# Properties

 Label 384.10.d.a.193.2 Level $384$ Weight $10$ Character 384.193 Analytic conductor $197.774$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$197.773761087$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225$$ x^8 + 13062*x^6 + 45211107*x^4 + 45928424926*x^2 + 852972309225 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 193.2 Root $$-4.34998i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.10.d.a.193.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-81.0000i q^{3} -1428.10i q^{5} +6382.13 q^{7} -6561.00 q^{9} +O(q^{10})$$ $$q-81.0000i q^{3} -1428.10i q^{5} +6382.13 q^{7} -6561.00 q^{9} +4073.61i q^{11} -154396. i q^{13} -115676. q^{15} +255714. q^{17} +1.01702e6i q^{19} -516952. i q^{21} -1.56457e6 q^{23} -86331.5 q^{25} +531441. i q^{27} +7.26136e6i q^{29} -611897. q^{31} +329962. q^{33} -9.11429e6i q^{35} -464986. i q^{37} -1.25061e7 q^{39} +2.04743e7 q^{41} +2.95237e7i q^{43} +9.36973e6i q^{45} +1.31748e6 q^{47} +377951. q^{49} -2.07129e7i q^{51} +5.76562e7i q^{53} +5.81750e6 q^{55} +8.23785e7 q^{57} +6.35883e7i q^{59} -5.01862e7i q^{61} -4.18731e7 q^{63} -2.20493e8 q^{65} +2.90029e8i q^{67} +1.26730e8i q^{69} -3.18003e8 q^{71} -3.49122e8 q^{73} +6.99285e6i q^{75} +2.59983e7i q^{77} -4.29993e8 q^{79} +4.30467e7 q^{81} +3.40898e8i q^{83} -3.65185e8i q^{85} +5.88170e8 q^{87} +5.24211e8 q^{89} -9.85376e8i q^{91} +4.95636e7i q^{93} +1.45240e9 q^{95} -9.61784e8 q^{97} -2.67270e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10})$$ 8 * q - 13632 * q^7 - 52488 * q^9 $$8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97}+O(q^{100})$$ 8 * q - 13632 * q^7 - 52488 * q^9 - 90720 * q^15 + 8304 * q^17 - 4612608 * q^23 - 4754904 * q^25 - 7499328 * q^31 - 6213024 * q^33 - 23211360 * q^39 - 43518896 * q^41 + 49382016 * q^47 - 74106808 * q^49 + 19030656 * q^55 + 38141280 * q^57 + 89439552 * q^63 - 110270336 * q^65 - 741751296 * q^71 - 1507903440 * q^73 + 1008373440 * q^79 + 344373768 * q^81 + 423468000 * q^87 - 1337034448 * q^89 - 543950208 * q^95 - 904817936 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ − 81.0000i − 0.577350i
$$4$$ 0 0
$$5$$ − 1428.10i − 1.02186i −0.859622 0.510931i $$-0.829301\pi$$
0.859622 0.510931i $$-0.170699\pi$$
$$6$$ 0 0
$$7$$ 6382.13 1.00467 0.502336 0.864672i $$-0.332474\pi$$
0.502336 + 0.864672i $$0.332474\pi$$
$$8$$ 0 0
$$9$$ −6561.00 −0.333333
$$10$$ 0 0
$$11$$ 4073.61i 0.0838904i 0.999120 + 0.0419452i $$0.0133555\pi$$
−0.999120 + 0.0419452i $$0.986645\pi$$
$$12$$ 0 0
$$13$$ − 154396.i − 1.49931i −0.661829 0.749655i $$-0.730219\pi$$
0.661829 0.749655i $$-0.269781\pi$$
$$14$$ 0 0
$$15$$ −115676. −0.589972
$$16$$ 0 0
$$17$$ 255714. 0.742566 0.371283 0.928520i $$-0.378918\pi$$
0.371283 + 0.928520i $$0.378918\pi$$
$$18$$ 0 0
$$19$$ 1.01702e6i 1.79035i 0.445716 + 0.895175i $$0.352949\pi$$
−0.445716 + 0.895175i $$0.647051\pi$$
$$20$$ 0 0
$$21$$ − 516952.i − 0.580048i
$$22$$ 0 0
$$23$$ −1.56457e6 −1.16579 −0.582894 0.812548i $$-0.698080\pi$$
−0.582894 + 0.812548i $$0.698080\pi$$
$$24$$ 0 0
$$25$$ −86331.5 −0.0442017
$$26$$ 0 0
$$27$$ 531441.i 0.192450i
$$28$$ 0 0
$$29$$ 7.26136e6i 1.90646i 0.302253 + 0.953228i $$0.402261\pi$$
−0.302253 + 0.953228i $$0.597739\pi$$
$$30$$ 0 0
$$31$$ −611897. −0.119001 −0.0595005 0.998228i $$-0.518951\pi$$
−0.0595005 + 0.998228i $$0.518951\pi$$
$$32$$ 0 0
$$33$$ 329962. 0.0484342
$$34$$ 0 0
$$35$$ − 9.11429e6i − 1.02664i
$$36$$ 0 0
$$37$$ − 464986.i − 0.0407880i −0.999792 0.0203940i $$-0.993508\pi$$
0.999792 0.0203940i $$-0.00649205\pi$$
$$38$$ 0 0
$$39$$ −1.25061e7 −0.865627
$$40$$ 0 0
$$41$$ 2.04743e7 1.13157 0.565785 0.824553i $$-0.308573\pi$$
0.565785 + 0.824553i $$0.308573\pi$$
$$42$$ 0 0
$$43$$ 2.95237e7i 1.31693i 0.752611 + 0.658465i $$0.228794\pi$$
−0.752611 + 0.658465i $$0.771206\pi$$
$$44$$ 0 0
$$45$$ 9.36973e6i 0.340621i
$$46$$ 0 0
$$47$$ 1.31748e6 0.0393824 0.0196912 0.999806i $$-0.493732\pi$$
0.0196912 + 0.999806i $$0.493732\pi$$
$$48$$ 0 0
$$49$$ 377951. 0.00936598
$$50$$ 0 0
$$51$$ − 2.07129e7i − 0.428721i
$$52$$ 0 0
$$53$$ 5.76562e7i 1.00370i 0.864954 + 0.501851i $$0.167347\pi$$
−0.864954 + 0.501851i $$0.832653\pi$$
$$54$$ 0 0
$$55$$ 5.81750e6 0.0857244
$$56$$ 0 0
$$57$$ 8.23785e7 1.03366
$$58$$ 0 0
$$59$$ 6.35883e7i 0.683193i 0.939847 + 0.341596i $$0.110968\pi$$
−0.939847 + 0.341596i $$0.889032\pi$$
$$60$$ 0 0
$$61$$ − 5.01862e7i − 0.464088i −0.972705 0.232044i $$-0.925459\pi$$
0.972705 0.232044i $$-0.0745413\pi$$
$$62$$ 0 0
$$63$$ −4.18731e7 −0.334891
$$64$$ 0 0
