# Properties

 Label 384.8.d.d.193.1 Level $384$ Weight $8$ Character 384.193 Analytic conductor $119.956$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$119.955849786$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 449 x^{6} + 50632 x^{4} + 69129 x^{2} + 18225$$ 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 193.1 Root $$15.2503i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.8.d.d.193.8

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-27.0000i q^{3} -522.419i q^{5} +1309.01 q^{7} -729.000 q^{9} +O(q^{10})$$ $$q-27.0000i q^{3} -522.419i q^{5} +1309.01 q^{7} -729.000 q^{9} +2794.57i q^{11} +7351.17i q^{13} -14105.3 q^{15} +5644.22 q^{17} -20285.7i q^{19} -35343.4i q^{21} +79560.7 q^{23} -194797. q^{25} +19683.0i q^{27} -140530. i q^{29} +321339. q^{31} +75453.4 q^{33} -683854. i q^{35} -504903. i q^{37} +198482. q^{39} +262029. q^{41} -503139. i q^{43} +380844. i q^{45} -107852. q^{47} +889976. q^{49} -152394. i q^{51} +1.13491e6i q^{53} +1.45994e6 q^{55} -547714. q^{57} +1.80023e6i q^{59} -1.00077e6i q^{61} -954271. q^{63} +3.84039e6 q^{65} -4.70272e6i q^{67} -2.14814e6i q^{69} -4.38848e6 q^{71} -1.51938e6 q^{73} +5.25951e6i q^{75} +3.65813e6i q^{77} +144735. q^{79} +531441. q^{81} +1.45912e6i q^{83} -2.94865e6i q^{85} -3.79430e6 q^{87} +8.58397e6 q^{89} +9.62279e6i q^{91} -8.67616e6i q^{93} -1.05976e7 q^{95} -8.08726e6 q^{97} -2.03724e6i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2880 q^{7} - 5832 q^{9} + O(q^{10})$$ $$8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97} + O(q^{100})$$

## 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
$$3$$ − 27.0000i − 0.577350i
$$4$$ 0 0
$$5$$ − 522.419i − 1.86906i −0.355880 0.934532i $$-0.615819\pi$$
0.355880 0.934532i $$-0.384181\pi$$
$$6$$ 0 0
$$7$$ 1309.01 1.44245 0.721226 0.692700i $$-0.243579\pi$$
0.721226 + 0.692700i $$0.243579\pi$$
$$8$$ 0 0
$$9$$ −729.000 −0.333333
$$10$$ 0 0
$$11$$ 2794.57i 0.633054i 0.948584 + 0.316527i $$0.102517\pi$$
−0.948584 + 0.316527i $$0.897483\pi$$
$$12$$ 0 0
$$13$$ 7351.17i 0.928015i 0.885831 + 0.464007i $$0.153589\pi$$
−0.885831 + 0.464007i $$0.846411\pi$$
$$14$$ 0 0
$$15$$ −14105.3 −1.07910
$$16$$ 0 0
$$17$$ 5644.22 0.278633 0.139316 0.990248i $$-0.455509\pi$$
0.139316 + 0.990248i $$0.455509\pi$$
$$18$$ 0 0
$$19$$ − 20285.7i − 0.678504i −0.940695 0.339252i $$-0.889826\pi$$
0.940695 0.339252i $$-0.110174\pi$$
$$20$$ 0 0
$$21$$ − 35343.4i − 0.832800i
$$22$$ 0 0
$$23$$ 79560.7 1.36349 0.681744 0.731591i $$-0.261222\pi$$
0.681744 + 0.731591i $$0.261222\pi$$
$$24$$ 0 0
$$25$$ −194797. −2.49340
$$26$$ 0 0
$$27$$ 19683.0i 0.192450i
$$28$$ 0 0
$$29$$ − 140530.i − 1.06998i −0.844859 0.534989i $$-0.820316\pi$$
0.844859 0.534989i $$-0.179684\pi$$
$$30$$ 0 0
$$31$$ 321339. 1.93730 0.968652 0.248423i $$-0.0799123\pi$$
0.968652 + 0.248423i $$0.0799123\pi$$
$$32$$ 0 0
$$33$$ 75453.4 0.365494
$$34$$ 0 0
$$35$$ − 683854.i − 2.69603i
$$36$$ 0 0
$$37$$ − 504903.i − 1.63871i −0.573288 0.819354i $$-0.694332\pi$$
0.573288 0.819354i $$-0.305668\pi$$
$$38$$ 0 0
$$39$$ 198482. 0.535790
$$40$$ 0 0
$$41$$ 262029. 0.593753 0.296876 0.954916i $$-0.404055\pi$$
0.296876 + 0.954916i $$0.404055\pi$$
$$42$$ 0 0
$$43$$ − 503139.i − 0.965046i −0.875883 0.482523i $$-0.839720\pi$$
0.875883 0.482523i $$-0.160280\pi$$
$$44$$ 0 0
$$45$$ 380844.i 0.623021i
$$46$$ 0 0
$$47$$ −107852. −0.151525 −0.0757625 0.997126i $$-0.524139\pi$$
−0.0757625 + 0.997126i $$0.524139\pi$$
$$48$$ 0 0
$$49$$ 889976. 1.08067
$$50$$ 0 0
$$51$$ − 152394.i − 0.160869i
$$52$$ 0 0
$$53$$ 1.13491e6i 1.04712i 0.851989 + 0.523560i $$0.175397\pi$$
−0.851989 + 0.523560i $$0.824603\pi$$
$$54$$ 0 0
$$55$$ 1.45994e6 1.18322
$$56$$ 0 0
$$57$$ −547714. −0.391735
$$58$$ 0 0
$$59$$ 1.80023e6i 1.14116i 0.821242 + 0.570579i $$0.193281\pi$$
−0.821242 + 0.570579i $$0.806719\pi$$
$$60$$ 0 0
$$61$$ − 1.00077e6i − 0.564518i −0.959338 0.282259i $$-0.908916\pi$$
0.959338 0.282259i $$-0.0910838\pi$$
$$62$$ 0 0
$$63$$ −954271. −0.480817
$$64$$ 0 0
$$65$$ 3.84039e6 1.73452
$$66$$ 0 0
$$67$$ − 4.70272e6i − 1.91024i −0.296222 0.955119i $$-0.595727\pi$$
0.296222 0.955119i $$-0.404273\pi$$
$$68$$ 0 0
$$69$$ − 2.14814e6i − 0.787210i
$$70$$ 0 0
$$71$$ −4.38848e6 −1.45516 −0.727579 0.686024i $$-0.759354\pi$$
−0.727579 + 0.686024i $$0.759354\pi$$
$$72$$ 0 0
$$73$$ −1.51938e6 −0.457127 −0.228563 0.973529i $$-0.573403\pi$$
−0.228563 + 0.973529i $$0.573403\pi$$
