$384$$ 0 0
$$385$$ 83.6266 4.26201
$$386$$ 0 0
$$387$$ −0.259104 −0.0131710
$$388$$ 0 0
$$389$$ −27.5101 −1.39482 −0.697409 0.716674i $$-0.745664\pi$$
−0.697409 + 0.716674i $$0.745664\pi$$
$$390$$ 0 0
$$391$$ 16.1597 0.817231
$$392$$ 0 0
$$393$$ −8.91983 −0.449946
$$394$$ 0 0
$$395$$ −45.1437 −2.27143
$$396$$ 0 0
$$397$$ 32.7251 1.64242 0.821212 0.570624i $$-0.193298\pi$$
0.821212 + 0.570624i $$0.193298\pi$$
$$398$$ 0 0
$$399$$ 9.97886 0.499568
$$400$$ 0 0
$$401$$ 13.5472 0.676515 0.338257 0.941054i $$-0.390163\pi$$
0.338257 + 0.941054i $$0.390163\pi$$
$$402$$ 0 0
$$403$$ 6.06397 0.302068
$$404$$ 0 0
$$405$$ 28.0963 1.39612
$$406$$ 0 0
$$407$$ −32.9396 −1.63276
$$408$$ 0 0
$$409$$ 27.0909 1.33956 0.669779 0.742561i $$-0.266389\pi$$
0.669779 + 0.742561i $$0.266389\pi$$
$$410$$ 0 0
$$411$$ 28.9995 1.43044
$$412$$ 0 0
$$413$$ −34.5058 −1.69792
$$414$$ 0 0
$$415$$ 21.0288 1.03226
$$416$$ 0 0
$$417$$ 50.5486 2.47538
$$418$$ 0 0
$$419$$ 12.0175 0.587092 0.293546 0.955945i $$-0.405165\pi$$
0.293546 + 0.955945i $$0.405165\pi$$
$$420$$ 0 0
$$421$$ −21.6253 −1.05395 −0.526977 0.849880i $$-0.676675\pi$$
−0.526977 + 0.849880i $$0.676675\pi$$
$$422$$ 0 0
$$423$$ −12.3647 −0.601194
$$424$$ 0 0
$$425$$ −14.1802 −0.687839
$$426$$ 0 0
$$427$$ 23.1306 1.11937
$$428$$ 0 0
$$429$$ 12.9069 0.623150
$$430$$ 0 0
$$431$$ 0.466340 0.0224628 0.0112314 0.999937i $$-0.496425\pi$$
0.0112314 + 0.999937i $$0.496425\pi$$
$$432$$ 0 0
$$433$$ −12.1789 −0.585281 −0.292640 0.956223i $$-0.594534\pi$$
−0.292640 + 0.956223i $$0.594534\pi$$
$$434$$ 0 0
$$435$$ −30.7337 −1.47357
$$436$$ 0 0
$$437$$ −6.16717 −0.295016
$$438$$ 0 0
$$439$$ −22.9016 −1.09303 −0.546517 0.837448i $$-0.684047\pi$$
−0.546517 + 0.837448i $$0.684047\pi$$
$$440$$ 0 0
$$441$$ 28.8994 1.37616
$$442$$ 0 0
$$443$$ 4.15864 0.197583 0.0987914 0.995108i $$-0.468502\pi$$
0.0987914 + 0.995108i $$0.468502\pi$$
$$444$$ 0 0
$$445$$ 49.6485 2.35356
$$446$$ 0 0
$$447$$ −50.4144 −2.38452
$$448$$ 0 0
$$449$$ −13.4154 −0.633112 −0.316556 0.948574i $$-0.602527\pi$$
−0.316556 + 0.948574i $$0.602527\pi$$
$$450$$ 0 0
$$451$$ −73.7391 −3.47224
$$452$$ 0 0
$$453$$ −32.5095 −1.52743
$$454$$ 0 0
$$455$$ −10.6352 −0.498588
$$456$$ 0 0
$$457$$ −23.9359 −1.11968 −0.559838 0.828602i $$-0.689136\pi$$
−0.559838 + 0.828602i $$0.689136\pi$$
$$458$$ 0 0
$$459$$ 0.611636 0.0285487
$$460$$ 0 0
$$461$$ 31.0335 1.44537 0.722686 0.691176i $$-0.242907\pi$$