$$65$$ −2.20493e8 −1.53209
$$66$$ 0 0
$$67$$ 2.90029e8i 1.75835i 0.476499 + 0.879175i $$0.341906\pi$$
−0.476499 + 0.879175i $$0.658094\pi$$
$$68$$ 0 0
$$69$$ 1.26730e8i 0.673068i
$$70$$ 0 0
$$71$$ −3.18003e8 −1.48514 −0.742572 0.669766i $$-0.766394\pi$$
−0.742572 + 0.669766i $$0.766394\pi$$
$$72$$ 0 0
$$73$$ −3.49122e8 −1.43888 −0.719440 0.694554i $$-0.755602\pi$$
−0.719440 + 0.694554i $$0.755602\pi$$
$$74$$ 0 0
$$75$$ 6.99285e6i 0.0255199i
$$76$$ 0 0
$$77$$ 2.59983e7i 0.0842824i
$$78$$ 0 0
$$79$$ −4.29993e8 −1.24205 −0.621025 0.783791i $$-0.713284\pi$$
−0.621025 + 0.783791i $$0.713284\pi$$
$$80$$ 0 0
$$81$$ 4.30467e7 0.111111
$$82$$ 0 0
$$83$$ 3.40898e8i 0.788447i 0.919015 + 0.394223i $$0.128986\pi$$
−0.919015 + 0.394223i $$0.871014\pi$$
$$84$$ 0 0
$$85$$ − 3.65185e8i − 0.758800i
$$86$$ 0 0
$$87$$ 5.88170e8 1.10069
$$88$$ 0 0
$$89$$ 5.24211e8 0.885628 0.442814 0.896614i $$-0.353980\pi$$
0.442814 + 0.896614i $$0.353980\pi$$
$$90$$ 0 0
$$91$$ − 9.85376e8i − 1.50632i
$$92$$ 0 0
$$93$$ 4.95636e7i 0.0687053i
$$94$$ 0 0
$$95$$ 1.45240e9 1.82949
$$96$$ 0 0
$$97$$ −9.61784e8 −1.10307 −0.551537 0.834150i $$-0.685958\pi$$
−0.551537 + 0.834150i $$0.685958\pi$$
$$98$$ 0 0
$$99$$ − 2.67270e7i − 0.0279635i
$$100$$ 0 0
$$101$$ 2.61811e8i 0.250346i 0.992135 + 0.125173i $$0.0399486\pi$$
−0.992135 + 0.125173i $$0.960051\pi$$
$$102$$ 0 0
$$103$$ −4.87298e8 −0.426606 −0.213303 0.976986i $$-0.568422\pi$$
−0.213303 + 0.976986i $$0.568422\pi$$
$$104$$ 0 0
$$105$$ −7.38257e8 −0.592729
$$106$$ 0 0
$$107$$ − 1.63558e9i − 1.20627i −0.797640 0.603134i $$-0.793918\pi$$
0.797640 0.603134i $$-0.206082\pi$$
$$108$$ 0 0
$$109$$ 1.81852e9i 1.23396i 0.786981 + 0.616978i $$0.211643\pi$$
−0.786981 + 0.616978i $$0.788357\pi$$
$$110$$ 0 0
$$111$$ −3.76639e7 −0.0235489
$$112$$ 0 0
$$113$$ −2.89653e9 −1.67119 −0.835594 0.549348i $$-0.814876\pi$$
−0.835594 + 0.549348i $$0.814876\pi$$
$$114$$ 0 0
$$115$$ 2.23435e9i 1.19127i
$$116$$ 0 0
$$117$$ 1.01299e9i 0.499770i
$$118$$ 0 0
$$119$$ 1.63200e9 0.746036
$$120$$ 0 0
$$121$$ 2.34135e9 0.992962
$$122$$ 0 0
$$123$$ − 1.65842e9i − 0.653312i
$$124$$ 0 0
$$125$$ − 2.66596e9i − 0.976694i
$$126$$ 0 0
$$127$$ −4.08260e8 −0.139258 −0.0696290 0.997573i $$-0.522182\pi$$
−0.0696290 + 0.997573i $$0.522182\pi$$
$$128$$ 0 0
$$129$$ 2.39142e9 0.760330
$$130$$ 0 0
$$131$$ − 1.67425e9i − 0.496706i −0.968670 0.248353i $$-0.920111\pi$$
0.968670 0.248353i $$-0.0798893\pi$$
$$132$$ 0 0
$$133$$ 6.49074e9i 1.79871i
$$134$$ 0 0
$$135$$ 7.58948e8 0.196657
$$136$$ 0 0
$$137$$ −1.57362e9 −0.381642 −0.190821 0.981625i $$-0.561115\pi$$
−0.190821 + 0.981625i $$0.561115\pi$$
$$138$$ 0 0
$$139$$ 5.36040e9i 1.21795i 0.793188 + 0.608977i $$0.208420\pi$$
−0.793188 + 0.608977i $$0.791580\pi$$
$$140$$ 0 0
$$141$$ − 1.06715e8i − 0.0227374i
$$142$$ 0 0
$$143$$ 6.28950e8 0.125778
$$144$$ 0 0
$$145$$ 1.03699e10 1.94813
$$146$$ 0 0
$$147$$ − 3.06140e7i − 0.00540745i
$$148$$ 0 0
$$149$$ − 1.83174e9i − 0.304457i −0.988345 0.152228i $$-0.951355\pi$$
0.988345 0.152228i $$-0.0486449\pi$$
$$150$$ 0 0
$$151$$ 4.73373e9 0.740981 0.370490 0.928836i $$-0.379190\pi$$
0.370490 + 0.928836i $$0.379190\pi$$
$$152$$ 0 0
$$153$$ −1.67774e9 −0.247522
$$154$$ 0 0
$$155$$ 8.73847e8i 0.121603i
$$156$$ 0 0
$$157$$ − 8.73612e8i − 0.114755i −0.998353 0.0573773i $$-0.981726\pi$$
0.998353 0.0573773i $$-0.0182738\pi$$
$$158$$ 0 0
$$159$$ 4.67015e9 0.579487
$$160$$ 0 0
$$161$$ −9.98528e9 −1.17123
$$162$$ 0 0
$$163$$ 8.32642e9i 0.923877i 0.886912 + 0.461939i $$0.152846\pi$$
−0.886912 + 0.461939i $$0.847154\pi$$
$$164$$ 0 0
$$165$$ − 4.71218e8i − 0.0494930i
$$166$$ 0 0
$$167$$ 1.58090e9 0.157283 0.0786414 0.996903i $$-0.474942\pi$$
0.0786414 + 0.996903i $$0.474942\pi$$
$$168$$ 0 0
$$169$$ −1.32337e10 −1.24793
$$170$$ 0 0
$$171$$ − 6.67266e9i − 0.596783i
$$172$$ 0 0
$$173$$ − 1.19275e10i − 1.01238i −0.862423 0.506189i $$-0.831054\pi$$
0.862423 0.506189i $$-0.168946\pi$$
$$174$$ 0 0
$$175$$ −5.50979e8 −0.0444082
$$176$$ 0 0
$$177$$ 5.15065e9 0.394441
$$178$$ 0 0
$$179$$ 1.27369e10i 0.927309i 0.886016 + 0.463654i $$0.153462\pi$$
−0.886016 + 0.463654i $$0.846538\pi$$
$$180$$ 0 0
$$181$$ − 8.46716e9i − 0.586387i −0.956053 0.293193i $$-0.905282\pi$$
0.956053 0.293193i $$-0.0947180\pi$$
$$182$$ 0 0
$$183$$ −4.06508e9 −0.267941
$$184$$ 0 0
$$185$$ −6.64044e8 −0.0416797
$$186$$ 0 0
$$187$$ 1.04168e9i 0.0622942i
$$188$$ 0 0
$$189$$ 3.39172e9i 0.193349i
$$190$$ 0 0
$$191$$ −3.90757e9 −0.212450 −0.106225 0.994342i $$-0.533876\pi$$
−0.106225 + 0.994342i $$0.533876\pi$$