$$74$$ 0 0
$$75$$ 5.25951e6i 1.43956i
$$76$$ 0 0
$$77$$ 3.65813e6i 0.913149i
$$78$$ 0 0
$$79$$ 144735. 0.0330278 0.0165139 0.999864i $$-0.494743\pi$$
0.0165139 + 0.999864i $$0.494743\pi$$
$$80$$ 0 0
$$81$$ 531441. 0.111111
$$82$$ 0 0
$$83$$ 1.45912e6i 0.280102i 0.990144 + 0.140051i $$0.0447267\pi$$
−0.990144 + 0.140051i $$0.955273\pi$$
$$84$$ 0 0
$$85$$ − 2.94865e6i − 0.520783i
$$86$$ 0 0
$$87$$ −3.79430e6 −0.617753
$$88$$ 0 0
$$89$$ 8.58397e6 1.29069 0.645346 0.763890i $$-0.276713\pi$$
0.645346 + 0.763890i $$0.276713\pi$$
$$90$$ 0 0
$$91$$ 9.62279e6i 1.33862i
$$92$$ 0 0
$$93$$ − 8.67616e6i − 1.11850i
$$94$$ 0 0
$$95$$ −1.05976e7 −1.26817
$$96$$ 0 0
$$97$$ −8.08726e6 −0.899705 −0.449853 0.893103i $$-0.648524\pi$$
−0.449853 + 0.893103i $$0.648524\pi$$
$$98$$ 0 0
$$99$$ − 2.03724e6i − 0.211018i
$$100$$ 0 0
$$101$$ − 3.92632e6i − 0.379193i −0.981862 0.189597i $$-0.939282\pi$$
0.981862 0.189597i $$-0.0607180\pi$$
$$102$$ 0 0
$$103$$ −9.56910e6 −0.862861 −0.431430 0.902146i $$-0.641991\pi$$
−0.431430 + 0.902146i $$0.641991\pi$$
$$104$$ 0 0
$$105$$ −1.84641e7 −1.55656
$$106$$ 0 0
$$107$$ 282494.i 0.0222928i 0.999938 + 0.0111464i $$0.00354809\pi$$
−0.999938 + 0.0111464i $$0.996452\pi$$
$$108$$ 0 0
$$109$$ − 6.12690e6i − 0.453156i −0.973993 0.226578i $$-0.927246\pi$$
0.973993 0.226578i $$-0.0727539\pi$$
$$110$$ 0 0
$$111$$ −1.36324e7 −0.946108
$$112$$ 0 0
$$113$$ −2.76431e6 −0.180224 −0.0901118 0.995932i $$-0.528722\pi$$
−0.0901118 + 0.995932i $$0.528722\pi$$
$$114$$ 0 0
$$115$$ − 4.15640e7i − 2.54844i
$$116$$ 0 0
$$117$$ − 5.35901e6i − 0.309338i
$$118$$ 0 0
$$119$$ 7.38836e6 0.401915
$$120$$ 0 0
$$121$$ 1.16776e7 0.599243
$$122$$ 0 0
$$123$$ − 7.07478e6i − 0.342803i
$$124$$ 0 0
$$125$$ 6.09515e7i 2.79126i
$$126$$ 0 0
$$127$$ 2.26911e7 0.982976 0.491488 0.870884i $$-0.336453\pi$$
0.491488 + 0.870884i $$0.336453\pi$$
$$128$$ 0 0
$$129$$ −1.35847e7 −0.557170
$$130$$ 0 0
$$131$$ 2.52080e6i 0.0979690i 0.998800 + 0.0489845i $$0.0155985\pi$$
−0.998800 + 0.0489845i $$0.984401\pi$$
$$132$$ 0 0
$$133$$ − 2.65543e7i − 0.978710i
$$134$$ 0 0
$$135$$ 1.02828e7 0.359701
$$136$$ 0 0
$$137$$ 5.09088e7 1.69149 0.845747 0.533584i $$-0.179155\pi$$
0.845747 + 0.533584i $$0.179155\pi$$
$$138$$ 0 0
$$139$$ − 5.49354e7i − 1.73500i −0.497434 0.867502i $$-0.665724\pi$$
0.497434 0.867502i $$-0.334276\pi$$
$$140$$ 0 0
$$141$$ 2.91200e6i 0.0874830i
$$142$$ 0 0
$$143$$ −2.05434e7 −0.587483
$$144$$ 0 0
$$145$$ −7.34154e7 −1.99986
$$146$$ 0 0
$$147$$ − 2.40293e7i − 0.623923i
$$148$$ 0 0
$$149$$ 7.39556e7i 1.83155i 0.401690 + 0.915776i $$0.368423\pi$$
−0.401690 + 0.915776i $$0.631577\pi$$
$$150$$ 0 0
$$151$$ −1.70345e7 −0.402634 −0.201317 0.979526i $$-0.564522\pi$$
−0.201317 + 0.979526i $$0.564522\pi$$
$$152$$ 0 0
$$153$$ −4.11463e6 −0.0928776
$$154$$ 0 0
$$155$$ − 1.67874e8i − 3.62094i
$$156$$ 0 0
$$157$$ 4.59236e7i 0.947080i 0.880772 + 0.473540i $$0.157024\pi$$
−0.880772 + 0.473540i $$0.842976\pi$$
$$158$$ 0 0
$$159$$ 3.06426e7 0.604556
$$160$$ 0 0
$$161$$ 1.04146e8 1.96676
$$162$$ 0 0
$$163$$ 6.55609e7i 1.18574i 0.805299 + 0.592868i $$0.202005\pi$$
−0.805299 + 0.592868i $$0.797995\pi$$
$$164$$ 0 0
$$165$$ − 3.94183e7i − 0.683131i
$$166$$ 0 0
$$167$$ −2.06517e6 −0.0343121 −0.0171561 0.999853i $$-0.505461\pi$$
−0.0171561 + 0.999853i $$0.505461\pi$$
$$168$$ 0 0
$$169$$ 8.70875e6 0.138788
$$170$$ 0 0
$$171$$ 1.47883e7i 0.226168i
$$172$$ 0 0
$$173$$ − 7.19310e7i − 1.05622i −0.849176 0.528111i $$-0.822901\pi$$
0.849176 0.528111i $$-0.177099\pi$$
$$174$$ 0 0
$$175$$ −2.54992e8 −3.59661
$$176$$ 0 0
$$177$$ 4.86062e7 0.658848
$$178$$ 0 0
$$179$$ 6.97037e7i 0.908386i 0.890903 + 0.454193i $$0.150072\pi$$
−0.890903 + 0.454193i $$0.849928\pi$$
$$180$$ 0 0
$$181$$ − 5.59928e7i − 0.701870i −0.936400 0.350935i $$-0.885864\pi$$
0.936400 0.350935i $$-0.114136\pi$$
$$182$$ 0 0
$$183$$ −2.70207e7 −0.325925
$$184$$ 0 0
$$185$$ −2.63771e8 −3.06285
$$186$$ 0 0
$$187$$ 1.57732e7i 0.176390i
$$188$$ 0 0
$$189$$ 2.57653e7i 0.277600i
$$190$$ 0 0
$$191$$ −9.89901e7 −1.02796 −0.513979 0.857803i $$-0.671829\pi$$
−0.513979 + 0.857803i $$0.671829\pi$$
$$192$$ 0 0
$$193$$ −6.30846e7 −0.631645 −0.315822 0.948818i $$-0.602280\pi$$
−0.315822 + 0.948818i $$0.602280\pi$$
$$194$$ 0 0
$$195$$ − 1.03691e8i − 1.00142i
$$196$$ 0 0
$$197$$ − 5.83443e7i − 0.543709i −0.962338 0.271855i $$-0.912363\pi$$
0.962338 0.271855i $$-0.0876370\pi$$
