Properties

 Label 384.7.b.d.319.2 Level $384$ Weight $7$ Character 384.319 Analytic conductor $88.341$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$88.3407681100$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{32}\cdot 3^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 319.2 Root $$-1.35234 - 0.780776i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.319 Dual form 384.7.b.d.319.3

$q$-expansion

 $$f(q)$$ $$=$$ $$q-15.5885 q^{3} -20.0000i q^{5} +155.727i q^{7} +243.000 q^{9} +O(q^{10})$$ $$q-15.5885 q^{3} -20.0000i q^{5} +155.727i q^{7} +243.000 q^{9} +2306.46 q^{11} -3221.18i q^{13} +311.769i q^{15} -6566.73 q^{17} -3926.40 q^{19} -2427.54i q^{21} -17392.0i q^{23} +15225.0 q^{25} -3788.00 q^{27} +44713.3i q^{29} +30636.4i q^{31} -35954.2 q^{33} +3114.54 q^{35} +44158.1i q^{37} +50213.3i q^{39} +9008.18 q^{41} +25270.3 q^{43} -4860.00i q^{45} -175606. i q^{47} +93398.1 q^{49} +102365. q^{51} -96206.3i q^{53} -46129.2i q^{55} +61206.5 q^{57} -35018.3 q^{59} +86135.7i q^{61} +37841.7i q^{63} -64423.6 q^{65} -424605. q^{67} +271114. i q^{69} -36572.4i q^{71} -375121. q^{73} -237334. q^{75} +359179. i q^{77} -520323. i q^{79} +59049.0 q^{81} +594208. q^{83} +131335. i q^{85} -697011. i q^{87} +901101. q^{89} +501625. q^{91} -477574. i q^{93} +78528.0i q^{95} -1.25744e6 q^{97} +560470. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 1944 q^{9} + O(q^{10})$$ $$8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97} + O(q^{100})$$

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
$$3$$ −15.5885 −0.577350
$$4$$ 0 0
$$5$$ − 20.0000i − 0.160000i −0.996795 0.0800000i $$-0.974508\pi$$
0.996795 0.0800000i $$-0.0254920\pi$$
$$6$$ 0 0
$$7$$ 155.727i 0.454015i 0.973893 + 0.227007i $$0.0728942\pi$$
−0.973893 + 0.227007i $$0.927106\pi$$
$$8$$ 0 0
$$9$$ 243.000 0.333333
$$10$$ 0 0
$$11$$ 2306.46 1.73288 0.866439 0.499282i $$-0.166403\pi$$
0.866439 + 0.499282i $$0.166403\pi$$
$$12$$ 0 0
$$13$$ − 3221.18i − 1.46617i −0.680136 0.733086i $$-0.738079\pi$$
0.680136 0.733086i $$-0.261921\pi$$
$$14$$ 0 0
$$15$$ 311.769i 0.0923760i
$$16$$ 0 0
$$17$$ −6566.73 −1.33660 −0.668301 0.743891i $$-0.732978\pi$$
−0.668301 + 0.743891i $$0.732978\pi$$
$$18$$ 0 0
$$19$$ −3926.40 −0.572445 −0.286223 0.958163i $$-0.592400\pi$$
−0.286223 + 0.958163i $$0.592400\pi$$
$$20$$ 0 0
$$21$$ − 2427.54i − 0.262126i
$$22$$ 0 0
$$23$$ − 17392.0i − 1.42944i −0.699411 0.714720i $$-0.746554\pi$$
0.699411 0.714720i $$-0.253446\pi$$
$$24$$ 0 0
$$25$$ 15225.0 0.974400
$$26$$ 0 0
$$27$$ −3788.00 −0.192450
$$28$$ 0 0
$$29$$ 44713.3i 1.83334i 0.399648 + 0.916669i $$0.369132\pi$$
−0.399648 + 0.916669i $$0.630868\pi$$
$$30$$ 0 0
$$31$$ 30636.4i 1.02838i 0.857677 + 0.514188i $$0.171907\pi$$
−0.857677 + 0.514188i $$0.828093\pi$$
$$32$$ 0 0
$$33$$ −35954.2 −1.00048
$$34$$ 0 0
$$35$$ 3114.54 0.0726424
$$36$$ 0 0
$$37$$ 44158.1i 0.871776i 0.900001 + 0.435888i $$0.143566\pi$$
−0.900001 + 0.435888i $$0.856434\pi$$
$$38$$ 0 0
$$39$$ 50213.3i 0.846495i
$$40$$ 0 0
$$41$$ 9008.18 0.130703 0.0653515 0.997862i $$-0.479183\pi$$
0.0653515 + 0.997862i $$0.479183\pi$$
$$42$$ 0 0
$$43$$ 25270.3 0.317838 0.158919 0.987292i $$-0.449199\pi$$
0.158919 + 0.987292i $$0.449199\pi$$
$$44$$ 0 0
$$45$$ − 4860.00i − 0.0533333i
$$46$$ 0 0
$$47$$ − 175606.i − 1.69140i −0.533658 0.845700i $$-0.679183\pi$$
0.533658 0.845700i $$-0.320817\pi$$
$$48$$ 0 0
$$49$$ 93398.1 0.793871
$$50$$ 0 0
$$51$$ 102365. 0.771688
$$52$$ 0 0
$$53$$ − 96206.3i − 0.646214i −0.946363 0.323107i $$-0.895273\pi$$
0.946363 0.323107i $$-0.104727\pi$$
$$54$$ 0 0
$$55$$ − 46129.2i − 0.277261i
$$56$$ 0 0
$$57$$ 61206.5 0.330501
$$58$$ 0 0
$$59$$ −35018.3 −0.170506 −0.0852529 0.996359i $$-0.527170\pi$$
−0.0852529 + 0.996359i $$0.527170\pi$$
$$60$$ 0 0
$$61$$ 86135.7i 0.379484i 0.981834 + 0.189742i $$0.0607652\pi$$
−0.981834 + 0.189742i $$0.939235\pi$$
$$62$$ 0 0
$$63$$ 37841.7i 0.151338i
$$64$$ 0 0
$$65$$ −64423.6 −0.234588
$$66$$ 0 0
$$67$$ −424605. −1.41176 −0.705880 0.708332i $$-0.749448\pi$$
−0.705880 + 0.708332i $$0.749448\pi$$
$$68$$ 0 0
$$69$$ 271114.i 0.825287i
$$70$$ 0 0
$$71$$ − 36572.4i − 0.102183i −0.998694 0.0510915i $$-0.983730\pi$$
0.998694 0.0510915i $$-0.0162700\pi$$
$$72$$ 0 0
$$73$$ −375121. −0.964280 −0.482140 0.876094i $$-0.660140\pi$$
−0.482140 + 0.876094i $$0.660140\pi$$
$$74$$ 0 0
$$75$$ −237334. −0.562570
$$76$$ 0 0
$$77$$ 359179.i 0.786753i
$$78$$ 0 0