$$192$$ 0 0
$$193$$ 1.82705e10 0.947858 0.473929 0.880563i $$-0.342835\pi$$
0.473929 + 0.880563i $$0.342835\pi$$
$$194$$ 0 0
$$195$$ 1.78599e10i 0.884552i
$$196$$ 0 0
$$197$$ − 1.05198e10i − 0.497633i −0.968551 0.248816i $$-0.919958\pi$$
0.968551 0.248816i $$-0.0800416\pi$$
$$198$$ 0 0
$$199$$ −3.02525e10 −1.36748 −0.683741 0.729725i $$-0.739648\pi$$
−0.683741 + 0.729725i $$0.739648\pi$$
$$200$$ 0 0
$$201$$ 2.34924e10 1.01518
$$202$$ 0 0
$$203$$ 4.63429e10i 1.91536i
$$204$$ 0 0
$$205$$ − 2.92392e10i − 1.15631i
$$206$$ 0 0
$$207$$ 1.02651e10 0.388596
$$208$$ 0 0
$$209$$ −4.14294e9 −0.150193
$$210$$ 0 0
$$211$$ 1.53444e10i 0.532940i 0.963843 + 0.266470i $$0.0858573\pi$$
−0.963843 + 0.266470i $$0.914143\pi$$
$$212$$ 0 0
$$213$$ 2.57582e10i 0.857448i
$$214$$ 0 0
$$215$$ 4.21626e10 1.34572
$$216$$ 0 0
$$217$$ −3.90520e9 −0.119557
$$218$$ 0 0
$$219$$ 2.82789e10i 0.830738i
$$220$$ 0 0
$$221$$ − 3.94814e10i − 1.11334i
$$222$$ 0 0
$$223$$ 6.26805e10 1.69731 0.848654 0.528948i $$-0.177414\pi$$
0.848654 + 0.528948i $$0.177414\pi$$
$$224$$ 0 0
$$225$$ 5.66421e8 0.0147339
$$226$$ 0 0
$$227$$ − 4.85806e10i − 1.21436i −0.794565 0.607179i $$-0.792301\pi$$
0.794565 0.607179i $$-0.207699\pi$$
$$228$$ 0 0
$$229$$ 6.64074e10i 1.59572i 0.602843 + 0.797860i $$0.294035\pi$$
−0.602843 + 0.797860i $$0.705965\pi$$
$$230$$ 0 0
$$231$$ 2.10586e9 0.0486605
$$232$$ 0 0
$$233$$ 1.53115e10 0.340342 0.170171 0.985415i $$-0.445568\pi$$
0.170171 + 0.985415i $$0.445568\pi$$
$$234$$ 0 0
$$235$$ − 1.88148e9i − 0.0402434i
$$236$$ 0 0
$$237$$ 3.48294e10i 0.717098i
$$238$$ 0 0
$$239$$ 5.49921e10 1.09021 0.545105 0.838368i $$-0.316490\pi$$
0.545105 + 0.838368i $$0.316490\pi$$
$$240$$ 0 0
$$241$$ −1.88950e10 −0.360802 −0.180401 0.983593i $$-0.557740\pi$$
−0.180401 + 0.983593i $$0.557740\pi$$
$$242$$ 0 0
$$243$$ − 3.48678e9i − 0.0641500i
$$244$$ 0 0
$$245$$ − 5.39750e8i − 0.00957074i
$$246$$ 0 0
$$247$$ 1.57024e11 2.68429
$$248$$ 0 0
$$249$$ 2.76127e10 0.455210
$$250$$ 0 0
$$251$$ 1.10372e11i 1.75520i 0.479395 + 0.877599i $$0.340856\pi$$
−0.479395 + 0.877599i $$0.659144\pi$$
$$252$$ 0 0
$$253$$ − 6.37344e9i − 0.0977984i
$$254$$ 0 0
$$255$$ −2.95800e10 −0.438093
$$256$$ 0 0
$$257$$ 2.31328e10 0.330772 0.165386 0.986229i $$-0.447113\pi$$
0.165386 + 0.986229i $$0.447113\pi$$
$$258$$ 0 0
$$259$$ − 2.96760e9i − 0.0409785i
$$260$$ 0 0
$$261$$ − 4.76418e10i − 0.635485i
$$262$$ 0 0
$$263$$ −1.88727e10 −0.243239 −0.121619 0.992577i $$-0.538809\pi$$
−0.121619 + 0.992577i $$0.538809\pi$$
$$264$$ 0 0
$$265$$ 8.23385e10 1.02564
$$266$$ 0 0
$$267$$ − 4.24611e10i − 0.511317i
$$268$$ 0 0
$$269$$ 6.86708e10i 0.799626i 0.916597 + 0.399813i $$0.130925\pi$$
−0.916597 + 0.399813i $$0.869075\pi$$
$$270$$ 0 0
$$271$$ 1.15508e11 1.30092 0.650460 0.759540i $$-0.274576\pi$$
0.650460 + 0.759540i $$0.274576\pi$$
$$272$$ 0 0
$$273$$ −7.98155e10 −0.869672
$$274$$ 0 0
$$275$$ − 3.51681e8i − 0.00370810i
$$276$$ 0 0
$$277$$ − 4.52642e10i − 0.461950i −0.972960 0.230975i $$-0.925808\pi$$
0.972960 0.230975i $$-0.0741916\pi$$
$$278$$ 0 0
$$279$$ 4.01465e9 0.0396670
$$280$$ 0 0
$$281$$ −9.00026e10 −0.861146 −0.430573 0.902556i $$-0.641688\pi$$
−0.430573 + 0.902556i $$0.641688\pi$$
$$282$$ 0 0
$$283$$ 9.87406e10i 0.915076i 0.889190 + 0.457538i $$0.151269\pi$$
−0.889190 + 0.457538i $$0.848731\pi$$
$$284$$ 0 0
$$285$$ − 1.17644e11i − 1.05626i
$$286$$ 0 0
$$287$$ 1.30670e11 1.13686
$$288$$ 0 0
$$289$$ −5.31980e10 −0.448595
$$290$$ 0 0
$$291$$ 7.79045e10i 0.636860i
$$292$$ 0 0
$$293$$ 8.54059e10i 0.676992i 0.940968 + 0.338496i $$0.109918\pi$$
−0.940968 + 0.338496i $$0.890082\pi$$
$$294$$ 0 0
$$295$$ 9.08101e10 0.698128
$$296$$ 0 0
$$297$$ −2.16488e9 −0.0161447
$$298$$ 0 0
$$299$$ 2.41563e11i 1.74788i
$$300$$ 0 0
$$301$$ 1.88424e11i 1.32308i
$$302$$ 0 0
$$303$$ 2.12067e10 0.144537
$$304$$ 0 0
$$305$$ −7.16707e10 −0.474234
$$306$$ 0 0
$$307$$ 1.83582e11i 1.17953i 0.807576 + 0.589763i $$0.200779\pi$$
−0.807576 + 0.589763i $$0.799221\pi$$
$$308$$ 0 0
$$309$$ 3.94711e10i 0.246301i
$$310$$ 0 0
$$311$$ −1.13932e11 −0.690598 −0.345299 0.938493i $$-0.612222\pi$$
−0.345299 + 0.938493i $$0.612222\pi$$
$$312$$ 0 0
$$313$$ −2.41679e11 −1.42328 −0.711640 0.702545i $$-0.752047\pi$$
−0.711640 + 0.702545i $$0.752047\pi$$
$$314$$ 0 0
$$315$$ 5.97988e10i 0.342212i
$$316$$ 0 0
$$317$$ − 1.06227e11i − 0.590840i −0.955367 0.295420i $$-0.904540\pi$$
0.955367 0.295420i $$-0.0954596\pi$$
$$318$$ 0 0
$$319$$ −2.95799e10 −0.159933
$$320$$ 0 0
$$321$$ −1.32482e11 −0.696440
$$322$$ 0 0