$$198$$ 0 0
$$199$$ −1.34085e8 −1.20613 −0.603067 0.797691i $$-0.706055\pi$$
−0.603067 + 0.797691i $$0.706055\pi$$
$$200$$ 0 0
$$201$$ −1.26974e8 −1.10288
$$202$$ 0 0
$$203$$ − 1.83955e8i − 1.54339i
$$204$$ 0 0
$$205$$ − 1.36889e8i − 1.10976i
$$206$$ 0 0
$$207$$ −5.79997e7 −0.454496
$$208$$ 0 0
$$209$$ 5.66898e7 0.429530
$$210$$ 0 0
$$211$$ − 6.48026e7i − 0.474902i −0.971400 0.237451i $$-0.923688\pi$$
0.971400 0.237451i $$-0.0763118\pi$$
$$212$$ 0 0
$$213$$ 1.18489e8i 0.840135i
$$214$$ 0 0
$$215$$ −2.62849e8 −1.80373
$$216$$ 0 0
$$217$$ 4.20637e8 2.79447
$$218$$ 0 0
$$219$$ 4.10233e7i 0.263922i
$$220$$ 0 0
$$221$$ 4.14916e7i 0.258576i
$$222$$ 0 0
$$223$$ −8.00971e6 −0.0483671 −0.0241835 0.999708i $$-0.507699\pi$$
−0.0241835 + 0.999708i $$0.507699\pi$$
$$224$$ 0 0
$$225$$ 1.42007e8 0.831133
$$226$$ 0 0
$$227$$ − 6.80971e7i − 0.386401i −0.981159 0.193200i $$-0.938113\pi$$
0.981159 0.193200i $$-0.0618867\pi$$
$$228$$ 0 0
$$229$$ 1.20257e8i 0.661740i 0.943676 + 0.330870i $$0.107342\pi$$
−0.943676 + 0.330870i $$0.892658\pi$$
$$230$$ 0 0
$$231$$ 9.87696e7 0.527207
$$232$$ 0 0
$$233$$ −6.83662e7 −0.354075 −0.177038 0.984204i $$-0.556651\pi$$
−0.177038 + 0.984204i $$0.556651\pi$$
$$234$$ 0 0
$$235$$ 5.63438e7i 0.283210i
$$236$$ 0 0
$$237$$ − 3.90785e6i − 0.0190686i
$$238$$ 0 0
$$239$$ −3.05674e8 −1.44832 −0.724162 0.689630i $$-0.757773\pi$$
−0.724162 + 0.689630i $$0.757773\pi$$
$$240$$ 0 0
$$241$$ −2.58964e7 −0.119174 −0.0595869 0.998223i $$-0.518978\pi$$
−0.0595869 + 0.998223i $$0.518978\pi$$
$$242$$ 0 0
$$243$$ − 1.43489e7i − 0.0641500i
$$244$$ 0 0
$$245$$ − 4.64940e8i − 2.01983i
$$246$$ 0 0
$$247$$ 1.49124e8 0.629662
$$248$$ 0 0
$$249$$ 3.93961e7 0.161717
$$250$$ 0 0
$$251$$ 4.32650e7i 0.172694i 0.996265 + 0.0863472i $$0.0275194\pi$$
−0.996265 + 0.0863472i $$0.972481\pi$$
$$252$$ 0 0
$$253$$ 2.22338e8i 0.863161i
$$254$$ 0 0
$$255$$ −7.96135e7 −0.300674
$$256$$ 0 0
$$257$$ −1.33105e8 −0.489133 −0.244567 0.969632i $$-0.578646\pi$$
−0.244567 + 0.969632i $$0.578646\pi$$
$$258$$ 0 0
$$259$$ − 6.60925e8i − 2.36376i
$$260$$ 0 0
$$261$$ 1.02446e8i 0.356660i
$$262$$ 0 0
$$263$$ 5.73164e8 1.94283 0.971413 0.237394i $$-0.0762934\pi$$
0.971413 + 0.237394i $$0.0762934\pi$$
$$264$$ 0 0
$$265$$ 5.92900e8 1.95714
$$266$$ 0 0
$$267$$ − 2.31767e8i − 0.745182i
$$268$$ 0 0
$$269$$ − 4.24355e8i − 1.32922i −0.747191 0.664609i $$-0.768598\pi$$
0.747191 0.664609i $$-0.231402\pi$$
$$270$$ 0 0
$$271$$ −5.45956e8 −1.66635 −0.833173 0.553013i $$-0.813478\pi$$
−0.833173 + 0.553013i $$0.813478\pi$$
$$272$$ 0 0
$$273$$ 2.59815e8 0.772851
$$274$$ 0 0
$$275$$ − 5.44373e8i − 1.57845i
$$276$$ 0 0
$$277$$ 4.35552e8i 1.23129i 0.788023 + 0.615645i $$0.211105\pi$$
−0.788023 + 0.615645i $$0.788895\pi$$
$$278$$ 0 0
$$279$$ −2.34256e8 −0.645768
$$280$$ 0 0
$$281$$ −4.75803e8 −1.27925 −0.639625 0.768687i $$-0.720910\pi$$
−0.639625 + 0.768687i $$0.720910\pi$$
$$282$$ 0 0
$$283$$ − 4.81622e8i − 1.26315i −0.775316 0.631573i $$-0.782409\pi$$
0.775316 0.631573i $$-0.217591\pi$$
$$284$$ 0 0
$$285$$ 2.86136e8i 0.732177i
$$286$$ 0 0
$$287$$ 3.42999e8 0.856459
$$288$$ 0 0
$$289$$ −3.78481e8 −0.922364
$$290$$ 0 0
$$291$$ 2.18356e8i 0.519445i
$$292$$ 0 0
$$293$$ 5.88523e8i 1.36687i 0.730012 + 0.683435i $$0.239515\pi$$
−0.730012 + 0.683435i $$0.760485\pi$$
$$294$$ 0 0
$$295$$ 9.40475e8 2.13290
$$296$$ 0 0
$$297$$ −5.50055e7 −0.121831
$$298$$ 0 0
$$299$$ 5.84864e8i 1.26534i
$$300$$ 0 0
$$301$$ − 6.58616e8i − 1.39203i
$$302$$ 0 0
$$303$$ −1.06011e8 −0.218927
$$304$$ 0 0
$$305$$ −5.22819e8 −1.05512
$$306$$ 0 0
$$307$$ − 4.44023e8i − 0.875834i −0.899016 0.437917i $$-0.855716\pi$$
0.899016 0.437917i $$-0.144284\pi$$
$$308$$ 0 0
$$309$$ 2.58366e8i 0.498173i
$$310$$ 0 0
$$311$$ −1.57481e8 −0.296871 −0.148435 0.988922i $$-0.547424\pi$$
−0.148435 + 0.988922i $$0.547424\pi$$
$$312$$ 0 0
$$313$$ 1.55060e8 0.285821 0.142911 0.989736i $$-0.454354\pi$$
0.142911 + 0.989736i $$0.454354\pi$$
$$314$$ 0 0
$$315$$ 4.98530e8i 0.898678i
$$316$$ 0 0
$$317$$ − 2.90492e8i − 0.512185i −0.966652 0.256092i $$-0.917565\pi$$
0.966652 0.256092i $$-0.0824352\pi$$
$$318$$ 0 0
$$319$$ 3.92720e8 0.677354
$$320$$ 0 0
$$321$$ 7.62733e6 0.0128708
$$322$$ 0 0
$$323$$ − 1.14497e8i − 0.189054i
$$324$$ 0 0
$$325$$ − 1.43198e9i − 2.31391i
$$326$$ 0 0
$$327$$ −1.65426e8 −0.261630
$$328$$ 0 0
$$329$$ −1.41179e8 −0.218568
$$330$$ 0 0
$$331$$ 4.85485e8i 0.735831i 0.929859 + 0.367915i $$0.119928\pi$$
