# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

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

## Embedding invariants

 Embedding label 193.2 Root $$-53.8781i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.10.d.b.193.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-81.0000i q^{3} -473.533i q^{5} +574.939 q^{7} -6561.00 q^{9} +O(q^{10})$$ $$q-81.0000i q^{3} -473.533i q^{5} +574.939 q^{7} -6561.00 q^{9} +72774.7i q^{11} +14829.2i q^{13} -38356.2 q^{15} +109932. q^{17} -497573. i q^{19} -46570.1i q^{21} +104764. q^{23} +1.72889e6 q^{25} +531441. i q^{27} +7.06390e6i q^{29} +2.30578e6 q^{31} +5.89475e6 q^{33} -272253. i q^{35} -5.20960e6i q^{37} +1.20116e6 q^{39} -1.08234e7 q^{41} -2.04081e7i q^{43} +3.10685e6i q^{45} -3.47476e7 q^{47} -4.00231e7 q^{49} -8.90445e6i q^{51} -9.34226e7i q^{53} +3.44612e7 q^{55} -4.03034e7 q^{57} +9.04786e7i q^{59} +2.00269e7i q^{61} -3.77218e6 q^{63} +7.02210e6 q^{65} -1.09151e8i q^{67} -8.48588e6i q^{69} -2.65931e8 q^{71} +5.05525e7 q^{73} -1.40040e8i q^{75} +4.18410e7i q^{77} +1.24238e7 q^{79} +4.30467e7 q^{81} +2.72920e8i q^{83} -5.20562e7i q^{85} +5.72176e8 q^{87} -2.70637e8 q^{89} +8.52588e6i q^{91} -1.86768e8i q^{93} -2.35617e8 q^{95} -4.06136e8 q^{97} -4.77475e8i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10})$$ 8 * q + 13632 * q^7 - 52488 * q^9 $$8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100})$$ 8 * q + 13632 * q^7 - 52488 * q^9 + 90720 * q^15 + 8304 * q^17 + 4612608 * q^23 - 4754904 * q^25 + 7499328 * q^31 - 6213024 * q^33 + 23211360 * q^39 - 43518896 * q^41 - 49382016 * q^47 - 74106808 * q^49 - 19030656 * q^55 + 38141280 * q^57 - 89439552 * q^63 - 110270336 * q^65 + 741751296 * q^71 - 1507903440 * q^73 - 1008373440 * q^79 + 344373768 * q^81 - 423468000 * q^87 - 1337034448 * q^89 + 543950208 * q^95 - 904817936 * q^97

## Character values

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

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

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 81.0000i − 0.577350i
$$4$$ 0 0
$$5$$ − 473.533i − 0.338833i −0.985545 0.169416i $$-0.945812\pi$$
0.985545 0.169416i $$-0.0541882\pi$$
$$6$$ 0 0
$$7$$ 574.939 0.0905067 0.0452534 0.998976i $$-0.485590\pi$$
0.0452534 + 0.998976i $$0.485590\pi$$
$$8$$ 0 0
$$9$$ −6561.00 −0.333333
$$10$$ 0 0
$$11$$ 72774.7i 1.49869i 0.662177 + 0.749347i $$0.269633\pi$$
−0.662177 + 0.749347i $$0.730367\pi$$
$$12$$ 0 0
$$13$$ 14829.2i 0.144003i 0.997405 + 0.0720016i $$0.0229387\pi$$
−0.997405 + 0.0720016i $$0.977061\pi$$
$$14$$ 0 0
$$15$$ −38356.2 −0.195625
$$16$$ 0 0
$$17$$ 109932. 0.319229 0.159614 0.987179i $$-0.448975\pi$$
0.159614 + 0.987179i $$0.448975\pi$$
$$18$$ 0 0
$$19$$ − 497573.i − 0.875922i −0.898994 0.437961i $$-0.855701\pi$$
0.898994 0.437961i $$-0.144299\pi$$
$$20$$ 0 0
$$21$$ − 46570.1i − 0.0522541i
$$22$$ 0 0
$$23$$ 104764. 0.0780614 0.0390307 0.999238i $$-0.487573\pi$$
0.0390307 + 0.999238i $$0.487573\pi$$
$$24$$ 0 0
$$25$$ 1.72889e6 0.885192
$$26$$ 0 0
$$27$$ 531441.i 0.192450i
$$28$$ 0 0
$$29$$ 7.06390e6i 1.85461i 0.374301 + 0.927307i $$0.377883\pi$$
−0.374301 + 0.927307i $$0.622117\pi$$
$$30$$ 0 0
$$31$$ 2.30578e6 0.448426 0.224213 0.974540i $$-0.428019\pi$$
0.224213 + 0.974540i $$0.428019\pi$$
$$32$$ 0 0
$$33$$ 5.89475e6 0.865272
$$34$$ 0 0
$$35$$ − 272253.i − 0.0306666i
$$36$$ 0 0
$$37$$ − 5.20960e6i − 0.456979i −0.973546 0.228490i $$-0.926621\pi$$
0.973546 0.228490i $$-0.0733787\pi$$
$$38$$ 0 0
$$39$$ 1.20116e6 0.0831402
$$40$$ 0 0
$$41$$ −1.08234e7 −0.598185 −0.299092 0.954224i $$-0.596684\pi$$
−0.299092 + 0.954224i $$0.596684\pi$$
$$42$$ 0 0
$$43$$ − 2.04081e7i − 0.910319i −0.890410 0.455160i $$-0.849582\pi$$
0.890410 0.455160i $$-0.150418\pi$$
$$44$$ 0 0
$$45$$ 3.10685e6i 0.112944i
$$46$$ 0 0
$$47$$ −3.47476e7 −1.03869 −0.519343 0.854566i $$-0.673823\pi$$
−0.519343 + 0.854566i $$0.673823\pi$$
$$48$$ 0 0
$$49$$ −4.00231e7 −0.991809
$$50$$ 0 0
$$51$$ − 8.90445e6i − 0.184307i
$$52$$ 0 0
$$53$$ − 9.34226e7i − 1.62634i −0.582028 0.813169i $$-0.697741\pi$$
0.582028 0.813169i $$-0.302259\pi$$
$$54$$ 0 0
$$55$$ 3.44612e7 0.507807
$$56$$ 0 0
$$57$$ −4.03034e7 −0.505714
$$58$$ 0 0
$$59$$ 9.04786e7i 0.972102i 0.873930 + 0.486051i $$0.161563\pi$$
−0.873930 + 0.486051i $$0.838437\pi$$
$$60$$ 0 0
$$61$$ 2.00269e7i 0.185195i 0.995704 + 0.0925976i $$0.0295170\pi$$
−0.995704 + 0.0925976i $$0.970483\pi$$
$$62$$ 0 0
$$63$$ −3.77218e6 −0.0301689
$$64$$ 0 0
$$65$$ 7.02210e6 0.0487930
$$66$$ 0 0
