# Properties

 Label 256.12.b.c.129.2 Level $256$ Weight $12$ Character 256.129 Analytic conductor $196.696$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$196.695854223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 129.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.129 Dual form 256.12.b.c.129.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+252.000i q^{3} +4830.00i q^{5} -16744.0 q^{7} +113643. q^{9} +O(q^{10})$$ $$q+252.000i q^{3} +4830.00i q^{5} -16744.0 q^{7} +113643. q^{9} -534612. i q^{11} +577738. i q^{13} -1.21716e6 q^{15} -6.90593e6 q^{17} +1.06614e7i q^{19} -4.21949e6i q^{21} +1.86433e7 q^{23} +2.54992e7 q^{25} +7.32791e7i q^{27} -1.28407e8i q^{29} +5.28432e7 q^{31} +1.34722e8 q^{33} -8.08735e7i q^{35} -1.82213e8i q^{37} -1.45590e8 q^{39} -3.08120e8 q^{41} +1.71257e7i q^{43} +5.48896e8i q^{45} -2.68735e9 q^{47} -1.69697e9 q^{49} -1.74030e9i q^{51} -1.59606e9i q^{53} +2.58218e9 q^{55} -2.68668e9 q^{57} +5.18920e9i q^{59} -6.95648e9i q^{61} -1.90284e9 q^{63} -2.79047e9 q^{65} -1.54818e10i q^{67} +4.69810e9i q^{69} +9.79149e9 q^{71} -1.46379e9 q^{73} +6.42580e9i q^{75} +8.95154e9i q^{77} -3.81168e10 q^{79} +1.66519e9 q^{81} -2.93351e10i q^{83} -3.33557e10i q^{85} +3.23585e10 q^{87} +2.49929e10 q^{89} -9.67365e9i q^{91} +1.33165e10i q^{93} -5.14947e10 q^{95} +7.50136e10 q^{97} -6.07549e10i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 33488q^{7} + 227286q^{9} + O(q^{10})$$ $$2q - 33488q^{7} + 227286q^{9} - 2434320q^{15} - 13811868q^{17} + 37286544q^{23} + 50998450q^{25} + 105686336q^{31} + 269444448q^{33} - 291179952q^{39} - 616240884q^{41} - 5374696992q^{47} - 3393930414q^{49} + 5164351920q^{55} - 5373355680q^{57} - 3805676784q^{63} - 5580949080q^{65} + 19582970544q^{71} - 2927582644q^{73} - 76233691360q^{79} + 3330376722q^{81} + 64716941520q^{87} + 49985834220q^{89} - 102989317200q^{95} + 150027137092q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$3$$ 252.000i 0.598734i 0.954138 + 0.299367i $$0.0967754\pi$$
−0.954138 + 0.299367i $$0.903225\pi$$
$$4$$ 0 0
$$5$$ 4830.00i 0.691213i 0.938379 + 0.345607i $$0.112327\pi$$
−0.938379 + 0.345607i $$0.887673\pi$$
$$6$$ 0 0
$$7$$ −16744.0 −0.376548 −0.188274 0.982117i $$-0.560289\pi$$
−0.188274 + 0.982117i $$0.560289\pi$$
$$8$$ 0 0
$$9$$ 113643. 0.641518
$$10$$ 0 0
$$11$$ − 534612.i − 1.00087i −0.865773 0.500436i $$-0.833173\pi$$
0.865773 0.500436i $$-0.166827\pi$$
$$12$$ 0 0
