# Properties

 Label 768.7.g.e.511.2 Level $768$ Weight $7$ Character 768.511 Analytic conductor $176.682$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 768.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$176.681536220$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 511.2 Root $$-0.780776 + 1.35234i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.511 Dual form 768.7.g.e.511.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-15.5885i q^{3} +20.0000 q^{5} +155.727i q^{7} -243.000 q^{9} +O(q^{10})$$ $$q-15.5885i q^{3} +20.0000 q^{5} +155.727i q^{7} -243.000 q^{9} -2306.46i q^{11} -3221.18 q^{13} -311.769i q^{15} -6566.73 q^{17} -3926.40i q^{19} +2427.54 q^{21} -17392.0i q^{23} -15225.0 q^{25} +3788.00i q^{27} +44713.3 q^{29} -30636.4i q^{31} -35954.2 q^{33} +3114.54i q^{35} -44158.1 q^{37} +50213.3i q^{39} -9008.18 q^{41} -25270.3i q^{43} -4860.00 q^{45} +175606. i q^{47} +93398.1 q^{49} +102365. i q^{51} +96206.3 q^{53} -46129.2i q^{55} -61206.5 q^{57} +35018.3i q^{59} +86135.7 q^{61} -37841.7i q^{63} -64423.6 q^{65} -424605. i q^{67} -271114. q^{69} -36572.4i q^{71} +375121. q^{73} +237334. i q^{75} +359179. q^{77} +520323. i q^{79} +59049.0 q^{81} +594208. i q^{83} -131335. q^{85} -697011. i q^{87} -901101. q^{89} -501625. i q^{91} -477574. q^{93} -78528.0i q^{95} -1.25744e6 q^{97} +560470. i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 80 q^{5} - 972 q^{9}+O(q^{10})$$ 4 * q + 80 * q^5 - 972 * q^9 $$4 q + 80 q^{5} - 972 q^{9} - 5760 q^{13} + 2232 q^{17} - 11664 q^{21} - 60900 q^{25} + 50608 q^{29} - 58320 q^{33} - 212256 q^{37} + 49464 q^{41} - 19440 q^{45} - 139388 q^{49} - 256400 q^{53} + 11664 q^{57} + 80928 q^{61} - 115200 q^{65} - 443232 q^{69} - 38456 q^{73} + 1180224 q^{77} + 236196 q^{81} + 44640 q^{85} - 3319416 q^{89} - 478224 q^{93} - 2464840 q^{97}+O(q^{100})$$ 4 * q + 80 * q^5 - 972 * q^9 - 5760 * q^13 + 2232 * q^17 - 11664 * q^21 - 60900 * q^25 + 50608 * q^29 - 58320 * q^33 - 212256 * q^37 + 49464 * q^41 - 19440 * q^45 - 139388 * q^49 - 256400 * q^53 + 11664 * q^57 + 80928 * q^61 - 115200 * q^65 - 443232 * q^69 - 38456 * q^73 + 1180224 * q^77 + 236196 * q^81 + 44640 * q^85 - 3319416 * q^89 - 478224 * q^93 - 2464840 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ − 15.5885i − 0.577350i
$$4$$ 0 0
$$5$$ 20.0000 0.160000 0.0800000 0.996795i $$-0.474508\pi$$
0.0800000 + 0.996795i $$0.474508\pi$$
$$6$$ 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
$$11$$ − 2306.46i − 1.73288i −0.499282 0.866439i $$-0.666403\pi$$
0.499282 0.866439i $$-0.333597\pi$$
$$12$$ 0 0
$$13$$ −3221.18 −1.46617 −0.733086 0.680136i $$-0.761921\pi$$
−0.733086 + 0.680136i $$0.761921\pi$$
$$14$$ 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
$$19$$ − 3926.40i − 0.572445i −0.958163 0.286223i $$-0.907600\pi$$
0.958163 0.286223i $$-0.0923997\pi$$
$$20$$ 0 0
$$21$$ 2427.54 0.262126
$$22$$ 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
$$27$$ 3788.00i 0.192450i
$$28$$ 0 0
$$29$$ 44713.3 1.83334 0.916669 0.399648i $$-0.130868\pi$$
0.916669 + 0.399648i $$0.130868\pi$$
$$30$$ 0 0
$$31$$ − 30636.4i − 1.02838i −0.857677 0.514188i $$-0.828093\pi$$
0.857677 0.514188i $$-0.171907\pi$$
$$32$$ 0 0
$$33$$ −35954.2 −1.00048
$$34$$ 0 0
$$35$$ 3114.54i 0.0726424i
$$36$$ 0 0
$$37$$ −44158.1 −0.871776 −0.435888 0.900001i $$-0.643566\pi$$
−0.435888 + 0.900001i $$0.643566\pi$$
$$38$$ 0 0
$$39$$ 50213.3i 0.846495i
$$40$$ 0 0
$$41$$ −9008.18 −0.130703 −0.0653515 0.997862i $$-0.520817\pi$$
−0.0653515 + 0.997862i $$0.520817\pi$$
$$42$$ 0 0
$$43$$ − 25270.3i − 0.317838i −0.987292 0.158919i $$-0.949199\pi$$
0.987292 0.158919i $$-0.0508008\pi$$
$$44$$ 0 0
$$45$$ −4860.00 −0.0533333
$$46$$ 0 0
$$47$$ 175606.i 1.69140i 0.533658 + 0.845700i $$0.320817\pi$$
−0.533658 + 0.845700i $$0.679183\pi$$
$$48$$ 0 0
$$49$$ 93398.1 0.793871
$$50$$ 0 0
$$51$$ 102365.i 0.771688i
$$52$$ 0 0