0.722686 + 0.691176i $$0.242907\pi$$
$$462$$ 0 0
$$463$$ −31.3472 −1.45683 −0.728415 0.685136i $$-0.759743\pi$$
−0.728415 + 0.685136i $$0.759743\pi$$
$$464$$ 0 0
$$465$$ −59.2396 −2.74717
$$466$$ 0 0
$$467$$ 7.14257 0.330519 0.165259 0.986250i $$-0.447154\pi$$
0.165259 + 0.986250i $$0.447154\pi$$
$$468$$ 0 0
$$469$$ 46.1926 2.13298
$$470$$ 0 0
$$471$$ −37.1957 −1.71389
$$472$$ 0 0
$$473$$ −0.536847 −0.0246843
$$474$$ 0 0
$$475$$ 5.41171 0.248306
$$476$$ 0 0
$$477$$ −13.0778 −0.598792
$$478$$ 0 0
$$479$$ −26.5447 −1.21286 −0.606430 0.795137i $$-0.707399\pi$$
−0.606430 + 0.795137i $$0.707399\pi$$
$$480$$ 0 0
$$481$$ 4.18911 0.191007
$$482$$ 0 0
$$483$$ −61.5413 −2.80023
$$484$$ 0 0
$$485$$ 42.8037 1.94362
$$486$$ 0 0
$$487$$ 9.74569 0.441619 0.220810 0.975317i $$-0.429130\pi$$
0.220810 + 0.975317i $$0.429130\pi$$
$$488$$ 0 0
$$489$$ −29.3431 −1.32694
$$490$$ 0 0
$$491$$ 25.6150 1.15599 0.577995 0.816040i $$-0.303835\pi$$
0.577995 + 0.816040i $$0.303835\pi$$
$$492$$ 0 0
$$493$$ 10.1095 0.455309
$$494$$ 0 0
$$495$$ −64.0225 −2.87760
$$496$$ 0 0
$$497$$ −4.45312 −0.199750
$$498$$ 0 0
$$499$$ −5.48801 −0.245677 −0.122838 0.992427i $$-0.539200\pi$$
−0.122838 + 0.992427i $$0.539200\pi$$
$$500$$ 0 0
$$501$$ −47.5028 −2.12227
$$502$$ 0 0
$$503$$ −9.00013 −0.401296 −0.200648 0.979663i $$-0.564305\pi$$
−0.200648 + 0.979663i $$0.564305\pi$$
$$504$$ 0 0
$$505$$ 8.91578 0.396747
$$506$$ 0 0
$$507$$ 30.4519 1.35241
$$508$$ 0 0
$$509$$ −28.2818 −1.25357 −0.626784 0.779193i $$-0.715629\pi$$
−0.626784 + 0.779193i $$0.715629\pi$$
$$510$$ 0 0
$$511$$ 29.3240 1.29722
$$512$$ 0 0
$$513$$ −0.233424 −0.0103059
$$514$$ 0 0
$$515$$ 1.70740 0.0752371
$$516$$ 0 0
$$517$$ −25.6189 −1.12672
$$518$$ 0 0
$$519$$ 3.78921 0.166328
$$520$$ 0 0
$$521$$ −4.19618 −0.183838 −0.0919190 0.995766i $$-0.529300\pi$$
−0.0919190 + 0.995766i $$0.529300\pi$$
$$522$$ 0 0
$$523$$ −33.6631 −1.47198 −0.735992 0.676990i $$-0.763284\pi$$
−0.735992 + 0.676990i $$0.763284\pi$$
$$524$$ 0 0
$$525$$ 54.0027 2.35687
$$526$$ 0 0
$$527$$ 19.4862 0.848833
$$528$$ 0 0
$$529$$ 15.0340 0.653651
$$530$$ 0 0
$$531$$ 26.4168 1.14639
$$532$$ 0 0
$$533$$ 9.37779 0.406197
$$534$$ 0 0
$$535$$ −0.683344 −0.0295435
$$536$$ 0 0
$$537$$ 9.45191 0.407880
$$538$$ 0 0
$$539$$ 59.8776 2.57911
$$540$$ 0 0
$$541$$ 18.3995 0.791056 0.395528 0.918454i $$-0.370562\pi$$
0.395528 + 0.918454i $$0.370562\pi$$
