# Properties

 Label 4864.2.a.bs.1.9 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $1$ 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: $$1$$ 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 1.9 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 $$-0.247492\pi$$
0.712657 + 0.701513i $$0.247492\pi$$
$$4$$ 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 $$-0.223287\pi$$
0.763890 + 0.645346i $$0.223287\pi$$
$$8$$ 0 0
$$9$$ 3.09455 1.03152
$$10$$ 0 0
$$11$$ −6.41171 −1.93320 −0.966601 0.256286i $$-0.917501\pi$$
−0.966601 + 0.256286i $$0.917501\pi$$
$$12$$ 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 $$-0.277701\pi$$
0.642972 + 0.765890i $$0.277701\pi$$
$$24$$ 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 $$-0.267221\pi$$
0.667835 + 0.744309i $$0.267221\pi$$
$$32$$ 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 $$-0.497968\pi$$
0.00638429 + 0.999980i $$0.497968\pi$$
$$44$$ 0 0
$$45$$ −9.98525 −1.48851
$$46$$ 0 0
$$47$$ 3.99564 0.582824 0.291412 0.956598i $$-0.405875\pi$$
0.291412 + 0.956598i $$0.405875\pi$$
$$48$$ 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 $$-0.687543\pi$$
−0.555682 + 0.831395i $$0.687543\pi$$
$$60$$ 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 $$-0.254045\pi$$
0.698063 + 0.716036i $$0.254045\pi$$
$$68$$ 0 0
$$69$$ 15.2250 1.83287
$$70$$ 0 0
$$71$$ −1.10168 −0.130745 −0.0653725 0.997861i $$-0.520824\pi$$
−0.0653725 + 0.997861i $$0.520824\pi$$
$$72$$ 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 $$-0.788384\pi$$
−0.787033 + 0.616911i $$0.788384\pi$$
$$80$$ 0 0
$$81$$ −8.70740 −0.967489
$$82$$ 0 0
$$83$$ 6.51708 0.715343 0.357671 0.933848i $$-0.383571\pi$$
0.357671 + 0.933848i $$0.383571\pi$$
$$84$$ 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 $$-0.491701\pi$$
0.0260691 + 0.999660i $$0.491701\pi$$
$$104$$ 0 0
$$105$$ −32.1990 −3.14230
$$106$$ 0 0
$$107$$ −0.211777 −0.0204732 −0.0102366 0.999948i $$-0.503258\pi$$
−0.0102366 + 0.999948i $$0.503258\pi$$
$$108$$ 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 $$-0.595189\pi$$
−0.294608 + 0.955618i $$0.595189\pi$$
$$128$$ 0 0
$$129$$ 0.206704 0.0181992
$$130$$ 0 0
$$131$$ −3.61315 −0.315682 −0.157841 0.987465i $$-0.550453\pi$$
−0.157841 + 0.987465i $$0.550453\pi$$
$$132$$ 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 $$-0.165173\pi$$
0.868362 + 0.495931i $$0.165173\pi$$
$$140$$ 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 $$-0.679998\pi$$
−0.535822 + 0.844331i $$0.679998\pi$$
$$152$$ 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 $$-0.654122\pi$$
−0.465491 + 0.885053i $$0.654122\pi$$
$$164$$ 0 0
$$165$$ 51.0747 3.97616
$$166$$ 0 0
$$167$$ −19.2419 −1.48898 −0.744491 0.667632i $$-0.767308\pi$$
−0.744491 + 0.667632i $$0.767308\pi$$
$$168$$ 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 $$-0.454298\pi$$
0.143084 + 0.989711i $$0.454298\pi$$
$$180$$ 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 $$-0.245110\pi$$
0.717885 + 0.696162i $$0.245110\pi$$
$$192$$ 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 $$-0.309218\pi$$
0.564113 + 0.825697i $$0.309218\pi$$
$$200$$ 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 $$-0.555137\pi$$
−0.172352 + 0.985035i $$0.555137\pi$$
$$212$$ 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 $$-0.410343\pi$$
0.277957 + 0.960594i $$0.410343\pi$$
$$224$$ 0 0
$$225$$ 16.7468 1.11645
$$226$$ 0 0
$$227$$ 26.8177 1.77995 0.889976 0.456007i $$-0.150721\pi$$
0.889976 + 0.456007i $$0.150721\pi$$
$$228$$ 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 $$-0.855704\pi$$
−0.898999 + 0.437951i $$0.855704\pi$$
$$240$$ 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 $$-0.343111\pi$$
0.473167 + 0.880973i $$0.343111\pi$$
$$252$$ 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 $$-0.196612\pi$$