$$79$$ − 520323.i − 1.05534i −0.849450 0.527670i $$-0.823066\pi$$
0.849450 0.527670i $$-0.176934\pi$$
$$80$$ 0 0
$$81$$ 59049.0 0.111111
$$82$$ 0 0
$$83$$ 594208. 1.03921 0.519606 0.854406i $$-0.326079\pi$$
0.519606 + 0.854406i $$0.326079\pi$$
$$84$$ 0 0
$$85$$ 131335.i 0.213856i
$$86$$ 0 0
$$87$$ − 697011.i − 1.05848i
$$88$$ 0 0
$$89$$ 901101. 1.27821 0.639107 0.769118i $$-0.279304\pi$$
0.639107 + 0.769118i $$0.279304\pi$$
$$90$$ 0 0
$$91$$ 501625. 0.665664
$$92$$ 0 0
$$93$$ − 477574.i − 0.593733i
$$94$$ 0 0
$$95$$ 78528.0i 0.0915912i
$$96$$ 0 0
$$97$$ −1.25744e6 −1.37775 −0.688875 0.724880i $$-0.741895\pi$$
−0.688875 + 0.724880i $$0.741895\pi$$
$$98$$ 0 0
$$99$$ 560470. 0.577626
$$100$$ 0 0
$$101$$ − 391040.i − 0.379539i −0.981829 0.189770i $$-0.939226\pi$$
0.981829 0.189770i $$-0.0607741\pi$$
$$102$$ 0 0
$$103$$ − 1.36080e6i − 1.24533i −0.782490 0.622663i $$-0.786051\pi$$
0.782490 0.622663i $$-0.213949\pi$$
$$104$$ 0 0
$$105$$ −48550.9 −0.0419401
$$106$$ 0 0
$$107$$ −1.28620e6 −1.04992 −0.524960 0.851127i $$-0.675920\pi$$
−0.524960 + 0.851127i $$0.675920\pi$$
$$108$$ 0 0
$$109$$ − 1.75763e6i − 1.35721i −0.734504 0.678605i $$-0.762585\pi$$
0.734504 0.678605i $$-0.237415\pi$$
$$110$$ 0 0
$$111$$ − 688357.i − 0.503320i
$$112$$ 0 0
$$113$$ −730749. −0.506446 −0.253223 0.967408i $$-0.581491\pi$$
−0.253223 + 0.967408i $$0.581491\pi$$
$$114$$ 0 0
$$115$$ −347840. −0.228710
$$116$$ 0 0
$$117$$ − 782747.i − 0.488724i
$$118$$ 0 0
$$119$$ − 1.02262e6i − 0.606837i
$$120$$ 0 0
$$121$$ 3.54820e6 2.00287
$$122$$ 0 0
$$123$$ −140424. −0.0754614
$$124$$ 0 0
$$125$$ − 617000.i − 0.315904i
$$126$$ 0 0
$$127$$ − 2.59678e6i − 1.26772i −0.773448 0.633860i $$-0.781469\pi$$
0.773448 0.633860i $$-0.218531\pi$$
$$128$$ 0 0
$$129$$ −393925. −0.183504
$$130$$ 0 0
$$131$$ 605543. 0.269359 0.134679 0.990889i $$-0.457000\pi$$
0.134679 + 0.990889i $$0.457000\pi$$
$$132$$ 0 0
$$133$$ − 611447.i − 0.259899i
$$134$$ 0 0
$$135$$ 75759.9i 0.0307920i
$$136$$ 0 0
$$137$$ −1.75932e6 −0.684199 −0.342100 0.939664i $$-0.611138\pi$$
−0.342100 + 0.939664i $$0.611138\pi$$
$$138$$ 0 0
$$139$$ 303680. 0.113076 0.0565382 0.998400i $$-0.481994\pi$$
0.0565382 + 0.998400i $$0.481994\pi$$
$$140$$ 0 0
$$141$$ 2.73743e6i 0.976531i
$$142$$ 0 0
$$143$$ − 7.42953e6i − 2.54070i
$$144$$ 0 0
$$145$$ 894265. 0.293334
$$146$$ 0 0
$$147$$ −1.45593e6 −0.458341
$$148$$ 0 0
$$149$$ 1.32218e6i 0.399696i 0.979827 + 0.199848i $$0.0640449\pi$$
−0.979827 + 0.199848i $$0.935955\pi$$
$$150$$ 0 0
$$151$$ − 4.67518e6i − 1.35790i −0.734184 0.678950i $$-0.762435\pi$$
0.734184 0.678950i $$-0.237565\pi$$
$$152$$ 0 0
$$153$$ −1.59571e6 −0.445534
$$154$$ 0 0
$$155$$ 612727. 0.164540
$$156$$ 0 0
$$157$$ 2.44769e6i 0.632495i 0.948677 + 0.316247i $$0.102423\pi$$
−0.948677 + 0.316247i $$0.897577\pi$$
$$158$$ 0 0
$$159$$ 1.49971e6i 0.373092i
$$160$$ 0 0
$$161$$ 2.70840e6 0.648987
$$162$$ 0 0
$$163$$ 6.62033e6 1.52868 0.764340 0.644813i $$-0.223065\pi$$
0.764340 + 0.644813i $$0.223065\pi$$
$$164$$ 0 0
$$165$$ 719084.i 0.160076i
$$166$$ 0 0
$$167$$ − 7.27537e6i − 1.56209i −0.624476 0.781044i $$-0.714687\pi$$
0.624476 0.781044i $$-0.285313\pi$$
$$168$$ 0 0
$$169$$ −5.54920e6 −1.14966
$$170$$ 0 0
$$171$$ −954116. −0.190815
$$172$$ 0 0
$$173$$ − 3.72601e6i − 0.719623i −0.933025 0.359812i $$-0.882841\pi$$
0.933025 0.359812i $$-0.117159\pi$$
$$174$$ 0 0
$$175$$ 2.37094e6i 0.442392i
$$176$$ 0 0
$$177$$ 545881. 0.0984415
$$178$$ 0 0
$$179$$ −201811. −0.0351873 −0.0175937 0.999845i $$-0.505601\pi$$
−0.0175937 + 0.999845i $$0.505601\pi$$
$$180$$ 0 0
$$181$$ − 5.43440e6i − 0.916466i −0.888832 0.458233i $$-0.848483\pi$$
0.888832 0.458233i $$-0.151517\pi$$
$$182$$ 0 0
$$183$$ − 1.34272e6i − 0.219095i
$$184$$ 0 0
$$185$$ 883162. 0.139484
$$186$$ 0 0
$$187$$ −1.51459e7 −2.31617
$$188$$ 0 0
$$189$$ − 589893.i − 0.0873752i
$$190$$ 0 0
$$191$$ − 6.82608e6i − 0.979650i −0.871821 0.489825i $$-0.837061\pi$$
0.871821 0.489825i $$-0.162939\pi$$
$$192$$ 0 0
$$193$$ 1.30841e6 0.182001 0.0910004 0.995851i $$-0.470994\pi$$
0.0910004 + 0.995851i $$0.470994\pi$$
$$194$$ 0 0
$$195$$ 1.00427e6 0.135439
$$196$$ 0 0
$$197$$ 7.61375e6i 0.995863i 0.867216 + 0.497932i $$0.165907\pi$$
−0.867216 + 0.497932i $$0.834093\pi$$
$$198$$ 0 0
$$199$$ − 6.47692e6i − 0.821882i −0.911662 0.410941i $$-0.865200\pi$$
0.911662 0.410941i $$-0.134800\pi$$
$$200$$ 0 0
$$201$$ 6.61894e6 0.815080
$$202$$ 0 0
$$203$$ −6.96307e6 −0.832362