$$13$$ 577738.i 0.431561i 0.976442 + 0.215781i $$0.0692296\pi$$
−0.976442 + 0.215781i $$0.930770\pi$$
$$14$$ 0 0
$$15$$ −1.21716e6 −0.413853
$$16$$ 0 0
$$17$$ −6.90593e6 −1.17965 −0.589825 0.807531i $$-0.700803\pi$$
−0.589825 + 0.807531i $$0.700803\pi$$
$$18$$ 0 0
$$19$$ 1.06614e7i 0.987803i 0.869518 + 0.493901i $$0.164430\pi$$
−0.869518 + 0.493901i $$0.835570\pi$$
$$20$$ 0 0
$$21$$ − 4.21949e6i − 0.225452i
$$22$$ 0 0
$$23$$ 1.86433e7 0.603975 0.301988 0.953312i $$-0.402350\pi$$
0.301988 + 0.953312i $$0.402350\pi$$
$$24$$ 0 0
$$25$$ 2.54992e7 0.522224
$$26$$ 0 0
$$27$$ 7.32791e7i 0.982832i
$$28$$ 0 0
$$29$$ − 1.28407e8i − 1.16251i −0.813720 0.581257i $$-0.802561\pi$$
0.813720 0.581257i $$-0.197439\pi$$
$$30$$ 0 0
$$31$$ 5.28432e7 0.331512 0.165756 0.986167i $$-0.446994\pi$$
0.165756 + 0.986167i $$0.446994\pi$$
$$32$$ 0 0
$$33$$ 1.34722e8 0.599256
$$34$$ 0 0
$$35$$ − 8.08735e7i − 0.260275i
$$36$$ 0 0
$$37$$ − 1.82213e8i − 0.431987i −0.976395 0.215993i $$-0.930701\pi$$
0.976395 0.215993i $$-0.0692990\pi$$
$$38$$ 0 0
$$39$$ −1.45590e8 −0.258390
$$40$$ 0 0
$$41$$ −3.08120e8 −0.415345 −0.207673 0.978198i $$-0.566589\pi$$
−0.207673 + 0.978198i $$0.566589\pi$$
$$42$$ 0 0
$$43$$ 1.71257e7i 0.0177653i 0.999961 + 0.00888264i $$0.00282747\pi$$
−0.999961 + 0.00888264i $$0.997173\pi$$
$$44$$ 0 0
$$45$$ 5.48896e8i 0.443426i
$$46$$ 0 0
$$47$$ −2.68735e9 −1.70917 −0.854586 0.519310i $$-0.826189\pi$$
−0.854586 + 0.519310i $$0.826189\pi$$
$$48$$ 0 0
$$49$$ −1.69697e9 −0.858212
$$50$$ 0 0
$$51$$ − 1.74030e9i − 0.706296i
$$52$$ 0 0
$$53$$ − 1.59606e9i − 0.524241i −0.965035 0.262120i $$-0.915578\pi$$
0.965035 0.262120i $$-0.0844217\pi$$
$$54$$ 0 0
$$55$$ 2.58218e9 0.691817
$$56$$ 0 0
$$57$$ −2.68668e9 −0.591431
$$58$$ 0 0
$$59$$ 5.18920e9i 0.944963i 0.881341 + 0.472481i $$0.156642\pi$$
−0.881341 + 0.472481i $$0.843358\pi$$
$$60$$ 0 0
$$61$$ − 6.95648e9i − 1.05457i −0.849689 0.527285i $$-0.823210\pi$$
0.849689 0.527285i $$-0.176790\pi$$
$$62$$ 0 0
$$63$$ −1.90284e9 −0.241562
$$64$$ 0 0
$$65$$ −2.79047e9 −0.298301
$$66$$ 0 0
$$67$$ − 1.54818e10i − 1.40091i −0.713696 0.700456i $$-0.752980\pi$$
0.713696 0.700456i $$-0.247020\pi$$
$$68$$ 0 0
$$69$$ 4.69810e9i 0.361620i
$$70$$ 0 0
$$71$$ 9.79149e9 0.644062 0.322031 0.946729i $$-0.395634\pi$$
0.322031 + 0.946729i $$0.395634\pi$$
$$72$$ 0 0
$$73$$ −1.46379e9 −0.0826425 −0.0413212 0.999146i $$-0.513157\pi$$
−0.0413212 + 0.999146i $$0.513157\pi$$
$$74$$ 0 0