−0.929859 + 0.367915i $$0.880072\pi$$
$$332$$ 0 0
$$333$$ 3.68074e8i 0.546236i
$$334$$ 0 0
$$335$$ −2.45679e9 −3.57036
$$336$$ 0 0
$$337$$ −2.59275e8 −0.369026 −0.184513 0.982830i $$-0.559071\pi$$
−0.184513 + 0.982830i $$0.559071\pi$$
$$338$$ 0 0
$$339$$ 7.46363e7i 0.104052i
$$340$$ 0 0
$$341$$ 8.98005e8i 1.22642i
$$342$$ 0 0
$$343$$ 8.69613e7 0.116358
$$344$$ 0 0
$$345$$ −1.12223e9 −1.47134
$$346$$ 0 0
$$347$$ 8.57834e8i 1.10217i 0.834448 + 0.551087i $$0.185787\pi$$
−0.834448 + 0.551087i $$0.814213\pi$$
$$348$$ 0 0
$$349$$ 1.37991e7i 0.0173765i 0.999962 + 0.00868827i $$0.00276560\pi$$
−0.999962 + 0.00868827i $$0.997234\pi$$
$$350$$ 0 0
$$351$$ −1.44693e8 −0.178597
$$352$$ 0 0
$$353$$ −1.74329e8 −0.210940 −0.105470 0.994422i $$-0.533635\pi$$
−0.105470 + 0.994422i $$0.533635\pi$$
$$354$$ 0 0
$$355$$ 2.29263e9i 2.71978i
$$356$$ 0 0
$$357$$ − 1.99486e8i − 0.232045i
$$358$$ 0 0
$$359$$ 1.45142e9 1.65563 0.827813 0.561004i $$-0.189585\pi$$
0.827813 + 0.561004i $$0.189585\pi$$
$$360$$ 0 0
$$361$$ 4.82362e8 0.539632
$$362$$ 0 0
$$363$$ − 3.15294e8i − 0.345973i
$$364$$ 0 0
$$365$$ 7.93753e8i 0.854399i
$$366$$ 0 0
$$367$$ −9.49474e8 −1.00266 −0.501328 0.865257i $$-0.667155\pi$$
−0.501328 + 0.865257i $$0.667155\pi$$
$$368$$ 0 0
$$369$$ −1.91019e8 −0.197918
$$370$$ 0 0
$$371$$ 1.48562e9i 1.51042i
$$372$$ 0 0
$$373$$ 6.16143e8i 0.614753i 0.951588 + 0.307377i $$0.0994511\pi$$
−0.951588 + 0.307377i $$0.900549\pi$$
$$374$$ 0 0
$$375$$ 1.64569e9 1.61153
$$376$$ 0 0
$$377$$ 1.03306e9 0.992956
$$378$$ 0 0
$$379$$ − 2.44694e8i − 0.230880i −0.993314 0.115440i $$-0.963172\pi$$
0.993314 0.115440i $$-0.0368278\pi$$
$$380$$ 0 0
$$381$$ − 6.12660e8i − 0.567521i
$$382$$ 0 0
$$383$$ 1.25530e9 1.14170 0.570848 0.821056i $$-0.306615\pi$$
0.570848 + 0.821056i $$0.306615\pi$$
$$384$$ 0 0
$$385$$ 1.91108e9 1.70673
$$386$$ 0 0
$$387$$ 3.66788e8i 0.321682i
$$388$$ 0 0
$$389$$ 9.17850e8i 0.790584i 0.918556 + 0.395292i $$0.129357\pi$$
−0.918556 + 0.395292i $$0.870643\pi$$
$$390$$ 0 0
$$391$$ 4.49058e8 0.379912
$$392$$ 0 0
$$393$$ 6.80616e7 0.0565625
$$394$$ 0 0
$$395$$ − 7.56124e7i − 0.0617310i
$$396$$ 0 0
$$397$$ 2.91845e8i 0.234091i 0.993127 + 0.117046i $$0.0373424\pi$$
−0.993127 + 0.117046i $$0.962658\pi$$
$$398$$ 0 0
$$399$$ −7.16966e8 −0.565058
$$400$$ 0 0
$$401$$ 2.30913e9 1.78831 0.894157 0.447754i $$-0.147776\pi$$
0.894157 + 0.447754i $$0.147776\pi$$
$$402$$ 0 0
$$403$$ 2.36222e9i 1.79785i
$$404$$ 0 0
$$405$$ − 2.77635e8i − 0.207674i
$$406$$ 0 0
$$407$$ 1.41099e9 1.03739
$$408$$ 0 0
$$409$$ −1.85520e9 −1.34078 −0.670392 0.742007i $$-0.733874\pi$$
−0.670392 + 0.742007i $$0.733874\pi$$
$$410$$ 0 0
$$411$$ − 1.37454e9i − 0.976585i
$$412$$ 0 0
$$413$$ 2.35653e9i 1.64607i
$$414$$ 0 0
$$415$$ 7.62270e8 0.523528
$$416$$ 0 0
$$417$$ −1.48326e9 −1.00170
$$418$$ 0 0
$$419$$ − 2.05244e9i − 1.36308i −0.731782 0.681539i $$-0.761311\pi$$
0.731782 0.681539i $$-0.238689\pi$$
$$420$$ 0 0
$$421$$ − 1.56223e9i − 1.02037i −0.860065 0.510184i $$-0.829577\pi$$
0.860065 0.510184i $$-0.170423\pi$$
$$422$$ 0 0
$$423$$ 7.86239e7 0.0505083
$$424$$ 0 0
$$425$$ −1.09947e9 −0.694743
$$426$$ 0 0
$$427$$ − 1.31002e9i − 0.814290i
$$428$$ 0 0
$$429$$ 5.54671e8i 0.339184i
$$430$$ 0 0
$$431$$ 1.14071e9 0.686283 0.343141 0.939284i $$-0.388509\pi$$
0.343141 + 0.939284i $$0.388509\pi$$
$$432$$ 0 0
$$433$$ 1.21428e9 0.718802 0.359401 0.933183i $$-0.382981\pi$$
0.359401 + 0.933183i $$0.382981\pi$$
$$434$$ 0 0
$$435$$ 1.98222e9i 1.15462i
$$436$$ 0 0
$$437$$ − 1.61395e9i − 0.925132i
$$438$$ 0 0
$$439$$ 2.88103e9 1.62526 0.812630 0.582781i $$-0.198035\pi$$
0.812630 + 0.582781i $$0.198035\pi$$
$$440$$ 0 0
$$441$$ −6.48792e8 −0.360222
$$442$$ 0 0
$$443$$ − 2.98450e8i − 0.163102i −0.996669 0.0815508i $$-0.974013\pi$$
0.996669 0.0815508i $$-0.0259873\pi$$
$$444$$ 0 0
$$445$$ − 4.48443e9i − 2.41239i
$$446$$ 0 0
$$447$$ 1.99680e9 1.05745
$$448$$ 0 0
$$449$$ −6.76245e7 −0.0352567 −0.0176284 0.999845i $$-0.505612\pi$$
−0.0176284 + 0.999845i $$0.505612\pi$$
$$450$$ 0 0
$$451$$ 7.32258e8i 0.375877i
$$452$$ 0 0
$$453$$ 4.59932e8i 0.232461i
$$454$$ 0 0
$$455$$ 5.02713e9 2.50196
$$456$$ 0 0
$$457$$ 2.61798e9 1.28310 0.641548 0.767083i $$-0.278293\pi$$
0.641548 + 0.767083i $$0.278293\pi$$
$$458$$ 0 0
$$459$$ 1.11095e8i 0.0536229i
$$460$$ 0 0
$$461$$ 2.33589e9i 1.11045i 0.831700 + 0.555225i $$0.187368\pi$$
−0.831700 + 0.555225i $$0.812632\pi$$