$$204$$ 0 0
$$205$$ − 180164.i − 0.0209125i
$$206$$ 0 0
$$207$$ − 4.22625e6i − 0.476480i
$$208$$ 0 0
$$209$$ −9.05609e6 −0.991978
$$210$$ 0 0
$$211$$ 1.58058e7 1.68256 0.841278 0.540603i $$-0.181804\pi$$
0.841278 + 0.540603i $$0.181804\pi$$
$$212$$ 0 0
$$213$$ 570108.i 0.0589954i
$$214$$ 0 0
$$215$$ − 505406.i − 0.0508540i
$$216$$ 0 0
$$217$$ −4.77091e6 −0.466898
$$218$$ 0 0
$$219$$ 5.84756e6 0.556727
$$220$$ 0 0
$$221$$ 2.11526e7i 1.95969i
$$222$$ 0 0
$$223$$ − 2.07099e7i − 1.86751i −0.357907 0.933757i $$-0.616509\pi$$
0.357907 0.933757i $$-0.383491\pi$$
$$224$$ 0 0
$$225$$ 3.69967e6 0.324800
$$226$$ 0 0
$$227$$ 7.06179e6 0.603722 0.301861 0.953352i $$-0.402392\pi$$
0.301861 + 0.953352i $$0.402392\pi$$
$$228$$ 0 0
$$229$$ 4.64766e6i 0.387015i 0.981099 + 0.193507i $$0.0619863\pi$$
−0.981099 + 0.193507i $$0.938014\pi$$
$$230$$ 0 0
$$231$$ − 5.59904e6i − 0.454232i
$$232$$ 0 0
$$233$$ 1.13777e7 0.899469 0.449735 0.893162i $$-0.351519\pi$$
0.449735 + 0.893162i $$0.351519\pi$$
$$234$$ 0 0
$$235$$ −3.51213e6 −0.270624
$$236$$ 0 0
$$237$$ 8.11104e6i 0.609300i
$$238$$ 0 0
$$239$$ 1.93765e7i 1.41932i 0.704542 + 0.709662i $$0.251152\pi$$
−0.704542 + 0.709662i $$0.748848\pi$$
$$240$$ 0 0
$$241$$ 6.84883e6 0.489289 0.244644 0.969613i $$-0.421329\pi$$
0.244644 + 0.969613i $$0.421329\pi$$
$$242$$ 0 0
$$243$$ −920483. −0.0641500
$$244$$ 0 0
$$245$$ − 1.86796e6i − 0.127019i
$$246$$ 0 0
$$247$$ 1.26477e7i 0.839304i
$$248$$ 0 0
$$249$$ −9.26279e6 −0.599989
$$250$$ 0 0
$$251$$ −129527. −0.00819104 −0.00409552 0.999992i $$-0.501304\pi$$
−0.00409552 + 0.999992i $$0.501304\pi$$
$$252$$ 0 0
$$253$$ − 4.01140e7i − 2.47705i
$$254$$ 0 0
$$255$$ − 2.04730e6i − 0.123470i
$$256$$ 0 0
$$257$$ 1.18895e7 0.700427 0.350213 0.936670i $$-0.386109\pi$$
0.350213 + 0.936670i $$0.386109\pi$$
$$258$$ 0 0
$$259$$ −6.87661e6 −0.395799
$$260$$ 0 0
$$261$$ 1.08653e7i 0.611113i
$$262$$ 0 0
$$263$$ 1.35674e7i 0.745811i 0.927869 + 0.372906i $$0.121638\pi$$
−0.927869 + 0.372906i $$0.878362\pi$$
$$264$$ 0 0
$$265$$ −1.92413e6 −0.103394
$$266$$ 0 0
$$267$$ −1.40468e7 −0.737977
$$268$$ 0 0
$$269$$ − 1.71231e7i − 0.879680i −0.898076 0.439840i $$-0.855035\pi$$
0.898076 0.439840i $$-0.144965\pi$$
$$270$$ 0 0
$$271$$ − 320201.i − 0.0160885i −0.999968 0.00804423i $$-0.997439\pi$$
0.999968 0.00804423i $$-0.00256058\pi$$
$$272$$ 0 0
$$273$$ −7.81956e6 −0.384321
$$274$$ 0 0
$$275$$ 3.51159e7 1.68852
$$276$$ 0 0
$$277$$ − 1.50949e7i − 0.710216i −0.934825 0.355108i $$-0.884444\pi$$
0.934825 0.355108i $$-0.115556\pi$$
$$278$$ 0 0
$$279$$ 7.44463e6i 0.342792i
$$280$$ 0 0
$$281$$ −4.32580e7 −1.94961 −0.974804 0.223063i $$-0.928394\pi$$
−0.974804 + 0.223063i $$0.928394\pi$$
$$282$$ 0 0
$$283$$ −3.22002e7 −1.42069 −0.710345 0.703854i $$-0.751461\pi$$
−0.710345 + 0.703854i $$0.751461\pi$$
$$284$$ 0 0
$$285$$ − 1.22413e6i − 0.0528802i
$$286$$ 0 0
$$287$$ 1.40282e6i 0.0593411i
$$288$$ 0 0
$$289$$ 1.89843e7 0.786505
$$290$$ 0 0
$$291$$ 1.96015e7 0.795444
$$292$$ 0 0
$$293$$ − 3.01307e7i − 1.19786i −0.800801 0.598931i $$-0.795592\pi$$
0.800801 0.598931i $$-0.204408\pi$$
$$294$$ 0 0
$$295$$ 700366.i 0.0272809i
$$296$$ 0 0
$$297$$ −8.73687e6 −0.333493
$$298$$ 0 0
$$299$$ −5.60228e7 −2.09581
$$300$$ 0 0
$$301$$ 3.93527e6i 0.144303i
$$302$$ 0 0
$$303$$ 6.09571e6i 0.219127i
$$304$$ 0 0
$$305$$ 1.72271e6 0.0607175
$$306$$ 0 0
$$307$$ 2.09501e7 0.724053 0.362026 0.932168i $$-0.382085\pi$$
0.362026 + 0.932168i $$0.382085\pi$$
$$308$$ 0 0
$$309$$ 2.12128e7i 0.718990i
$$310$$ 0 0
$$311$$ − 4.94101e6i − 0.164261i −0.996622 0.0821305i $$-0.973828\pi$$
0.996622 0.0821305i $$-0.0261724\pi$$
$$312$$ 0 0
$$313$$ 2.67449e7 0.872182 0.436091 0.899903i $$-0.356363\pi$$
0.436091 + 0.899903i $$0.356363\pi$$
$$314$$ 0 0
$$315$$ 756834. 0.0242141
$$316$$ 0 0
$$317$$ 4.74499e7i 1.48956i 0.667311 + 0.744779i $$0.267445\pi$$
−0.667311 + 0.744779i $$0.732555\pi$$
$$318$$ 0 0
$$319$$ 1.03129e8i 3.17695i
$$320$$ 0 0
$$321$$ 2.00498e7 0.606172
$$322$$ 0 0
$$323$$ 2.57836e7 0.765131
$$324$$ 0 0
$$325$$ − 4.90425e7i − 1.42864i
$$326$$ 0 0
$$327$$ 2.73987e7i 0.783585i
$$328$$ 0 0
$$329$$ 2.73467e7 0.767921
$$330$$ 0 0
$$331$$ 4.84318e7 1.33551 0.667754 0.744382i $$-0.267256\pi$$
0.667754 + 0.744382i $$0.267256\pi$$
$$332$$ 0 0
$$333$$ 1.07304e7i 0.290592i
$$334$$ 0 0
$$335$$ 8.49210e6i 0.225881i
$$336$$ 0 0
$$337$$ −5.96066e7 −1.55742 −0.778709 0.627386i $$-0.784125\pi$$