$$53$$ 96206.3 0.646214 0.323107 0.946363i $$-0.395273\pi$$
0.323107 + 0.946363i $$0.395273\pi$$
$$54$$ 0 0
$$55$$ − 46129.2i − 0.277261i
$$56$$ 0 0
$$57$$ −61206.5 −0.330501
$$58$$ 0 0
$$59$$ 35018.3i 0.170506i 0.996359 + 0.0852529i $$0.0271698\pi$$
−0.996359 + 0.0852529i $$0.972830\pi$$
$$60$$ 0 0
$$61$$ 86135.7 0.379484 0.189742 0.981834i $$-0.439235\pi$$
0.189742 + 0.981834i $$0.439235\pi$$
$$62$$ 0 0
$$63$$ − 37841.7i − 0.151338i
$$64$$ 0 0
$$65$$ −64423.6 −0.234588
$$66$$ 0 0
$$67$$ − 424605.i − 1.41176i −0.708332 0.705880i $$-0.750552\pi$$
0.708332 0.705880i $$-0.249448\pi$$
$$68$$ 0 0
$$69$$ −271114. −0.825287
$$70$$ 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 $$-0.339860\pi$$
0.482140 + 0.876094i $$0.339860\pi$$
$$74$$ 0 0
$$75$$ 237334.i 0.562570i
$$76$$ 0 0
$$77$$ 359179. 0.786753
$$78$$ 0 0
$$79$$ 520323.i 1.05534i 0.849450 + 0.527670i $$0.176934\pi$$
−0.849450 + 0.527670i $$0.823066\pi$$
$$80$$ 0 0
$$81$$ 59049.0 0.111111
$$82$$ 0 0
$$83$$ 594208.i 1.03921i 0.854406 + 0.519606i $$0.173921\pi$$
−0.854406 + 0.519606i $$0.826079\pi$$
$$84$$ 0 0
$$85$$ −131335. −0.213856
$$86$$ 0 0
$$87$$ − 697011.i − 1.05848i
$$88$$ 0 0
$$89$$ −901101. −1.27821 −0.639107 0.769118i $$-0.720696\pi$$
−0.639107 + 0.769118i $$0.720696\pi$$
$$90$$ 0 0
$$91$$ − 501625.i − 0.665664i
$$92$$ 0 0
$$93$$ −477574. −0.593733
$$94$$ 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.i 0.577626i
$$100$$ 0 0
$$101$$ 391040. 0.379539 0.189770 0.981829i $$-0.439226\pi$$
0.189770 + 0.981829i $$0.439226\pi$$
$$102$$ 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
$$107$$ 1.28620e6i 1.04992i 0.851127 + 0.524960i $$0.175920\pi$$
−0.851127 + 0.524960i $$0.824080\pi$$
$$108$$ 0 0
$$109$$ −1.75763e6 −1.35721 −0.678605 0.734504i $$-0.737415\pi$$
−0.678605 + 0.734504i $$0.737415\pi$$
$$110$$ 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.i − 0.228710i
$$116$$ 0 0
$$117$$ 782747. 0.488724
$$118$$ 0 0
$$119$$ − 1.02262e6i − 0.606837i
$$120$$ 0 0
$$121$$ −3.54820e6 −2.00287
$$122$$ 0 0
$$123$$ 140424.i 0.0754614i
$$124$$ 0 0
$$125$$ −617000. −0.315904
$$126$$ 0 0
$$127$$ 2.59678e6i 1.26772i 0.773448 + 0.633860i $$0.218531\pi$$
−0.773448 + 0.633860i $$0.781469\pi$$
$$128$$ 0 0
$$129$$ −393925. −0.183504
$$130$$ 0 0
$$131$$ 605543.i 0.269359i 0.990889 + 0.134679i $$0.0430005\pi$$
−0.990889 + 0.134679i $$0.957000\pi$$
$$132$$ 0 0
$$133$$ 611447. 0.259899
$$134$$ 0 0
$$135$$ 75759.9i 0.0307920i
$$136$$ 0 0
$$137$$ 1.75932e6 0.684199 0.342100 0.939664i $$-0.388862\pi$$
0.342100 + 0.939664i $$0.388862\pi$$
$$138$$ 0 0
$$139$$ − 303680.i − 0.113076i −0.998400 0.0565382i $$-0.981994\pi$$
0.998400 0.0565382i $$-0.0180063\pi$$
$$140$$ 0 0
$$141$$ 2.73743e6 0.976531
$$142$$ 0 0
$$143$$ 7.42953e6i 2.54070i
$$144$$ 0 0
$$145$$ 894265. 0.293334
$$146$$ 0 0
$$147$$ − 1.45593e6i − 0.458341i
$$148$$ 0 0
$$149$$ −1.32218e6 −0.399696 −0.199848 0.979827i $$-0.564045\pi$$
−0.199848 + 0.979827i $$0.564045\pi$$
$$150$$ 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.i − 0.164540i
$$156$$ 0 0
$$157$$ 2.44769e6 0.632495 0.316247 0.948677i $$-0.397577\pi$$
0.316247 + 0.948677i $$0.397577\pi$$
$$158$$ 0 0
$$159$$ − 1.49971e6i − 0.373092i
$$160$$ 0 0
$$161$$ 2.70840e6 0.648987
$$162$$ 0 0
$$163$$ 6.62033e6i 1.52868i 0.644813 + 0.764340i $$0.276935\pi$$
−0.644813 + 0.764340i $$0.723065\pi$$
$$164$$ 0 0
$$165$$ −719084. −0.160076
$$166$$ 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.i 0.190815i
$$172$$ 0 0
$$173$$ −3.72601e6 −0.719623 −0.359812 0.933025i $$-0.617159\pi$$
−0.359812 + 0.933025i $$0.617159\pi$$
$$174$$ 0 0
$$175$$ − 2.37094e6i − 0.442392i
$$176$$ 0 0
$$177$$ 545881. 0.0984415
$$178$$ 0 0
$$179$$ − 201811.i − 0.0351873i −0.999845 0.0175937i $$-0.994399\pi$$
0.999845 0.0175937i $$-0.00560053\pi$$
$$180$$ 0 0
$$181$$ 5.43440e6 0.916466 0.458233 0.888832i $$-0.348483\pi$$
0.458233 + 0.888832i $$0.348483\pi$$
$$182$$ 0 0