$$67$$ − 1.09151e8i − 0.661745i −0.943676 0.330872i $$-0.892657\pi$$
0.943676 0.330872i $$-0.107343\pi$$
$$68$$ 0 0
$$69$$ − 8.48588e6i − 0.0450688i
$$70$$ 0 0
$$71$$ −2.65931e8 −1.24196 −0.620978 0.783828i $$-0.713264\pi$$
−0.620978 + 0.783828i $$0.713264\pi$$
$$72$$ 0 0
$$73$$ 5.05525e7 0.208348 0.104174 0.994559i $$-0.466780\pi$$
0.104174 + 0.994559i $$0.466780\pi$$
$$74$$ 0 0
$$75$$ − 1.40040e8i − 0.511066i
$$76$$ 0 0
$$77$$ 4.18410e7i 0.135642i
$$78$$ 0 0
$$79$$ 1.24238e7 0.0358865 0.0179433 0.999839i $$-0.494288\pi$$
0.0179433 + 0.999839i $$0.494288\pi$$
$$80$$ 0 0
$$81$$ 4.30467e7 0.111111
$$82$$ 0 0
$$83$$ 2.72920e8i 0.631225i 0.948888 + 0.315613i $$0.102210\pi$$
−0.948888 + 0.315613i $$0.897790\pi$$
$$84$$ 0 0
$$85$$ − 5.20562e7i − 0.108165i
$$86$$ 0 0
$$87$$ 5.72176e8 1.07076
$$88$$ 0 0
$$89$$ −2.70637e8 −0.457227 −0.228613 0.973517i $$-0.573419\pi$$
−0.228613 + 0.973517i $$0.573419\pi$$
$$90$$ 0 0
$$91$$ 8.52588e6i 0.0130333i
$$92$$ 0 0
$$93$$ − 1.86768e8i − 0.258899i
$$94$$ 0 0
$$95$$ −2.35617e8 −0.296791
$$96$$ 0 0
$$97$$ −4.06136e8 −0.465799 −0.232899 0.972501i $$-0.574821\pi$$
−0.232899 + 0.972501i $$0.574821\pi$$
$$98$$ 0 0
$$99$$ − 4.77475e8i − 0.499565i
$$100$$ 0 0
$$101$$ − 3.58458e8i − 0.342762i −0.985205 0.171381i $$-0.945177\pi$$
0.985205 0.171381i $$-0.0548229\pi$$
$$102$$ 0 0
$$103$$ −3.12727e8 −0.273777 −0.136889 0.990586i $$-0.543710\pi$$
−0.136889 + 0.990586i $$0.543710\pi$$
$$104$$ 0 0
$$105$$ −2.20525e7 −0.0177054
$$106$$ 0 0
$$107$$ 9.28960e8i 0.685125i 0.939495 + 0.342562i $$0.111295\pi$$
−0.939495 + 0.342562i $$0.888705\pi$$
$$108$$ 0 0
$$109$$ − 1.54301e9i − 1.04701i −0.852024 0.523503i $$-0.824625\pi$$
0.852024 0.523503i $$-0.175375\pi$$
$$110$$ 0 0
$$111$$ −4.21977e8 −0.263837
$$112$$ 0 0
$$113$$ −5.98385e8 −0.345245 −0.172623 0.984988i $$-0.555224\pi$$
−0.172623 + 0.984988i $$0.555224\pi$$
$$114$$ 0 0
$$115$$ − 4.96092e7i − 0.0264498i
$$116$$ 0 0
$$117$$ − 9.72942e7i − 0.0480010i
$$118$$ 0 0
$$119$$ 6.32039e7 0.0288924
$$120$$ 0 0
$$121$$ −2.93820e9 −1.24609
$$122$$ 0 0
$$123$$ 8.76693e8i 0.345362i
$$124$$ 0 0
$$125$$ − 1.74356e9i − 0.638765i
$$126$$ 0 0
$$127$$ 2.52986e9 0.862940 0.431470 0.902127i $$-0.357995\pi$$
0.431470 + 0.902127i $$0.357995\pi$$
$$128$$ 0 0
$$129$$ −1.65305e9 −0.525573
$$130$$ 0 0
$$131$$ − 1.70203e8i − 0.0504948i −0.999681 0.0252474i $$-0.991963\pi$$
0.999681 0.0252474i $$-0.00803734\pi$$
$$132$$ 0 0
$$133$$ − 2.86074e8i − 0.0792768i
$$134$$ 0 0
$$135$$ 2.51655e8 0.0652084
$$136$$ 0 0
$$137$$ 2.59035e9 0.628226 0.314113 0.949386i $$-0.398293\pi$$
0.314113 + 0.949386i $$0.398293\pi$$
$$138$$ 0 0
$$139$$ 2.75040e9i 0.624928i 0.949930 + 0.312464i $$0.101154\pi$$
−0.949930 + 0.312464i $$0.898846\pi$$
$$140$$ 0 0
$$141$$ 2.81456e9i 0.599686i
$$142$$ 0 0
$$143$$ −1.07919e9 −0.215817
$$144$$ 0 0
$$145$$ 3.34499e9 0.628404
$$146$$ 0 0
$$147$$ 3.24187e9i 0.572621i
$$148$$ 0 0
$$149$$ − 2.97566e9i − 0.494590i −0.968940 0.247295i $$-0.920458\pi$$
0.968940 0.247295i $$-0.0795416\pi$$
$$150$$ 0 0
$$151$$ −1.10799e10 −1.73435 −0.867177 0.498000i $$-0.834068\pi$$
−0.867177 + 0.498000i $$0.834068\pi$$
$$152$$ 0 0
$$153$$ −7.21261e8 −0.106410
$$154$$ 0 0
$$155$$ − 1.09186e9i − 0.151941i
$$156$$ 0 0
$$157$$ 1.20320e10i 1.58048i 0.612797 + 0.790240i $$0.290044\pi$$
−0.612797 + 0.790240i $$0.709956\pi$$
$$158$$ 0 0
$$159$$ −7.56723e9 −0.938966
$$160$$ 0 0
$$161$$ 6.02329e7 0.00706508
$$162$$ 0 0
$$163$$ 1.21499e9i 0.134812i 0.997726 + 0.0674062i $$0.0214724\pi$$
−0.997726 + 0.0674062i $$0.978528\pi$$
$$164$$ 0 0
$$165$$ − 2.79136e9i − 0.293182i
$$166$$ 0 0
$$167$$ 1.01298e10 1.00780 0.503902 0.863761i $$-0.331897\pi$$
0.503902 + 0.863761i $$0.331897\pi$$
$$168$$ 0 0
$$169$$ 1.03846e10 0.979263
$$170$$ 0 0
$$171$$ 3.26458e9i 0.291974i
$$172$$ 0 0
$$173$$ 1.12366e10i 0.953738i 0.878974 + 0.476869i $$0.158228\pi$$
−0.878974 + 0.476869i $$0.841772\pi$$
$$174$$ 0 0
$$175$$ 9.94008e8 0.0801159
$$176$$ 0 0
$$177$$ 7.32877e9 0.561244
$$178$$ 0 0
$$179$$ − 3.72939e9i − 0.271518i −0.990742 0.135759i $$-0.956653\pi$$
0.990742 0.135759i $$-0.0433473\pi$$
$$180$$ 0 0
$$181$$ − 1.47052e10i − 1.01840i −0.860649 0.509198i $$-0.829942\pi$$
0.860649 0.509198i $$-0.170058\pi$$
$$182$$ 0 0
$$183$$ 1.62218e9 0.106922
$$184$$ 0 0
$$185$$ −2.46692e9 −0.154839
$$186$$ 0 0
$$187$$ 8.00023e9i 0.478426i
$$188$$ 0 0
$$189$$ 3.05546e8i 0.0174180i
$$190$$ 0 0
$$191$$ −2.16788e10 −1.17865 −0.589325 0.807896i $$-0.700606\pi$$
−0.589325 + 0.807896i $$0.700606\pi$$