−0.778709 + 0.627386i $$0.784125\pi$$
$$338$$ 0 0
$$339$$ 1.13913e7 0.292397
$$340$$ 0 0
$$341$$ 7.06616e7i 1.78205i
$$342$$ 0 0
$$343$$ 3.28657e7i 0.814444i
$$344$$ 0 0
$$345$$ 5.42229e6 0.132046
$$346$$ 0 0
$$347$$ −2.95157e7 −0.706423 −0.353211 0.935544i $$-0.614910\pi$$
−0.353211 + 0.935544i $$0.614910\pi$$
$$348$$ 0 0
$$349$$ − 5.55247e7i − 1.30620i −0.757272 0.653100i $$-0.773468\pi$$
0.757272 0.653100i $$-0.226532\pi$$
$$350$$ 0 0
$$351$$ 1.22018e7i 0.282165i
$$352$$ 0 0
$$353$$ −1.81396e7 −0.412386 −0.206193 0.978511i $$-0.566107\pi$$
−0.206193 + 0.978511i $$0.566107\pi$$
$$354$$ 0 0
$$355$$ −731449. −0.0163493
$$356$$ 0 0
$$357$$ 1.59410e7i 0.350358i
$$358$$ 0 0
$$359$$ 8.83261e6i 0.190900i 0.995434 + 0.0954499i $$0.0304290\pi$$
−0.995434 + 0.0954499i $$0.969571\pi$$
$$360$$ 0 0
$$361$$ −3.16293e7 −0.672307
$$362$$ 0 0
$$363$$ −5.53110e7 −1.15636
$$364$$ 0 0
$$365$$ 7.50242e6i 0.154285i
$$366$$ 0 0
$$367$$ − 4.83247e7i − 0.977623i −0.872389 0.488811i $$-0.837431\pi$$
0.872389 0.488811i $$-0.162569\pi$$
$$368$$ 0 0
$$369$$ 2.18899e6 0.0435677
$$370$$ 0 0
$$371$$ 1.49819e7 0.293391
$$372$$ 0 0
$$373$$ − 8.98844e7i − 1.73204i −0.500010 0.866020i $$-0.666670\pi$$
0.500010 0.866020i $$-0.333330\pi$$
$$374$$ 0 0
$$375$$ 9.61808e6i 0.182387i
$$376$$ 0 0
$$377$$ 1.44030e8 2.68799
$$378$$ 0 0
$$379$$ −8.60279e7 −1.58024 −0.790118 0.612955i $$-0.789981\pi$$
−0.790118 + 0.612955i $$0.789981\pi$$
$$380$$ 0 0
$$381$$ 4.04798e7i 0.731919i
$$382$$ 0 0
$$383$$ 6.65042e7i 1.18373i 0.806037 + 0.591865i $$0.201608\pi$$
−0.806037 + 0.591865i $$0.798392\pi$$
$$384$$ 0 0
$$385$$ 7.18357e6 0.125880
$$386$$ 0 0
$$387$$ 6.14069e6 0.105946
$$388$$ 0 0
$$389$$ 5.66889e7i 0.963050i 0.876432 + 0.481525i $$0.159917\pi$$
−0.876432 + 0.481525i $$0.840083\pi$$
$$390$$ 0 0
$$391$$ 1.14208e8i 1.91059i
$$392$$ 0 0
$$393$$ −9.43949e6 −0.155514
$$394$$ 0 0
$$395$$ −1.04065e7 −0.168854
$$396$$ 0 0
$$397$$ − 3.21267e7i − 0.513446i −0.966485 0.256723i $$-0.917357\pi$$
0.966485 0.256723i $$-0.0826429\pi$$
$$398$$ 0 0
$$399$$ 9.53152e6i 0.150053i
$$400$$ 0 0
$$401$$ −7.28625e7 −1.12998 −0.564991 0.825097i $$-0.691120\pi$$
−0.564991 + 0.825097i $$0.691120\pi$$
$$402$$ 0 0
$$403$$ 9.86853e7 1.50778
$$404$$ 0 0
$$405$$ − 1.18098e6i − 0.0177778i
$$406$$ 0 0
$$407$$ 1.01849e8i 1.51068i
$$408$$ 0 0
$$409$$ −2.07185e7 −0.302823 −0.151412 0.988471i $$-0.548382\pi$$
−0.151412 + 0.988471i $$0.548382\pi$$
$$410$$ 0 0
$$411$$ 2.74251e7 0.395023
$$412$$ 0 0
$$413$$ − 5.45330e6i − 0.0774121i
$$414$$ 0 0
$$415$$ − 1.18842e7i − 0.166274i
$$416$$ 0 0
$$417$$ −4.73390e6 −0.0652846
$$418$$ 0 0
$$419$$ 6.79561e7 0.923818 0.461909 0.886927i $$-0.347165\pi$$
0.461909 + 0.886927i $$0.347165\pi$$
$$420$$ 0 0
$$421$$ 522580.i 0.00700336i 0.999994 + 0.00350168i $$0.00111462\pi$$
−0.999994 + 0.00350168i $$0.998885\pi$$
$$422$$ 0 0
$$423$$ − 4.26723e7i − 0.563800i
$$424$$ 0 0
$$425$$ −9.99784e7 −1.30239
$$426$$ 0 0
$$427$$ −1.34137e7 −0.172291
$$428$$ 0 0
$$429$$ 1.15815e8i 1.46687i
$$430$$ 0 0
$$431$$ 1.10533e8i 1.38057i 0.723538 + 0.690285i $$0.242515\pi$$
−0.723538 + 0.690285i $$0.757485\pi$$
$$432$$ 0 0
$$433$$ 6.65239e7 0.819435 0.409717 0.912213i $$-0.365627\pi$$
0.409717 + 0.912213i $$0.365627\pi$$
$$434$$ 0 0
$$435$$ −1.39402e7 −0.169356
$$436$$ 0 0
$$437$$ 6.82879e7i 0.818276i
$$438$$ 0 0
$$439$$ − 1.06995e8i − 1.26465i −0.774702 0.632327i $$-0.782100\pi$$
0.774702 0.632327i $$-0.217900\pi$$
$$440$$ 0 0
$$441$$ 2.26957e7 0.264624
$$442$$ 0 0
$$443$$ 1.01597e7 0.116861 0.0584304 0.998291i $$-0.481390\pi$$
0.0584304 + 0.998291i $$0.481390\pi$$
$$444$$ 0 0
$$445$$ − 1.80220e7i − 0.204514i
$$446$$ 0 0
$$447$$ − 2.06107e7i − 0.230765i
$$448$$ 0 0
$$449$$ −4.39511e7 −0.485546 −0.242773 0.970083i $$-0.578057\pi$$
−0.242773 + 0.970083i $$0.578057\pi$$
$$450$$ 0 0
$$451$$ 2.07770e7 0.226492
$$452$$ 0 0
$$453$$ 7.28789e7i 0.783984i
$$454$$ 0 0
$$455$$ − 1.00325e7i − 0.106506i
$$456$$ 0 0
$$457$$ −5.92987e6 −0.0621293 −0.0310646 0.999517i $$-0.509890\pi$$
−0.0310646 + 0.999517i $$0.509890\pi$$
$$458$$ 0 0
$$459$$ 2.48747e7 0.257229
$$460$$ 0 0
$$461$$ 4.31145e7i 0.440069i 0.975492 + 0.220035i $$0.0706170\pi$$
−0.975492 + 0.220035i $$0.929383\pi$$
$$462$$ 0 0
$$463$$ 8.09555e7i 0.815649i 0.913060 + 0.407825i $$0.133712\pi$$
−0.913060 + 0.407825i $$0.866288\pi$$
$$464$$ 0 0
$$465$$ −9.55147e6 −0.0949973
$$466$$ 0 0
$$467$$ −1.20309e8 −1.18127 −0.590633 0.806940i $$-0.701122\pi$$