$$542$$ 0 0
$$543$$ 14.5571 0.624705
$$544$$ 0 0
$$545$$ 31.4253 1.34611
$$546$$ 0 0
$$547$$ −5.83156 −0.249340 −0.124670 0.992198i $$-0.539787\pi$$
−0.124670 + 0.992198i $$0.539787\pi$$
$$548$$ 0 0
$$549$$ −17.7082 −0.755770
$$550$$ 0 0
$$551$$ −3.85818 −0.164364
$$552$$ 0 0
$$553$$ −56.5518 −2.40483
$$554$$ 0 0
$$555$$ −40.9239 −1.73712
$$556$$ 0 0
$$557$$ 30.9517 1.31146 0.655732 0.754993i $$-0.272360\pi$$
0.655732 + 0.754993i $$0.272360\pi$$
$$558$$ 0 0
$$559$$ 0.0682737 0.00288767
$$560$$ 0 0
$$561$$ 41.4755 1.75110
$$562$$ 0 0
$$563$$ 43.3007 1.82490 0.912452 0.409183i $$-0.134186\pi$$
0.912452 + 0.409183i $$0.134186\pi$$
$$564$$ 0 0
$$565$$ 19.6594 0.827076
$$566$$ 0 0
$$567$$ 35.1964 1.47811
$$568$$ 0 0
$$569$$ −8.32798 −0.349127 −0.174563 0.984646i $$-0.555851\pi$$
−0.174563 + 0.984646i $$0.555851\pi$$
$$570$$ 0 0
$$571$$ 21.5676 0.902574 0.451287 0.892379i $$-0.350965\pi$$
0.451287 + 0.892379i $$0.350965\pi$$
$$572$$ 0 0
$$573$$ 48.9861 2.04642
$$574$$ 0 0
$$575$$ −33.3749 −1.39183
$$576$$ 0 0
$$577$$ −31.5716 −1.31434 −0.657172 0.753741i $$-0.728247\pi$$
−0.657172 + 0.753741i $$0.728247\pi$$
$$578$$ 0 0
$$579$$ 8.94324 0.371669
$$580$$ 0 0
$$581$$ 26.3429 1.09289
$$582$$ 0 0
$$583$$ −27.0964 −1.12222
$$584$$ 0 0
$$585$$ 8.14208 0.336634
$$586$$ 0 0
$$587$$ 28.9679 1.19563 0.597817 0.801633i $$-0.296035\pi$$
0.597817 + 0.801633i $$0.296035\pi$$
$$588$$ 0 0
$$589$$ −7.43670 −0.306424
$$590$$ 0 0
$$591$$ −32.3871 −1.33223
$$592$$ 0 0
$$593$$ 23.8431 0.979117 0.489559 0.871970i $$-0.337158\pi$$
0.489559 + 0.871970i $$0.337158\pi$$
$$594$$ 0 0
$$595$$ −34.1758 −1.40107
$$596$$ 0 0
$$597$$ 39.2911 1.60808
$$598$$ 0 0
$$599$$ 24.0716 0.983537 0.491769 0.870726i $$-0.336351\pi$$
0.491769 + 0.870726i $$0.336351\pi$$
$$600$$ 0 0
$$601$$ −1.72226 −0.0702523 −0.0351262 0.999383i $$-0.511183\pi$$
−0.0351262 + 0.999383i $$0.511183\pi$$
$$602$$ 0 0
$$603$$ −35.3639 −1.44013
$$604$$ 0 0
$$605$$ −97.1564 −3.94997
$$606$$ 0 0
$$607$$ 17.2198 0.698928 0.349464 0.936950i $$-0.386364\pi$$
0.349464 + 0.936950i $$0.386364\pi$$
$$608$$ 0 0
$$609$$ −38.5002 −1.56011
$$610$$ 0 0
$$611$$ 3.25809 0.131808
$$612$$ 0 0
$$613$$ −32.5924 −1.31639 −0.658197 0.752845i $$-0.728681\pi$$
−0.658197 + 0.752845i $$0.728681\pi$$
$$614$$ 0 0
$$615$$ −91.6128 −3.69418
$$616$$ 0 0
$$617$$ 6.44448 0.259445 0.129722 0.991550i $$-0.458591\pi$$
0.129722 + 0.991550i $$0.458591\pi$$