$$75$$ 6.42580e9i 0.312673i
$$76$$ 0 0
$$77$$ 8.95154e9i 0.376876i
$$78$$ 0 0
$$79$$ −3.81168e10 −1.39370 −0.696848 0.717219i $$-0.745415\pi$$
−0.696848 + 0.717219i $$0.745415\pi$$
$$80$$ 0 0
$$81$$ 1.66519e9 0.0530635
$$82$$ 0 0
$$83$$ − 2.93351e10i − 0.817444i −0.912659 0.408722i $$-0.865975\pi$$
0.912659 0.408722i $$-0.134025\pi$$
$$84$$ 0 0
$$85$$ − 3.33557e10i − 0.815390i
$$86$$ 0 0
$$87$$ 3.23585e10 0.696037
$$88$$ 0 0
$$89$$ 2.49929e10 0.474430 0.237215 0.971457i $$-0.423765\pi$$
0.237215 + 0.971457i $$0.423765\pi$$
$$90$$ 0 0
$$91$$ − 9.67365e9i − 0.162503i
$$92$$ 0 0
$$93$$ 1.33165e10i 0.198488i
$$94$$ 0 0
$$95$$ −5.14947e10 −0.682782
$$96$$ 0 0
$$97$$ 7.50136e10 0.886942 0.443471 0.896289i $$-0.353747\pi$$
0.443471 + 0.896289i $$0.353747\pi$$
$$98$$ 0 0
$$99$$ − 6.07549e10i − 0.642078i
$$100$$ 0 0
$$101$$ 8.17430e10i 0.773896i 0.922101 + 0.386948i $$0.126471\pi$$
−0.922101 + 0.386948i $$0.873529\pi$$
$$102$$ 0 0
$$103$$ −2.25755e11 −1.91881 −0.959407 0.282025i $$-0.908994\pi$$
−0.959407 + 0.282025i $$0.908994\pi$$
$$104$$ 0 0
$$105$$ 2.03801e10 0.155835
$$106$$ 0 0
$$107$$ − 9.02413e10i − 0.622006i −0.950409 0.311003i $$-0.899335\pi$$
0.950409 0.311003i $$-0.100665\pi$$
$$108$$ 0 0
$$109$$ − 7.34827e10i − 0.457445i −0.973492 0.228723i $$-0.926545\pi$$
0.973492 0.228723i $$-0.0734549\pi$$
$$110$$ 0 0
$$111$$ 4.59178e10 0.258645
$$112$$ 0 0
$$113$$ −8.51469e10 −0.434748 −0.217374 0.976088i $$-0.569749\pi$$
−0.217374 + 0.976088i $$0.569749\pi$$
$$114$$ 0 0
$$115$$ 9.00470e10i 0.417476i
$$116$$ 0 0
$$117$$ 6.56559e10i 0.276854i
$$118$$ 0 0
$$119$$ 1.15633e11 0.444195
$$120$$ 0 0
$$121$$ −4.98320e8 −0.00174658
$$122$$ 0 0
$$123$$ − 7.76464e10i − 0.248681i
$$124$$ 0 0
$$125$$ 3.59001e11i 1.05218i
$$126$$ 0 0
$$127$$ 2.62717e11 0.705615 0.352808 0.935696i $$-0.385227\pi$$
0.352808 + 0.935696i $$0.385227\pi$$
$$128$$ 0 0
$$129$$ −4.31568e9 −0.0106367
$$130$$ 0 0
$$131$$ 6.31529e11i 1.43021i 0.699015 + 0.715107i $$0.253622\pi$$
−0.699015 + 0.715107i $$0.746378\pi$$
$$132$$ 0 0
$$133$$ − 1.78515e11i − 0.371955i
$$134$$ 0 0
$$135$$ −3.53938e11 −0.679347
$$136$$ 0 0
$$137$$ 2.97199e11 0.526119 0.263059 0.964780i $$-0.415268\pi$$
0.263059 + 0.964780i $$0.415268\pi$$
$$138$$ 0 0
$$139$$ − 5.96794e11i − 0.975535i −0.872974 0.487767i $$-0.837811\pi$$
0.872974 0.487767i $$-0.162189\pi$$
$$140$$ 0 0
$$141$$ − 6.77212e11i − 1.02334i
$$142$$ 0 0
$$143$$ 3.08866e11 0.431938
$$144$$ 0 0
$$145$$ 6.20204e11 0.803546