$$183$$ − 1.34272e6i − 0.219095i
$$184$$ 0 0
$$185$$ −883162. −0.139484
$$186$$ 0 0
$$187$$ 1.51459e7i 2.31617i
$$188$$ 0 0
$$189$$ −589893. −0.0873752
$$190$$ 0 0
$$191$$ 6.82608e6i 0.979650i 0.871821 + 0.489825i $$0.162939\pi$$
−0.871821 + 0.489825i $$0.837061\pi$$
$$192$$ 0 0
$$193$$ 1.30841e6 0.182001 0.0910004 0.995851i $$-0.470994\pi$$
0.0910004 + 0.995851i $$0.470994\pi$$
$$194$$ 0 0
$$195$$ 1.00427e6i 0.135439i
$$196$$ 0 0
$$197$$ −7.61375e6 −0.995863 −0.497932 0.867216i $$-0.665907\pi$$
−0.497932 + 0.867216i $$0.665907\pi$$
$$198$$ 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
$$203$$ 6.96307e6i 0.832362i
$$204$$ 0 0
$$205$$ −180164. −0.0209125
$$206$$ 0 0
$$207$$ 4.22625e6i 0.476480i
$$208$$ 0 0
$$209$$ −9.05609e6 −0.991978
$$210$$ 0 0
$$211$$ 1.58058e7i 1.68256i 0.540603 + 0.841278i $$0.318196\pi$$
−0.540603 + 0.841278i $$0.681804\pi$$
$$212$$ 0 0
$$213$$ −570108. −0.0589954
$$214$$ 0 0
$$215$$ − 505406.i − 0.0508540i
$$216$$ 0 0
$$217$$ 4.77091e6 0.466898
$$218$$ 0 0
$$219$$ − 5.84756e6i − 0.556727i
$$220$$ 0 0
$$221$$ 2.11526e7 1.95969
$$222$$ 0 0
$$223$$ 2.07099e7i 1.86751i 0.357907 + 0.933757i $$0.383491\pi$$
−0.357907 + 0.933757i $$0.616509\pi$$
$$224$$ 0 0
$$225$$ 3.69967e6 0.324800
$$226$$ 0 0
$$227$$ 7.06179e6i 0.603722i 0.953352 + 0.301861i $$0.0976079\pi$$
−0.953352 + 0.301861i $$0.902392\pi$$
$$228$$ 0 0
$$229$$ −4.64766e6 −0.387015 −0.193507 0.981099i $$-0.561986\pi$$
−0.193507 + 0.981099i $$0.561986\pi$$
$$230$$ 0 0
$$231$$ − 5.59904e6i − 0.454232i
$$232$$ 0 0
$$233$$ −1.13777e7 −0.899469 −0.449735 0.893162i $$-0.648481\pi$$
−0.449735 + 0.893162i $$0.648481\pi$$
$$234$$ 0 0
$$235$$ 3.51213e6i 0.270624i
$$236$$ 0 0
$$237$$ 8.11104e6 0.609300
$$238$$ 0 0
$$239$$ − 1.93765e7i − 1.41932i −0.704542 0.709662i $$-0.748848\pi$$
0.704542 0.709662i $$-0.251152\pi$$
$$240$$ 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.i − 0.0641500i
$$244$$ 0 0
$$245$$ 1.86796e6 0.127019
$$246$$ 0 0
$$247$$ 1.26477e7i 0.839304i
$$248$$ 0 0
$$249$$ 9.26279e6 0.599989
$$250$$ 0 0
$$251$$ 129527.i 0.00819104i 0.999992 + 0.00409552i $$0.00130365\pi$$
−0.999992 + 0.00409552i $$0.998696\pi$$
$$252$$ 0 0
$$253$$ −4.01140e7 −2.47705
$$254$$ 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
$$259$$ − 6.87661e6i − 0.395799i
$$260$$ 0 0
$$261$$ −1.08653e7 −0.611113
$$262$$ 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
$$267$$ 1.40468e7i 0.737977i
$$268$$ 0 0
$$269$$ −1.71231e7 −0.879680 −0.439840 0.898076i $$-0.644965\pi$$
−0.439840 + 0.898076i $$0.644965\pi$$
$$270$$ 0 0
$$271$$ 320201.i 0.0160885i 0.999968 + 0.00804423i $$0.00256058\pi$$
−0.999968 + 0.00804423i $$0.997439\pi$$
$$272$$ 0 0
$$273$$ −7.81956e6 −0.384321
$$274$$ 0 0
$$275$$ 3.51159e7i 1.68852i
$$276$$ 0 0
$$277$$ 1.50949e7 0.710216 0.355108 0.934825i $$-0.384444\pi$$
0.355108 + 0.934825i $$0.384444\pi$$
$$278$$ 0 0
$$279$$ 7.44463e6i 0.342792i
$$280$$ 0 0
$$281$$ 4.32580e7 1.94961 0.974804 0.223063i $$-0.0716056\pi$$
0.974804 + 0.223063i $$0.0716056\pi$$
$$282$$ 0 0
$$283$$ 3.22002e7i 1.42069i 0.703854 + 0.710345i $$0.251461\pi$$
−0.703854 + 0.710345i $$0.748539\pi$$
$$284$$ 0 0
$$285$$ −1.22413e6 −0.0528802
$$286$$ 0 0
$$287$$ − 1.40282e6i − 0.0593411i
$$288$$ 0 0
$$289$$ 1.89843e7 0.786505
$$290$$ 0 0
$$291$$ 1.96015e7i 0.795444i
$$292$$ 0 0
$$293$$ 3.01307e7 1.19786 0.598931 0.800801i $$-0.295592\pi$$
0.598931 + 0.800801i $$0.295592\pi$$
$$294$$ 0 0
$$295$$ 700366.i 0.0272809i
$$296$$ 0 0
$$297$$ 8.73687e6 0.333493
$$298$$ 0 0
$$299$$ 5.60228e7i 2.09581i
$$300$$ 0 0
$$301$$ 3.93527e6 0.144303
$$302$$ 0 0
$$303$$ − 6.09571e6i − 0.219127i
$$304$$ 0 0
$$305$$ 1.72271e6 0.0607175
$$306$$ 0 0
$$307$$ 2.09501e7i 0.724053i 0.932168 + 0.362026i $$0.117915\pi$$
−0.932168 + 0.362026i $$0.882085\pi$$
$$308$$ 0 0
$$309$$ −2.12128e7 −0.718990
$$310$$ 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 $$-0.643637\pi$$