−0.590633 + 0.806940i $$0.701122\pi$$
$$468$$ 0 0
$$469$$ − 6.61225e7i − 0.640960i
$$470$$ 0 0
$$471$$ − 3.81557e7i − 0.365171i
$$472$$ 0 0
$$473$$ 5.82850e7 0.550774
$$474$$ 0 0
$$475$$ −5.97795e7 −0.557791
$$476$$ 0 0
$$477$$ − 2.33781e7i − 0.215405i
$$478$$ 0 0
$$479$$ − 1.54238e8i − 1.40341i −0.712468 0.701705i $$-0.752422\pi$$
0.712468 0.701705i $$-0.247578\pi$$
$$480$$ 0 0
$$481$$ 1.42241e8 1.27817
$$482$$ 0 0
$$483$$ −4.22198e7 −0.374693
$$484$$ 0 0
$$485$$ 2.51487e7i 0.220440i
$$486$$ 0 0
$$487$$ − 6.53601e7i − 0.565882i −0.959137 0.282941i $$-0.908690\pi$$
0.959137 0.282941i $$-0.0913101\pi$$
$$488$$ 0 0
$$489$$ −1.03201e8 −0.882584
$$490$$ 0 0
$$491$$ −4.51212e7 −0.381185 −0.190593 0.981669i $$-0.561041\pi$$
−0.190593 + 0.981669i $$0.561041\pi$$
$$492$$ 0 0
$$493$$ − 2.93620e8i − 2.45044i
$$494$$ 0 0
$$495$$ − 1.12094e7i − 0.0924202i
$$496$$ 0 0
$$497$$ 5.69532e6 0.0463926
$$498$$ 0 0
$$499$$ 5.03581e7 0.405292 0.202646 0.979252i $$-0.435046\pi$$
0.202646 + 0.979252i $$0.435046\pi$$
$$500$$ 0 0
$$501$$ 1.13412e8i 0.901872i
$$502$$ 0 0
$$503$$ 4.41054e6i 0.0346567i 0.999850 + 0.0173284i $$0.00551607\pi$$
−0.999850 + 0.0173284i $$0.994484\pi$$
$$504$$ 0 0
$$505$$ −7.82080e6 −0.0607263
$$506$$ 0 0
$$507$$ 8.65035e7 0.663758
$$508$$ 0 0
$$509$$ 2.44005e7i 0.185031i 0.995711 + 0.0925156i $$0.0294908\pi$$
−0.995711 + 0.0925156i $$0.970509\pi$$
$$510$$ 0 0
$$511$$ − 5.84165e7i − 0.437797i
$$512$$ 0 0
$$513$$ 1.48732e7 0.110167
$$514$$ 0 0
$$515$$ −2.72160e7 −0.199252
$$516$$ 0 0
$$517$$ − 4.05029e8i − 2.93099i
$$518$$ 0 0
$$519$$ 5.80827e7i 0.415475i
$$520$$ 0 0
$$521$$ 6.53917e7 0.462391 0.231195 0.972907i $$-0.425736\pi$$
0.231195 + 0.972907i $$0.425736\pi$$
$$522$$ 0 0
$$523$$ −1.19428e8 −0.834833 −0.417417 0.908715i $$-0.637064\pi$$
−0.417417 + 0.908715i $$0.637064\pi$$
$$524$$ 0 0
$$525$$ − 3.69594e7i − 0.255415i
$$526$$ 0 0
$$527$$ − 2.01181e8i − 1.37453i
$$528$$ 0 0
$$529$$ −1.54446e8 −1.04330
$$530$$ 0 0
$$531$$ −8.50945e6 −0.0568352
$$532$$ 0 0
$$533$$ − 2.90170e7i − 0.191633i
$$534$$ 0 0
$$535$$ 2.57240e7i 0.167987i
$$536$$ 0 0
$$537$$ 3.14593e6 0.0203154
$$538$$ 0 0
$$539$$ 2.15419e8 1.37568
$$540$$ 0 0
$$541$$ − 3.06540e7i − 0.193596i −0.995304 0.0967979i $$-0.969140\pi$$
0.995304 0.0967979i $$-0.0308600\pi$$
$$542$$ 0 0
$$543$$ 8.47140e7i 0.529122i
$$544$$ 0 0
$$545$$ −3.51525e7 −0.217153
$$546$$ 0 0
$$547$$ −1.60733e8 −0.982071 −0.491035 0.871140i $$-0.663381\pi$$
−0.491035 + 0.871140i $$0.663381\pi$$
$$548$$ 0 0
$$549$$ 2.09310e7i 0.126495i
$$550$$ 0 0
$$551$$ − 1.75562e8i − 1.04949i
$$552$$ 0 0
$$553$$ 8.10284e7 0.479140
$$554$$ 0 0
$$555$$ −1.37671e7 −0.0805313
$$556$$ 0 0
$$557$$ 3.11683e8i 1.80363i 0.432125 + 0.901814i $$0.357764\pi$$
−0.432125 + 0.901814i $$0.642236\pi$$
$$558$$ 0 0
$$559$$ − 8.14003e7i − 0.466005i
$$560$$ 0 0
$$561$$ 2.36101e8 1.33724
$$562$$ 0 0
$$563$$ 2.81653e8 1.57830 0.789149 0.614201i $$-0.210522\pi$$
0.789149 + 0.614201i $$0.210522\pi$$
$$564$$ 0 0
$$565$$ 1.46150e7i 0.0810313i
$$566$$ 0 0
$$567$$ 9.19553e6i 0.0504461i
$$568$$ 0 0
$$569$$ 8.82677e7 0.479143 0.239571 0.970879i $$-0.422993\pi$$
0.239571 + 0.970879i $$0.422993\pi$$
$$570$$ 0 0
$$571$$ 1.66103e8 0.892217 0.446108 0.894979i $$-0.352810\pi$$
0.446108 + 0.894979i $$0.352810\pi$$
$$572$$ 0 0
$$573$$ 1.06408e8i 0.565601i
$$574$$ 0 0
$$575$$ − 2.64793e8i − 1.39285i
$$576$$ 0 0
$$577$$ −5.26455e7 −0.274053 −0.137026 0.990567i $$-0.543754\pi$$
−0.137026 + 0.990567i $$0.543754\pi$$
$$578$$ 0 0
$$579$$ −2.03962e7 −0.105078
$$580$$ 0 0
$$581$$ 9.25343e7i 0.471818i
$$582$$ 0 0
$$583$$ − 2.21896e8i − 1.11981i
$$584$$ 0 0
$$585$$ −1.56549e7 −0.0781959
$$586$$ 0 0
$$587$$ −3.88069e7 −0.191865 −0.0959323 0.995388i $$-0.530583\pi$$
−0.0959323 + 0.995388i $$0.530583\pi$$
$$588$$ 0 0
$$589$$ − 1.20291e8i − 0.588689i
$$590$$ 0 0
$$591$$ − 1.18687e8i − 0.574962i
$$592$$ 0 0
$$593$$ −3.26100e8 −1.56382 −0.781909 0.623392i $$-0.785754\pi$$
−0.781909 + 0.623392i $$0.785754\pi$$
$$594$$ 0 0
$$595$$ −2.04523e7 −0.0970939
$$596$$ 0 0
$$597$$ 1.00965e8i 0.474514i
$$598$$ 0 0
$$599$$ − 3.03039e8i − 1.41000i −0.709210 0.704998i $$-0.750948\pi$$
0.709210 0.704998i $$-0.249052\pi$$
$$600$$ 0 0
$$601$$ −3.98916e8 −1.83763 −0.918815 0.394689i $$-0.870852\pi$$
−0.918815 + 0.394689i $$0.870852\pi$$
$$602$$ 0 0
$$603$$ −1.03179e8 −0.470586
$$604$$ 0 0
$$605$$ − 7.09641e7i − 0.320459i