$$323$$ 2.60066e11i 1.32945i
$$324$$ 0 0
$$325$$ 1.33293e10i 0.0662721i
$$326$$ 0 0
$$327$$ 1.47300e11 0.712424
$$328$$ 0 0
$$329$$ 8.40830e9 0.0395664
$$330$$ 0 0
$$331$$ − 6.82287e10i − 0.312422i −0.987724 0.156211i $$-0.950072\pi$$
0.987724 0.156211i $$-0.0499279\pi$$
$$332$$ 0 0
$$333$$ 3.05077e9i 0.0135960i
$$334$$ 0 0
$$335$$ 4.14190e11 1.79679
$$336$$ 0 0
$$337$$ 3.87135e11 1.63504 0.817518 0.575903i $$-0.195349\pi$$
0.817518 + 0.575903i $$0.195349\pi$$
$$338$$ 0 0
$$339$$ 2.34619e11i 0.964861i
$$340$$ 0 0
$$341$$ − 2.49263e9i − 0.00998304i
$$342$$ 0 0
$$343$$ −2.55130e11 −0.995262
$$344$$ 0 0
$$345$$ 1.80983e11 0.687782
$$346$$ 0 0
$$347$$ − 2.81800e11i − 1.04342i −0.853124 0.521709i $$-0.825295\pi$$
0.853124 0.521709i $$-0.174705\pi$$
$$348$$ 0 0
$$349$$ − 1.80174e11i − 0.650096i −0.945697 0.325048i $$-0.894619\pi$$
0.945697 0.325048i $$-0.105381\pi$$
$$350$$ 0 0
$$351$$ 8.20525e10 0.288542
$$352$$ 0 0
$$353$$ 1.18821e11 0.407294 0.203647 0.979044i $$-0.434721\pi$$
0.203647 + 0.979044i $$0.434721\pi$$
$$354$$ 0 0
$$355$$ 4.54139e11i 1.51761i
$$356$$ 0 0
$$357$$ − 1.32192e11i − 0.430724i
$$358$$ 0 0
$$359$$ 4.95475e11 1.57433 0.787167 0.616740i $$-0.211547\pi$$
0.787167 + 0.616740i $$0.211547\pi$$
$$360$$ 0 0
$$361$$ −7.11639e11 −2.20535
$$362$$ 0 0
$$363$$ − 1.89650e11i − 0.573287i
$$364$$ 0 0
$$365$$ 4.98580e11i 1.47034i
$$366$$ 0 0
$$367$$ 5.16234e11 1.48542 0.742709 0.669614i $$-0.233540\pi$$
0.742709 + 0.669614i $$0.233540\pi$$
$$368$$ 0 0
$$369$$ −1.34332e11 −0.377190
$$370$$ 0 0
$$371$$ 3.67969e11i 1.00839i
$$372$$ 0 0
$$373$$ − 2.26879e11i − 0.606881i −0.952850 0.303441i $$-0.901865\pi$$
0.952850 0.303441i $$-0.0981354\pi$$
$$374$$ 0 0
$$375$$ −2.15943e11 −0.563894
$$376$$ 0 0
$$377$$ 1.12113e12 2.85837
$$378$$ 0 0
$$379$$ 5.96753e11i 1.48566i 0.669482 + 0.742828i $$0.266516\pi$$
−0.669482 + 0.742828i $$0.733484\pi$$
$$380$$ 0 0
$$381$$ 3.30691e10i 0.0804007i
$$382$$ 0 0
$$383$$ −3.71694e11 −0.882654 −0.441327 0.897346i $$-0.645492\pi$$
−0.441327 + 0.897346i $$0.645492\pi$$
$$384$$ 0 0
$$385$$ 3.71281e10 0.0861249
$$386$$ 0 0
$$387$$ − 1.93705e11i − 0.438977i
$$388$$ 0 0
$$389$$ 2.71241e11i 0.600596i 0.953845 + 0.300298i $$0.0970862\pi$$
−0.953845 + 0.300298i $$0.902914\pi$$
$$390$$ 0 0
$$391$$ −4.00083e11 −0.865674
$$392$$ 0 0
$$393$$ −1.35614e11 −0.286773
$$394$$ 0 0
$$395$$ 6.14071e11i 1.26920i
$$396$$ 0 0
$$397$$ 3.00982e11i 0.608112i 0.952654 + 0.304056i $$0.0983410\pi$$
−0.952654 + 0.304056i $$0.901659\pi$$
$$398$$ 0 0
$$399$$ 5.25750e11 1.03849
$$400$$ 0 0
$$401$$ 3.10723e11 0.600101 0.300050 0.953923i $$-0.402997\pi$$
0.300050 + 0.953923i $$0.402997\pi$$
$$402$$ 0 0
$$403$$ 9.44746e10i 0.178419i
$$404$$ 0 0
$$405$$ − 6.14748e10i − 0.113540i
$$406$$ 0 0
$$407$$ 1.89417e9 0.00342172
$$408$$ 0 0
$$409$$ −4.46642e11 −0.789231 −0.394616 0.918846i $$-0.629122\pi$$
−0.394616 + 0.918846i $$0.629122\pi$$
$$410$$ 0 0
$$411$$ 1.27463e11i 0.220341i
$$412$$ 0 0
$$413$$ 4.05829e11i 0.686384i
$$414$$ 0 0
$$415$$ 4.86834e11 0.805684
$$416$$ 0 0
$$417$$ 4.34193e11 0.703186
$$418$$ 0 0
$$419$$ − 8.23618e11i − 1.30546i −0.757592 0.652728i $$-0.773624\pi$$
0.757592 0.652728i $$-0.226376\pi$$
$$420$$ 0 0
$$421$$ − 6.04524e11i − 0.937872i −0.883232 0.468936i $$-0.844637\pi$$
0.883232 0.468936i $$-0.155363\pi$$
$$422$$ 0 0
$$423$$ −8.64395e9 −0.0131275
$$424$$ 0 0
$$425$$ −2.20762e10 −0.0328227
$$426$$ 0 0
$$427$$ − 3.20295e11i − 0.466256i
$$428$$ 0 0
$$429$$ − 5.09450e10i − 0.0726179i
$$430$$ 0 0
$$431$$ 1.20184e12 1.67765 0.838823 0.544404i $$-0.183244\pi$$
0.838823 + 0.544404i $$0.183244\pi$$
$$432$$ 0 0
$$433$$ 1.53930e11 0.210440 0.105220 0.994449i $$-0.466445\pi$$
0.105220 + 0.994449i $$0.466445\pi$$
$$434$$ 0 0
$$435$$ − 8.39963e11i − 1.12476i
$$436$$ 0 0
$$437$$ − 1.59120e12i − 2.08717i
$$438$$ 0 0
$$439$$ 4.56899e11 0.587124 0.293562 0.955940i $$-0.405159\pi$$
0.293562 + 0.955940i $$0.405159\pi$$
$$440$$ 0 0
$$441$$ −2.47974e9 −0.00312199
$$442$$ 0 0
$$443$$ − 1.43507e12i − 1.77034i −0.465271 0.885168i $$-0.654043\pi$$
0.465271 0.885168i $$-0.345957\pi$$
$$444$$ 0 0
$$445$$ − 7.48623e11i − 0.904989i
$$446$$ 0 0
$$447$$ −1.48371e11 −0.175778
$$448$$ 0 0
$$449$$ −9.54602e11 −1.10844 −0.554222 0.832369i $$-0.686984\pi$$
−0.554222 + 0.832369i $$0.686984\pi$$
$$450$$ 0 0
$$451$$ 8.34043e10i 0.0949279i
$$452$$ 0 0
$$453$$ − 3.83432e11i − 0.427805i
$$454$$ 0 0
$$455$$ −1.40721e12 −1.53925
$$456$$ 0 0
$$457$$ 4.57857e11 0.491029 0.245515 0.969393i $$-0.421043\pi$$
0.245515 + 0.969393i $$0.421043\pi$$