−0.436091 + 0.899903i $$0.643637\pi$$
$$314$$ 0 0
$$315$$ − 756834.i − 0.0242141i
$$316$$ 0 0
$$317$$ 4.74499e7 1.48956 0.744779 0.667311i $$-0.232555\pi$$
0.744779 + 0.667311i $$0.232555\pi$$
$$318$$ 0 0
$$319$$ − 1.03129e8i − 3.17695i
$$320$$ 0 0
$$321$$ 2.00498e7 0.606172
$$322$$ 0 0
$$323$$ 2.57836e7i 0.765131i
$$324$$ 0 0
$$325$$ 4.90425e7 1.42864
$$326$$ 0 0
$$327$$ 2.73987e7i 0.783585i
$$328$$ 0 0
$$329$$ −2.73467e7 −0.767921
$$330$$ 0 0
$$331$$ − 4.84318e7i − 1.33551i −0.744382 0.667754i $$-0.767256\pi$$
0.744382 0.667754i $$-0.232744\pi$$
$$332$$ 0 0
$$333$$ 1.07304e7 0.290592
$$334$$ 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
$$339$$ 1.13913e7i 0.292397i
$$340$$ 0 0
$$341$$ −7.06616e7 −1.78205
$$342$$ 0 0
$$343$$ 3.28657e7i 0.814444i
$$344$$ 0 0
$$345$$ −5.42229e6 −0.132046
$$346$$ 0 0
$$347$$ 2.95157e7i 0.706423i 0.935544 + 0.353211i $$0.114910\pi$$
−0.935544 + 0.353211i $$0.885090\pi$$
$$348$$ 0 0
$$349$$ −5.55247e7 −1.30620 −0.653100 0.757272i $$-0.726532\pi$$
−0.653100 + 0.757272i $$0.726532\pi$$
$$350$$ 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.i − 0.0163493i
$$356$$ 0 0
$$357$$ −1.59410e7 −0.350358
$$358$$ 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
$$363$$ 5.53110e7i 1.15636i
$$364$$ 0 0
$$365$$ 7.50242e6 0.154285
$$366$$ 0 0
$$367$$ 4.83247e7i 0.977623i 0.872389 + 0.488811i $$0.162569\pi$$
−0.872389 + 0.488811i $$0.837431\pi$$
$$368$$ 0 0
$$369$$ 2.18899e6 0.0435677
$$370$$ 0 0
$$371$$ 1.49819e7i 0.293391i
$$372$$ 0 0
$$373$$ 8.98844e7 1.73204 0.866020 0.500010i $$-0.166670\pi$$
0.866020 + 0.500010i $$0.166670\pi$$
$$374$$ 0 0
$$375$$ 9.61808e6i 0.182387i
$$376$$ 0 0
$$377$$ −1.44030e8 −2.68799
$$378$$ 0 0
$$379$$ 8.60279e7i 1.58024i 0.612955 + 0.790118i $$0.289981\pi$$
−0.612955 + 0.790118i $$0.710019\pi$$
$$380$$ 0 0
$$381$$ 4.04798e7 0.731919
$$382$$ 0 0
$$383$$ − 6.65042e7i − 1.18373i −0.806037 0.591865i $$-0.798392\pi$$
0.806037 0.591865i $$-0.201608\pi$$
$$384$$ 0 0
$$385$$ 7.18357e6 0.125880
$$386$$ 0 0
$$387$$ 6.14069e6i 0.105946i
$$388$$ 0 0
$$389$$ −5.66889e7 −0.963050 −0.481525 0.876432i $$-0.659917\pi$$
−0.481525 + 0.876432i $$0.659917\pi$$
$$390$$ 0 0
$$391$$ 1.14208e8i 1.91059i
$$392$$ 0 0
$$393$$ 9.43949e6 0.155514
$$394$$ 0 0
$$395$$ 1.04065e7i 0.168854i
$$396$$ 0 0
$$397$$ −3.21267e7 −0.513446 −0.256723 0.966485i $$-0.582643\pi$$
−0.256723 + 0.966485i $$0.582643\pi$$
$$398$$ 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
$$403$$ 9.86853e7i 1.50778i
$$404$$ 0 0
$$405$$ 1.18098e6 0.0177778
$$406$$ 0 0
$$407$$ 1.01849e8i 1.51068i
$$408$$ 0 0
$$409$$ 2.07185e7 0.302823 0.151412 0.988471i $$-0.451618\pi$$
0.151412 + 0.988471i $$0.451618\pi$$
$$410$$ 0 0
$$411$$ − 2.74251e7i − 0.395023i
$$412$$ 0 0
$$413$$ −5.45330e6 −0.0774121
$$414$$ 0 0
$$415$$ 1.18842e7i 0.166274i
$$416$$ 0 0
$$417$$ −4.73390e6 −0.0652846
$$418$$ 0 0
$$419$$ 6.79561e7i 0.923818i 0.886927 + 0.461909i $$0.152835\pi$$
−0.886927 + 0.461909i $$0.847165\pi$$
$$420$$ 0 0
$$421$$ −522580. −0.00700336 −0.00350168 0.999994i $$-0.501115\pi$$
−0.00350168 + 0.999994i $$0.501115\pi$$
$$422$$ 0 0
$$423$$ − 4.26723e7i − 0.563800i
$$424$$ 0 0
$$425$$ 9.99784e7 1.30239
$$426$$ 0 0
$$427$$ 1.34137e7i 0.172291i
$$428$$ 0 0
$$429$$ 1.15815e8 1.46687
$$430$$ 0 0
$$431$$ − 1.10533e8i − 1.38057i −0.723538 0.690285i $$-0.757485\pi$$
0.723538 0.690285i $$-0.242515\pi$$
$$432$$ 0 0
$$433$$ 6.65239e7 0.819435 0.409717 0.912213i $$-0.365627\pi$$
0.409717 + 0.912213i $$0.365627\pi$$
$$434$$ 0 0
$$435$$ − 1.39402e7i − 0.169356i
$$436$$ 0 0
$$437$$ −6.82879e7 −0.818276
$$438$$ 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
$$443$$ − 1.01597e7i − 0.116861i −0.998291 0.0584304i $$-0.981390\pi$$
0.998291 0.0584304i $$-0.0186096\pi$$
$$444$$ 0 0
$$445$$ −1.80220e7 −0.204514
$$446$$ 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
$$451$$ 2.07770e7i 0.226492i
$$452$$ 0 0