$$606$$ 0 0
$$607$$ − 2.46327e8i − 1.10140i −0.834703 0.550701i $$-0.814361\pi$$
0.834703 0.550701i $$-0.185639\pi$$
$$608$$ 0 0
$$609$$ 1.08543e8 0.480565
$$610$$ 0 0
$$611$$ −5.65660e8 −2.47989
$$612$$ 0 0
$$613$$ 2.16670e8i 0.940628i 0.882499 + 0.470314i $$0.155859\pi$$
−0.882499 + 0.470314i $$0.844141\pi$$
$$614$$ 0 0
$$615$$ 2.80847e6i 0.0120738i
$$616$$ 0 0
$$617$$ 1.27942e8 0.544699 0.272350 0.962198i $$-0.412199\pi$$
0.272350 + 0.962198i $$0.412199\pi$$
$$618$$ 0 0
$$619$$ 1.96738e8 0.829499 0.414749 0.909936i $$-0.363869\pi$$
0.414749 + 0.909936i $$0.363869\pi$$
$$620$$ 0 0
$$621$$ 6.58808e7i 0.275096i
$$622$$ 0 0
$$623$$ 1.40326e8i 0.580328i
$$624$$ 0 0
$$625$$ 2.25551e8 0.923855
$$626$$ 0 0
$$627$$ 1.41171e8 0.572719
$$628$$ 0 0
$$629$$ − 2.89974e8i − 1.16522i
$$630$$ 0 0
$$631$$ − 9.81163e7i − 0.390529i −0.980751 0.195264i $$-0.937443\pi$$
0.980751 0.195264i $$-0.0625565\pi$$
$$632$$ 0 0
$$633$$ −2.46388e8 −0.971424
$$634$$ 0 0
$$635$$ −5.19355e7 −0.202835
$$636$$ 0 0
$$637$$ − 3.00852e8i − 1.16395i
$$638$$ 0 0
$$639$$ − 8.88710e6i − 0.0340610i
$$640$$ 0 0
$$641$$ 2.85929e8 1.08564 0.542819 0.839850i $$-0.317357\pi$$
0.542819 + 0.839850i $$0.317357\pi$$
$$642$$ 0 0
$$643$$ −2.98146e8 −1.12149 −0.560745 0.827988i $$-0.689485\pi$$
−0.560745 + 0.827988i $$0.689485\pi$$
$$644$$ 0 0
$$645$$ 7.87850e6i 0.0293606i
$$646$$ 0 0
$$647$$ 3.84615e7i 0.142008i 0.997476 + 0.0710042i $$0.0226204\pi$$
−0.997476 + 0.0710042i $$0.977380\pi$$
$$648$$ 0 0
$$649$$ −8.07684e7 −0.295466
$$650$$ 0 0
$$651$$ 7.43711e7 0.269564
$$652$$ 0 0
$$653$$ 9.67190e7i 0.347354i 0.984803 + 0.173677i $$0.0555648\pi$$
−0.984803 + 0.173677i $$0.944435\pi$$
$$654$$ 0 0
$$655$$ − 1.21109e7i − 0.0430974i
$$656$$ 0 0
$$657$$ −9.11545e7 −0.321427
$$658$$ 0 0
$$659$$ −6.49525e7 −0.226955 −0.113477 0.993541i $$-0.536199\pi$$
−0.113477 + 0.993541i $$0.536199\pi$$
$$660$$ 0 0
$$661$$ − 3.29739e8i − 1.14174i −0.821042 0.570868i $$-0.806607\pi$$
0.821042 0.570868i $$-0.193393\pi$$
$$662$$ 0 0
$$663$$ − 3.29737e8i − 1.13143i
$$664$$ 0 0
$$665$$ −1.22289e7 −0.0415838
$$666$$ 0 0
$$667$$ 7.77653e8 2.62065
$$668$$ 0 0
$$669$$ 3.22836e8i 1.07821i
$$670$$ 0 0
$$671$$ 1.98669e8i 0.657600i
$$672$$ 0 0
$$673$$ 5.17571e8 1.69795 0.848975 0.528433i $$-0.177220\pi$$
0.848975 + 0.528433i $$0.177220\pi$$
$$674$$ 0 0
$$675$$ −5.76722e7 −0.187523
$$676$$ 0 0
$$677$$ 3.25473e8i 1.04894i 0.851430 + 0.524469i $$0.175736\pi$$
−0.851430 + 0.524469i $$0.824264\pi$$
$$678$$ 0 0
$$679$$ − 1.95817e8i − 0.625519i
$$680$$ 0 0
$$681$$ −1.10082e8 −0.348559
$$682$$ 0 0
$$683$$ 5.37538e8 1.68713 0.843563 0.537031i $$-0.180454\pi$$
0.843563 + 0.537031i $$0.180454\pi$$
$$684$$ 0 0
$$685$$ 3.51864e7i 0.109472i
$$686$$ 0 0
$$687$$ − 7.24498e7i − 0.223443i
$$688$$ 0 0
$$689$$ −3.09898e8 −0.947461
$$690$$ 0 0
$$691$$ −4.64528e8 −1.40792 −0.703960 0.710239i $$-0.748587\pi$$
−0.703960 + 0.710239i $$0.748587\pi$$
$$692$$ 0 0
$$693$$ 8.72804e7i 0.262251i
$$694$$ 0 0
$$695$$ − 6.07360e6i − 0.0180922i
$$696$$ 0 0
$$697$$ −5.91543e7 −0.174698
$$698$$ 0 0
$$699$$ −1.77361e8 −0.519309
$$700$$ 0 0
$$701$$ 3.59130e7i 0.104255i 0.998640 + 0.0521275i $$0.0166002\pi$$
−0.998640 + 0.0521275i $$0.983400\pi$$
$$702$$ 0 0
$$703$$ − 1.73382e8i − 0.499044i
$$704$$ 0 0
$$705$$ 5.47486e7 0.156245
$$706$$ 0 0
$$707$$ 6.08955e7 0.172317
$$708$$ 0 0
$$709$$ 1.86788e8i 0.524094i 0.965055 + 0.262047i $$0.0843976\pi$$
−0.965055 + 0.262047i $$0.915602\pi$$
$$710$$ 0 0
$$711$$ − 1.26439e8i − 0.351780i
$$712$$ 0 0
$$713$$ 5.32827e8 1.47000
$$714$$ 0 0
$$715$$ −1.48591e8 −0.406512
$$716$$ 0 0
$$717$$ − 3.02050e8i − 0.819447i
$$718$$ 0 0
$$719$$ − 3.76368e7i − 0.101257i −0.998718 0.0506286i $$-0.983878\pi$$
0.998718 0.0506286i $$-0.0161225\pi$$
$$720$$ 0 0
$$721$$ 2.11914e8 0.565397
$$722$$ 0 0
$$723$$ −1.06763e8 −0.282491
$$724$$ 0 0
$$725$$ 6.80760e8i 1.78640i
$$726$$ 0 0
$$727$$ 2.58872e8i 0.673724i 0.941554 + 0.336862i $$0.109366\pi$$
−0.941554 + 0.336862i $$0.890634\pi$$
$$728$$ 0 0
$$729$$ 1.43489e7 0.0370370
$$730$$ 0 0
$$731$$ −1.65943e8 −0.424822
$$732$$ 0 0
$$733$$ 5.31872e8i 1.35050i 0.737588 + 0.675251i $$0.235965\pi$$
−0.737588 + 0.675251i $$0.764035\pi$$
$$734$$ 0 0
$$735$$ 2.91186e7i 0.0733346i
$$736$$ 0 0
$$737$$ −9.79335e8 −2.44641
$$738$$ 0 0
$$739$$ −1.70943e8 −0.423564 −0.211782 0.977317i $$-0.567927\pi$$
−0.211782 + 0.977317i $$0.567927\pi$$
$$740$$ 0 0