$$146$$ 0 0
$$147$$ − 4.27635e11i − 0.513840i
$$148$$ 0 0
$$149$$ − 1.11543e12i − 1.24428i −0.782905 0.622142i $$-0.786263\pi$$
0.782905 0.622142i $$-0.213737\pi$$
$$150$$ 0 0
$$151$$ −8.24447e11 −0.854653 −0.427326 0.904097i $$-0.640544\pi$$
−0.427326 + 0.904097i $$0.640544\pi$$
$$152$$ 0 0
$$153$$ −7.84811e11 −0.756767
$$154$$ 0 0
$$155$$ 2.55233e11i 0.229146i
$$156$$ 0 0
$$157$$ − 1.31512e12i − 1.10031i −0.835062 0.550156i $$-0.814568\pi$$
0.835062 0.550156i $$-0.185432\pi$$
$$158$$ 0 0
$$159$$ 4.02206e11 0.313881
$$160$$ 0 0
$$161$$ −3.12163e11 −0.227425
$$162$$ 0 0
$$163$$ − 3.57833e11i − 0.243584i −0.992556 0.121792i $$-0.961136\pi$$
0.992556 0.121792i $$-0.0388640\pi$$
$$164$$ 0 0
$$165$$ 6.50708e11i 0.414214i
$$166$$ 0 0
$$167$$ 2.75483e12 1.64117 0.820587 0.571521i $$-0.193646\pi$$
0.820587 + 0.571521i $$0.193646\pi$$
$$168$$ 0 0
$$169$$ 1.45838e12 0.813755
$$170$$ 0 0
$$171$$ 1.21160e12i 0.633693i
$$172$$ 0 0
$$173$$ 9.50387e11i 0.466280i 0.972443 + 0.233140i $$0.0749001\pi$$
−0.972443 + 0.233140i $$0.925100\pi$$
$$174$$ 0 0
$$175$$ −4.26959e11 −0.196642
$$176$$ 0 0
$$177$$ −1.30768e12 −0.565781
$$178$$ 0 0
$$179$$ 1.68138e12i 0.683873i 0.939723 + 0.341936i $$0.111083\pi$$
−0.939723 + 0.341936i $$0.888917\pi$$
$$180$$ 0 0
$$181$$ − 9.96774e11i − 0.381386i −0.981650 0.190693i $$-0.938927\pi$$
0.981650 0.190693i $$-0.0610735\pi$$
$$182$$ 0 0
$$183$$ 1.75303e12 0.631406
$$184$$ 0 0
$$185$$ 8.80090e11 0.298595
$$186$$ 0 0
$$187$$ 3.69200e12i 1.18068i
$$188$$ 0 0
$$189$$ − 1.22698e12i − 0.370083i
$$190$$ 0 0
$$191$$ −2.76240e12 −0.786328 −0.393164 0.919468i $$-0.628619\pi$$
−0.393164 + 0.919468i $$0.628619\pi$$
$$192$$ 0 0
$$193$$ 5.44239e12 1.46293 0.731466 0.681878i $$-0.238836\pi$$
0.731466 + 0.681878i $$0.238836\pi$$
$$194$$ 0 0
$$195$$ − 7.03200e11i − 0.178603i
$$196$$ 0 0
$$197$$ − 2.87609e12i − 0.690619i −0.938489 0.345309i $$-0.887774\pi$$
0.938489 0.345309i $$-0.112226\pi$$
$$198$$ 0 0
$$199$$ 7.28391e11 0.165452 0.0827262 0.996572i $$-0.473637\pi$$
0.0827262 + 0.996572i $$0.473637\pi$$
$$200$$ 0 0
$$201$$ 3.90142e12 0.838773
$$202$$ 0 0
$$203$$ 2.15004e12i 0.437742i
$$204$$ 0 0
$$205$$ − 1.48822e12i − 0.287092i
$$206$$ 0 0
$$207$$ 2.11868e12 0.387461
$$208$$ 0 0
$$209$$ 5.69972e12 0.988665
$$210$$ 0 0
$$211$$ − 6.79317e12i − 1.11820i −0.829101 0.559099i $$-0.811147\pi$$
0.829101 0.559099i $$-0.188853\pi$$
$$212$$ 0 0
$$213$$ 2.46745e12i 0.385622i