$$462$$ 0 0
$$463$$ −3.00500e9 −1.40706 −0.703528 0.710667i $$-0.748393\pi$$
−0.703528 + 0.710667i $$0.748393\pi$$
$$464$$ 0 0
$$465$$ −4.53259e9 −2.09055
$$466$$ 0 0
$$467$$ 1.57079e9i 0.713690i 0.934164 + 0.356845i $$0.116148\pi$$
−0.934164 + 0.356845i $$0.883852\pi$$
$$468$$ 0 0
$$469$$ − 6.15593e9i − 2.75543i
$$470$$ 0 0
$$471$$ 1.23994e9 0.546797
$$472$$ 0 0
$$473$$ 1.40606e9 0.610926
$$474$$ 0 0
$$475$$ 3.95159e9i 1.69178i
$$476$$ 0 0
$$477$$ − 8.27351e8i − 0.349040i
$$478$$ 0 0
$$479$$ −1.47580e9 −0.613556 −0.306778 0.951781i $$-0.599251\pi$$
−0.306778 + 0.951781i $$0.599251\pi$$
$$480$$ 0 0
$$481$$ 3.71163e9 1.52075
$$482$$ 0 0
$$483$$ − 2.81194e9i − 1.13551i
$$484$$ 0 0
$$485$$ 4.22494e9i 1.68161i
$$486$$ 0 0
$$487$$ 3.56484e9 1.39859 0.699293 0.714836i $$-0.253498\pi$$
0.699293 + 0.714836i $$0.253498\pi$$
$$488$$ 0 0
$$489$$ 1.77014e9 0.684585
$$490$$ 0 0
$$491$$ 1.80845e9i 0.689480i 0.938698 + 0.344740i $$0.112033\pi$$
−0.938698 + 0.344740i $$0.887967\pi$$
$$492$$ 0 0
$$493$$ − 7.93180e8i − 0.298131i
$$494$$ 0 0
$$495$$ −1.06429e9 −0.394406
$$496$$ 0 0
$$497$$ −5.74458e9 −2.09899
$$498$$ 0 0
$$499$$ − 3.73339e9i − 1.34509i −0.740055 0.672546i $$-0.765201\pi$$
0.740055 0.672546i $$-0.234799\pi$$
$$500$$ 0 0
$$501$$ 5.57595e7i 0.0198101i
$$502$$ 0 0
$$503$$ −3.16793e8 −0.110991 −0.0554955 0.998459i $$-0.517674\pi$$
−0.0554955 + 0.998459i $$0.517674\pi$$
$$504$$ 0 0
$$505$$ −2.05118e9 −0.708736
$$506$$ 0 0
$$507$$ − 2.35136e8i − 0.0801294i
$$508$$ 0 0
$$509$$ − 4.48810e9i − 1.50852i −0.656578 0.754258i $$-0.727997\pi$$
0.656578 0.754258i $$-0.272003\pi$$
$$510$$ 0 0
$$511$$ −1.98889e9 −0.659383
$$512$$ 0 0
$$513$$ 3.99284e8 0.130578
$$514$$ 0 0
$$515$$ 4.99908e9i 1.61274i
$$516$$ 0 0
$$517$$ − 3.01399e8i − 0.0959235i
$$518$$ 0 0
$$519$$ −1.94214e9 −0.609810
$$520$$ 0 0
$$521$$ 4.33409e9 1.34266 0.671329 0.741159i $$-0.265724\pi$$
0.671329 + 0.741159i $$0.265724\pi$$
$$522$$ 0 0
$$523$$ 5.74767e9i 1.75686i 0.477875 + 0.878428i $$0.341407\pi$$
−0.477875 + 0.878428i $$0.658593\pi$$
$$524$$ 0 0
$$525$$ 6.88478e9i 2.07650i
$$526$$ 0 0
$$527$$ 1.81371e9 0.539796
$$528$$ 0 0
$$529$$ 2.92508e9 0.859097
$$530$$ 0 0
$$531$$ − 1.31237e9i − 0.380386i
$$532$$ 0 0
$$533$$ 1.92622e9i 0.551011i
$$534$$ 0 0
$$535$$ 1.47580e8 0.0416667
$$536$$ 0 0
$$537$$ 1.88200e9 0.524457
$$538$$ 0 0
$$539$$ 2.48710e9i 0.684120i
$$540$$ 0 0
$$541$$ 4.47248e9i 1.21439i 0.794553 + 0.607195i $$0.207705\pi$$
−0.794553 + 0.607195i $$0.792295\pi$$
$$542$$ 0 0
$$543$$ −1.51180e9 −0.405225
$$544$$ 0 0
$$545$$ −3.20081e9 −0.846978
$$546$$ 0 0
$$547$$ − 8.14796e8i − 0.212860i −0.994320 0.106430i $$-0.966058\pi$$
0.994320 0.106430i $$-0.0339420\pi$$
$$548$$ 0 0
$$549$$ 7.29558e8i 0.188173i
$$550$$ 0 0
$$551$$ −2.85075e9 −0.725985
$$552$$ 0 0
$$553$$ 1.89460e8 0.0476410
$$554$$ 0 0
$$555$$ 7.12181e9i 1.76834i
$$556$$ 0 0
$$557$$ 5.85518e8i 0.143565i 0.997420 + 0.0717823i $$0.0228687\pi$$
−0.997420 + 0.0717823i $$0.977131\pi$$
$$558$$ 0 0
$$559$$ 3.69866e9 0.895578
$$560$$ 0 0
$$561$$ 4.25875e8 0.101839
$$562$$ 0 0
$$563$$ − 4.19201e9i − 0.990018i −0.868888 0.495009i $$-0.835165\pi$$
0.868888 0.495009i $$-0.164835\pi$$
$$564$$ 0 0
$$565$$ 1.44413e9i 0.336849i
$$566$$ 0 0
$$567$$ 6.95664e8 0.160272
$$568$$ 0 0
$$569$$ −7.74762e9 −1.76309 −0.881547 0.472096i $$-0.843497\pi$$
−0.881547 + 0.472096i $$0.843497\pi$$
$$570$$ 0 0
$$571$$ − 4.48502e9i − 1.00818i −0.863651 0.504090i $$-0.831828\pi$$
0.863651 0.504090i $$-0.168172\pi$$
$$572$$ 0 0
$$573$$ 2.67273e9i 0.593492i
$$574$$ 0 0
$$575$$ −1.54982e10 −3.39972
$$576$$ 0 0
$$577$$ 6.51505e9 1.41190 0.705948 0.708264i $$-0.250521\pi$$
0.705948 + 0.708264i $$0.250521\pi$$
$$578$$ 0 0
$$579$$ 1.70328e9i 0.364680i
$$580$$ 0 0
$$581$$ 1.91000e9i 0.404034i
$$582$$ 0 0
$$583$$ −3.17159e9 −0.662884
$$584$$ 0 0
$$585$$ −2.79965e9 −0.578173
$$586$$ 0 0
$$587$$ − 1.46081e9i − 0.298099i −0.988830 0.149050i $$-0.952379\pi$$
0.988830 0.149050i $$-0.0476214\pi$$
$$588$$ 0 0
$$589$$ − 6.51859e9i − 1.31447i
$$590$$ 0 0
$$591$$ −1.57530e9 −0.313911
$$592$$ 0 0
$$593$$ −6.59477e9 −1.29870 −0.649349 0.760490i $$-0.724959\pi$$
−0.649349 + 0.760490i $$0.724959\pi$$
$$594$$ 0 0
$$595$$ − 3.85982e9i − 0.751204i
$$596$$ 0 0
$$597$$ 3.62030e9i 0.696362i
$$598$$ 0 0
$$599$$ −1.13100e9 −0.215015 −0.107508 0.994204i $$-0.534287\pi$$
−0.107508 + 0.994204i $$0.534287\pi$$
$$600$$ 0 0