$$741$$ − 1.97157e8i − 0.484572i
$$742$$ 0 0
$$743$$ 4.23377e8i 1.03219i 0.856531 + 0.516096i $$0.172615\pi$$
−0.856531 + 0.516096i $$0.827385\pi$$
$$744$$ 0 0
$$745$$ 2.64435e7 0.0639514
$$746$$ 0 0
$$747$$ 1.44393e8 0.346404
$$748$$ 0 0
$$749$$ − 2.00296e8i − 0.476679i
$$750$$ 0 0
$$751$$ − 3.04170e8i − 0.718118i −0.933315 0.359059i $$-0.883098\pi$$
0.933315 0.359059i $$-0.116902\pi$$
$$752$$ 0 0
$$753$$ 2.01912e6 0.00472910
$$754$$ 0 0
$$755$$ −9.35037e7 −0.217264
$$756$$ 0 0
$$757$$ − 3.06435e8i − 0.706401i −0.935548 0.353201i $$-0.885093\pi$$
0.935548 0.353201i $$-0.114907\pi$$
$$758$$ 0 0
$$759$$ 6.25315e8i 1.43012i
$$760$$ 0 0
$$761$$ 4.23923e8 0.961907 0.480954 0.876746i $$-0.340291\pi$$
0.480954 + 0.876746i $$0.340291\pi$$
$$762$$ 0 0
$$763$$ 2.73710e8 0.616193
$$764$$ 0 0
$$765$$ 3.19143e7i 0.0712854i
$$766$$ 0 0
$$767$$ 1.12800e8i 0.249991i
$$768$$ 0 0
$$769$$ −5.19668e8 −1.14274 −0.571369 0.820693i $$-0.693588\pi$$
−0.571369 + 0.820693i $$0.693588\pi$$
$$770$$ 0 0
$$771$$ −1.85338e8 −0.404391
$$772$$ 0 0
$$773$$ − 1.89126e7i − 0.0409462i −0.999790 0.0204731i $$-0.993483\pi$$
0.999790 0.0204731i $$-0.00651724\pi$$
$$774$$ 0 0
$$775$$ 4.66439e8i 1.00205i
$$776$$ 0 0
$$777$$ 1.07196e8 0.228515
$$778$$ 0 0
$$779$$ −3.53697e7 −0.0748203
$$780$$ 0 0
$$781$$ − 8.43529e7i − 0.177071i
$$782$$ 0 0
$$783$$ − 1.69374e8i − 0.352826i
$$784$$ 0 0
$$785$$ 4.89537e7 0.101199
$$786$$ 0 0
$$787$$ 2.65036e8 0.543726 0.271863 0.962336i $$-0.412360\pi$$
0.271863 + 0.962336i $$0.412360\pi$$
$$788$$ 0 0
$$789$$ − 2.11495e8i − 0.430594i
$$790$$ 0 0
$$791$$ − 1.13797e8i − 0.229934i
$$792$$ 0 0
$$793$$ 2.77459e8 0.556390
$$794$$ 0 0
$$795$$ 2.99942e7 0.0596947
$$796$$ 0 0
$$797$$ − 6.82061e8i − 1.34725i −0.739073 0.673626i $$-0.764736\pi$$
0.739073 0.673626i $$-0.235264\pi$$
$$798$$ 0 0
$$799$$ 1.15316e9i 2.26073i
$$800$$ 0 0
$$801$$ 2.18968e8 0.426071
$$802$$ 0 0
$$803$$ −8.65203e8 −1.67098
$$804$$ 0 0
$$805$$ − 5.41681e7i − 0.103838i
$$806$$ 0 0
$$807$$ 2.66922e8i 0.507884i
$$808$$ 0 0
$$809$$ 5.50511e8 1.03973 0.519865 0.854248i $$-0.325982\pi$$
0.519865 + 0.854248i $$0.325982\pi$$
$$810$$ 0 0
$$811$$ −3.11549e8 −0.584068 −0.292034 0.956408i $$-0.594332\pi$$
−0.292034 + 0.956408i $$0.594332\pi$$
$$812$$ 0 0
$$813$$ 4.99143e6i 0.00928867i
$$814$$ 0 0
$$815$$ − 1.32407e8i − 0.244589i
$$816$$ 0 0
$$817$$ −9.92214e7 −0.181945
$$818$$ 0 0
$$819$$ 1.21895e8 0.221888
$$820$$ 0 0
$$821$$ − 7.17579e8i − 1.29670i −0.761342 0.648351i $$-0.775459\pi$$
0.761342 0.648351i $$-0.224541\pi$$
$$822$$ 0 0
$$823$$ − 5.41195e8i − 0.970854i −0.874277 0.485427i $$-0.838664\pi$$
0.874277 0.485427i $$-0.161336\pi$$
$$824$$ 0 0
$$825$$ −5.47402e8 −0.974866
$$826$$ 0 0
$$827$$ 8.79442e8 1.55486 0.777429 0.628971i $$-0.216524\pi$$
0.777429 + 0.628971i $$0.216524\pi$$
$$828$$ 0 0
$$829$$ − 4.29123e8i − 0.753213i −0.926373 0.376607i $$-0.877091\pi$$
0.926373 0.376607i $$-0.122909\pi$$
$$830$$ 0 0
$$831$$ 2.35306e8i 0.410043i
$$832$$ 0 0
$$833$$ −6.13320e8 −1.06109
$$834$$ 0 0
$$835$$ −1.45507e8 −0.249934
$$836$$ 0 0
$$837$$ − 1.16050e8i − 0.197911i
$$838$$ 0 0
$$839$$ − 5.86636e8i − 0.993305i −0.867949 0.496653i $$-0.834562\pi$$
0.867949 0.496653i $$-0.165438\pi$$
$$840$$ 0 0
$$841$$ −1.40445e9 −2.36113
$$842$$ 0 0
$$843$$ 6.74325e8 1.12561
$$844$$ 0 0
$$845$$ 1.10984e8i 0.183946i
$$846$$ 0 0
$$847$$ 5.52552e8i 0.909332i
$$848$$ 0 0
$$849$$ 5.01951e8 0.820236
$$850$$ 0 0
$$851$$ 7.67997e8 1.24615
$$852$$ 0 0
$$853$$ 7.00029e8i 1.12790i 0.825810 + 0.563948i $$0.190718\pi$$
−0.825810 + 0.563948i $$0.809282\pi$$
$$854$$ 0 0
$$855$$ 1.90823e7i 0.0305304i
$$856$$ 0 0
$$857$$ 9.52114e8 1.51268 0.756339 0.654180i $$-0.226986\pi$$
0.756339 + 0.654180i $$0.226986\pi$$
$$858$$ 0 0
$$859$$ 4.16968e8 0.657844 0.328922 0.944357i $$-0.393315\pi$$
0.328922 + 0.944357i $$0.393315\pi$$
$$860$$ 0 0
$$861$$ − 2.18678e7i − 0.0342606i
$$862$$ 0 0
$$863$$ 4.62686e8i 0.719869i 0.932978 + 0.359935i $$0.117201\pi$$
−0.932978 + 0.359935i $$0.882799\pi$$
$$864$$ 0 0
$$865$$ −7.45201e7 −0.115140
$$866$$ 0 0
$$867$$ −2.95936e8 −0.454089
$$868$$ 0 0
$$869$$ − 1.20011e9i − 1.82877i
$$870$$ 0 0
$$871$$ 1.36773e9i 2.06988i
$$872$$ 0 0
$$873$$ −3.05557e8 −0.459250
$$874$$ 0 0
$$875$$ 9.60836e7 0.143425
$$876$$ 0 0
$$877$$ 6.73329e8i 0.998225i 0.866537 + 0.499113i $$0.166341\pi$$
−0.866537 + 0.499113i $$0.833659\pi$$
$$878$$ 0 0
$$879$$ 4.69692e8i 0.691586i
$$880$$ 0 0