$$618$$ 0 0
$$619$$ −31.8637 −1.28071 −0.640356 0.768078i $$-0.721213\pi$$
−0.640356 + 0.768078i $$0.721213\pi$$
$$620$$ 0 0
$$621$$ 1.43957 0.0577678
$$622$$ 0 0
$$623$$ 62.1949 2.49179
$$624$$ 0 0
$$625$$ −22.7720 −0.910879
$$626$$ 0 0
$$627$$ −15.8287 −0.632136
$$628$$ 0 0
$$629$$ 13.4615 0.536743
$$630$$ 0 0
$$631$$ −7.36348 −0.293135 −0.146568 0.989201i $$-0.546823\pi$$
−0.146568 + 0.989201i $$0.546823\pi$$
$$632$$ 0 0
$$633$$ −12.3612 −0.491312
$$634$$ 0 0
$$635$$ −21.4258 −0.850257
$$636$$ 0 0
$$637$$ −7.61496 −0.301716
$$638$$ 0 0
$$639$$ 3.40920 0.134866
$$640$$ 0 0
$$641$$ 39.4715 1.55903 0.779514 0.626384i $$-0.215466\pi$$
0.779514 + 0.626384i $$0.215466\pi$$
$$642$$ 0 0
$$643$$ 15.7411 0.620768 0.310384 0.950611i $$-0.399542\pi$$
0.310384 + 0.950611i $$0.399542\pi$$
$$644$$ 0 0
$$645$$ −0.666974 −0.0262621
$$646$$ 0 0
$$647$$ 17.8689 0.702499 0.351250 0.936282i $$-0.385757\pi$$
0.351250 + 0.936282i $$0.385757\pi$$
$$648$$ 0 0
$$649$$ 54.7338 2.14849
$$650$$ 0 0
$$651$$ −74.2098 −2.90851
$$652$$ 0 0
$$653$$ −1.83114 −0.0716581 −0.0358291 0.999358i $$-0.511407\pi$$
−0.0358291 + 0.999358i $$0.511407\pi$$
$$654$$ 0 0
$$655$$ −11.6586 −0.455540
$$656$$ 0 0
$$657$$ −22.4497 −0.875846
$$658$$ 0 0
$$659$$ 17.9446 0.699021 0.349510 0.936932i $$-0.386348\pi$$
0.349510 + 0.936932i $$0.386348\pi$$
$$660$$ 0 0
$$661$$ 0.256100 0.00996114 0.00498057 0.999988i $$-0.498415\pi$$
0.00498057 + 0.999988i $$0.498415\pi$$
$$662$$ 0 0
$$663$$ −5.27466 −0.204851
$$664$$ 0 0
$$665$$ 13.0428 0.505778
$$666$$ 0 0
$$667$$ 23.7940 0.921309
$$668$$ 0 0
$$669$$ 20.4942 0.792350
$$670$$ 0 0
$$671$$ −36.6903 −1.41641
$$672$$ 0 0
$$673$$ −3.46127 −0.133422 −0.0667111 0.997772i $$-0.521251\pi$$
−0.0667111 + 0.997772i $$0.521251\pi$$
$$674$$ 0 0
$$675$$ −1.26322 −0.0486215
$$676$$ 0 0
$$677$$ 13.7402 0.528080 0.264040 0.964512i $$-0.414945\pi$$
0.264040 + 0.964512i $$0.414945\pi$$
$$678$$ 0 0
$$679$$ 53.6205 2.05777
$$680$$ 0 0
$$681$$ 66.2052 2.53699
$$682$$ 0 0
$$683$$ −5.31127 −0.203230 −0.101615 0.994824i $$-0.532401\pi$$
−0.101615 + 0.994824i $$0.532401\pi$$
$$684$$ 0 0
$$685$$ 37.9036 1.44822
$$686$$ 0 0
$$687$$ 15.1852 0.579353
$$688$$ 0 0
$$689$$ 3.44599 0.131282
$$690$$ 0 0
$$691$$ 14.0583 0.534802 0.267401 0.963585i $$-0.413835\pi$$
0.267401 + 0.963585i $$0.413835\pi$$
$$692$$ 0 0
$$693$$ −80.2013 −3.04660
$$694$$ 0 0
$$695$$ 66.0692 2.50615
$$696$$ 0 0