0.815228 + 0.579140i $$0.196612\pi$$
$$264$$ 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 $$-0.639959\pi$$
−0.425663 + 0.904882i $$0.639959\pi$$
$$272$$ 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 $$-0.480379\pi$$
0.0616018 + 0.998101i $$0.480379\pi$$
$$284$$ 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 $$-0.553653\pi$$
−0.167758 + 0.985828i $$0.553653\pi$$
$$308$$ 0 0
$$309$$ 1.30631 0.0743132
$$310$$ 0 0
$$311$$ −12.6861 −0.719365 −0.359683 0.933075i $$-0.617115\pi$$
−0.359683 + 0.933075i $$0.617115\pi$$
$$312$$ 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 $$-0.683212\pi$$
−0.544318 + 0.838879i $$0.683212\pi$$
$$332$$ 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 $$-0.262386\pi$$
0.679063 + 0.734080i $$0.262386\pi$$
$$348$$ 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 $$-0.261909\pi$$
0.680164 + 0.733060i $$0.261909\pi$$
$$360$$ 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 $$-0.532601\pi$$
−0.102242 + 0.994760i $$0.532601\pi$$
$$368$$ 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 $$-0.424824\pi$$
0.233982 + 0.972241i $$0.424824\pi$$
$$380$$ 0 0
$$381$$ −16.3926 −0.839816
$$382$$ 0 0
$$383$$ −28.4774 −1.45513 −0.727564 0.686040i $$-0.759348\pi$$
−0.727564 + 0.686040i $$0.759348\pi$$
$$384$$ 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 $$-0.594835\pi$$
−0.293546 + 0.955945i $$0.594835\pi$$
$$420$$ 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 $$-0.503575\pi$$
−0.0112314 + 0.999937i $$0.503575\pi$$
$$432$$ 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 $$-0.315953\pi$$
0.546517 + 0.837448i $$0.315953\pi$$
$$440$$ 0 0
$$441$$ 28.8994 1.37616
$$442$$ 0 0
$$443$$ −4.15864 −0.197583 −0.0987914 0.995108i $$-0.531498\pi$$
−0.0987914 + 0.995108i $$0.531498\pi$$
$$444$$ 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 $$-0.240257\pi$$
0.728415 + 0.685136i $$0.240257\pi$$
$$464$$ 0 0
$$465$$ −59.2396 −2.74717
$$466$$ 0 0
$$467$$ −7.14257 −0.330519 −0.165259 0.986250i $$-0.552846\pi$$
−0.165259 + 0.986250i $$0.552846\pi$$
$$468$$ 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 $$-0.292601\pi$$
0.606430 + 0.795137i $$0.292601\pi$$
$$480$$ 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 $$-0.570870\pi$$
−0.220810 + 0.975317i $$0.570870\pi$$
$$488$$ 0 0
$$489$$ −29.3431 −1.32694
$$490$$ 0 0
$$491$$ −25.6150 −1.15599 −0.577995 0.816040i $$-0.696165\pi$$
−0.577995 + 0.816040i $$0.696165\pi$$
$$492$$ 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 $$-0.460800\pi$$
0.122838 + 0.992427i $$0.460800\pi$$
$$500$$ 0 0
$$501$$ −47.5028 −2.12227
$$502$$ 0 0
$$503$$ 9.00013 0.401296 0.200648 0.979663i $$-0.435695\pi$$
0.200648 + 0.979663i $$0.435695\pi$$
$$504$$ 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 $$-0.236716\pi$$
0.735992 + 0.676990i $$0.236716\pi$$
$$524$$ 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 $$-0.460213\pi$$
0.124670 + 0.992198i $$0.460213\pi$$
$$548$$ 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 $$-0.865814\pi$$
−0.912452 + 0.409183i $$0.865814\pi$$
$$564$$ 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 $$-0.649035\pi$$
−0.451287 + 0.892379i $$0.649035\pi$$
$$572$$ 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 $$-0.703965\pi$$
−0.597817 + 0.801633i $$0.703965\pi$$
$$588$$ 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 $$-0.663649\pi$$
−0.491769 + 0.870726i $$0.663649\pi$$
$$600$$ 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 $$-0.613636\pi$$
−0.349464 + 0.936950i $$0.613636\pi$$
$$608$$ 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 $$-0.278787\pi$$
0.640356 + 0.768078i $$0.278787\pi$$
$$620$$ 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 $$-0.453177\pi$$
0.146568 + 0.989201i $$0.453177\pi$$