$$214$$ 0 0
$$215$$ −8.27172e10 −0.0122796
$$216$$ 0 0
$$217$$ −8.84806e11 −0.124830
$$218$$ 0 0
$$219$$ − 3.68875e11i − 0.0494808i
$$220$$ 0 0
$$221$$ − 3.98982e12i − 0.509092i
$$222$$ 0 0
$$223$$ −7.33486e12 −0.890667 −0.445333 0.895365i $$-0.646915\pi$$
−0.445333 + 0.895365i $$0.646915\pi$$
$$224$$ 0 0
$$225$$ 2.89781e12 0.335016
$$226$$ 0 0
$$227$$ − 1.35984e12i − 0.149743i −0.997193 0.0748713i $$-0.976145\pi$$
0.997193 0.0748713i $$-0.0238546\pi$$
$$228$$ 0 0
$$229$$ − 1.18244e13i − 1.24075i −0.784305 0.620375i $$-0.786980\pi$$
0.784305 0.620375i $$-0.213020\pi$$
$$230$$ 0 0
$$231$$ −2.25579e12 −0.225649
$$232$$ 0 0
$$233$$ 1.75634e13 1.67552 0.837761 0.546038i $$-0.183865\pi$$
0.837761 + 0.546038i $$0.183865\pi$$
$$234$$ 0 0
$$235$$ − 1.29799e13i − 1.18140i
$$236$$ 0 0
$$237$$ − 9.60545e12i − 0.834452i
$$238$$ 0 0
$$239$$ 7.13958e12 0.592221 0.296111 0.955154i $$-0.404310\pi$$
0.296111 + 0.955154i $$0.404310\pi$$
$$240$$ 0 0
$$241$$ −2.31307e11 −0.0183271 −0.00916357 0.999958i $$-0.502917\pi$$
−0.00916357 + 0.999958i $$0.502917\pi$$
$$242$$ 0 0
$$243$$ 1.34008e13i 1.01460i
$$244$$ 0 0
$$245$$ − 8.19634e12i − 0.593207i
$$246$$ 0 0
$$247$$ −6.15951e12 −0.426297
$$248$$ 0 0
$$249$$ 7.39245e12 0.489431
$$250$$ 0 0
$$251$$ − 1.29831e13i − 0.822567i −0.911507 0.411284i $$-0.865081\pi$$
0.911507 0.411284i $$-0.134919\pi$$
$$252$$ 0 0
$$253$$ − 9.96692e12i − 0.604502i
$$254$$ 0 0
$$255$$ 8.40563e12 0.488201
$$256$$ 0 0
$$257$$ 2.39612e13 1.33314 0.666571 0.745442i $$-0.267761\pi$$
0.666571 + 0.745442i $$0.267761\pi$$
$$258$$ 0 0
$$259$$ 3.05098e12i 0.162664i
$$260$$ 0 0
$$261$$ − 1.45925e13i − 0.745774i
$$262$$ 0 0
$$263$$ −2.42737e13 −1.18954 −0.594771 0.803895i $$-0.702757\pi$$
−0.594771 + 0.803895i $$0.702757\pi$$
$$264$$ 0 0
$$265$$ 7.70895e12 0.362362
$$266$$ 0 0
$$267$$ 6.29822e12i 0.284057i
$$268$$ 0 0
$$269$$ − 2.58377e13i − 1.11845i −0.829016 0.559225i $$-0.811099\pi$$
0.829016 0.559225i $$-0.188901\pi$$
$$270$$ 0 0
$$271$$ 3.76793e12 0.156593 0.0782964 0.996930i $$-0.475052\pi$$
0.0782964 + 0.996930i $$0.475052\pi$$
$$272$$ 0 0
$$273$$ 2.43776e12 0.0972963
$$274$$ 0 0
$$275$$ − 1.36322e13i − 0.522680i
$$276$$ 0 0
$$277$$ − 1.64189e13i − 0.604931i −0.953160 0.302466i $$-0.902190\pi$$
0.953160 0.302466i $$-0.0978098\pi$$
$$278$$ 0 0
$$279$$ 6.00526e12 0.212671
$$280$$ 0 0
$$281$$ −2.10357e13 −0.716263 −0.358132 0.933671i $$-0.616586\pi$$
−0.358132 + 0.933671i $$0.616586\pi$$