$$601$$ 7.83427e9 1.47210 0.736051 0.676926i $$-0.236688\pi$$
0.736051 + 0.676926i $$0.236688\pi$$
$$602$$ 0 0
$$603$$ 3.42829e9i 0.636746i
$$604$$ 0 0
$$605$$ − 6.10058e9i − 1.12002i
$$606$$ 0 0
$$607$$ −1.94589e9 −0.353149 −0.176575 0.984287i $$-0.556502\pi$$
−0.176575 + 0.984287i $$0.556502\pi$$
$$608$$ 0 0
$$609$$ −4.96680e9 −0.891078
$$610$$ 0 0
$$611$$ − 7.92836e8i − 0.140617i
$$612$$ 0 0
$$613$$ − 7.86592e9i − 1.37923i −0.724174 0.689617i $$-0.757779\pi$$
0.724174 0.689617i $$-0.242221\pi$$
$$614$$ 0 0
$$615$$ −3.69600e9 −0.640721
$$616$$ 0 0
$$617$$ −7.05302e9 −1.20886 −0.604432 0.796657i $$-0.706600\pi$$
−0.604432 + 0.796657i $$0.706600\pi$$
$$618$$ 0 0
$$619$$ − 5.09288e9i − 0.863070i −0.902096 0.431535i $$-0.857972\pi$$
0.902096 0.431535i $$-0.142028\pi$$
$$620$$ 0 0
$$621$$ 1.56599e9i 0.262403i
$$622$$ 0 0
$$623$$ 1.12365e10 1.86176
$$624$$ 0 0
$$625$$ 1.66237e10 2.72363
$$626$$ 0 0
$$627$$ − 1.53063e9i − 0.247989i
$$628$$ 0 0
$$629$$ − 2.84978e9i − 0.456598i
$$630$$ 0 0
$$631$$ 6.95936e9 1.10272 0.551362 0.834266i $$-0.314108\pi$$
0.551362 + 0.834266i $$0.314108\pi$$
$$632$$ 0 0
$$633$$ −1.74967e9 −0.274185
$$634$$ 0 0
$$635$$ − 1.18543e10i − 1.83724i
$$636$$ 0 0
$$637$$ 6.54237e9i 1.00288i
$$638$$ 0 0
$$639$$ 3.19920e9 0.485052
$$640$$ 0 0
$$641$$ −4.26493e9 −0.639601 −0.319801 0.947485i $$-0.603616\pi$$
−0.319801 + 0.947485i $$0.603616\pi$$
$$642$$ 0 0
$$643$$ − 7.80345e9i − 1.15757i −0.815479 0.578787i $$-0.803526\pi$$
0.815479 0.578787i $$-0.196474\pi$$
$$644$$ 0 0
$$645$$ 7.09693e9i 1.04139i
$$646$$ 0 0
$$647$$ 4.27680e9 0.620803 0.310401 0.950606i $$-0.399537\pi$$
0.310401 + 0.950606i $$0.399537\pi$$
$$648$$ 0 0
$$649$$ −5.03087e9 −0.722415
$$650$$ 0 0
$$651$$ − 1.13572e10i − 1.61339i
$$652$$ 0 0
$$653$$ 4.79185e9i 0.673452i 0.941603 + 0.336726i $$0.109320\pi$$
−0.941603 + 0.336726i $$0.890680\pi$$
$$654$$ 0 0
$$655$$ 1.31691e9 0.183110
$$656$$ 0 0
$$657$$ 1.10763e9 0.152376
$$658$$ 0 0
$$659$$ 4.04254e9i 0.550244i 0.961409 + 0.275122i $$0.0887183\pi$$
−0.961409 + 0.275122i $$0.911282\pi$$
$$660$$ 0 0
$$661$$ 4.02211e8i 0.0541688i 0.999633 + 0.0270844i $$0.00862228\pi$$
−0.999633 + 0.0270844i $$0.991378\pi$$
$$662$$ 0 0
$$663$$ 1.12027e9 0.149289
$$664$$ 0 0
$$665$$ −1.38725e10 −1.82927
$$666$$ 0 0
$$667$$ − 1.11806e10i − 1.45890i
$$668$$ 0 0
$$669$$ 2.16262e8i 0.0279247i
$$670$$ 0 0
$$671$$ 2.79671e9 0.357370
$$672$$ 0 0
$$673$$ 4.17076e9 0.527427 0.263713 0.964601i $$-0.415053\pi$$
0.263713 + 0.964601i $$0.415053\pi$$
$$674$$ 0 0
$$675$$ − 3.83418e9i − 0.479855i
$$676$$ 0 0
$$677$$ − 7.81235e8i − 0.0967657i −0.998829 0.0483828i $$-0.984593\pi$$
0.998829 0.0483828i $$-0.0154068\pi$$
$$678$$ 0 0
$$679$$ −1.05863e10 −1.29778
$$680$$ 0 0
$$681$$ −1.83862e9 −0.223089
$$682$$ 0 0
$$683$$ − 1.16917e10i − 1.40412i −0.712117 0.702061i $$-0.752263\pi$$
0.712117 0.702061i $$-0.247737\pi$$
$$684$$ 0 0
$$685$$ − 2.65957e10i − 3.16151i
$$686$$ 0 0
$$687$$ 3.24695e9 0.382056
$$688$$ 0 0
$$689$$ −8.34294e9 −0.971744
$$690$$ 0 0
$$691$$ 2.33629e9i 0.269373i 0.990888 + 0.134687i $$0.0430028\pi$$
−0.990888 + 0.134687i $$0.956997\pi$$
$$692$$ 0 0
$$693$$ − 2.66678e9i − 0.304383i
$$694$$ 0 0
$$695$$ −2.86993e10 −3.24283
$$696$$ 0 0
$$697$$ 1.47895e9 0.165439
$$698$$ 0 0
$$699$$ 1.84589e9i 0.204426i
$$700$$ 0 0
$$701$$ − 1.38336e10i − 1.51678i −0.651801 0.758390i $$-0.725986\pi$$
0.651801 0.758390i $$-0.274014\pi$$
$$702$$ 0 0
$$703$$ −1.02423e10 −1.11187
$$704$$ 0 0
$$705$$ 1.52128e9 0.163511
$$706$$ 0 0
$$707$$ − 5.13961e9i − 0.546968i
$$708$$ 0 0
$$709$$ − 4.96747e8i − 0.0523448i −0.999657 0.0261724i $$-0.991668\pi$$
0.999657 0.0261724i $$-0.00833188\pi$$
$$710$$ 0 0
$$711$$ −1.05512e8 −0.0110093
$$712$$ 0 0
$$713$$ 2.55660e10 2.64149
$$714$$ 0 0
$$715$$ 1.07322e10i 1.09804i
$$716$$ 0 0
$$717$$ 8.25320e9i 0.836191i
$$718$$ 0 0
$$719$$ −2.84708e9 −0.285660 −0.142830 0.989747i $$-0.545620\pi$$
−0.142830 + 0.989747i $$0.545620\pi$$
$$720$$ 0 0
$$721$$ −1.25261e10 −1.24463
$$722$$ 0 0
$$723$$ 6.99204e8i 0.0688050i
$$724$$ 0 0
$$725$$ 2.73747e10i 2.66788i
$$726$$ 0 0
$$727$$ −9.04361e8 −0.0872914 −0.0436457 0.999047i $$-0.513897\pi$$
−0.0436457 + 0.999047i $$0.513897\pi$$
$$728$$ 0 0
$$729$$ −3.87420e8 −0.0370370
$$730$$ 0 0
$$731$$ − 2.83982e9i − 0.268894i
$$732$$ 0 0
$$733$$ 1.59471e10i 1.49561i 0.663921 + 0.747803i $$0.268891\pi$$
−0.663921 + 0.747803i $$0.731109\pi$$
$$734$$ 0 0