$$458$$ 0 0
$$459$$ 1.35897e11i 0.142907i
$$460$$ 0 0
$$461$$ − 1.58756e12i − 1.63710i −0.574437 0.818549i $$-0.694779\pi$$
0.574437 0.818549i $$-0.305221\pi$$
$$462$$ 0 0
$$463$$ 7.26819e11 0.735041 0.367521 0.930015i $$-0.380207\pi$$
0.367521 + 0.930015i $$0.380207\pi$$
$$464$$ 0 0
$$465$$ 7.07816e10 0.0702073
$$466$$ 0 0
$$467$$ 8.73721e11i 0.850054i 0.905181 + 0.425027i $$0.139735\pi$$
−0.905181 + 0.425027i $$0.860265\pi$$
$$468$$ 0 0
$$469$$ 1.85100e12i 1.76657i
$$470$$ 0 0
$$471$$ −7.07626e10 −0.0662536
$$472$$ 0 0
$$473$$ −1.20268e11 −0.110478
$$474$$ 0 0
$$475$$ − 8.78007e10i − 0.0791365i
$$476$$ 0 0
$$477$$ − 3.78282e11i − 0.334567i
$$478$$ 0 0
$$479$$ 6.89800e11 0.598706 0.299353 0.954142i $$-0.403229\pi$$
0.299353 + 0.954142i $$0.403229\pi$$
$$480$$ 0 0
$$481$$ −7.17921e10 −0.0611538
$$482$$ 0 0
$$483$$ 8.08807e11i 0.676212i
$$484$$ 0 0
$$485$$ 1.37352e12i 1.12719i
$$486$$ 0 0
$$487$$ −2.05185e12 −1.65297 −0.826485 0.562959i $$-0.809663\pi$$
−0.826485 + 0.562959i $$0.809663\pi$$
$$488$$ 0 0
$$489$$ 6.74440e11 0.533401
$$490$$ 0 0
$$491$$ 5.51164e11i 0.427971i 0.976837 + 0.213985i $$0.0686445\pi$$
−0.976837 + 0.213985i $$0.931356\pi$$
$$492$$ 0 0
$$493$$ 1.85683e12i 1.41567i
$$494$$ 0 0
$$495$$ −3.81686e10 −0.0285748
$$496$$ 0 0
$$497$$ −2.02954e12 −1.49208
$$498$$ 0 0
$$499$$ − 1.99167e12i − 1.43802i −0.695000 0.719009i $$-0.744596\pi$$
0.695000 0.719009i $$-0.255404\pi$$
$$500$$ 0 0
$$501$$ − 1.28053e11i − 0.0908073i
$$502$$ 0 0
$$503$$ −2.42693e11 −0.169044 −0.0845221 0.996422i $$-0.526936\pi$$
−0.0845221 + 0.996422i $$0.526936\pi$$
$$504$$ 0 0
$$505$$ 3.73891e11 0.255819
$$506$$ 0 0
$$507$$ 1.07193e12i 0.720494i
$$508$$ 0 0
$$509$$ 7.46973e11i 0.493259i 0.969110 + 0.246629i $$0.0793230\pi$$
−0.969110 + 0.246629i $$0.920677\pi$$
$$510$$ 0 0
$$511$$ −2.22814e12 −1.44560
$$512$$ 0 0
$$513$$ −5.40485e11 −0.344553
$$514$$ 0 0
$$515$$ 6.95908e11i 0.435933i
$$516$$ 0 0
$$517$$ 5.36688e9i 0.00330381i
$$518$$ 0 0
$$519$$ −9.66128e11 −0.584496
$$520$$ 0 0
$$521$$ 1.38178e12 0.821619 0.410809 0.911721i $$-0.365246\pi$$
0.410809 + 0.911721i $$0.365246\pi$$
$$522$$ 0 0
$$523$$ 7.83681e10i 0.0458017i 0.999738 + 0.0229008i $$0.00729020\pi$$
−0.999738 + 0.0229008i $$0.992710\pi$$
$$524$$ 0 0
$$525$$ 4.46293e10i 0.0256391i
$$526$$ 0 0
$$527$$ −1.56471e11 −0.0883661
$$528$$ 0 0
$$529$$ 6.46722e11 0.359060
$$530$$ 0 0
$$531$$ − 4.17203e11i − 0.227731i
$$532$$ 0 0
$$533$$ − 3.16115e12i − 1.69658i
$$534$$ 0 0
$$535$$ −2.33576e12 −1.23264
$$536$$ 0 0
$$537$$ 1.03169e12 0.535382
$$538$$ 0 0
$$539$$ 1.53963e9i 0 0.000785716i
$$540$$ 0 0
$$541$$ 1.88658e12i 0.946865i 0.880830 + 0.473433i $$0.156985\pi$$
−0.880830 + 0.473433i $$0.843015\pi$$
$$542$$ 0 0
$$543$$ −6.85840e11 −0.338551
$$544$$ 0 0
$$545$$ 2.59702e12 1.26093
$$546$$ 0 0
$$547$$ − 3.29535e12i − 1.57383i −0.617061 0.786915i $$-0.711677\pi$$
0.617061 0.786915i $$-0.288323\pi$$
$$548$$ 0 0
$$549$$ 3.29272e11i 0.154696i
$$550$$ 0 0
$$551$$ −7.38493e12 −3.41322
$$552$$ 0 0
$$553$$ −2.74427e12 −1.24785
$$554$$ 0 0
$$555$$ 5.37876e10i 0.0240638i
$$556$$ 0 0
$$557$$ − 1.26338e12i − 0.556143i −0.960560 0.278072i $$-0.910305\pi$$
0.960560 0.278072i $$-0.0896952\pi$$
$$558$$ 0 0
$$559$$ 4.55835e12 1.97449
$$560$$ 0 0
$$561$$ 8.43762e10 0.0359656
$$562$$ 0 0
$$563$$ − 2.18128e12i − 0.915004i −0.889209 0.457502i $$-0.848744\pi$$
0.889209 0.457502i $$-0.151256\pi$$
$$564$$ 0 0
$$565$$ 4.13652e12i 1.70772i
$$566$$ 0 0
$$567$$ 2.74730e11 0.111630
$$568$$ 0 0
$$569$$ −3.77066e12 −1.50804 −0.754020 0.656852i $$-0.771888\pi$$
−0.754020 + 0.656852i $$0.771888\pi$$
$$570$$ 0 0
$$571$$ 4.82244e10i 0.0189847i 0.999955 + 0.00949236i $$0.00302156\pi$$
−0.999955 + 0.00949236i $$0.996978\pi$$
$$572$$ 0 0
$$573$$ 3.16513e11i 0.122658i
$$574$$ 0 0
$$575$$ 1.35072e11 0.0515298
$$576$$ 0 0
$$577$$ −3.41311e12 −1.28192 −0.640958 0.767576i $$-0.721463\pi$$
−0.640958 + 0.767576i $$0.721463\pi$$
$$578$$ 0 0
$$579$$ − 1.47991e12i − 0.547246i
$$580$$ 0 0
$$581$$ 2.17565e12i 0.792131i
$$582$$ 0 0
$$583$$ −2.34869e11 −0.0842009
$$584$$ 0 0
$$585$$ 1.44665e12 0.510696
$$586$$ 0 0
$$587$$ − 5.13080e12i − 1.78366i −0.452366 0.891832i $$-0.649420\pi$$
0.452366 0.891832i $$-0.350580\pi$$
$$588$$ 0 0
$$589$$ − 6.22310e11i − 0.213053i
$$590$$ 0 0
$$591$$ −8.52103e11 −0.287308
$$592$$ 0 0
$$593$$ 3.28338e12 1.09037 0.545186 0.838315i $$-0.316459\pi$$
0.545186 + 0.838315i $$0.316459\pi$$
$$594$$ 0 0
$$595$$ − 2.33066e12i − 0.762345i
$$596$$ 0 0
$$597$$ 2.45045e12i 0.789516i