$$192$$ 0 0
$$193$$ −2.45321e10 −1.27270 −0.636350 0.771400i $$-0.719557\pi$$
−0.636350 + 0.771400i $$0.719557\pi$$
$$194$$ 0 0
$$195$$ − 5.68790e8i − 0.0281706i
$$196$$ 0 0
$$197$$ − 1.44518e10i − 0.683632i −0.939767 0.341816i $$-0.888958\pi$$
0.939767 0.341816i $$-0.111042\pi$$
$$198$$ 0 0
$$199$$ −2.71655e10 −1.22794 −0.613972 0.789328i $$-0.710429\pi$$
−0.613972 + 0.789328i $$0.710429\pi$$
$$200$$ 0 0
$$201$$ −8.84122e9 −0.382058
$$202$$ 0 0
$$203$$ 4.06132e9i 0.167855i
$$204$$ 0 0
$$205$$ 5.12522e9i 0.202684i
$$206$$ 0 0
$$207$$ −6.87356e8 −0.0260205
$$208$$ 0 0
$$209$$ 3.62107e10 1.31274
$$210$$ 0 0
$$211$$ 3.52449e10i 1.22412i 0.790810 + 0.612062i $$0.209660\pi$$
−0.790810 + 0.612062i $$0.790340\pi$$
$$212$$ 0 0
$$213$$ 2.15404e10i 0.717044i
$$214$$ 0 0
$$215$$ −9.66389e9 −0.308446
$$216$$ 0 0
$$217$$ 1.32568e9 0.0405855
$$218$$ 0 0
$$219$$ − 4.09475e9i − 0.120290i
$$220$$ 0 0
$$221$$ 1.63019e9i 0.0459699i
$$222$$ 0 0
$$223$$ −1.45878e10 −0.395019 −0.197510 0.980301i $$-0.563285\pi$$
−0.197510 + 0.980301i $$0.563285\pi$$
$$224$$ 0 0
$$225$$ −1.13433e10 −0.295064
$$226$$ 0 0
$$227$$ 6.30500e10i 1.57605i 0.615646 + 0.788023i $$0.288895\pi$$
−0.615646 + 0.788023i $$0.711105\pi$$
$$228$$ 0 0
$$229$$ 4.91705e10i 1.18153i 0.806843 + 0.590765i $$0.201174\pi$$
−0.806843 + 0.590765i $$0.798826\pi$$
$$230$$ 0 0
$$231$$ 3.38912e9 0.0783129
$$232$$ 0 0
$$233$$ 3.30380e10 0.734366 0.367183 0.930149i $$-0.380322\pi$$
0.367183 + 0.930149i $$0.380322\pi$$
$$234$$ 0 0
$$235$$ 1.64541e10i 0.351941i
$$236$$ 0 0
$$237$$ − 1.00633e9i − 0.0207191i
$$238$$ 0 0
$$239$$ −7.33801e10 −1.45475 −0.727374 0.686241i $$-0.759260\pi$$
−0.727374 + 0.686241i $$0.759260\pi$$
$$240$$ 0 0
$$241$$ 5.53816e10 1.05752 0.528760 0.848771i $$-0.322657\pi$$
0.528760 + 0.848771i $$0.322657\pi$$
$$242$$ 0 0
$$243$$ − 3.48678e9i − 0.0641500i
$$244$$ 0 0
$$245$$ 1.89522e10i 0.336057i
$$246$$ 0 0
$$247$$ 7.37860e9 0.126136
$$248$$ 0 0
$$249$$ 2.21065e10 0.364438
$$250$$ 0 0
$$251$$ 9.51298e10i 1.51281i 0.654103 + 0.756406i $$0.273046\pi$$
−0.654103 + 0.756406i $$0.726954\pi$$
$$252$$ 0 0
$$253$$ 7.62416e9i 0.116990i
$$254$$ 0 0
$$255$$ −4.21655e9 −0.0624492
$$256$$ 0 0
$$257$$ −5.82111e10 −0.832351 −0.416176 0.909284i $$-0.636630\pi$$
−0.416176 + 0.909284i $$0.636630\pi$$
$$258$$ 0 0
$$259$$ − 2.99520e9i − 0.0413597i
$$260$$ 0 0
$$261$$ − 4.63463e10i − 0.618205i
$$262$$ 0 0
$$263$$ −1.26131e11 −1.62563 −0.812814 0.582524i $$-0.802065\pi$$
−0.812814 + 0.582524i $$0.802065\pi$$
$$264$$ 0 0
$$265$$ −4.42387e10 −0.551056
$$266$$ 0 0
$$267$$ 2.19216e10i 0.263980i
$$268$$ 0 0
$$269$$ 1.23726e10i 0.144071i 0.997402 + 0.0720356i $$0.0229495\pi$$
−0.997402 + 0.0720356i $$0.977050\pi$$
$$270$$ 0 0
$$271$$ −6.40678e10 −0.721569 −0.360785 0.932649i $$-0.617491\pi$$
−0.360785 + 0.932649i $$0.617491\pi$$
$$272$$ 0 0
$$273$$ 6.90596e8 0.00752475
$$274$$ 0 0
$$275$$ 1.25820e11i 1.32663i
$$276$$ 0 0
$$277$$ − 1.56675e11i − 1.59897i −0.600688 0.799484i $$-0.705107\pi$$
0.600688 0.799484i $$-0.294893\pi$$
$$278$$ 0 0
$$279$$ −1.51282e10 −0.149475
$$280$$ 0 0
$$281$$ −9.55568e10 −0.914288 −0.457144 0.889393i $$-0.651128\pi$$
−0.457144 + 0.889393i $$0.651128\pi$$
$$282$$ 0 0
$$283$$ 7.99388e10i 0.740830i 0.928866 + 0.370415i $$0.120785\pi$$
−0.928866 + 0.370415i $$0.879215\pi$$
$$284$$ 0 0
$$285$$ 1.90850e10i 0.171352i
$$286$$ 0 0
$$287$$ −6.22278e9 −0.0541397
$$288$$ 0 0
$$289$$ −1.06503e11 −0.898093
$$290$$ 0 0
$$291$$ 3.28970e10i 0.268929i
$$292$$ 0 0
$$293$$ − 9.87325e10i − 0.782629i −0.920257 0.391315i $$-0.872020\pi$$
0.920257 0.391315i $$-0.127980\pi$$
$$294$$ 0 0
$$295$$ 4.28446e10 0.329380
$$296$$ 0 0
$$297$$ −3.86754e10 −0.288424
$$298$$ 0 0
$$299$$ 1.55356e9i 0.0112411i
$$300$$ 0 0
$$301$$ − 1.17334e10i − 0.0823900i
$$302$$ 0 0
$$303$$ −2.90351e10 −0.197894
$$304$$ 0 0
$$305$$ 9.48340e9 0.0627502
$$306$$ 0 0
$$307$$ 2.01628e11i 1.29547i 0.761864 + 0.647737i $$0.224284\pi$$
−0.761864 + 0.647737i $$0.775716\pi$$
$$308$$ 0 0
$$309$$ 2.53309e10i 0.158065i
$$310$$ 0 0
$$311$$ −1.25577e11 −0.761182 −0.380591 0.924744i $$-0.624279\pi$$
−0.380591 + 0.924744i $$0.624279\pi$$
$$312$$ 0 0
$$313$$ 9.72056e10 0.572455 0.286228 0.958162i $$-0.407599\pi$$
0.286228 + 0.958162i $$0.407599\pi$$
$$314$$ 0 0
$$315$$ 1.78625e9i 0.0102222i
$$316$$ 0 0
$$317$$ − 3.24299e10i − 0.180376i −0.995925 0.0901880i $$-0.971253\pi$$
0.995925 0.0901880i $$-0.0287468\pi$$
$$318$$ 0 0
$$319$$ −5.14073e11 −2.77950
$$320$$ 0 0
$$321$$ 7.52457e10 0.395557