$$881$$ 1.77155e8 0.259074 0.129537 0.991575i $$-0.458651\pi$$
0.129537 + 0.991575i $$0.458651\pi$$
$$882$$ 0 0
$$883$$ 1.35707e8 0.197115 0.0985574 0.995131i $$-0.468577\pi$$
0.0985574 + 0.995131i $$0.468577\pi$$
$$884$$ 0 0
$$885$$ − 1.09176e7i − 0.0157506i
$$886$$ 0 0
$$887$$ 1.09553e9i 1.56984i 0.619598 + 0.784919i $$0.287296\pi$$
−0.619598 + 0.784919i $$0.712704\pi$$
$$888$$ 0 0
$$889$$ 4.04389e8 0.575564
$$890$$ 0 0
$$891$$ 1.36194e8 0.192542
$$892$$ 0 0
$$893$$ 6.89501e8i 0.968234i
$$894$$ 0 0
$$895$$ 4.03623e6i 0.00562997i
$$896$$ 0 0
$$897$$ 8.73309e8 1.21001
$$898$$ 0 0
$$899$$ −1.36985e9 −1.88536
$$900$$ 0 0
$$901$$ 6.31761e8i 0.863731i
$$902$$ 0 0
$$903$$ − 6.13448e7i − 0.0833133i
$$904$$ 0 0
$$905$$ −1.08688e8 −0.146635
$$906$$ 0 0
$$907$$ −1.34159e9 −1.79803 −0.899015 0.437917i $$-0.855716\pi$$
−0.899015 + 0.437917i $$0.855716\pi$$
$$908$$ 0 0
$$909$$ − 9.50227e7i − 0.126513i
$$910$$ 0 0
$$911$$ 1.16107e9i 1.53569i 0.640638 + 0.767843i $$0.278670\pi$$
−0.640638 + 0.767843i $$0.721330\pi$$
$$912$$ 0 0
$$913$$ 1.37052e9 1.80083
$$914$$ 0 0
$$915$$ −2.68545e7 −0.0350553
$$916$$ 0 0
$$917$$ 9.42995e7i 0.122293i
$$918$$ 0 0
$$919$$ 1.03538e9i 1.33399i 0.745061 + 0.666996i $$0.232420\pi$$
−0.745061 + 0.666996i $$0.767580\pi$$
$$920$$ 0 0
$$921$$ −3.26579e8 −0.418032
$$922$$ 0 0
$$923$$ −1.17806e8 −0.149818
$$924$$ 0 0
$$925$$ 6.72307e8i 0.849459i
$$926$$ 0 0
$$927$$ − 3.30675e8i − 0.415109i
$$928$$ 0 0
$$929$$ −1.47698e9 −1.84217 −0.921083 0.389367i $$-0.872694\pi$$
−0.921083 + 0.389367i $$0.872694\pi$$
$$930$$ 0 0
$$931$$ −3.66718e8 −0.454447
$$932$$ 0 0
$$933$$ 7.70227e7i 0.0948361i
$$934$$ 0 0
$$935$$ 3.02918e8i 0.370587i
$$936$$ 0 0
$$937$$ 4.90868e8 0.596686 0.298343 0.954459i $$-0.403566\pi$$
0.298343 + 0.954459i $$0.403566\pi$$
$$938$$ 0 0
$$939$$ −4.16911e8 −0.503555
$$940$$ 0 0
$$941$$ − 1.54536e8i − 0.185464i −0.995691 0.0927321i $$-0.970440\pi$$
0.995691 0.0927321i $$-0.0295600\pi$$
$$942$$ 0 0
$$943$$ − 1.56670e8i − 0.186832i
$$944$$ 0 0
$$945$$ −1.17979e7 −0.0139800
$$946$$ 0 0
$$947$$ 1.24667e9 1.46792 0.733961 0.679192i $$-0.237669\pi$$
0.733961 + 0.679192i $$0.237669\pi$$
$$948$$ 0 0
$$949$$ 1.20833e9i 1.41380i
$$950$$ 0 0
$$951$$ − 7.39671e8i − 0.859997i
$$952$$ 0 0
$$953$$ −4.21385e8 −0.486856 −0.243428 0.969919i $$-0.578272\pi$$
−0.243428 + 0.969919i $$0.578272\pi$$
$$954$$ 0 0
$$955$$ −1.36522e8 −0.156744
$$956$$ 0 0
$$957$$ − 1.60763e9i − 1.83421i
$$958$$ 0 0
$$959$$ − 2.73973e8i − 0.310637i
$$960$$ 0 0
$$961$$ −5.10826e7 −0.0575576
$$962$$ 0 0
$$963$$ −3.12546e8 −0.349973
$$964$$ 0 0
$$965$$ − 2.61683e7i − 0.0291201i
$$966$$ 0 0
$$967$$ − 1.35010e9i − 1.49309i −0.665334 0.746546i $$-0.731711\pi$$
0.665334 0.746546i $$-0.268289\pi$$
$$968$$ 0 0
$$969$$ −4.01927e8 −0.441749
$$970$$ 0 0
$$971$$ 2.75696e8 0.301143 0.150571 0.988599i $$-0.451889\pi$$
0.150571 + 0.988599i $$0.451889\pi$$
$$972$$ 0 0
$$973$$ 4.72912e7i 0.0513383i
$$974$$ 0 0
$$975$$ 7.64497e8i 0.824825i
$$976$$ 0 0
$$977$$ −4.86639e8 −0.521823 −0.260911 0.965363i $$-0.584023\pi$$
−0.260911 + 0.965363i $$0.584023\pi$$
$$978$$ 0 0
$$979$$ 2.07836e9 2.21499
$$980$$ 0 0
$$981$$ − 4.27103e8i − 0.452403i
$$982$$ 0 0
$$983$$ 1.34935e9i 1.42058i 0.703909 + 0.710290i $$0.251436\pi$$
−0.703909 + 0.710290i $$0.748564\pi$$
$$984$$ 0 0
$$985$$ 1.52275e8 0.159338
$$986$$ 0 0
$$987$$ −4.26292e8 −0.443359
$$988$$ 0 0
$$989$$ − 4.39501e8i − 0.454330i
$$990$$ 0 0
$$991$$ − 5.76213e8i − 0.592056i −0.955179 0.296028i $$-0.904338\pi$$
0.955179 0.296028i $$-0.0956621\pi$$
$$992$$ 0 0
$$993$$ −7.54976e8 −0.771055
$$994$$ 0 0
$$995$$ −1.29538e8 −0.131501
$$996$$ 0 0
$$997$$ 8.84682e7i 0.0892692i 0.999003 + 0.0446346i $$0.0142124\pi$$
−0.999003 + 0.0446346i $$0.985788\pi$$
$$998$$ 0 0
$$999$$ − 1.67271e8i − 0.167773i
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 384.7.b.d.319.2 8
4.3 odd 2 inner 384.7.b.d.319.5 yes 8
8.3 odd 2 inner 384.7.b.d.319.3 yes 8
8.5 even 2 inner 384.7.b.d.319.8 yes 8
16.3 odd 4 768.7.g.c.511.4 4
16.5 even 4 768.7.g.e.511.3 4
16.11 odd 4 768.7.g.e.511.2 4
16.13 even 4 768.7.g.c.511.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.d.319.2 8 1.1 even 1 trivial
384.7.b.d.319.3 yes 8 8.3 odd 2 inner
384.7.b.d.319.5 yes 8 4.3 odd 2 inner
384.7.b.d.319.8 yes 8 8.5 even 2 inner
768.7.g.c.511.1 4 16.13 even 4
768.7.g.c.511.4 4 16.3 odd 4
768.7.g.e.511.2 4 16.11 odd 4
768.7.g.e.511.3 4 16.5 even 4