$$282$$ 0 0
$$283$$ − 1.67132e13i − 0.547310i −0.961828 0.273655i $$-0.911767\pi$$
0.961828 0.273655i $$-0.0882327\pi$$
$$284$$ 0 0
$$285$$ − 1.29767e13i − 0.408805i
$$286$$ 0 0
$$287$$ 5.15917e12 0.156397
$$288$$ 0 0
$$289$$ 1.34200e13 0.391575
$$290$$ 0 0
$$291$$ 1.89034e13i 0.531042i
$$292$$ 0 0
$$293$$ − 2.39269e13i − 0.647312i −0.946175 0.323656i $$-0.895088\pi$$
0.946175 0.323656i $$-0.104912\pi$$
$$294$$ 0 0
$$295$$ −2.50639e13 −0.653171
$$296$$ 0 0
$$297$$ 3.91759e13 0.983690
$$298$$ 0 0
$$299$$ 1.07709e13i 0.260652i
$$300$$ 0 0
$$301$$ − 2.86753e11i − 0.00668947i
$$302$$ 0 0
$$303$$ −2.05992e13 −0.463358
$$304$$ 0 0
$$305$$ 3.35998e13 0.728933
$$306$$ 0 0
$$307$$ 1.53111e13i 0.320439i 0.987081 + 0.160219i $$0.0512202\pi$$
−0.987081 + 0.160219i $$0.948780\pi$$
$$308$$ 0 0
$$309$$ − 5.68903e13i − 1.14886i
$$310$$ 0 0
$$311$$ 4.98752e13 0.972080 0.486040 0.873936i $$-0.338441\pi$$
0.486040 + 0.873936i $$0.338441\pi$$
$$312$$ 0 0
$$313$$ 9.94808e13 1.87174 0.935870 0.352345i $$-0.114616\pi$$
0.935870 + 0.352345i $$0.114616\pi$$
$$314$$ 0 0
$$315$$ − 9.19071e12i − 0.166971i
$$316$$ 0 0
$$317$$ − 8.33692e13i − 1.46278i −0.681958 0.731392i $$-0.738871\pi$$
0.681958 0.731392i $$-0.261129\pi$$
$$318$$ 0 0
$$319$$ −6.86477e13 −1.16353
$$320$$ 0 0
$$321$$ 2.27408e13 0.372416
$$322$$ 0 0
$$323$$ − 7.36271e13i − 1.16526i
$$324$$ 0 0
$$325$$ 1.47319e13i 0.225372i
$$326$$ 0 0
$$327$$ 1.85176e13 0.273888
$$328$$ 0 0
$$329$$ 4.49970e13 0.643585
$$330$$ 0 0
$$331$$ 6.35840e13i 0.879618i 0.898091 + 0.439809i $$0.144954\pi$$
−0.898091 + 0.439809i $$0.855046\pi$$
$$332$$ 0 0
$$333$$ − 2.07073e13i − 0.277127i
$$334$$ 0 0
$$335$$ 7.47772e13 0.968329
$$336$$ 0 0
$$337$$ 1.21001e14 1.51644 0.758221 0.651997i $$-0.226069\pi$$
0.758221 + 0.651997i $$0.226069\pi$$
$$338$$ 0 0
$$339$$ − 2.14570e13i − 0.260298i
$$340$$ 0 0
$$341$$ − 2.82506e13i − 0.331802i
$$342$$ 0 0
$$343$$ 6.15223e13 0.699705
$$344$$ 0 0
$$345$$ −2.26918e13 −0.249957
$$346$$ 0 0
$$347$$ 1.55662e14i 1.66100i 0.557020 + 0.830499i $$0.311945\pi$$
−0.557020 + 0.830499i $$0.688055\pi$$
$$348$$ 0 0
$$349$$ 2.56430e13i 0.265112i 0.991176 + 0.132556i $$0.0423184\pi$$
−0.991176 + 0.132556i $$0.957682\pi$$
$$350$$ 0 0
$$351$$ −4.23361e13 −0.424152
$$352$$ 0 0
$$353$$ 2.49098e13 0.241885 0.120943 0.992659i $$-0.461408\pi$$
0.120943 + 0.992659i $$0.461408\pi$$
$$354$$ 0 0
$$355$$ 4.72929e13i 0.445184i
$$356$$ 0 0
$$357$$ 2.91395e13i 0.265954i