$$322$$ 0 0
$$323$$ − 5.46989e10i − 0.279620i
$$324$$ 0 0
$$325$$ 2.56380e10i 0.127470i
$$326$$ 0 0
$$327$$ −1.24984e11 −0.604490
$$328$$ 0 0
$$329$$ −1.99778e10 −0.0940081
$$330$$ 0 0
$$331$$ 2.13971e11i 0.979781i 0.871784 + 0.489890i $$0.162963\pi$$
−0.871784 + 0.489890i $$0.837037\pi$$
$$332$$ 0 0
$$333$$ 3.41802e10i 0.152326i
$$334$$ 0 0
$$335$$ −5.16865e10 −0.224221
$$336$$ 0 0
$$337$$ −1.67278e11 −0.706489 −0.353244 0.935531i $$-0.614922\pi$$
−0.353244 + 0.935531i $$0.614922\pi$$
$$338$$ 0 0
$$339$$ 4.84692e10i 0.199328i
$$340$$ 0 0
$$341$$ 1.67802e11i 0.672053i
$$342$$ 0 0
$$343$$ −4.62117e10 −0.180272
$$344$$ 0 0
$$345$$ −4.01834e9 −0.0152708
$$346$$ 0 0
$$347$$ 1.93452e11i 0.716293i 0.933665 + 0.358146i $$0.116591\pi$$
−0.933665 + 0.358146i $$0.883409\pi$$
$$348$$ 0 0
$$349$$ − 1.79870e11i − 0.649000i −0.945886 0.324500i $$-0.894804\pi$$
0.945886 0.324500i $$-0.105196\pi$$
$$350$$ 0 0
$$351$$ −7.88083e9 −0.0277134
$$352$$ 0 0
$$353$$ 2.13154e11 0.730646 0.365323 0.930881i $$-0.380959\pi$$
0.365323 + 0.930881i $$0.380959\pi$$
$$354$$ 0 0
$$355$$ 1.25927e11i 0.420815i
$$356$$ 0 0
$$357$$ − 5.11952e9i − 0.0166810i
$$358$$ 0 0
$$359$$ 1.85269e11 0.588677 0.294339 0.955701i $$-0.404901\pi$$
0.294339 + 0.955701i $$0.404901\pi$$
$$360$$ 0 0
$$361$$ 7.51089e10 0.232760
$$362$$ 0 0
$$363$$ 2.37995e11i 0.719428i
$$364$$ 0 0
$$365$$ − 2.39383e10i − 0.0705952i
$$366$$ 0 0
$$367$$ −3.99652e11 −1.14997 −0.574983 0.818166i $$-0.694991\pi$$
−0.574983 + 0.818166i $$0.694991\pi$$
$$368$$ 0 0
$$369$$ 7.10122e10 0.199395
$$370$$ 0 0
$$371$$ − 5.37123e10i − 0.147194i
$$372$$ 0 0
$$373$$ 3.17540e11i 0.849394i 0.905335 + 0.424697i $$0.139619\pi$$
−0.905335 + 0.424697i $$0.860381\pi$$
$$374$$ 0 0
$$375$$ −1.41228e11 −0.368791
$$376$$ 0 0
$$377$$ −1.04752e11 −0.267070
$$378$$ 0 0
$$379$$ 2.10387e11i 0.523771i 0.965099 + 0.261886i $$0.0843443\pi$$
−0.965099 + 0.261886i $$0.915656\pi$$
$$380$$ 0 0
$$381$$ − 2.04919e11i − 0.498219i
$$382$$ 0 0
$$383$$ −6.26458e11 −1.48764 −0.743819 0.668381i $$-0.766988\pi$$
−0.743819 + 0.668381i $$0.766988\pi$$
$$384$$ 0 0
$$385$$ 1.98131e10 0.0459599
$$386$$ 0 0
$$387$$ 1.33897e11i 0.303440i
$$388$$ 0 0
$$389$$ 4.71840e11i 1.04477i 0.852709 + 0.522386i $$0.174958\pi$$
−0.852709 + 0.522386i $$0.825042\pi$$
$$390$$ 0 0
$$391$$ 1.15169e10 0.0249194
$$392$$ 0 0
$$393$$ −1.37864e10 −0.0291532
$$394$$ 0 0
$$395$$ − 5.88307e9i − 0.0121595i
$$396$$ 0 0
$$397$$ − 4.08614e11i − 0.825573i −0.910828 0.412786i $$-0.864556\pi$$
0.910828 0.412786i $$-0.135444\pi$$
$$398$$ 0 0
$$399$$ −2.31720e10 −0.0457705
$$400$$ 0 0
$$401$$ −4.75227e11 −0.917807 −0.458903 0.888486i $$-0.651758\pi$$
−0.458903 + 0.888486i $$0.651758\pi$$
$$402$$ 0 0
$$403$$ 3.41928e10i 0.0645747i
$$404$$ 0 0
$$405$$ − 2.03840e10i − 0.0376481i
$$406$$ 0 0
$$407$$ 3.79127e11 0.684872
$$408$$ 0 0
$$409$$ 1.47969e11 0.261466 0.130733 0.991418i $$-0.458267\pi$$
0.130733 + 0.991418i $$0.458267\pi$$
$$410$$ 0 0
$$411$$ − 2.09818e11i − 0.362706i
$$412$$ 0 0
$$413$$ 5.20197e10i 0.0879818i
$$414$$ 0 0
$$415$$ 1.29237e11 0.213880
$$416$$ 0 0
$$417$$ 2.22783e11 0.360802
$$418$$ 0 0
$$419$$ 8.99050e11i 1.42502i 0.701663 + 0.712509i $$0.252441\pi$$
−0.701663 + 0.712509i $$0.747559\pi$$
$$420$$ 0 0
$$421$$ − 9.67201e10i − 0.150054i −0.997182 0.0750269i $$-0.976096\pi$$
0.997182 0.0750269i $$-0.0239043\pi$$
$$422$$ 0 0
$$423$$ 2.27979e11 0.346229
$$424$$ 0 0
$$425$$ 1.90060e11 0.282579
$$426$$ 0 0
$$427$$ 1.15143e10i 0.0167614i
$$428$$ 0 0
$$429$$ 8.74143e10i 0.124602i
$$430$$ 0 0
$$431$$ −4.91630e11 −0.686263 −0.343131 0.939287i $$-0.611488\pi$$
−0.343131 + 0.939287i $$0.611488\pi$$
$$432$$ 0 0
$$433$$ −2.39033e10 −0.0326786 −0.0163393 0.999867i $$-0.505201\pi$$
−0.0163393 + 0.999867i $$0.505201\pi$$
$$434$$ 0 0
$$435$$ − 2.70944e11i − 0.362809i
$$436$$ 0 0
$$437$$ − 5.21277e10i − 0.0683757i
$$438$$ 0 0
$$439$$ 8.29279e11 1.06564 0.532820 0.846229i $$-0.321132\pi$$
0.532820 + 0.846229i $$0.321132\pi$$
$$440$$ 0 0
$$441$$ 2.62591e11 0.330603
$$442$$ 0 0
$$443$$ 5.71843e11i 0.705440i 0.935729 + 0.352720i $$0.114743\pi$$
−0.935729 + 0.352720i $$0.885257\pi$$
$$444$$ 0 0
$$445$$ 1.28155e11i 0.154923i
$$446$$ 0 0
$$447$$ −2.41028e11 −0.285551
$$448$$ 0 0
$$449$$ 1.02914e12 1.19500 0.597500 0.801869i $$-0.296161\pi$$
0.597500 + 0.801869i $$0.296161\pi$$
$$450$$ 0 0
$$451$$ − 7.87668e11i − 0.896496i
$$452$$ 0 0
$$453$$ 8.97468e11i 1.00133i
$$454$$ 0 0
$$455$$ 4.03728e9 0.00441609
$$456$$ 0 0