$$735$$ −1.25534e10 −1.16615
$$736$$ 0 0
$$737$$ 1.31421e10 1.20928
$$738$$ 0 0
$$739$$ 1.23619e9i 0.112675i 0.998412 + 0.0563376i $$0.0179423\pi$$
−0.998412 + 0.0563376i $$0.982058\pi$$
$$740$$ 0 0
$$741$$ − 4.02634e9i − 0.363536i
$$742$$ 0 0
$$743$$ 1.28465e10 1.14901 0.574505 0.818501i $$-0.305194\pi$$
0.574505 + 0.818501i $$0.305194\pi$$
$$744$$ 0 0
$$745$$ 3.86358e10 3.42329
$$746$$ 0 0
$$747$$ − 1.06370e9i − 0.0933673i
$$748$$ 0 0
$$749$$ 3.69788e8i 0.0321563i
$$750$$ 0 0
$$751$$ 6.66117e9 0.573866 0.286933 0.957951i $$-0.407364\pi$$
0.286933 + 0.957951i $$0.407364\pi$$
$$752$$ 0 0
$$753$$ 1.16815e9 0.0997052
$$754$$ 0 0
$$755$$ 8.89916e9i 0.752549i
$$756$$ 0 0
$$757$$ − 1.22593e10i − 1.02714i −0.858049 0.513569i $$-0.828323\pi$$
0.858049 0.513569i $$-0.171677\pi$$
$$758$$ 0 0
$$759$$ 6.00312e9 0.498346
$$760$$ 0 0
$$761$$ −4.82752e8 −0.0397080 −0.0198540 0.999803i $$-0.506320\pi$$
−0.0198540 + 0.999803i $$0.506320\pi$$
$$762$$ 0 0
$$763$$ − 8.02020e9i − 0.653656i
$$764$$ 0 0
$$765$$ 2.14956e9i 0.173594i
$$766$$ 0 0
$$767$$ −1.32338e10 −1.05901
$$768$$ 0 0
$$769$$ −1.09177e10 −0.865745 −0.432872 0.901455i $$-0.642500\pi$$
−0.432872 + 0.901455i $$0.642500\pi$$
$$770$$ 0 0
$$771$$ 3.59383e9i 0.282401i
$$772$$ 0 0
$$773$$ 1.93076e10i 1.50349i 0.659453 + 0.751746i $$0.270788\pi$$
−0.659453 + 0.751746i $$0.729212\pi$$
$$774$$ 0 0
$$775$$ −6.25958e10 −4.83047
$$776$$ 0 0
$$777$$ −1.78450e10 −1.36472
$$778$$ 0 0
$$779$$ − 5.31544e9i − 0.402864i
$$780$$ 0 0
$$781$$ − 1.22639e10i − 0.921193i
$$782$$ 0 0
$$783$$ 2.76605e9 0.205918
$$784$$ 0 0
$$785$$ 2.39913e10 1.77015
$$786$$ 0 0
$$787$$ 2.09108e10i 1.52918i 0.644517 + 0.764590i $$0.277058\pi$$
−0.644517 + 0.764590i $$0.722942\pi$$
$$788$$ 0 0
$$789$$ − 1.54754e10i − 1.12169i
$$790$$ 0 0
$$791$$ −3.61852e9 −0.259964
$$792$$ 0 0
$$793$$ 7.35680e9 0.523881
$$794$$ 0 0
$$795$$ − 1.60083e10i − 1.12995i
$$796$$ 0 0
$$797$$ 9.05983e9i 0.633893i 0.948443 + 0.316946i $$0.102658\pi$$
−0.948443 + 0.316946i $$0.897342\pi$$
$$798$$ 0 0
$$799$$ −6.08738e8 −0.0422199
$$800$$ 0 0
$$801$$ −6.25771e9 −0.430231
$$802$$ 0 0
$$803$$ − 4.24601e9i − 0.289386i
$$804$$ 0 0
$$805$$ − 5.44079e10i − 3.67601i
$$806$$ 0 0
$$807$$ −1.14576e10 −0.767424
$$808$$ 0 0
$$809$$ −2.28791e10 −1.51921 −0.759606 0.650384i $$-0.774608\pi$$
−0.759606 + 0.650384i $$0.774608\pi$$
$$810$$ 0 0
$$811$$ 5.84918e9i 0.385054i 0.981292 + 0.192527i $$0.0616683\pi$$
−0.981292 + 0.192527i $$0.938332\pi$$
$$812$$ 0 0
$$813$$ 1.47408e10i 0.962065i
$$814$$ 0 0
$$815$$ 3.42503e10 2.21622
$$816$$ 0 0
$$817$$ −1.02065e10 −0.654788
$$818$$ 0 0
$$819$$ − 7.01502e9i − 0.446206i
$$820$$ 0 0
$$821$$ 1.60574e10i 1.01268i 0.862333 + 0.506342i $$0.169003\pi$$
−0.862333 + 0.506342i $$0.830997\pi$$
$$822$$ 0 0
$$823$$ 1.90253e8 0.0118969 0.00594844 0.999982i $$-0.498107\pi$$
0.00594844 + 0.999982i $$0.498107\pi$$
$$824$$ 0 0
$$825$$ −1.46981e10 −0.911321
$$826$$ 0 0
$$827$$ 1.27195e10i 0.781989i 0.920393 + 0.390994i $$0.127869\pi$$
−0.920393 + 0.390994i $$0.872131\pi$$
$$828$$ 0 0
$$829$$ − 5.37231e9i − 0.327506i −0.986501 0.163753i $$-0.947640\pi$$
0.986501 0.163753i $$-0.0523601\pi$$
$$830$$ 0 0
$$831$$ 1.17599e10 0.710886
$$832$$ 0 0
$$833$$ 5.02322e9 0.301109
$$834$$ 0 0
$$835$$ 1.07888e9i 0.0641316i
$$836$$ 0 0
$$837$$ 6.32492e9i 0.372834i
$$838$$ 0 0
$$839$$ −1.73272e10 −1.01289 −0.506445 0.862272i $$-0.669041\pi$$
−0.506445 + 0.862272i $$0.669041\pi$$
$$840$$ 0 0
$$841$$ −2.49873e9 −0.144855
$$842$$ 0 0
$$843$$ 1.28467e10i 0.738575i
$$844$$ 0 0
$$845$$ − 4.54962e9i − 0.259404i
$$846$$ 0 0
$$847$$ 1.52861e10 0.864379
$$848$$ 0 0
$$849$$ −1.30038e10 −0.729278
$$850$$ 0 0
$$851$$ − 4.01704e10i − 2.23436i
$$852$$ 0 0
$$853$$ 2.20248e10i 1.21504i 0.794305 + 0.607520i $$0.207835\pi$$
−0.794305 + 0.607520i $$0.792165\pi$$
$$854$$ 0 0
$$855$$ 7.72568e9 0.422723
$$856$$ 0 0
$$857$$ 2.96218e9 0.160760 0.0803800 0.996764i $$-0.474387\pi$$
0.0803800 + 0.996764i $$0.474387\pi$$
$$858$$ 0 0
$$859$$ − 3.47029e9i − 0.186806i −0.995628 0.0934028i $$-0.970226\pi$$
0.995628 0.0934028i $$-0.0297744\pi$$
$$860$$ 0 0
$$861$$ − 9.26099e9i − 0.494477i
$$862$$ 0 0
$$863$$ 9.13684e9 0.483903 0.241951 0.970288i $$-0.422213\pi$$
0.241951 + 0.970288i $$0.422213\pi$$
$$864$$ 0 0
$$865$$ −3.75781e10 −1.97414
$$866$$ 0 0
$$867$$ 1.02190e10i 0.532527i
$$868$$ 0 0
$$869$$ 4.04472e8i 0.0209083i
$$870$$ 0 0
$$871$$ 3.45705e10 1.77273