$$697$$ 30.1350 1.14144
$$698$$ 0 0
$$699$$ −19.2536 −0.728239
$$700$$ 0 0
$$701$$ 47.7193 1.80233 0.901167 0.433473i $$-0.142712\pi$$
0.901167 + 0.433473i $$0.142712\pi$$
$$702$$ 0 0
$$703$$ −5.13742 −0.193761
$$704$$ 0 0
$$705$$ −31.8287 −1.19874
$$706$$ 0 0
$$707$$ 11.1689 0.420048
$$708$$ 0 0
$$709$$ −12.8747 −0.483519 −0.241759 0.970336i $$-0.577724\pi$$
−0.241759 + 0.970336i $$0.577724\pi$$
$$710$$ 0 0
$$711$$ 43.2947 1.62368
$$712$$ 0 0
$$713$$ 45.8634 1.71760
$$714$$ 0 0
$$715$$ 16.8698 0.630896
$$716$$ 0 0
$$717$$ −68.6213 −2.56271
$$718$$ 0 0
$$719$$ −6.30671 −0.235201 −0.117600 0.993061i $$-0.537520\pi$$
−0.117600 + 0.993061i $$0.537520\pi$$
$$720$$ 0 0
$$721$$ 2.13887 0.0796557
$$722$$ 0 0
$$723$$ 13.6606 0.508042
$$724$$ 0 0
$$725$$ −20.8793 −0.775439
$$726$$ 0 0
$$727$$ −24.8092 −0.920123 −0.460061 0.887887i $$-0.652173\pi$$
−0.460061 + 0.887887i $$0.652173\pi$$
$$728$$ 0 0
$$729$$ −28.6743 −1.06201
$$730$$ 0 0
$$731$$ 0.219394 0.00811457
$$732$$ 0 0
$$733$$ 20.8961 0.771816 0.385908 0.922537i $$-0.373888\pi$$
0.385908 + 0.922537i $$0.373888\pi$$
$$734$$ 0 0
$$735$$ 74.3914 2.74397
$$736$$ 0 0
$$737$$ −73.2716 −2.69900
$$738$$ 0 0
$$739$$ 9.57101 0.352075 0.176038 0.984383i $$-0.443672\pi$$
0.176038 + 0.984383i $$0.443672\pi$$
$$740$$ 0 0
$$741$$ 2.01302 0.0739500
$$742$$ 0 0
$$743$$ −10.3553 −0.379900 −0.189950 0.981794i $$-0.560833\pi$$
−0.189950 + 0.981794i $$0.560833\pi$$
$$744$$ 0 0
$$745$$ −65.8939 −2.41416
$$746$$ 0 0
$$747$$ −20.1675 −0.737889
$$748$$ 0 0
$$749$$ −0.856029 −0.0312786
$$750$$ 0 0
$$751$$ −30.4868 −1.11248 −0.556239 0.831022i $$-0.687756\pi$$
−0.556239 + 0.831022i $$0.687756\pi$$
$$752$$ 0 0
$$753$$ 37.0128 1.34882
$$754$$ 0 0
$$755$$ −42.4913 −1.54642
$$756$$ 0 0
$$757$$ −13.0841 −0.475549 −0.237774 0.971320i $$-0.576418\pi$$
−0.237774 + 0.971320i $$0.576418\pi$$
$$758$$ 0 0
$$759$$ 97.6181 3.54331
$$760$$ 0 0
$$761$$ 32.3638 1.17319 0.586594 0.809881i $$-0.300468\pi$$
0.586594 + 0.809881i $$0.300468\pi$$
$$762$$ 0 0
$$763$$ 39.3666 1.42517
$$764$$ 0 0
$$765$$ 26.1641 0.945965
$$766$$ 0 0
$$767$$ −6.96079 −0.251340
$$768$$ 0 0
$$769$$ 10.9234 0.393908 0.196954 0.980413i $$-0.436895\pi$$
0.196954 + 0.980413i $$0.436895\pi$$
$$770$$ 0 0
$$771$$ −33.5753 −1.20919
$$772$$ 0 0
$$773$$ −34.9008 −1.25530 −0.627648 0.778497i $$-0.715982\pi$$
−0.627648 + 0.778497i $$0.715982\pi$$
$$774$$ 0 0
$$775$$ −40.2452 −1.44565