$$457$$ −7.44956e11 −0.798928 −0.399464 0.916749i $$-0.630804\pi$$
−0.399464 + 0.916749i $$0.630804\pi$$
$$458$$ 0 0
$$459$$ 5.84221e10i 0.0614356i
$$460$$ 0 0
$$461$$ 1.35785e11i 0.140023i 0.997546 + 0.0700115i $$0.0223036\pi$$
−0.997546 + 0.0700115i $$0.977696\pi$$
$$462$$ 0 0
$$463$$ −6.39280e10 −0.0646512 −0.0323256 0.999477i $$-0.510291\pi$$
−0.0323256 + 0.999477i $$0.510291\pi$$
$$464$$ 0 0
$$465$$ −8.84409e10 −0.0877233
$$466$$ 0 0
$$467$$ − 8.08293e11i − 0.786399i −0.919453 0.393199i $$-0.871368\pi$$
0.919453 0.393199i $$-0.128632\pi$$
$$468$$ 0 0
$$469$$ − 6.27551e10i − 0.0598923i
$$470$$ 0 0
$$471$$ 9.74591e11 0.912491
$$472$$ 0 0
$$473$$ 1.48519e12 1.36429
$$474$$ 0 0
$$475$$ − 8.60250e11i − 0.775360i
$$476$$ 0 0
$$477$$ 6.12946e11i 0.542112i
$$478$$ 0 0
$$479$$ −1.38621e12 −1.20315 −0.601573 0.798818i $$-0.705459\pi$$
−0.601573 + 0.798818i $$0.705459\pi$$
$$480$$ 0 0
$$481$$ 7.72540e10 0.0658064
$$482$$ 0 0
$$483$$ − 4.87886e9i − 0.00407903i
$$484$$ 0 0
$$485$$ 1.92319e11i 0.157828i
$$486$$ 0 0
$$487$$ 1.20417e11 0.0970078 0.0485039 0.998823i $$-0.484555\pi$$
0.0485039 + 0.998823i $$0.484555\pi$$
$$488$$ 0 0
$$489$$ 9.84146e10 0.0778340
$$490$$ 0 0
$$491$$ − 1.43283e11i − 0.111257i −0.998452 0.0556287i $$-0.982284\pi$$
0.998452 0.0556287i $$-0.0177163\pi$$
$$492$$ 0 0
$$493$$ 7.76545e11i 0.592046i
$$494$$ 0 0
$$495$$ −2.26100e11 −0.169269
$$496$$ 0 0
$$497$$ −1.52894e11 −0.112405
$$498$$ 0 0
$$499$$ 1.93465e12i 1.39685i 0.715683 + 0.698426i $$0.246116\pi$$
−0.715683 + 0.698426i $$0.753884\pi$$
$$500$$ 0 0
$$501$$ − 8.20513e11i − 0.581856i
$$502$$ 0 0
$$503$$ 2.14780e12 1.49602 0.748010 0.663687i $$-0.231009\pi$$
0.748010 + 0.663687i $$0.231009\pi$$
$$504$$ 0 0
$$505$$ −1.69742e11 −0.116139
$$506$$ 0 0
$$507$$ − 8.41152e11i − 0.565378i
$$508$$ 0 0
$$509$$ 4.93770e10i 0.0326058i 0.999867 + 0.0163029i $$0.00518961\pi$$
−0.999867 + 0.0163029i $$0.994810\pi$$
$$510$$ 0 0
$$511$$ 2.90646e10 0.0188569
$$512$$ 0 0
$$513$$ 2.64431e11 0.168571
$$514$$ 0 0
$$515$$ 1.48086e11i 0.0927646i
$$516$$ 0 0
$$517$$ − 2.52875e12i − 1.55667i
$$518$$ 0 0
$$519$$ 9.10168e11 0.550641
$$520$$ 0 0
$$521$$ −3.06642e12 −1.82331 −0.911657 0.410951i $$-0.865197\pi$$
−0.911657 + 0.410951i $$0.865197\pi$$
$$522$$ 0 0
$$523$$ − 1.36512e12i − 0.797836i −0.916987 0.398918i $$-0.869386\pi$$
0.916987 0.398918i $$-0.130614\pi$$
$$524$$ 0 0
$$525$$ − 8.05146e10i − 0.0462549i
$$526$$ 0 0
$$527$$ 2.53478e11 0.143150
$$528$$ 0 0
$$529$$ −1.79018e12 −0.993906
$$530$$ 0 0
$$531$$ − 5.93630e11i − 0.324034i
$$532$$ 0 0
$$533$$ − 1.60502e11i − 0.0861405i
$$534$$ 0 0
$$535$$ 4.39893e11 0.232143
$$536$$ 0 0
$$537$$ −3.02081e11 −0.156761
$$538$$ 0 0
$$539$$ − 2.91266e12i − 1.48642i
$$540$$ 0 0
$$541$$ − 2.34837e12i − 1.17863i −0.807902 0.589317i $$-0.799397\pi$$
0.807902 0.589317i $$-0.200603\pi$$
$$542$$ 0 0
$$543$$ −1.19112e12 −0.587971
$$544$$ 0 0
$$545$$ −7.30666e11 −0.354760
$$546$$ 0 0
$$547$$ − 3.90038e12i − 1.86279i −0.364007 0.931396i $$-0.618592\pi$$
0.364007 0.931396i $$-0.381408\pi$$
$$548$$ 0 0
$$549$$ − 1.31397e11i − 0.0617317i
$$550$$ 0 0
$$551$$ 3.51481e12 1.62450
$$552$$ 0 0
$$553$$ 7.14292e9 0.00324797
$$554$$ 0 0
$$555$$ 1.99820e11i 0.0893966i
$$556$$ 0 0
$$557$$ − 9.85631e11i − 0.433876i −0.976185 0.216938i $$-0.930393\pi$$
0.976185 0.216938i $$-0.0696070\pi$$
$$558$$ 0 0
$$559$$ 3.02635e11 0.131089
$$560$$ 0 0
$$561$$ 6.48019e11 0.276220
$$562$$ 0 0
$$563$$ − 3.71992e12i − 1.56044i −0.625508 0.780218i $$-0.715108\pi$$
0.625508 0.780218i $$-0.284892\pi$$
$$564$$ 0 0
$$565$$ 2.83355e11i 0.116980i
$$566$$ 0 0
$$567$$ 2.47493e10 0.0100563
$$568$$ 0 0
$$569$$ 2.34608e12 0.938293 0.469146 0.883120i $$-0.344562\pi$$
0.469146 + 0.883120i $$0.344562\pi$$
$$570$$ 0 0
$$571$$ 1.58941e12i 0.625710i 0.949801 + 0.312855i $$0.101285\pi$$
−0.949801 + 0.312855i $$0.898715\pi$$
$$572$$ 0 0
$$573$$ 1.75598e12i 0.680493i
$$574$$ 0 0
$$575$$ 1.81125e11 0.0690994
$$576$$ 0 0
$$577$$ −4.20772e12 −1.58036 −0.790179 0.612876i $$-0.790013\pi$$
−0.790179 + 0.612876i $$0.790013\pi$$
$$578$$ 0 0
$$579$$ 1.98710e12i 0.734794i
$$580$$ 0 0
$$581$$ 1.56913e11i 0.0571302i
$$582$$ 0 0
$$583$$ 6.79880e12 2.43738
$$584$$ 0 0
$$585$$ −4.60720e10 −0.0162643
$$586$$ 0 0
$$587$$ − 1.63748e12i − 0.569252i −0.958639 0.284626i $$-0.908131\pi$$
0.958639 0.284626i $$-0.0918694\pi$$
$$588$$ 0 0
$$589$$ − 1.14729e12i − 0.392786i
$$590$$ 0 0
$$591$$ −1.17059e12 −0.394695
$$592$$ 0 0
$$593$$ 5.53148e12 1.83694 0.918471 0.395489i $$-0.129425\pi$$