$$632$$ 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 $$-0.600458\pi$$
−0.310384 + 0.950611i $$0.600458\pi$$
$$644$$ 0 0
$$645$$ −0.666974 −0.0262621
$$646$$ 0 0
$$647$$ −17.8689 −0.702499 −0.351250 0.936282i $$-0.614243\pi$$
−0.351250 + 0.936282i $$0.614243\pi$$
$$648$$ 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 $$-0.613652\pi$$
−0.349510 + 0.936932i $$0.613652\pi$$
$$660$$ 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 $$-0.467599\pi$$
0.101615 + 0.994824i $$0.467599\pi$$
$$684$$ 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 $$-0.586165\pi$$
−0.267401 + 0.963585i $$0.586165\pi$$
$$692$$ 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 $$-0.462480\pi$$
0.117600 + 0.993061i $$0.462480\pi$$
$$720$$ 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 $$-0.347827\pi$$
0.460061 + 0.887887i $$0.347827\pi$$
$$728$$ 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 $$-0.556328\pi$$
−0.176038 + 0.984383i $$0.556328\pi$$
$$740$$ 0 0
$$741$$ 2.01302 0.0739500
$$742$$ 0 0
$$743$$ 10.3553 0.379900 0.189950 0.981794i $$-0.439167\pi$$
0.189950 + 0.981794i $$0.439167\pi$$
$$744$$ 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 $$-0.312244\pi$$
0.556239 + 0.831022i $$0.312244\pi$$
$$752$$ 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 $$-0.467450\pi$$
0.102080 + 0.994776i $$0.467450\pi$$
$$788$$ 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 $$-0.384397\pi$$
0.355246 + 0.934773i $$0.384397\pi$$
$$812$$ 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 $$-0.468866\pi$$
0.0976559 + 0.995220i $$0.468866\pi$$
$$824$$ 0 0
$$825$$ −85.6601 −2.98230
$$826$$ 0 0
$$827$$ −28.3875 −0.987129 −0.493565 0.869709i $$-0.664306\pi$$
−0.493565 + 0.869709i $$0.664306\pi$$
$$828$$ 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 $$-0.216801\pi$$
0.776881 + 0.629647i $$0.216801\pi$$
$$840$$ 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 $$-0.541246\pi$$
−0.129215 + 0.991617i $$0.541246\pi$$
$$860$$ 0 0
$$861$$ −114.764 −3.91114
$$862$$ 0 0
$$863$$ −34.6063 −1.17801 −0.589007 0.808128i $$-0.700481\pi$$
−0.589007 + 0.808128i $$0.700481\pi$$
$$864$$ 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 $$-0.639969\pi$$
−0.425691 + 0.904869i $$0.639969\pi$$
$$884$$ 0 0
$$885$$ 68.0008 2.28582
$$886$$ 0 0
$$887$$ −1.92340 −0.0645815 −0.0322908 0.999479i $$-0.510280\pi$$
−0.0322908 + 0.999479i $$0.510280\pi$$
$$888$$ 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 $$-0.347837\pi$$
0.460036 + 0.887900i $$0.347837\pi$$
$$908$$ 0 0
$$909$$ −8.55060 −0.283605
$$910$$ 0 0
$$911$$ 51.4735 1.70539 0.852696 0.522407i $$-0.174966\pi$$
0.852696 + 0.522407i $$0.174966\pi$$
$$912$$ 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 $$-0.637688\pi$$
−0.419195 + 0.907896i $$0.637688\pi$$
$$920$$ 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 $$-0.593297\pi$$
−0.288924 + 0.957352i $$0.593297\pi$$
$$948$$ 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 $$-0.633636\pi$$
−0.407606 + 0.913158i $$0.633636\pi$$
$$968$$ 0 0
$$969$$ 6.46871 0.207805
$$970$$ 0 0
$$971$$ −45.3901 −1.45664 −0.728319 0.685238i $$-0.759698\pi$$
−0.728319 + 0.685238i $$0.759698\pi$$
$$972$$ 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 $$-0.524632\pi$$
−0.0773056 + 0.997007i $$0.524632\pi$$
$$984$$ 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 $$-0.352724\pi$$
0.446348 + 0.894859i $$0.352724\pi$$
$$992$$ 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.bs.1.9 10
4.3 odd 2 4864.2.a.bt.1.1 10
8.3 odd 2 inner 4864.2.a.bs.1.10 10
8.5 even 2 4864.2.a.bt.1.2 10
16.3 odd 4 2432.2.c.j.1217.4 yes 20
16.5 even 4 2432.2.c.j.1217.3 20
16.11 odd 4 2432.2.c.j.1217.17 yes 20
16.13 even 4 2432.2.c.j.1217.18 yes 20

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