$$872$$ 0 0
$$873$$ 5.89561e9 0.299902
$$874$$ 0 0
$$875$$ 7.97864e10i 4.02625i
$$876$$ 0 0
$$877$$ − 4.62260e9i − 0.231413i −0.993283 0.115706i $$-0.963087\pi$$
0.993283 0.115706i $$-0.0369132\pi$$
$$878$$ 0 0
$$879$$ 1.58901e10 0.789163
$$880$$ 0 0
$$881$$ 8.53164e9 0.420356 0.210178 0.977663i $$-0.432596\pi$$
0.210178 + 0.977663i $$0.432596\pi$$
$$882$$ 0 0
$$883$$ − 2.09924e10i − 1.02612i −0.858352 0.513061i $$-0.828511\pi$$
0.858352 0.513061i $$-0.171489\pi$$
$$884$$ 0 0
$$885$$ − 2.53928e10i − 1.23143i
$$886$$ 0 0
$$887$$ −1.79154e10 −0.861973 −0.430986 0.902358i $$-0.641834\pi$$
−0.430986 + 0.902358i $$0.641834\pi$$
$$888$$ 0 0
$$889$$ 2.97030e10 1.41790
$$890$$ 0 0
$$891$$ 1.48515e9i 0.0703393i
$$892$$ 0 0
$$893$$ 2.18785e9i 0.102810i
$$894$$ 0 0
$$895$$ 3.64146e10 1.69783
$$896$$ 0 0
$$897$$ 1.57913e10 0.730542
$$898$$ 0 0
$$899$$ − 4.51577e10i − 2.07287i
$$900$$ 0 0
$$901$$ 6.40569e9i 0.291762i
$$902$$ 0 0
$$903$$ −1.77826e10 −0.803691
$$904$$ 0 0
$$905$$ −2.92517e10 −1.31184
$$906$$ 0 0
$$907$$ − 2.75245e10i − 1.22488i −0.790516 0.612441i $$-0.790188\pi$$
0.790516 0.612441i $$-0.209812\pi$$
$$908$$ 0 0
$$909$$ 2.86229e9i 0.126398i
$$910$$ 0 0
$$911$$ −1.40208e10 −0.614409 −0.307204 0.951644i $$-0.599393\pi$$
−0.307204 + 0.951644i $$0.599393\pi$$
$$912$$ 0 0
$$913$$ −4.07760e9 −0.177320
$$914$$ 0 0
$$915$$ 1.41161e10i 0.609174i
$$916$$ 0 0
$$917$$ 3.29976e9i 0.141316i
$$918$$ 0 0
$$919$$ 3.39157e10 1.44144 0.720721 0.693226i $$-0.243811\pi$$
0.720721 + 0.693226i $$0.243811\pi$$
$$920$$ 0 0
$$921$$ −1.19886e10 −0.505663
$$922$$ 0 0
$$923$$ − 3.22605e10i − 1.35041i
$$924$$ 0 0
$$925$$ 9.83534e10i 4.08595i
$$926$$ 0 0
$$927$$ 6.97587e9 0.287620
$$928$$ 0 0
$$929$$ −9.41796e9 −0.385391 −0.192696 0.981259i $$-0.561723\pi$$
−0.192696 + 0.981259i $$0.561723\pi$$
$$930$$ 0 0
$$931$$ − 1.80538e10i − 0.733237i
$$932$$ 0 0
$$933$$ 4.25199e9i 0.171398i
$$934$$ 0 0
$$935$$ 8.24020e9 0.329683
$$936$$ 0 0
$$937$$ −2.54198e10 −1.00945 −0.504723 0.863281i $$-0.668405\pi$$
−0.504723 + 0.863281i $$0.668405\pi$$
$$938$$ 0 0
$$939$$ − 4.18662e9i − 0.165019i
$$940$$ 0 0
$$941$$ − 8.70006e9i − 0.340376i −0.985412 0.170188i $$-0.945563\pi$$
0.985412 0.170188i $$-0.0544375\pi$$
$$942$$ 0 0
$$943$$ 2.08472e10 0.809574
$$944$$ 0 0
$$945$$ 1.34603e10 0.518852
$$946$$ 0 0
$$947$$ 2.27301e10i 0.869716i 0.900499 + 0.434858i $$0.143201\pi$$
−0.900499 + 0.434858i $$0.856799\pi$$
$$948$$ 0 0
$$949$$ − 1.11692e10i − 0.424220i
$$950$$ 0 0
$$951$$ −7.84328e9 −0.295710
$$952$$ 0 0
$$953$$ 3.29285e10 1.23238 0.616192 0.787596i $$-0.288674\pi$$
0.616192 + 0.787596i $$0.288674\pi$$
$$954$$ 0 0
$$955$$ 5.17143e10i 1.92132i
$$956$$ 0 0
$$957$$ − 1.06034e10i − 0.391071i
$$958$$ 0 0
$$959$$ 6.66403e10 2.43990
$$960$$ 0 0
$$961$$ 7.57462e10 2.75314
$$962$$ 0 0
$$963$$ − 2.05938e8i − 0.00743095i
$$964$$ 0 0
$$965$$ 3.29566e10i 1.18058i
$$966$$ 0 0
$$967$$ 1.34709e10 0.479074 0.239537 0.970887i $$-0.423004\pi$$
0.239537 + 0.970887i $$0.423004\pi$$
$$968$$ 0 0
$$969$$ −3.09142e9 −0.109150
$$970$$ 0 0
$$971$$ 3.31485e10i 1.16198i 0.813912 + 0.580988i $$0.197334\pi$$
−0.813912 + 0.580988i $$0.802666\pi$$
$$972$$ 0 0
$$973$$ − 7.19112e10i − 2.50266i
$$974$$ 0 0
$$975$$ −3.86636e10 −1.33594
$$976$$ 0 0
$$977$$ 2.46700e10 0.846326 0.423163 0.906053i $$-0.360920\pi$$
0.423163 + 0.906053i $$0.360920\pi$$
$$978$$ 0 0
$$979$$ 2.39885e10i 0.817078i
$$980$$ 0 0
$$981$$ 4.46651e9i 0.151052i
$$982$$ 0 0
$$983$$ 7.13043e9 0.239430 0.119715 0.992808i $$-0.461802\pi$$
0.119715 + 0.992808i $$0.461802\pi$$
$$984$$ 0 0
$$985$$ −3.04802e10 −1.01623
$$986$$ 0 0
$$987$$ 3.81184e9i 0.126190i
$$988$$ 0 0
$$989$$ − 4.00301e10i − 1.31583i
$$990$$ 0 0
$$991$$ 3.96102e10 1.29285 0.646426 0.762976i $$-0.276263\pi$$
0.646426 + 0.762976i $$0.276263\pi$$
$$992$$ 0 0
$$993$$ 1.31081e10 0.424832
$$994$$ 0 0
$$995$$ 7.00487e10i 2.25434i
$$996$$ 0 0
$$997$$ 4.99492e9i 0.159623i 0.996810 + 0.0798116i $$0.0254319\pi$$
−0.996810 + 0.0798116i $$0.974568\pi$$
$$998$$ 0 0
$$999$$ 9.93800e9 0.315369
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 384.8.d.d.193.1 yes 8
4.3 odd 2 384.8.d.c.193.5 yes 8
8.3 odd 2 384.8.d.c.193.4 8
8.5 even 2 inner 384.8.d.d.193.8 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.c.193.4 8 8.3 odd 2
384.8.d.c.193.5 yes 8 4.3 odd 2
384.8.d.d.193.1 yes 8 1.1 even 1 trivial
384.8.d.d.193.8 yes 8 8.5 even 2 inner