$$598$$ 0 0
$$599$$ −2.83692e12 −0.900382 −0.450191 0.892932i $$-0.648644\pi$$
−0.450191 + 0.892932i $$0.648644\pi$$
$$600$$ 0 0
$$601$$ −2.83184e11 −0.0885389 −0.0442694 0.999020i $$-0.514096\pi$$
−0.0442694 + 0.999020i $$0.514096\pi$$
$$602$$ 0 0
$$603$$ − 1.90288e12i − 0.586117i
$$604$$ 0 0
$$605$$ − 3.34368e12i − 1.01467i
$$606$$ 0 0
$$607$$ 2.38906e12 0.714294 0.357147 0.934048i $$-0.383749\pi$$
0.357147 + 0.934048i $$0.383749\pi$$
$$608$$ 0 0
$$609$$ 3.75378e12 1.10584
$$610$$ 0 0
$$611$$ − 2.03413e11i − 0.0590464i
$$612$$ 0 0
$$613$$ − 6.27251e11i − 0.179419i −0.995968 0.0897097i $$-0.971406\pi$$
0.995968 0.0897097i $$-0.0285939\pi$$
$$614$$ 0 0
$$615$$ −2.36838e12 −0.667595
$$616$$ 0 0
$$617$$ −6.57062e12 −1.82525 −0.912626 0.408794i $$-0.865949\pi$$
−0.912626 + 0.408794i $$0.865949\pi$$
$$618$$ 0 0
$$619$$ − 1.27976e12i − 0.350366i −0.984536 0.175183i $$-0.943948\pi$$
0.984536 0.175183i $$-0.0560517\pi$$
$$620$$ 0 0
$$621$$ − 8.31476e11i − 0.224356i
$$622$$ 0 0
$$623$$ 3.34558e12 0.889765
$$624$$ 0 0
$$625$$ −3.97586e12 −1.04225
$$626$$ 0 0
$$627$$ 3.35578e11i 0.0867141i
$$628$$ 0 0
$$629$$ − 1.18904e11i − 0.0302878i
$$630$$ 0 0
$$631$$ −6.41367e12 −1.61055 −0.805276 0.592900i $$-0.797983\pi$$
−0.805276 + 0.592900i $$0.797983\pi$$
$$632$$ 0 0
$$633$$ 1.24289e12 0.307693
$$634$$ 0 0
$$635$$ 5.83035e11i 0.142302i
$$636$$ 0 0
$$637$$ − 5.83542e10i − 0.0140425i
$$638$$ 0 0
$$639$$ 2.08642e12 0.495048
$$640$$ 0 0
$$641$$ 2.19384e11 0.0513266 0.0256633 0.999671i $$-0.491830\pi$$
0.0256633 + 0.999671i $$0.491830\pi$$
$$642$$ 0 0
$$643$$ 1.43315e12i 0.330629i 0.986241 + 0.165315i $$0.0528639\pi$$
−0.986241 + 0.165315i $$0.947136\pi$$
$$644$$ 0 0
$$645$$ − 3.41517e12i − 0.776952i
$$646$$ 0 0
$$647$$ −2.24505e12 −0.503682 −0.251841 0.967769i $$-0.581036\pi$$
−0.251841 + 0.967769i $$0.581036\pi$$
$$648$$ 0 0
$$649$$ −2.59034e11 −0.0573133
$$650$$ 0 0
$$651$$ 3.16322e11i 0.0690262i
$$652$$ 0 0
$$653$$ − 4.25852e12i − 0.916535i −0.888814 0.458267i $$-0.848470\pi$$
0.888814 0.458267i $$-0.151530\pi$$
$$654$$ 0 0
$$655$$ −2.39099e12 −0.507565
$$656$$ 0 0
$$657$$ 2.29059e12 0.479627
$$658$$ 0 0
$$659$$ − 2.98594e12i − 0.616733i −0.951268 0.308367i $$-0.900218\pi$$
0.951268 0.308367i $$-0.0997823\pi$$
$$660$$ 0 0
$$661$$ 3.60046e11i 0.0733587i 0.999327 + 0.0366794i $$0.0116780\pi$$
−0.999327 + 0.0366794i $$0.988322\pi$$
$$662$$ 0 0
$$663$$ −3.19799e12 −0.642786
$$664$$ 0 0
$$665$$ 9.26940e12 1.83804
$$666$$ 0 0
$$667$$ − 1.13609e13i − 2.22252i
$$668$$ 0 0
$$669$$ − 5.07712e12i − 0.979941i
$$670$$ 0 0
$$671$$ 2.04439e11 0.0389325
$$672$$ 0 0
$$673$$ −9.00012e12 −1.69114 −0.845572 0.533862i $$-0.820740\pi$$
−0.845572 + 0.533862i $$0.820740\pi$$
$$674$$ 0 0
$$675$$ − 4.58801e10i − 0.00850662i
$$676$$ 0 0
$$677$$ − 3.09265e12i − 0.565825i −0.959146 0.282913i $$-0.908699\pi$$
0.959146 0.282913i $$-0.0913006\pi$$
$$678$$ 0 0
$$679$$ −6.13823e12 −1.10823
$$680$$ 0 0
$$681$$ −3.93503e12 −0.701109
$$682$$ 0 0
$$683$$ − 9.11567e12i − 1.60286i −0.598090 0.801429i $$-0.704073\pi$$
0.598090 0.801429i $$-0.295927\pi$$
$$684$$ 0 0
$$685$$ 2.24727e12i 0.389985i
$$686$$ 0 0
$$687$$ 5.37900e12 0.921289
$$688$$ 0 0
$$689$$ 8.90190e12 1.50486
$$690$$ 0 0
$$691$$ 8.66620e12i 1.44603i 0.690832 + 0.723016i $$0.257245\pi$$
−0.690832 + 0.723016i $$0.742755\pi$$
$$692$$ 0 0
$$693$$ − 1.70575e11i − 0.0280941i
$$694$$ 0 0
$$695$$ 7.65517e12 1.24458
$$696$$ 0 0
$$697$$ 5.23557e12 0.840266
$$698$$ 0 0
$$699$$ − 1.24023e12i − 0.196497i
$$700$$ 0 0
$$701$$ 1.21255e13i 1.89657i 0.317417 + 0.948286i $$0.397184\pi$$
−0.317417 + 0.948286i $$0.602816\pi$$
$$702$$ 0 0
$$703$$ 4.72899e11 0.0730247
$$704$$ 0 0
$$705$$ −1.52400e11 −0.0232345
$$706$$ 0 0
$$707$$ 1.67091e12i 0.251516i
$$708$$ 0 0
$$709$$ 6.84961e12i 1.01802i 0.860759 + 0.509012i $$0.169989\pi$$
−0.860759 + 0.509012i $$0.830011\pi$$
$$710$$ 0 0
$$711$$ 2.82118e12 0.414017
$$712$$ 0 0
$$713$$ 9.57354e11 0.138730
$$714$$ 0 0
$$715$$ − 8.98201e11i − 0.128528i
$$716$$ 0 0
$$717$$ − 4.45436e12i − 0.629433i
$$718$$ 0 0
$$719$$ −3.22078e12 −0.449450 −0.224725 0.974422i $$-0.572148\pi$$
−0.224725 + 0.974422i $$0.572148\pi$$
$$720$$ 0 0
$$721$$ −3.11000e12 −0.428599
$$722$$ 0 0
$$723$$ 1.53049e12i 0.208309i
$$724$$ 0 0
$$725$$ − 6.26884e11i − 0.0842686i
$$726$$ 0 0
$$727$$ −1.26034e13 −1.67334 −0.836669 0.547708i $$-0.815500\pi$$
−0.836669 + 0.547708i $$0.815500\pi$$
$$728$$ 0 0
$$729$$ −2.82430e11 −0.0370370
$$730$$ 0 0
$$731$$ 7.54964e12i 0.977908i
$$732$$ 0 0