$$776$$ 0 0
$$777$$ −51.2656 −1.83914
$$778$$ 0 0
$$779$$ −11.5007 −0.412055
$$780$$ 0 0
$$781$$ 7.06363 0.252756
$$782$$ 0 0
$$783$$ 0.900592 0.0321845
$$784$$ 0 0
$$785$$ −48.6164 −1.73519
$$786$$ 0 0
$$787$$ −5.72739 −0.204159 −0.102080 0.994776i $$-0.532550\pi$$
−0.102080 + 0.994776i $$0.532550\pi$$
$$788$$ 0 0
$$789$$ 65.2766 2.32391
$$790$$ 0 0
$$791$$ 24.6274 0.875650
$$792$$ 0 0
$$793$$ 4.66610 0.165698
$$794$$ 0 0
$$795$$ −33.6643 −1.19395
$$796$$ 0 0
$$797$$ 45.1622 1.59973 0.799863 0.600183i $$-0.204906\pi$$
0.799863 + 0.600183i $$0.204906\pi$$
$$798$$ 0 0
$$799$$ 10.4697 0.370391
$$800$$ 0 0
$$801$$ −47.6149 −1.68239
$$802$$ 0 0
$$803$$ −46.5143 −1.64145
$$804$$ 0 0
$$805$$ −80.4372 −2.83504
$$806$$ 0 0
$$807$$ 48.4928 1.70703
$$808$$ 0 0
$$809$$ 35.4870 1.24766 0.623828 0.781562i $$-0.285577\pi$$
0.623828 + 0.781562i $$0.285577\pi$$
$$810$$ 0 0
$$811$$ −20.2334 −0.710492 −0.355246 0.934773i $$-0.615603\pi$$
−0.355246 + 0.934773i $$0.615603\pi$$
$$812$$ 0 0
$$813$$ −34.5981 −1.21341
$$814$$ 0 0
$$815$$ −38.3527 −1.34344
$$816$$ 0 0
$$817$$ −0.0837292 −0.00292931
$$818$$ 0 0
$$819$$ 10.1996 0.356404
$$820$$ 0 0
$$821$$ 52.0305 1.81588 0.907938 0.419104i $$-0.137656\pi$$
0.907938 + 0.419104i $$0.137656\pi$$
$$822$$ 0 0
$$823$$ −5.60310 −0.195312 −0.0976559 0.995220i $$-0.531134\pi$$
−0.0976559 + 0.995220i $$0.531134\pi$$
$$824$$ 0 0
$$825$$ −85.6601 −2.98230
$$826$$ 0 0
$$827$$ 28.3875 0.987129 0.493565 0.869709i $$-0.335694\pi$$
0.493565 + 0.869709i $$0.335694\pi$$
$$828$$ 0 0
$$829$$ 32.5931 1.13201 0.566003 0.824403i $$-0.308489\pi$$
0.566003 + 0.824403i $$0.308489\pi$$
$$830$$ 0 0
$$831$$ 12.1604 0.421841
$$832$$ 0 0
$$833$$ −24.4702 −0.847844
$$834$$ 0 0
$$835$$ −62.0882 −2.14865
$$836$$ 0 0
$$837$$ 1.73591 0.0600017
$$838$$ 0 0
$$839$$ −45.0055 −1.55376 −0.776881 0.629647i $$-0.783199\pi$$
−0.776881 + 0.629647i $$0.783199\pi$$
$$840$$ 0 0
$$841$$ −14.1145 −0.486706
$$842$$ 0 0
$$843$$ −12.1082 −0.417029
$$844$$ 0 0
$$845$$ 39.8019 1.36923
$$846$$ 0 0
$$847$$ −121.708 −4.18195
$$848$$ 0 0
$$849$$ 5.11667 0.175604
$$850$$ 0 0
$$851$$ 31.6833 1.08609
$$852$$ 0 0
$$853$$ −21.6381 −0.740875 −0.370438 0.928857i $$-0.620792\pi$$
−0.370438 + 0.928857i $$0.620792\pi$$
$$854$$ 0 0
$$855$$ −9.98525 −0.341488
$$856$$ 0 0
$$857$$ −11.4432 −0.390892 −0.195446 0.980714i $$-0.562616\pi$$
−0.195446 + 0.980714i $$0.562616\pi$$