$$453$$ −7.28789e7 −0.783984
$$454$$ 0 0
$$455$$ − 1.00325e7i − 0.106506i
$$456$$ 0 0
$$457$$ 5.92987e6 0.0621293 0.0310646 0.999517i $$-0.490110\pi$$
0.0310646 + 0.999517i $$0.490110\pi$$
$$458$$ 0 0
$$459$$ − 2.48747e7i − 0.257229i
$$460$$ 0 0
$$461$$ 4.31145e7 0.440069 0.220035 0.975492i $$-0.429383\pi$$
0.220035 + 0.975492i $$0.429383\pi$$
$$462$$ 0 0
$$463$$ − 8.09555e7i − 0.815649i −0.913060 0.407825i $$-0.866288\pi$$
0.913060 0.407825i $$-0.133712\pi$$
$$464$$ 0 0
$$465$$ −9.55147e6 −0.0949973
$$466$$ 0 0
$$467$$ − 1.20309e8i − 1.18127i −0.806940 0.590633i $$-0.798878\pi$$
0.806940 0.590633i $$-0.201122\pi$$
$$468$$ 0 0
$$469$$ 6.61225e7 0.640960
$$470$$ 0 0
$$471$$ − 3.81557e7i − 0.365171i
$$472$$ 0 0
$$473$$ −5.82850e7 −0.550774
$$474$$ 0 0
$$475$$ 5.97795e7i 0.557791i
$$476$$ 0 0
$$477$$ −2.33781e7 −0.215405
$$478$$ 0 0
$$479$$ 1.54238e8i 1.40341i 0.712468 + 0.701705i $$0.247578\pi$$
−0.712468 + 0.701705i $$0.752422\pi$$
$$480$$ 0 0
$$481$$ 1.42241e8 1.27817
$$482$$ 0 0
$$483$$ − 4.22198e7i − 0.374693i
$$484$$ 0 0
$$485$$ −2.51487e7 −0.220440
$$486$$ 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
$$491$$ 4.51212e7i 0.381185i 0.981669 + 0.190593i $$0.0610410\pi$$
−0.981669 + 0.190593i $$0.938959\pi$$
$$492$$ 0 0
$$493$$ −2.93620e8 −2.45044
$$494$$ 0 0
$$495$$ 1.12094e7i 0.0924202i
$$496$$ 0 0
$$497$$ 5.69532e6 0.0463926
$$498$$ 0 0
$$499$$ 5.03581e7i 0.405292i 0.979252 + 0.202646i $$0.0649540\pi$$
−0.979252 + 0.202646i $$0.935046\pi$$
$$500$$ 0 0
$$501$$ −1.13412e8 −0.901872
$$502$$ 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
$$507$$ − 8.65035e7i − 0.663758i
$$508$$ 0 0
$$509$$ 2.44005e7 0.185031 0.0925156 0.995711i $$-0.470509\pi$$
0.0925156 + 0.995711i $$0.470509\pi$$
$$510$$ 0 0
$$511$$ 5.84165e7i 0.437797i
$$512$$ 0 0
$$513$$ 1.48732e7 0.110167
$$514$$ 0 0
$$515$$ − 2.72160e7i − 0.199252i
$$516$$ 0 0
$$517$$ 4.05029e8 2.93099
$$518$$ 0 0
$$519$$ 5.80827e7i 0.415475i
$$520$$ 0 0
$$521$$ −6.53917e7 −0.462391 −0.231195 0.972907i $$-0.574264\pi$$
−0.231195 + 0.972907i $$0.574264\pi$$
$$522$$ 0 0
$$523$$ 1.19428e8i 0.834833i 0.908715 + 0.417417i $$0.137064\pi$$
−0.908715 + 0.417417i $$0.862936\pi$$
$$524$$ 0 0
$$525$$ −3.69594e7 −0.255415
$$526$$ 0 0
$$527$$ 2.01181e8i 1.37453i
$$528$$ 0 0
$$529$$ −1.54446e8 −1.04330
$$530$$ 0 0
$$531$$ − 8.50945e6i − 0.0568352i
$$532$$ 0 0
$$533$$ 2.90170e7 0.191633
$$534$$ 0 0
$$535$$ 2.57240e7i 0.167987i
$$536$$ 0 0
$$537$$ −3.14593e6 −0.0203154
$$538$$ 0 0
$$539$$ − 2.15419e8i − 1.37568i
$$540$$ 0 0
$$541$$ −3.06540e7 −0.193596 −0.0967979 0.995304i $$-0.530860\pi$$
−0.0967979 + 0.995304i $$0.530860\pi$$
$$542$$ 0 0
$$543$$ − 8.47140e7i − 0.529122i
$$544$$ 0 0
$$545$$ −3.51525e7 −0.217153
$$546$$ 0 0
$$547$$ − 1.60733e8i − 0.982071i −0.871140 0.491035i $$-0.836619\pi$$
0.871140 0.491035i $$-0.163381\pi$$
$$548$$ 0 0
$$549$$ −2.09310e7 −0.126495
$$550$$ 0 0
$$551$$ − 1.75562e8i − 1.04949i
$$552$$ 0 0
$$553$$ −8.10284e7 −0.479140
$$554$$ 0 0
$$555$$ 1.37671e7i 0.0805313i
$$556$$ 0 0
$$557$$ 3.11683e8 1.80363 0.901814 0.432125i $$-0.142236\pi$$
0.901814 + 0.432125i $$0.142236\pi$$
$$558$$ 0 0
$$559$$ 8.14003e7i 0.466005i
$$560$$ 0 0
$$561$$ 2.36101e8 1.33724
$$562$$ 0 0
$$563$$ 2.81653e8i 1.57830i 0.614201 + 0.789149i $$0.289478\pi$$
−0.614201 + 0.789149i $$0.710522\pi$$
$$564$$ 0 0
$$565$$ −1.46150e7 −0.0810313
$$566$$ 0 0
$$567$$ 9.19553e6i 0.0504461i
$$568$$ 0 0
$$569$$ −8.82677e7 −0.479143 −0.239571 0.970879i $$-0.577007\pi$$
−0.239571 + 0.970879i $$0.577007\pi$$
$$570$$ 0 0
$$571$$ − 1.66103e8i − 0.892217i −0.894979 0.446108i $$-0.852810\pi$$
0.894979 0.446108i $$-0.147190\pi$$
$$572$$ 0 0
$$573$$ 1.06408e8 0.565601
$$574$$ 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
$$579$$ − 2.03962e7i − 0.105078i
$$580$$ 0 0
$$581$$ −9.25343e7 −0.471818
$$582$$ 0 0
$$583$$ − 2.21896e8i − 1.11981i
$$584$$ 0 0
$$585$$ 1.56549e7 0.0781959