$$733$$ 7.29094e12i 0.932858i 0.884559 + 0.466429i $$0.154460\pi$$
−0.884559 + 0.466429i $$0.845540\pi$$
$$734$$ 0 0
$$735$$ −4.37198e10 −0.00552567
$$736$$ 0 0
$$737$$ −1.18147e12 −0.147509
$$738$$ 0 0
$$739$$ 3.39916e10i 0.00419248i 0.999998 + 0.00209624i $$0.000667255\pi$$
−0.999998 + 0.00209624i $$0.999333\pi$$
$$740$$ 0 0
$$741$$ − 1.27189e13i − 1.54978i
$$742$$ 0 0
$$743$$ −9.08965e12 −1.09420 −0.547101 0.837067i $$-0.684269\pi$$
−0.547101 + 0.837067i $$0.684269\pi$$
$$744$$ 0 0
$$745$$ −2.61590e12 −0.311113
$$746$$ 0 0
$$747$$ − 2.23663e12i − 0.262816i
$$748$$ 0 0
$$749$$ − 1.04385e13i − 1.21190i
$$750$$ 0 0
$$751$$ 7.25185e12 0.831896 0.415948 0.909388i $$-0.363450\pi$$
0.415948 + 0.909388i $$0.363450\pi$$
$$752$$ 0 0
$$753$$ 8.94012e12 1.01336
$$754$$ 0 0
$$755$$ − 6.76021e12i − 0.757180i
$$756$$ 0 0
$$757$$ 1.12343e13i 1.24341i 0.783253 + 0.621703i $$0.213559\pi$$
−0.783253 + 0.621703i $$0.786441\pi$$
$$758$$ 0 0
$$759$$ −5.16249e11 −0.0564639
$$760$$ 0 0
$$761$$ 1.63034e13 1.76217 0.881083 0.472962i $$-0.156815\pi$$
0.881083 + 0.472962i $$0.156815\pi$$
$$762$$ 0 0
$$763$$ 1.16060e13i 1.23972i
$$764$$ 0 0
$$765$$ 2.39598e12i 0.252933i
$$766$$ 0 0
$$767$$ 9.81779e12 1.02432
$$768$$ 0 0
$$769$$ 4.19893e12 0.432982 0.216491 0.976285i $$-0.430539\pi$$
0.216491 + 0.976285i $$0.430539\pi$$
$$770$$ 0 0
$$771$$ − 1.87375e12i − 0.190971i
$$772$$ 0 0
$$773$$ − 1.11644e13i − 1.12467i −0.826908 0.562337i $$-0.809902\pi$$
0.826908 0.562337i $$-0.190098\pi$$
$$774$$ 0 0
$$775$$ 5.28260e10 0.00526005
$$776$$ 0 0
$$777$$ −2.40376e11 −0.0236590
$$778$$ 0 0
$$779$$ 2.08227e13i 2.02591i
$$780$$ 0 0
$$781$$ − 1.29542e12i − 0.124589i
$$782$$ 0 0
$$783$$ −3.85898e12 −0.366898
$$784$$ 0 0
$$785$$ −1.24760e12 −0.117263
$$786$$ 0 0
$$787$$ 3.40070e12i 0.315996i 0.987439 + 0.157998i $$0.0505040\pi$$
−0.987439 + 0.157998i $$0.949496\pi$$
$$788$$ 0 0
$$789$$ 1.52869e12i 0.140434i
$$790$$ 0 0
$$791$$ −1.84860e13 −1.67900
$$792$$ 0 0
$$793$$ −7.74856e12 −0.695812
$$794$$ 0 0
$$795$$ − 6.66942e12i − 0.592156i
$$796$$ 0 0
$$797$$ 8.35775e11i 0.0733714i 0.999327 + 0.0366857i $$0.0116800\pi$$
−0.999327 + 0.0366857i $$0.988320\pi$$
$$798$$ 0 0
$$799$$ 3.36898e11 0.0292440
$$800$$ 0 0
$$801$$ −3.43935e12 −0.295209
$$802$$ 0 0
$$803$$ − 1.42219e12i − 0.120708i
$$804$$ 0 0
$$805$$ 1.42599e13i 1.19684i
$$806$$ 0 0
$$807$$ 5.56234e12 0.461664
$$808$$ 0 0
$$809$$ 1.21985e13 1.00124 0.500622 0.865666i $$-0.333105\pi$$
0.500622 + 0.865666i $$0.333105\pi$$
$$810$$ 0 0
$$811$$ 1.61845e13i 1.31373i 0.754009 + 0.656864i $$0.228117\pi$$
−0.754009 + 0.656864i $$0.771883\pi$$
$$812$$ 0 0
$$813$$ − 9.35616e12i − 0.751087i
$$814$$ 0 0
$$815$$ 1.18909e13 0.944075
$$816$$ 0 0
$$817$$ −3.00261e13 −2.35776
$$818$$ 0 0
$$819$$ 6.46506e12i 0.502105i
$$820$$ 0 0
$$821$$ − 3.85044e12i − 0.295778i −0.989004 0.147889i $$-0.952752\pi$$
0.989004 0.147889i $$-0.0472479\pi$$
$$822$$ 0 0
$$823$$ 8.54699e12 0.649403 0.324701 0.945817i $$-0.394736\pi$$
0.324701 + 0.945817i $$0.394736\pi$$
$$824$$ 0 0
$$825$$ −2.84861e10 −0.00214087
$$826$$ 0 0
$$827$$ 4.46202e12i 0.331709i 0.986150 + 0.165854i $$0.0530382\pi$$
−0.986150 + 0.165854i $$0.946962\pi$$
$$828$$ 0 0
$$829$$ 5.99259e12i 0.440675i 0.975424 + 0.220338i $$0.0707159\pi$$
−0.975424 + 0.220338i $$0.929284\pi$$
$$830$$ 0 0
$$831$$ −3.66640e12 −0.266707
$$832$$ 0 0
$$833$$ 9.66476e10 0.00695486
$$834$$ 0 0
$$835$$ − 2.25768e12i − 0.160721i
$$836$$ 0 0
$$837$$ − 3.25187e11i − 0.0229018i
$$838$$ 0 0
$$839$$ 9.87709e12 0.688177 0.344088 0.938937i $$-0.388188\pi$$
0.344088 + 0.938937i $$0.388188\pi$$
$$840$$ 0 0
$$841$$ −3.82201e13 −2.63457
$$842$$ 0 0
$$843$$ 7.29021e12i 0.497183i
$$844$$ 0 0
$$845$$ 1.88990e13i 1.27521i
$$846$$ 0 0
$$847$$ 1.49428e13 0.997602
$$848$$ 0 0
$$849$$ 7.99799e12 0.528319
$$850$$ 0 0
$$851$$ 7.27502e11i 0.0475501i
$$852$$ 0 0
$$853$$ − 2.87375e13i − 1.85857i −0.369365 0.929284i $$-0.620425\pi$$
0.369365 0.929284i $$-0.379575\pi$$
$$854$$ 0 0
$$855$$ −9.52919e12 −0.609830
$$856$$ 0 0
$$857$$ −4.90239e12 −0.310452 −0.155226 0.987879i $$-0.549611\pi$$
−0.155226 + 0.987879i $$0.549611\pi$$
$$858$$ 0 0
$$859$$ 4.08811e12i 0.256185i 0.991762 + 0.128092i $$0.0408854\pi$$
−0.991762 + 0.128092i $$0.959115\pi$$
$$860$$ 0 0
$$861$$ − 1.05842e13i − 0.656365i
$$862$$ 0 0
$$863$$ 2.90112e13 1.78040 0.890200 0.455570i $$-0.150565\pi$$
0.890200 + 0.455570i $$0.150565\pi$$
$$864$$ 0 0
$$865$$ −1.70336e13 −1.03451
$$866$$ 0 0
$$867$$ 4.30904e12i 0.258997i
$$868$$ 0 0