$$358$$ 0 0
$$359$$ 1.57584e14 1.39474 0.697370 0.716712i $$-0.254354\pi$$
0.697370 + 0.716712i $$0.254354\pi$$
$$360$$ 0 0
$$361$$ 2.82438e12 0.0242457
$$362$$ 0 0
$$363$$ − 1.25577e11i − 0.00104574i
$$364$$ 0 0
$$365$$ − 7.07011e12i − 0.0571236i
$$366$$ 0 0
$$367$$ 1.77901e14 1.39481 0.697406 0.716676i $$-0.254338\pi$$
0.697406 + 0.716676i $$0.254338\pi$$
$$368$$ 0 0
$$369$$ −3.50157e13 −0.266452
$$370$$ 0 0
$$371$$ 2.67244e13i 0.197402i
$$372$$ 0 0
$$373$$ − 5.51617e13i − 0.395585i −0.980244 0.197792i $$-0.936623\pi$$
0.980244 0.197792i $$-0.0633772\pi$$
$$374$$ 0 0
$$375$$ −9.04683e13 −0.629976
$$376$$ 0 0
$$377$$ 7.41854e13 0.501696
$$378$$ 0 0
$$379$$ − 1.46463e14i − 0.962083i −0.876698 0.481042i $$-0.840259\pi$$
0.876698 0.481042i $$-0.159741\pi$$
$$380$$ 0 0
$$381$$ 6.62047e13i 0.422476i
$$382$$ 0 0
$$383$$ −2.31450e14 −1.43504 −0.717519 0.696539i $$-0.754722\pi$$
−0.717519 + 0.696539i $$0.754722\pi$$
$$384$$ 0 0
$$385$$ −4.32360e13 −0.260502
$$386$$ 0 0
$$387$$ 1.94622e12i 0.0113967i
$$388$$ 0 0
$$389$$ − 1.49872e14i − 0.853093i −0.904466 0.426547i $$-0.859730\pi$$
0.904466 0.426547i $$-0.140270\pi$$
$$390$$ 0 0
$$391$$ −1.28749e14 −0.712480
$$392$$ 0 0
$$393$$ −1.59145e14 −0.856317
$$394$$ 0 0
$$395$$ − 1.84104e14i − 0.963341i
$$396$$ 0 0
$$397$$ − 2.08111e14i − 1.05912i −0.848271 0.529562i $$-0.822356\pi$$
0.848271 0.529562i $$-0.177644\pi$$
$$398$$ 0 0
$$399$$ 4.49857e13 0.222702
$$400$$ 0 0
$$401$$ −1.33408e14 −0.642521 −0.321261 0.946991i $$-0.604107\pi$$
−0.321261 + 0.946991i $$0.604107\pi$$
$$402$$ 0 0
$$403$$ 3.05295e13i 0.143068i
$$404$$ 0 0
$$405$$ 8.04286e12i 0.0366782i
$$406$$ 0 0
$$407$$ −9.74134e13 −0.432364
$$408$$ 0 0
$$409$$ 2.06168e14 0.890722 0.445361 0.895351i $$-0.353075\pi$$
0.445361 + 0.895351i $$0.353075\pi$$
$$410$$ 0 0
$$411$$ 7.48941e13i 0.315005i
$$412$$ 0 0
$$413$$ − 8.68880e13i − 0.355824i
$$414$$ 0 0
$$415$$ 1.41689e14 0.565028
$$416$$ 0 0
$$417$$ 1.50392e14 0.584085
$$418$$ 0 0
$$419$$ 7.34035e13i 0.277677i 0.990315 + 0.138838i $$0.0443369\pi$$
−0.990315 + 0.138838i $$0.955663\pi$$
$$420$$ 0 0
$$421$$ 1.71112e14i 0.630563i 0.948998 + 0.315282i $$0.102099\pi$$
−0.948998 + 0.315282i $$0.897901\pi$$
$$422$$ 0 0
$$423$$ −3.05398e14 −1.09646
$$424$$ 0 0
$$425$$ −1.76096e14 −0.616042
$$426$$ 0 0
$$427$$ 1.16479e14i 0.397096i
$$428$$ 0 0
$$429$$ 7.78341e13i 0.258616i
$$430$$ 0 0
$$431$$ 7.17758e13 0.232463 0.116231 0.993222i $$-0.462919\pi$$