0.918471 + 0.395489i $$0.129425\pi$$
$$594$$ 0 0
$$595$$ − 2.99292e10i − 0.00978967i
$$596$$ 0 0
$$597$$ 2.20040e12i 0.708954i
$$598$$ 0 0
$$599$$ 4.16661e12 1.32240 0.661199 0.750211i $$-0.270048\pi$$
0.661199 + 0.750211i $$0.270048\pi$$
$$600$$ 0 0
$$601$$ −2.90464e12 −0.908149 −0.454075 0.890964i $$-0.650030\pi$$
−0.454075 + 0.890964i $$0.650030\pi$$
$$602$$ 0 0
$$603$$ 7.16139e11i 0.220582i
$$604$$ 0 0
$$605$$ 1.39134e12i 0.422214i
$$606$$ 0 0
$$607$$ −4.66728e12 −1.39545 −0.697726 0.716365i $$-0.745805\pi$$
−0.697726 + 0.716365i $$0.745805\pi$$
$$608$$ 0 0
$$609$$ 3.28967e11 0.0969112
$$610$$ 0 0
$$611$$ − 5.15278e11i − 0.149574i
$$612$$ 0 0
$$613$$ 3.12221e11i 0.0893078i 0.999003 + 0.0446539i $$0.0142185\pi$$
−0.999003 + 0.0446539i $$0.985781\pi$$
$$614$$ 0 0
$$615$$ 4.15143e11 0.117020
$$616$$ 0 0
$$617$$ 3.59628e12 0.999011 0.499505 0.866311i $$-0.333515\pi$$
0.499505 + 0.866311i $$0.333515\pi$$
$$618$$ 0 0
$$619$$ − 4.85670e12i − 1.32964i −0.747004 0.664819i $$-0.768509\pi$$
0.747004 0.664819i $$-0.231491\pi$$
$$620$$ 0 0
$$621$$ 5.56758e10i 0.0150229i
$$622$$ 0 0
$$623$$ −1.55600e11 −0.0413821
$$624$$ 0 0
$$625$$ 2.55111e12 0.668758
$$626$$ 0 0
$$627$$ − 2.93307e12i − 0.757911i
$$628$$ 0 0
$$629$$ − 5.72699e11i − 0.145881i
$$630$$ 0 0
$$631$$ 6.19329e12 1.55521 0.777606 0.628752i $$-0.216434\pi$$
0.777606 + 0.628752i $$0.216434\pi$$
$$632$$ 0 0
$$633$$ 2.85484e12 0.706748
$$634$$ 0 0
$$635$$ − 1.19797e12i − 0.292392i
$$636$$ 0 0
$$637$$ − 5.93509e11i − 0.142824i
$$638$$ 0 0
$$639$$ 1.74477e12 0.413985
$$640$$ 0 0
$$641$$ −5.96307e12 −1.39511 −0.697555 0.716531i $$-0.745729\pi$$
−0.697555 + 0.716531i $$0.745729\pi$$
$$642$$ 0 0
$$643$$ − 7.48485e11i − 0.172677i −0.996266 0.0863383i $$-0.972483\pi$$
0.996266 0.0863383i $$-0.0275166\pi$$
$$644$$ 0 0
$$645$$ 7.82775e11i 0.178081i
$$646$$ 0 0
$$647$$ 5.21646e12 1.17032 0.585162 0.810916i $$-0.301031\pi$$
0.585162 + 0.810916i $$0.301031\pi$$
$$648$$ 0 0
$$649$$ −6.58455e12 −1.45688
$$650$$ 0 0
$$651$$ − 1.07380e11i − 0.0234321i
$$652$$ 0 0
$$653$$ 5.05968e12i 1.08896i 0.838772 + 0.544482i $$0.183274\pi$$
−0.838772 + 0.544482i $$0.816726\pi$$
$$654$$ 0 0
$$655$$ −8.05967e10 −0.0171093
$$656$$ 0 0
$$657$$ −3.31675e11 −0.0694494
$$658$$ 0 0
$$659$$ − 1.64097e12i − 0.338936i −0.985536 0.169468i $$-0.945795\pi$$
0.985536 0.169468i $$-0.0542049\pi$$
$$660$$ 0 0
$$661$$ − 9.39337e12i − 1.91388i −0.290284 0.956941i $$-0.593750\pi$$
0.290284 0.956941i $$-0.406250\pi$$
$$662$$ 0 0
$$663$$ 1.32046e11 0.0265408
$$664$$ 0 0
$$665$$ −1.35466e11 −0.0268616
$$666$$ 0 0
$$667$$ 7.40042e11i 0.144774i
$$668$$ 0 0
$$669$$ 1.18161e12i 0.228065i
$$670$$ 0 0
$$671$$ −1.45745e12 −0.277551
$$672$$ 0 0
$$673$$ 6.84794e11 0.128674 0.0643372 0.997928i $$-0.479507\pi$$
0.0643372 + 0.997928i $$0.479507\pi$$
$$674$$ 0 0
$$675$$ 9.18804e11i 0.170355i
$$676$$ 0 0
$$677$$ − 2.15344e12i − 0.393988i −0.980405 0.196994i $$-0.936882\pi$$
0.980405 0.196994i $$-0.0631180\pi$$
$$678$$ 0 0
$$679$$ −2.33503e11 −0.0421579
$$680$$ 0 0
$$681$$ 5.10705e12 0.909930
$$682$$ 0 0
$$683$$ 3.32483e12i 0.584623i 0.956323 + 0.292311i $$0.0944244\pi$$
−0.956323 + 0.292311i $$0.905576\pi$$
$$684$$ 0 0
$$685$$ − 1.22662e12i − 0.212863i
$$686$$ 0 0
$$687$$ 3.98281e12 0.682157
$$688$$ 0 0
$$689$$ 1.38538e12 0.234198
$$690$$ 0 0
$$691$$ − 1.50122e12i − 0.250491i −0.992126 0.125246i $$-0.960028\pi$$
0.992126 0.125246i $$-0.0399719\pi$$
$$692$$ 0 0
$$693$$ − 2.74519e11i − 0.0452140i
$$694$$ 0 0
$$695$$ 1.30241e12 0.211746
$$696$$ 0 0
$$697$$ −1.18983e12 −0.190958
$$698$$ 0 0
$$699$$ − 2.67608e12i − 0.423987i
$$700$$ 0 0
$$701$$ 3.02156e12i 0.472607i 0.971679 + 0.236304i $$0.0759360\pi$$
−0.971679 + 0.236304i $$0.924064\pi$$
$$702$$ 0 0
$$703$$ −2.59215e12 −0.400278
$$704$$ 0 0
$$705$$ 1.33279e12 0.203193
$$706$$ 0 0
$$707$$ − 2.06092e11i − 0.0310223i
$$708$$ 0 0
$$709$$ − 3.67440e12i − 0.546108i −0.961999 0.273054i $$-0.911966\pi$$
0.961999 0.273054i $$-0.0880337\pi$$
$$710$$ 0 0
$$711$$ −8.15124e10 −0.0119622
$$712$$ 0 0
$$713$$ 2.41563e11 0.0350047
$$714$$ 0 0
$$715$$ 5.11031e11i 0.0731257i
$$716$$ 0 0
$$717$$ 5.94379e12i 0.839899i
$$718$$ 0 0
$$719$$ −9.45558e12 −1.31950 −0.659748 0.751487i $$-0.729337\pi$$
−0.659748 + 0.751487i $$0.729337\pi$$
$$720$$ 0 0
$$721$$ −1.79799e11 −0.0247787
$$722$$ 0 0
$$723$$ − 4.48591e12i − 0.610560i
$$724$$ 0 0
$$725$$ 1.22127e13i 1.64169i
$$726$$ 0 0
$$727$$ 1.44431e13 1.91760 0.958798 0.284090i $$-0.0916913\pi$$
0.958798 + 0.284090i $$0.0916913\pi$$