$$858$$ 0 0
$$859$$ 7.57422 0.258429 0.129215 0.991617i $$-0.458754\pi$$
0.129215 + 0.991617i $$0.458754\pi$$
$$860$$ 0 0
$$861$$ −114.764 −3.91114
$$862$$ 0 0
$$863$$ 34.6063 1.17801 0.589007 0.808128i $$-0.299519\pi$$
0.589007 + 0.808128i $$0.299519\pi$$
$$864$$ 0 0
$$865$$ 4.95266 0.168396
$$866$$ 0 0
$$867$$ 25.0183 0.849667
$$868$$ 0 0
$$869$$ 89.7036 3.04299
$$870$$ 0 0
$$871$$ 9.31834 0.315740
$$872$$ 0 0
$$873$$ −41.0505 −1.38935
$$874$$ 0 0
$$875$$ 5.36981 0.181533
$$876$$ 0 0
$$877$$ −29.4493 −0.994434 −0.497217 0.867626i $$-0.665645\pi$$
−0.497217 + 0.867626i $$0.665645\pi$$
$$878$$ 0 0
$$879$$ −25.8243 −0.871033
$$880$$ 0 0
$$881$$ 51.5583 1.73704 0.868522 0.495651i $$-0.165070\pi$$
0.868522 + 0.495651i $$0.165070\pi$$
$$882$$ 0 0
$$883$$ 25.2991 0.851382 0.425691 0.904869i $$-0.360031\pi$$
0.425691 + 0.904869i $$0.360031\pi$$
$$884$$ 0 0
$$885$$ 68.0008 2.28582
$$886$$ 0 0
$$887$$ 1.92340 0.0645815 0.0322908 0.999479i $$-0.489720\pi$$
0.0322908 + 0.999479i $$0.489720\pi$$
$$888$$ 0 0
$$889$$ −26.8402 −0.900192
$$890$$ 0 0
$$891$$ −55.8293 −1.87035
$$892$$ 0 0
$$893$$ −3.99564 −0.133709
$$894$$ 0 0
$$895$$ 12.3541 0.412950
$$896$$ 0 0
$$897$$ −12.4146 −0.414512
$$898$$ 0 0
$$899$$ 28.6921 0.956936
$$900$$ 0 0
$$901$$ 11.0735 0.368912
$$902$$ 0 0
$$903$$ −0.835522 −0.0278044
$$904$$ 0 0
$$905$$ 19.0268 0.632471
$$906$$ 0 0
$$907$$ −27.7093 −0.920071 −0.460036 0.887900i $$-0.652163\pi$$
−0.460036 + 0.887900i $$0.652163\pi$$
$$908$$ 0 0
$$909$$ −8.55060 −0.283605
$$910$$ 0 0
$$911$$ −51.4735 −1.70539 −0.852696 0.522407i $$-0.825034\pi$$
−0.852696 + 0.522407i $$0.825034\pi$$
$$912$$ 0 0
$$913$$ −41.7856 −1.38290
$$914$$ 0 0
$$915$$ −45.5837 −1.50695
$$916$$ 0 0
$$917$$ −14.6048 −0.482293
$$918$$ 0 0
$$919$$ 25.4158 0.838390 0.419195 0.907896i $$-0.362312\pi$$
0.419195 + 0.907896i $$0.362312\pi$$
$$920$$ 0 0
$$921$$ −14.5129 −0.478216
$$922$$ 0 0
$$923$$ −0.898319 −0.0295685
$$924$$ 0 0
$$925$$ −27.8022 −0.914131
$$926$$ 0 0
$$927$$ −1.63747 −0.0537815
$$928$$ 0 0
$$929$$ −41.2485 −1.35332 −0.676661 0.736295i $$-0.736574\pi$$
−0.676661 + 0.736295i $$0.736574\pi$$
$$930$$ 0 0
$$931$$ 9.33880 0.306067
$$932$$ 0 0
$$933$$ −31.3185 −1.02532
$$934$$ 0 0
$$935$$ 54.2103 1.77287
$$936$$ 0 0
$$937$$ −36.1849 −1.18211 −0.591054 0.806632i $$-0.701288\pi$$
−0.591054 + 0.806632i $$0.701288\pi$$
$$938$$ 0 0
$$939$$ −15.4317 −0.503593
$$940$$ 0 0