$$586$$ 0 0
$$587$$ 3.88069e7i 0.191865i 0.995388 + 0.0959323i $$0.0305832\pi$$
−0.995388 + 0.0959323i $$0.969417\pi$$
$$588$$ 0 0
$$589$$ −1.20291e8 −0.588689
$$590$$ 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
$$595$$ − 2.04523e7i − 0.0970939i
$$596$$ 0 0
$$597$$ −1.00965e8 −0.474514
$$598$$ 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 $$-0.129148\pi$$
0.918815 + 0.394689i $$0.129148\pi$$
$$602$$ 0 0
$$603$$ 1.03179e8i 0.470586i
$$604$$ 0 0
$$605$$ −7.09641e7 −0.320459
$$606$$ 0 0
$$607$$ 2.46327e8i 1.10140i 0.834703 + 0.550701i $$0.185639\pi$$
−0.834703 + 0.550701i $$0.814361\pi$$
$$608$$ 0 0
$$609$$ 1.08543e8 0.480565
$$610$$ 0 0
$$611$$ − 5.65660e8i − 2.47989i
$$612$$ 0 0
$$613$$ −2.16670e8 −0.940628 −0.470314 0.882499i $$-0.655859\pi$$
−0.470314 + 0.882499i $$0.655859\pi$$
$$614$$ 0 0
$$615$$ 2.80847e6i 0.0120738i
$$616$$ 0 0
$$617$$ −1.27942e8 −0.544699 −0.272350 0.962198i $$-0.587801\pi$$
−0.272350 + 0.962198i $$0.587801\pi$$
$$618$$ 0 0
$$619$$ − 1.96738e8i − 0.829499i −0.909936 0.414749i $$-0.863869\pi$$
0.909936 0.414749i $$-0.136131\pi$$
$$620$$ 0 0
$$621$$ 6.58808e7 0.275096
$$622$$ 0 0
$$623$$ − 1.40326e8i − 0.580328i
$$624$$ 0 0
$$625$$ 2.25551e8 0.923855
$$626$$ 0 0
$$627$$ 1.41171e8i 0.572719i
$$628$$ 0 0
$$629$$ 2.89974e8 1.16522
$$630$$ 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
$$635$$ 5.19355e7i 0.202835i
$$636$$ 0 0
$$637$$ −3.00852e8 −1.16395
$$638$$ 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
$$643$$ − 2.98146e8i − 1.12149i −0.827988 0.560745i $$-0.810515\pi$$
0.827988 0.560745i $$-0.189485\pi$$
$$644$$ 0 0
$$645$$ −7.87850e6 −0.0293606
$$646$$ 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
$$651$$ − 7.43711e7i − 0.269564i
$$652$$ 0 0
$$653$$ 9.67190e7 0.347354 0.173677 0.984803i $$-0.444435\pi$$
0.173677 + 0.984803i $$0.444435\pi$$
$$654$$ 0 0
$$655$$ 1.21109e7i 0.0430974i
$$656$$ 0 0
$$657$$ −9.11545e7 −0.321427
$$658$$ 0 0
$$659$$ − 6.49525e7i − 0.226955i −0.993541 0.113477i $$-0.963801\pi$$
0.993541 0.113477i $$-0.0361990\pi$$
$$660$$ 0 0
$$661$$ 3.29739e8 1.14174 0.570868 0.821042i $$-0.306607\pi$$
0.570868 + 0.821042i $$0.306607\pi$$
$$662$$ 0 0
$$663$$ − 3.29737e8i − 1.13143i
$$664$$ 0 0
$$665$$ 1.22289e7 0.0415838
$$666$$ 0 0
$$667$$ − 7.77653e8i − 2.62065i
$$668$$ 0 0
$$669$$ 3.22836e8 1.07821
$$670$$ 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
$$675$$ − 5.76722e7i − 0.187523i
$$676$$ 0 0
$$677$$ −3.25473e8 −1.04894 −0.524469 0.851430i $$-0.675736\pi$$
−0.524469 + 0.851430i $$0.675736\pi$$
$$678$$ 0 0
$$679$$ − 1.95817e8i − 0.625519i
$$680$$ 0 0
$$681$$ 1.10082e8 0.348559
$$682$$ 0 0
$$683$$ − 5.37538e8i − 1.68713i −0.537031 0.843563i $$-0.680454\pi$$
0.537031 0.843563i $$-0.319546\pi$$
$$684$$ 0 0
$$685$$ 3.51864e7 0.109472
$$686$$ 0 0
$$687$$ 7.24498e7i 0.223443i
$$688$$ 0 0
$$689$$ −3.09898e8 −0.947461
$$690$$ 0 0
$$691$$ − 4.64528e8i − 1.40792i −0.710239 0.703960i $$-0.751413\pi$$
0.710239 0.703960i $$-0.248587\pi$$
$$692$$ 0 0
$$693$$ −8.72804e7 −0.262251
$$694$$ 0 0
$$695$$ − 6.07360e6i − 0.0180922i
$$696$$ 0 0
$$697$$ 5.91543e7 0.174698
$$698$$ 0 0
$$699$$ 1.77361e8i 0.519309i
$$700$$ 0 0
$$701$$ 3.59130e7 0.104255 0.0521275 0.998640i $$-0.483400\pi$$
0.0521275 + 0.998640i $$0.483400\pi$$
$$702$$ 0 0
$$703$$ 1.73382e8i 0.499044i
$$704$$ 0 0
$$705$$ 5.47486e7 0.156245
$$706$$ 0 0
$$707$$ 6.08955e7i 0.172317i
$$708$$ 0 0
$$709$$ −1.86788e8 −0.524094 −0.262047 0.965055i $$-0.584398\pi$$
−0.262047 + 0.965055i $$0.584398\pi$$
$$710$$ 0 0
$$711$$ − 1.26439e8i − 0.351780i
$$712$$ 0 0
$$713$$ −5.32827e8 −1.47000
$$714$$ 0 0
$$715$$ 1.48591e8i 0.406512i
$$716$$ 0 0
$$717$$ −3.02050e8 −0.819447
$$718$$ 0 0
$$719$$ 3.76368e7i 0.101257i 0.998718 + 0.0506286i $$0.0161225\pi$$
−0.998718 + 0.0506286i $$0.983878\pi$$
$$720$$ 0 0
$$721$$ 2.11914e8 0.565397
$$722$$ 0 0
$$723$$ − 1.06763e8i − 0.282491i
$$724$$ 0 0