$$728$$ 0 0
$$729$$ −2.82430e11 −0.0370370
$$730$$ 0 0
$$731$$ − 2.24349e12i − 0.290600i
$$732$$ 0 0
$$733$$ 4.09211e12i 0.523576i 0.965125 + 0.261788i $$0.0843121\pi$$
−0.965125 + 0.261788i $$0.915688\pi$$
$$734$$ 0 0
$$735$$ 1.53513e12 0.194023
$$736$$ 0 0
$$737$$ 7.94342e12 0.991753
$$738$$ 0 0
$$739$$ − 9.48921e12i − 1.17039i −0.810893 0.585195i $$-0.801018\pi$$
0.810893 0.585195i $$-0.198982\pi$$
$$740$$ 0 0
$$741$$ − 5.97666e11i − 0.0728244i
$$742$$ 0 0
$$743$$ −2.23279e12 −0.268781 −0.134390 0.990928i $$-0.542908\pi$$
−0.134390 + 0.990928i $$0.542908\pi$$
$$744$$ 0 0
$$745$$ −1.40907e12 −0.167583
$$746$$ 0 0
$$747$$ − 1.79063e12i − 0.210408i
$$748$$ 0 0
$$749$$ 5.34095e11i 0.0620084i
$$750$$ 0 0
$$751$$ −1.33252e13 −1.52860 −0.764298 0.644863i $$-0.776914\pi$$
−0.764298 + 0.644863i $$0.776914\pi$$
$$752$$ 0 0
$$753$$ 7.70552e12 0.873422
$$754$$ 0 0
$$755$$ 5.24667e12i 0.587656i
$$756$$ 0 0
$$757$$ − 2.46238e12i − 0.272536i −0.990672 0.136268i $$-0.956489\pi$$
0.990672 0.136268i $$-0.0435108\pi$$
$$758$$ 0 0
$$759$$ 6.17557e11 0.0675443
$$760$$ 0 0
$$761$$ 7.93783e12 0.857968 0.428984 0.903312i $$-0.358872\pi$$
0.428984 + 0.903312i $$0.358872\pi$$
$$762$$ 0 0
$$763$$ − 8.87137e11i − 0.0947611i
$$764$$ 0 0
$$765$$ 3.41541e11i 0.0360550i
$$766$$ 0 0
$$767$$ −1.34172e12 −0.139986
$$768$$ 0 0
$$769$$ −1.58369e13 −1.63305 −0.816527 0.577307i $$-0.804104\pi$$
−0.816527 + 0.577307i $$0.804104\pi$$
$$770$$ 0 0
$$771$$ 4.71510e12i 0.480558i
$$772$$ 0 0
$$773$$ 1.21012e13i 1.21904i 0.792769 + 0.609522i $$0.208639\pi$$
−0.792769 + 0.609522i $$0.791361\pi$$
$$774$$ 0 0
$$775$$ 3.98645e12 0.396943
$$776$$ 0 0
$$777$$ −2.42611e11 −0.0238790
$$778$$ 0 0
$$779$$ 5.38542e12i 0.523963i
$$780$$ 0 0
$$781$$ − 1.93530e13i − 1.86131i
$$782$$ 0 0
$$783$$ −3.75405e12 −0.356921
$$784$$ 0 0
$$785$$ 5.69754e12 0.535518
$$786$$ 0 0
$$787$$ 1.64934e13i 1.53258i 0.642495 + 0.766290i $$0.277899\pi$$
−0.642495 + 0.766290i $$0.722101\pi$$
$$788$$ 0 0
$$789$$ 1.02166e13i 0.938557i
$$790$$ 0 0
$$791$$ −3.44035e11 −0.0312470
$$792$$ 0 0
$$793$$ −2.96983e11 −0.0266687
$$794$$ 0 0
$$795$$ 3.58333e12i 0.318152i
$$796$$ 0 0
$$797$$ − 9.14539e12i − 0.802860i −0.915890 0.401430i $$-0.868513\pi$$
0.915890 0.401430i $$-0.131487\pi$$
$$798$$ 0 0
$$799$$ −3.81986e12 −0.331579
$$800$$ 0 0
$$801$$ 1.77565e12 0.152409
$$802$$ 0 0
$$803$$ 3.67894e12i 0.312250i
$$804$$ 0 0
$$805$$ − 2.85223e10i − 0.00239388i
$$806$$ 0 0
$$807$$ 1.00218e12 0.0831796
$$808$$ 0 0
$$809$$ 1.10523e12 0.0907160 0.0453580 0.998971i $$-0.485557\pi$$
0.0453580 + 0.998971i $$0.485557\pi$$
$$810$$ 0 0
$$811$$ − 5.17595e12i − 0.420142i −0.977686 0.210071i $$-0.932631\pi$$
0.977686 0.210071i $$-0.0673695\pi$$
$$812$$ 0 0
$$813$$ 5.18949e12i 0.416598i
$$814$$ 0 0
$$815$$ 5.75340e11 0.0456789
$$816$$ 0 0
$$817$$ −1.01545e13 −0.797369
$$818$$ 0 0
$$819$$ − 5.59383e10i − 0.00434442i
$$820$$ 0 0
$$821$$ 1.06367e13i 0.817076i 0.912741 + 0.408538i $$0.133961\pi$$
−0.912741 + 0.408538i $$0.866039\pi$$
$$822$$ 0 0
$$823$$ −1.89127e13 −1.43699 −0.718495 0.695532i $$-0.755169\pi$$
−0.718495 + 0.695532i $$0.755169\pi$$
$$824$$ 0 0
$$825$$ 1.01914e13 0.765932
$$826$$ 0 0
$$827$$ 6.62866e11i 0.0492777i 0.999696 + 0.0246389i $$0.00784359\pi$$
−0.999696 + 0.0246389i $$0.992156\pi$$
$$828$$ 0 0
$$829$$ − 2.27273e13i − 1.67129i −0.549267 0.835647i $$-0.685093\pi$$
0.549267 0.835647i $$-0.314907\pi$$
$$830$$ 0 0
$$831$$ −1.26906e13 −0.923164
$$832$$ 0 0
$$833$$ −4.39979e12 −0.316614
$$834$$ 0 0
$$835$$ − 4.79679e12i − 0.341477i
$$836$$ 0 0
$$837$$ 1.22539e12i 0.0862996i
$$838$$ 0 0
$$839$$ −2.12551e12 −0.148093 −0.0740463 0.997255i $$-0.523591\pi$$
−0.0740463 + 0.997255i $$0.523591\pi$$
$$840$$ 0 0
$$841$$ −3.53916e13 −2.43960
$$842$$ 0 0
$$843$$ 7.74010e12i 0.527865i
$$844$$ 0 0
$$845$$ − 4.91745e12i − 0.331806i
$$846$$ 0 0
$$847$$ −1.68929e12 −0.112779
$$848$$ 0 0
$$849$$ 6.47504e12 0.427718
$$850$$ 0 0
$$851$$ − 5.45778e11i − 0.0356724i
$$852$$ 0 0
$$853$$ − 2.52949e13i − 1.63592i −0.575276 0.817960i $$-0.695105\pi$$
0.575276 0.817960i $$-0.304895\pi$$
$$854$$ 0 0
$$855$$ 1.54588e12 0.0989303
$$856$$ 0 0
$$857$$ 2.42903e12 0.153822 0.0769112 0.997038i $$-0.475494\pi$$
0.0769112 + 0.997038i $$0.475494\pi$$
$$858$$ 0 0
$$859$$ 2.92636e13i 1.83383i 0.399083 + 0.916915i $$0.369329\pi$$
−0.399083 + 0.916915i $$0.630671\pi$$
$$860$$ 0 0
$$861$$ 5.04046e11i 0.0312576i
$$862$$ 0 0
$$863$$ −1.25262e13 −0.768724 −0.384362 0.923182i $$-0.625579\pi$$