$$941$$ −17.2262 −0.561559 −0.280779 0.959772i $$-0.590593\pi$$
−0.280779 + 0.959772i $$0.590593\pi$$
$$942$$ 0 0
$$943$$ 70.9267 2.30969
$$944$$ 0 0
$$945$$ −3.04451 −0.0990378
$$946$$ 0 0
$$947$$ 17.7823 0.577847 0.288924 0.957352i $$-0.406703\pi$$
0.288924 + 0.957352i $$0.406703\pi$$
$$948$$ 0 0
$$949$$ 5.91546 0.192024
$$950$$ 0 0
$$951$$ 11.6765 0.378636
$$952$$ 0 0
$$953$$ −47.3478 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$954$$ 0 0
$$955$$ 64.0269 2.07186
$$956$$ 0 0
$$957$$ 61.0698 1.97411
$$958$$ 0 0
$$959$$ 47.4821 1.53328
$$960$$ 0 0
$$961$$ 24.3045 0.784017
$$962$$ 0 0
$$963$$ 0.655354 0.0211185
$$964$$ 0 0
$$965$$ 11.6892 0.376289
$$966$$ 0 0
$$967$$ 25.3504 0.815213 0.407606 0.913158i $$-0.366364\pi$$
0.407606 + 0.913158i $$0.366364\pi$$
$$968$$ 0 0
$$969$$ 6.46871 0.207805
$$970$$ 0 0
$$971$$ 45.3901 1.45664 0.728319 0.685238i $$-0.240302\pi$$
0.728319 + 0.685238i $$0.240302\pi$$
$$972$$ 0 0
$$973$$ 82.7653 2.65333
$$974$$ 0 0
$$975$$ 10.8939 0.348883
$$976$$ 0 0
$$977$$ 27.3398 0.874679 0.437339 0.899297i $$-0.355921\pi$$
0.437339 + 0.899297i $$0.355921\pi$$
$$978$$ 0 0
$$979$$ −98.6549 −3.15302
$$980$$ 0 0
$$981$$ −30.1381 −0.962236
$$982$$ 0 0
$$983$$ 4.84750 0.154611 0.0773056 0.997007i $$-0.475368\pi$$
0.0773056 + 0.997007i $$0.475368\pi$$
$$984$$ 0 0
$$985$$ −42.3313 −1.34879
$$986$$ 0 0
$$987$$ −39.8720 −1.26914
$$988$$ 0 0
$$989$$ 0.516372 0.0164197
$$990$$ 0 0
$$991$$ −28.1022 −0.892696 −0.446348 0.894859i $$-0.647276\pi$$
−0.446348 + 0.894859i $$0.647276\pi$$
$$992$$ 0 0
$$993$$ −48.8954 −1.55165
$$994$$ 0 0
$$995$$ 51.3551 1.62807
$$996$$ 0 0
$$997$$ −6.99262 −0.221458 −0.110729 0.993851i $$-0.535319\pi$$
−0.110729 + 0.993851i $$0.535319\pi$$
$$998$$ 0 0
$$999$$ 1.19920 0.0379409
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 4864.2.a.bt.1.1 10
4.3 odd 2 4864.2.a.bs.1.9 10
8.3 odd 2 inner 4864.2.a.bt.1.2 10
8.5 even 2 4864.2.a.bs.1.10 10
16.3 odd 4 2432.2.c.j.1217.18 yes 20
16.5 even 4 2432.2.c.j.1217.17 yes 20
16.11 odd 4 2432.2.c.j.1217.3 20
16.13 even 4 2432.2.c.j.1217.4 yes 20

By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.3 20 16.11 odd 4
2432.2.c.j.1217.4 yes 20 16.13 even 4
2432.2.c.j.1217.17 yes 20 16.5 even 4
2432.2.c.j.1217.18 yes 20 16.3 odd 4
4864.2.a.bs.1.9 10 4.3 odd 2
4864.2.a.bs.1.10 10 8.5 even 2
4864.2.a.bt.1.1 10 1.1 even 1 trivial
4864.2.a.bt.1.2 10 8.3 odd 2 inner