$$725$$ −6.80760e8 −1.78640
$$726$$ 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
$$731$$ 1.65943e8i 0.424822i
$$732$$ 0 0
$$733$$ 5.31872e8 1.35050 0.675251 0.737588i $$-0.264035\pi$$
0.675251 + 0.737588i $$0.264035\pi$$
$$734$$ 0 0
$$735$$ − 2.91186e7i − 0.0733346i
$$736$$ 0 0
$$737$$ −9.79335e8 −2.44641
$$738$$ 0 0
$$739$$ − 1.70943e8i − 0.423564i −0.977317 0.211782i $$-0.932073\pi$$
0.977317 0.211782i $$-0.0679267\pi$$
$$740$$ 0 0
$$741$$ 1.97157e8 0.484572
$$742$$ 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
$$747$$ − 1.44393e8i − 0.346404i
$$748$$ 0 0
$$749$$ −2.00296e8 −0.476679
$$750$$ 0 0
$$751$$ 3.04170e8i 0.718118i 0.933315 + 0.359059i $$0.116902\pi$$
−0.933315 + 0.359059i $$0.883098\pi$$
$$752$$ 0 0
$$753$$ 2.01912e6 0.00472910
$$754$$ 0 0
$$755$$ − 9.35037e7i − 0.217264i
$$756$$ 0 0
$$757$$ 3.06435e8 0.706401 0.353201 0.935548i $$-0.385093\pi$$
0.353201 + 0.935548i $$0.385093\pi$$
$$758$$ 0 0
$$759$$ 6.25315e8i 1.43012i
$$760$$ 0 0
$$761$$ −4.23923e8 −0.961907 −0.480954 0.876746i $$-0.659709\pi$$
−0.480954 + 0.876746i $$0.659709\pi$$
$$762$$ 0 0
$$763$$ − 2.73710e8i − 0.616193i
$$764$$ 0 0
$$765$$ 3.19143e7 0.0712854
$$766$$ 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
$$771$$ − 1.85338e8i − 0.404391i
$$772$$ 0 0
$$773$$ 1.89126e7 0.0409462 0.0204731 0.999790i $$-0.493483\pi$$
0.0204731 + 0.999790i $$0.493483\pi$$
$$774$$ 0 0
$$775$$ 4.66439e8i 1.00205i
$$776$$ 0 0
$$777$$ −1.07196e8 −0.228515
$$778$$ 0 0
$$779$$ 3.53697e7i 0.0748203i
$$780$$ 0 0
$$781$$ −8.43529e7 −0.177071
$$782$$ 0 0
$$783$$ 1.69374e8i 0.352826i
$$784$$ 0 0
$$785$$ 4.89537e7 0.101199
$$786$$ 0 0
$$787$$ 2.65036e8i 0.543726i 0.962336 + 0.271863i $$0.0876398\pi$$
−0.962336 + 0.271863i $$0.912360\pi$$
$$788$$ 0 0
$$789$$ 2.11495e8 0.430594
$$790$$ 0 0
$$791$$ − 1.13797e8i − 0.229934i
$$792$$ 0 0
$$793$$ −2.77459e8 −0.556390
$$794$$ 0 0
$$795$$ − 2.99942e7i − 0.0596947i
$$796$$ 0 0
$$797$$ −6.82061e8 −1.34725 −0.673626 0.739073i $$-0.735264\pi$$
−0.673626 + 0.739073i $$0.735264\pi$$
$$798$$ 0 0
$$799$$ − 1.15316e9i − 2.26073i
$$800$$ 0 0
$$801$$ 2.18968e8 0.426071
$$802$$ 0 0
$$803$$ − 8.65203e8i − 1.67098i
$$804$$ 0 0
$$805$$ 5.41681e7 0.103838
$$806$$ 0 0
$$807$$ 2.66922e8i 0.507884i
$$808$$ 0 0
$$809$$ −5.50511e8 −1.03973 −0.519865 0.854248i $$-0.674018\pi$$
−0.519865 + 0.854248i $$0.674018\pi$$
$$810$$ 0 0
$$811$$ 3.11549e8i 0.584068i 0.956408 + 0.292034i $$0.0943321\pi$$
−0.956408 + 0.292034i $$0.905668\pi$$
$$812$$ 0 0
$$813$$ 4.99143e6 0.00928867
$$814$$ 0 0
$$815$$ 1.32407e8i 0.244589i
$$816$$ 0 0
$$817$$ −9.92214e7 −0.181945
$$818$$ 0 0
$$819$$ 1.21895e8i 0.221888i
$$820$$ 0 0
$$821$$ 7.17579e8 1.29670 0.648351 0.761342i $$-0.275459\pi$$
0.648351 + 0.761342i $$0.275459\pi$$
$$822$$ 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
$$827$$ − 8.79442e8i − 1.55486i −0.628971 0.777429i $$-0.716524\pi$$
0.628971 0.777429i $$-0.283476\pi$$
$$828$$ 0 0
$$829$$ −4.29123e8 −0.753213 −0.376607 0.926373i $$-0.622909\pi$$
−0.376607 + 0.926373i $$0.622909\pi$$
$$830$$ 0 0
$$831$$ − 2.35306e8i − 0.410043i
$$832$$ 0 0
$$833$$ −6.13320e8 −1.06109
$$834$$ 0 0
$$835$$ − 1.45507e8i − 0.249934i
$$836$$ 0 0
$$837$$ 1.16050e8 0.197911
$$838$$ 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
$$843$$ − 6.74325e8i − 1.12561i
$$844$$ 0 0
$$845$$ 1.10984e8 0.183946
$$846$$ 0 0
$$847$$ − 5.52552e8i − 0.909332i
$$848$$ 0 0
$$849$$ 5.01951e8 0.820236
$$850$$ 0 0
$$851$$ 7.67997e8i 1.24615i
$$852$$ 0 0
$$853$$ −7.00029e8 −1.12790 −0.563948 0.825810i $$-0.690718\pi$$
−0.563948 + 0.825810i $$0.690718\pi$$
$$854$$ 0 0
$$855$$ 1.90823e7i 0.0305304i
$$856$$ 0 0
$$857$$ −9.52114e8 −1.51268 −0.756339 0.654180i $$-0.773014\pi$$
−0.756339 + 0.654180i $$0.773014\pi$$
$$858$$ 0 0
$$859$$ − 4.16968e8i − 0.657844i −0.944357 0.328922i $$-0.893315\pi$$
0.944357 0.328922i $$-0.106685\pi$$
$$860$$ 0 0
$$861$$ −2.18678e7 −0.0342606