$$869$$ − 1.75162e12i − 0.104196i
$$870$$ 0 0
$$871$$ 4.47794e13 2.63631
$$872$$ 0 0
$$873$$ 6.31026e12 0.367691
$$874$$ 0 0
$$875$$ − 1.70145e13i − 0.981257i
$$876$$ 0 0
$$877$$ 3.38408e13i 1.93171i 0.259082 + 0.965855i $$0.416580\pi$$
−0.259082 + 0.965855i $$0.583420\pi$$
$$878$$ 0 0
$$879$$ 6.91788e12 0.390862
$$880$$ 0 0
$$881$$ 2.28323e13 1.27690 0.638452 0.769662i $$-0.279575\pi$$
0.638452 + 0.769662i $$0.279575\pi$$
$$882$$ 0 0
$$883$$ − 2.70857e12i − 0.149940i −0.997186 0.0749699i $$-0.976114\pi$$
0.997186 0.0749699i $$-0.0238861\pi$$
$$884$$ 0 0
$$885$$ − 7.35562e12i − 0.403065i
$$886$$ 0 0
$$887$$ 1.71218e13 0.928735 0.464368 0.885643i $$-0.346282\pi$$
0.464368 + 0.885643i $$0.346282\pi$$
$$888$$ 0 0
$$889$$ −2.60557e12 −0.139909
$$890$$ 0 0
$$891$$ 1.75356e11i 0.00932116i
$$892$$ 0 0
$$893$$ 1.33990e12i 0.0705082i
$$894$$ 0 0
$$895$$ 1.81895e13 0.947582
$$896$$ 0 0
$$897$$ 1.95666e13 1.00914
$$898$$ 0 0
$$899$$ − 4.44320e12i − 0.226870i
$$900$$ 0 0
$$901$$ 1.47435e13i 0.745315i
$$902$$ 0 0
$$903$$ 1.52623e13 0.763882
$$904$$ 0 0
$$905$$ −1.20919e13 −0.599206
$$906$$ 0 0
$$907$$ − 2.45440e13i − 1.20424i −0.798406 0.602119i $$-0.794323\pi$$
0.798406 0.602119i $$-0.205677\pi$$
$$908$$ 0 0
$$909$$ − 1.71774e12i − 0.0834488i
$$910$$ 0 0
$$911$$ −1.04402e13 −0.502198 −0.251099 0.967961i $$-0.580792\pi$$
−0.251099 + 0.967961i $$0.580792\pi$$
$$912$$ 0 0
$$913$$ −1.38868e12 −0.0661432
$$914$$ 0 0
$$915$$ 5.80533e12i 0.273799i
$$916$$ 0 0
$$917$$ − 1.06853e13i − 0.499027i
$$918$$ 0 0
$$919$$ −1.14054e13 −0.527464 −0.263732 0.964596i $$-0.584953\pi$$
−0.263732 + 0.964596i $$0.584953\pi$$
$$920$$ 0 0
$$921$$ 1.48702e13 0.681000
$$922$$ 0 0
$$923$$ 4.90985e13i 2.22669i
$$924$$ 0 0
$$925$$ 4.01429e10i 0.00180290i
$$926$$ 0 0
$$927$$ 3.19716e12 0.142202
$$928$$ 0 0
$$929$$ 1.77202e13 0.780544 0.390272 0.920700i $$-0.372381\pi$$
0.390272 + 0.920700i $$0.372381\pi$$
$$930$$ 0 0
$$931$$ 3.84383e11i 0.0167684i
$$932$$ 0 0
$$933$$ 9.22852e12i 0.398717i
$$934$$ 0 0
$$935$$ 1.48762e12 0.0636561
$$936$$ 0 0
$$937$$ 1.13938e13 0.482880 0.241440 0.970416i $$-0.422380\pi$$
0.241440 + 0.970416i $$0.422380\pi$$
$$938$$ 0 0
$$939$$ 1.95760e13i 0.821731i
$$940$$ 0 0
$$941$$ − 3.57056e12i − 0.148451i −0.997241 0.0742255i $$-0.976352\pi$$
0.997241 0.0742255i $$-0.0236484\pi$$
$$942$$ 0 0
$$943$$ −3.20334e13 −1.31917
$$944$$ 0 0
$$945$$ 4.84371e12 0.197576
$$946$$ 0 0
$$947$$ 3.48838e13i 1.40945i 0.709482 + 0.704724i $$0.248929\pi$$
−0.709482 + 0.704724i $$0.751071\pi$$
$$948$$ 0 0
$$949$$ 5.39032e13i 2.15733i
$$950$$ 0 0
$$951$$ −8.60442e12 −0.341122
$$952$$ 0 0
$$953$$ −2.48299e13 −0.975117 −0.487558 0.873090i $$-0.662112\pi$$
−0.487558 + 0.873090i $$0.662112\pi$$
$$954$$ 0 0
$$955$$ 5.58038e12i 0.217094i
$$956$$ 0 0
$$957$$ 2.39597e12i 0.0923376i
$$958$$ 0 0
$$959$$ −1.00430e13 −0.383425
$$960$$ 0 0
$$961$$ −2.60652e13 −0.985839
$$962$$ 0 0
$$963$$ 1.07310e13i 0.402090i
$$964$$ 0 0
$$965$$ − 2.60921e13i − 0.968580i
$$966$$ 0 0
$$967$$ −1.43858e13 −0.529071 −0.264535 0.964376i $$-0.585219\pi$$
−0.264535 + 0.964376i $$0.585219\pi$$
$$968$$ 0 0
$$969$$ 2.10654e13 0.767560
$$970$$ 0 0
$$971$$ 9.63124e12i 0.347693i 0.984773 + 0.173846i $$0.0556196\pi$$
−0.984773 + 0.173846i $$0.944380\pi$$
$$972$$ 0 0
$$973$$ 3.42108e13i 1.22364i
$$974$$ 0 0
$$975$$ 1.07967e12 0.0382622
$$976$$ 0 0
$$977$$ −9.87031e12 −0.346581 −0.173291 0.984871i $$-0.555440\pi$$
−0.173291 + 0.984871i $$0.555440\pi$$
$$978$$ 0 0
$$979$$ 2.13543e12i 0.0742957i
$$980$$ 0 0
$$981$$ − 1.19313e13i − 0.411318i
$$982$$ 0 0
$$983$$ 5.82371e13 1.98934 0.994669 0.103116i $$-0.0328812\pi$$
0.994669 + 0.103116i $$0.0328812\pi$$
$$984$$ 0 0
$$985$$ −1.50233e13 −0.508512
$$986$$ 0 0
$$987$$ − 6.81072e11i − 0.0228437i
$$988$$ 0 0
$$989$$ − 4.61918e13i − 1.53526i
$$990$$ 0 0
$$991$$ 6.04603e13 1.99131 0.995655 0.0931229i $$-0.0296849\pi$$
0.995655 + 0.0931229i $$0.0296849\pi$$
$$992$$ 0 0
$$993$$ −5.52652e12 −0.180377
$$994$$ 0 0
$$995$$ 4.32034e13i 1.39738i
$$996$$ 0 0
$$997$$ 4.77299e13i 1.52990i 0.644091 + 0.764949i $$0.277236\pi$$
−0.644091 + 0.764949i $$0.722764\pi$$
$$998$$ 0 0
$$999$$ 2.47113e11 0.00784965
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 384.10.d.a.193.2 8
4.3 odd 2 384.10.d.b.193.6 yes 8
8.3 odd 2 384.10.d.b.193.3 yes 8
8.5 even 2 inner 384.10.d.a.193.7 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.a.193.2 8 1.1 even 1 trivial
384.10.d.a.193.7 yes 8 8.5 even 2 inner
384.10.d.b.193.3 yes 8 8.3 odd 2
384.10.d.b.193.6 yes 8 4.3 odd 2