−0.384362 + 0.923182i $$0.625579\pi$$
$$864$$ 0 0
$$865$$ 5.32092e12 0.323157
$$866$$ 0 0
$$867$$ 8.62674e12i 0.518514i
$$868$$ 0 0
$$869$$ 9.04136e11i 0.0537830i
$$870$$ 0 0
$$871$$ 1.61862e12 0.0952933
$$872$$ 0 0
$$873$$ 2.66466e12 0.155266
$$874$$ 0 0
$$875$$ − 1.00244e12i − 0.0578125i
$$876$$ 0 0
$$877$$ 6.98911e11i 0.0398955i 0.999801 + 0.0199477i $$0.00634998\pi$$
−0.999801 + 0.0199477i $$0.993650\pi$$
$$878$$ 0 0
$$879$$ −7.99733e12 −0.451851
$$880$$ 0 0
$$881$$ −1.80581e13 −1.00991 −0.504953 0.863147i $$-0.668490\pi$$
−0.504953 + 0.863147i $$0.668490\pi$$
$$882$$ 0 0
$$883$$ 2.54103e13i 1.40665i 0.710867 + 0.703326i $$0.248303\pi$$
−0.710867 + 0.703326i $$0.751697\pi$$
$$884$$ 0 0
$$885$$ − 3.47041e12i − 0.190168i
$$886$$ 0 0
$$887$$ 2.87381e12 0.155884 0.0779421 0.996958i $$-0.475165\pi$$
0.0779421 + 0.996958i $$0.475165\pi$$
$$888$$ 0 0
$$889$$ 1.45452e12 0.0781019
$$890$$ 0 0
$$891$$ 3.13271e12i 0.166522i
$$892$$ 0 0
$$893$$ 1.72895e13i 0.909809i
$$894$$ 0 0
$$895$$ −1.76599e12 −0.0919993
$$896$$ 0 0
$$897$$ 1.25839e11 0.00649004
$$898$$ 0 0
$$899$$ 1.62878e13i 0.831657i
$$900$$ 0 0
$$901$$ − 1.02701e13i − 0.519174i
$$902$$ 0 0
$$903$$ −9.50405e11 −0.0475679
$$904$$ 0 0
$$905$$ −6.96339e12 −0.345066
$$906$$ 0 0
$$907$$ 9.81693e10i 0.00481662i 0.999997 + 0.00240831i $$0.000766590\pi$$
−0.999997 + 0.00240831i $$0.999233\pi$$
$$908$$ 0 0
$$909$$ 2.35185e12i 0.114254i
$$910$$ 0 0
$$911$$ −3.70352e12 −0.178148 −0.0890741 0.996025i $$-0.528391\pi$$
−0.0890741 + 0.996025i $$0.528391\pi$$
$$912$$ 0 0
$$913$$ −1.98617e13 −0.946014
$$914$$ 0 0
$$915$$ − 7.68156e11i − 0.0362288i
$$916$$ 0 0
$$917$$ − 9.78564e10i − 0.00457012i
$$918$$ 0 0
$$919$$ −3.79299e13 −1.75413 −0.877066 0.480370i $$-0.840502\pi$$
−0.877066 + 0.480370i $$0.840502\pi$$
$$920$$ 0 0
$$921$$ 1.63319e13 0.747942
$$922$$ 0 0
$$923$$ − 3.94354e12i − 0.178846i
$$924$$ 0 0
$$925$$ − 9.00683e12i − 0.404515i
$$926$$ 0 0
$$927$$ 2.05180e12 0.0912591
$$928$$ 0 0
$$929$$ 3.49219e13 1.53825 0.769124 0.639099i $$-0.220693\pi$$
0.769124 + 0.639099i $$0.220693\pi$$
$$930$$ 0 0
$$931$$ 1.99144e13i 0.868747i
$$932$$ 0 0
$$933$$ 1.01717e13i 0.439468i
$$934$$ 0 0
$$935$$ 3.78837e12 0.162106
$$936$$ 0 0
$$937$$ −2.39866e13 −1.01658 −0.508288 0.861187i $$-0.669722\pi$$
−0.508288 + 0.861187i $$0.669722\pi$$
$$938$$ 0 0
$$939$$ − 7.87365e12i − 0.330507i
$$940$$ 0 0
$$941$$ − 8.10164e12i − 0.336837i −0.985716 0.168418i $$-0.946134\pi$$
0.985716 0.168418i $$-0.0538660\pi$$
$$942$$ 0 0
$$943$$ −1.13390e12 −0.0466951
$$944$$ 0 0
$$945$$ 1.44686e11 0.00590180
$$946$$ 0 0
$$947$$ 1.66583e12i 0.0673061i 0.999434 + 0.0336531i $$0.0107141\pi$$
−0.999434 + 0.0336531i $$0.989286\pi$$
$$948$$ 0 0
$$949$$ 7.49652e11i 0.0300028i
$$950$$ 0 0
$$951$$ −2.62682e12 −0.104140
$$952$$ 0 0
$$953$$ 3.98680e13 1.56569 0.782846 0.622215i $$-0.213767\pi$$
0.782846 + 0.622215i $$0.213767\pi$$
$$954$$ 0 0
$$955$$ 1.02656e13i 0.399365i
$$956$$ 0 0
$$957$$ 4.16399e13i 1.60475i
$$958$$ 0 0
$$959$$ 1.48929e12 0.0568587
$$960$$ 0 0
$$961$$ −2.11230e13 −0.798914
$$962$$ 0 0
$$963$$ − 6.09490e12i − 0.228375i
$$964$$ 0 0
$$965$$ 1.16167e13i 0.431232i
$$966$$ 0 0
$$967$$ −3.53153e13 −1.29880 −0.649402 0.760446i $$-0.724981\pi$$
−0.649402 + 0.760446i $$0.724981\pi$$
$$968$$ 0 0
$$969$$ −4.43061e12 −0.161438
$$970$$ 0 0
$$971$$ 2.43725e13i 0.879862i 0.898032 + 0.439931i $$0.144997\pi$$
−0.898032 + 0.439931i $$0.855003\pi$$
$$972$$ 0 0
$$973$$ 1.58132e12i 0.0565602i
$$974$$ 0 0
$$975$$ 2.07668e12 0.0735951
$$976$$ 0 0
$$977$$ 3.11200e13 1.09273 0.546366 0.837546i $$-0.316011\pi$$
0.546366 + 0.837546i $$0.316011\pi$$
$$978$$ 0 0
$$979$$ − 1.96955e13i − 0.685243i
$$980$$ 0 0
$$981$$ 1.01237e13i 0.349002i
$$982$$ 0 0
$$983$$ −2.69352e13 −0.920088 −0.460044 0.887896i $$-0.652166\pi$$
−0.460044 + 0.887896i $$0.652166\pi$$
$$984$$ 0 0
$$985$$ −6.84338e12 −0.231637
$$986$$ 0 0
$$987$$ 1.61820e12i 0.0542756i
$$988$$ 0 0
$$989$$ − 2.13803e12i − 0.0710608i
$$990$$ 0 0
$$991$$ 3.56596e13 1.17448 0.587240 0.809413i $$-0.300215\pi$$
0.587240 + 0.809413i $$0.300215\pi$$
$$992$$ 0 0
$$993$$ 1.73317e13 0.565677
$$994$$ 0 0
$$995$$ 1.28638e13i 0.416068i
$$996$$ 0 0
$$997$$ − 3.95731e13i − 1.26844i −0.773151 0.634222i $$-0.781320\pi$$
0.773151 0.634222i $$-0.218680\pi$$
$$998$$ 0 0
$$999$$ 2.76859e12 0.0879457
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

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

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