$$862$$ 0 0
$$863$$ − 4.62686e8i − 0.719869i −0.932978 0.359935i $$-0.882799\pi$$
0.932978 0.359935i $$-0.117201\pi$$
$$864$$ 0 0
$$865$$ −7.45201e7 −0.115140
$$866$$ 0 0
$$867$$ − 2.95936e8i − 0.454089i
$$868$$ 0 0
$$869$$ 1.20011e9 1.82877
$$870$$ 0 0
$$871$$ 1.36773e9i 2.06988i
$$872$$ 0 0
$$873$$ 3.05557e8 0.459250
$$874$$ 0 0
$$875$$ − 9.60836e7i − 0.143425i
$$876$$ 0 0
$$877$$ 6.73329e8 0.998225 0.499113 0.866537i $$-0.333659\pi$$
0.499113 + 0.866537i $$0.333659\pi$$
$$878$$ 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
$$883$$ 1.35707e8i 0.197115i 0.995131 + 0.0985574i $$0.0314228\pi$$
−0.995131 + 0.0985574i $$0.968577\pi$$
$$884$$ 0 0
$$885$$ 1.09176e7 0.0157506
$$886$$ 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
$$891$$ − 1.36194e8i − 0.192542i
$$892$$ 0 0
$$893$$ 6.89501e8 0.968234
$$894$$ 0 0
$$895$$ − 4.03623e6i − 0.00562997i
$$896$$ 0 0
$$897$$ 8.73309e8 1.21001
$$898$$ 0 0
$$899$$ − 1.36985e9i − 1.88536i
$$900$$ 0 0
$$901$$ −6.31761e8 −0.863731
$$902$$ 0 0
$$903$$ − 6.13448e7i − 0.0833133i
$$904$$ 0 0
$$905$$ 1.08688e8 0.146635
$$906$$ 0 0
$$907$$ 1.34159e9i 1.79803i 0.437917 + 0.899015i $$0.355716\pi$$
−0.437917 + 0.899015i $$0.644284\pi$$
$$908$$ 0 0
$$909$$ −9.50227e7 −0.126513
$$910$$ 0 0
$$911$$ − 1.16107e9i − 1.53569i −0.640638 0.767843i $$-0.721330\pi$$
0.640638 0.767843i $$-0.278670\pi$$
$$912$$ 0 0
$$913$$ 1.37052e9 1.80083
$$914$$ 0 0
$$915$$ − 2.68545e7i − 0.0350553i
$$916$$ 0 0
$$917$$ −9.42995e7 −0.122293
$$918$$ 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
$$923$$ 1.17806e8i 0.149818i
$$924$$ 0 0
$$925$$ 6.72307e8 0.849459
$$926$$ 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
$$931$$ − 3.66718e8i − 0.454447i
$$932$$ 0 0
$$933$$ −7.70227e7 −0.0948361
$$934$$ 0 0
$$935$$ 3.02918e8i 0.370587i
$$936$$ 0 0
$$937$$ −4.90868e8 −0.596686 −0.298343 0.954459i $$-0.596434\pi$$
−0.298343 + 0.954459i $$0.596434\pi$$
$$938$$ 0 0
$$939$$ 4.16911e8i 0.503555i
$$940$$ 0 0
$$941$$ −1.54536e8 −0.185464 −0.0927321 0.995691i $$-0.529560\pi$$
−0.0927321 + 0.995691i $$0.529560\pi$$
$$942$$ 0 0
$$943$$ 1.56670e8i 0.186832i
$$944$$ 0 0
$$945$$ −1.17979e7 −0.0139800
$$946$$ 0 0
$$947$$ 1.24667e9i 1.46792i 0.679192 + 0.733961i $$0.262331\pi$$
−0.679192 + 0.733961i $$0.737669\pi$$
$$948$$ 0 0
$$949$$ −1.20833e9 −1.41380
$$950$$ 0 0
$$951$$ − 7.39671e8i − 0.859997i
$$952$$ 0 0
$$953$$ 4.21385e8 0.486856 0.243428 0.969919i $$-0.421728\pi$$
0.243428 + 0.969919i $$0.421728\pi$$
$$954$$ 0 0
$$955$$ 1.36522e8i 0.156744i
$$956$$ 0 0
$$957$$ −1.60763e9 −1.83421
$$958$$ 0 0
$$959$$ 2.73973e8i 0.310637i
$$960$$ 0 0
$$961$$ −5.10826e7 −0.0575576
$$962$$ 0 0
$$963$$ − 3.12546e8i − 0.349973i
$$964$$ 0 0
$$965$$ 2.61683e7 0.0291201
$$966$$ 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
$$971$$ − 2.75696e8i − 0.301143i −0.988599 0.150571i $$-0.951889\pi$$
0.988599 0.150571i $$-0.0481113\pi$$
$$972$$ 0 0
$$973$$ 4.72912e7 0.0513383
$$974$$ 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
$$979$$ 2.07836e9i 2.21499i
$$980$$ 0 0
$$981$$ 4.27103e8 0.452403
$$982$$ 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
$$987$$ 4.26292e8i 0.443359i
$$988$$ 0 0
$$989$$ −4.39501e8 −0.454330
$$990$$ 0 0
$$991$$ 5.76213e8i 0.592056i 0.955179 + 0.296028i $$0.0956621\pi$$
−0.955179 + 0.296028i $$0.904338\pi$$
$$992$$ 0 0
$$993$$ −7.54976e8 −0.771055
$$994$$ 0 0
$$995$$ − 1.29538e8i − 0.131501i
$$996$$ 0 0
$$997$$ −8.84682e7 −0.0892692 −0.0446346 0.999003i $$-0.514212\pi$$
−0.0446346 + 0.999003i $$0.514212\pi$$
$$998$$ 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 768.7.g.e.511.2 4
4.3 odd 2 inner 768.7.g.e.511.3 4
8.3 odd 2 768.7.g.c.511.1 4
8.5 even 2 768.7.g.c.511.4 4
16.3 odd 4 384.7.b.d.319.2 8
16.5 even 4 384.7.b.d.319.3 yes 8
16.11 odd 4 384.7.b.d.319.8 yes 8
16.13 even 4 384.7.b.d.319.5 yes 8

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