0.116231 + 0.993222i $$0.462919\pi$$
$$432$$ 0 0
$$433$$ 9.98812e13 0.315356 0.157678 0.987491i $$-0.449599\pi$$
0.157678 + 0.987491i $$0.449599\pi$$
$$434$$ 0 0
$$435$$ 1.56291e14i 0.481110i
$$436$$ 0 0
$$437$$ 1.98764e14i 0.596608i
$$438$$ 0 0
$$439$$ −2.90312e13 −0.0849788 −0.0424894 0.999097i $$-0.513529\pi$$
−0.0424894 + 0.999097i $$0.513529\pi$$
$$440$$ 0 0
$$441$$ −1.92848e14 −0.550558
$$442$$ 0 0
$$443$$ − 3.28370e14i − 0.914414i −0.889360 0.457207i $$-0.848850\pi$$
0.889360 0.457207i $$-0.151150\pi$$
$$444$$ 0 0
$$445$$ 1.20716e14i 0.327932i
$$446$$ 0 0
$$447$$ 2.81089e14 0.744994
$$448$$ 0 0
$$449$$ −6.12368e14 −1.58364 −0.791822 0.610752i $$-0.790867\pi$$
−0.791822 + 0.610752i $$0.790867\pi$$
$$450$$ 0 0
$$451$$ 1.64725e14i 0.415708i
$$452$$ 0 0
$$453$$ − 2.07761e14i − 0.511709i
$$454$$ 0 0
$$455$$ 4.67237e13 0.112325
$$456$$ 0 0
$$457$$ −3.03483e14 −0.712189 −0.356095 0.934450i $$-0.615892\pi$$
−0.356095 + 0.934450i $$0.615892\pi$$
$$458$$ 0 0
$$459$$ − 5.06060e14i − 1.15940i
$$460$$ 0 0
$$461$$ 7.29308e14i 1.63138i 0.578487 + 0.815691i $$0.303643\pi$$
−0.578487 + 0.815691i $$0.696357\pi$$
$$462$$ 0 0
$$463$$ −1.22188e14 −0.266891 −0.133445 0.991056i $$-0.542604\pi$$
−0.133445 + 0.991056i $$0.542604\pi$$
$$464$$ 0 0
$$465$$ −6.43186e13 −0.137197
$$466$$ 0 0
$$467$$ − 6.17381e14i − 1.28621i −0.765780 0.643103i $$-0.777647\pi$$
0.765780 0.643103i $$-0.222353\pi$$
$$468$$ 0 0
$$469$$ 2.59228e14i 0.527510i
$$470$$ 0 0
$$471$$ 3.31409e14 0.658794
$$472$$ 0 0
$$473$$ 9.15561e12 0.0177808
$$474$$ 0 0
$$475$$ 2.71858e14i 0.515854i
$$476$$ 0 0
$$477$$ − 1.81381e14i − 0.336310i
$$478$$ 0 0
$$479$$ −1.05084e15 −1.90410 −0.952052 0.305938i $$-0.901030\pi$$
−0.952052 + 0.305938i $$0.901030\pi$$
$$480$$ 0 0
$$481$$ 1.05272e14 0.186429
$$482$$ 0 0
$$483$$ − 7.86651e13i − 0.136167i
$$484$$ 0 0
$$485$$ 3.62316e14i 0.613066i
$$486$$ 0 0
$$487$$ −2.19910e14 −0.363777 −0.181889 0.983319i $$-0.558221\pi$$
−0.181889 + 0.983319i $$0.558221\pi$$
$$488$$ 0 0
$$489$$ 9.01739e13 0.145842
$$490$$ 0 0
$$491$$ 4.83863e14i 0.765199i 0.923914 + 0.382599i $$0.124971\pi$$
−0.923914 + 0.382599i $$0.875029\pi$$
$$492$$ 0 0
$$493$$ 8.86768e14i 1.37136i
$$494$$ 0 0
$$495$$ 2.93446e14 0.443813
$$496$$ 0 0
$$497$$ −1.63949e14 −0.242520
$$498$$ 0 0
$$499$$ − 1.08878e14i − 0.157538i −0.996893 0.0787691i $$-0.974901\pi$$
0.996893 0.0787691i $$-0.0250990\pi$$
$$500$$ 0 0
$$501$$ 6.94218e14i 0.982626i
$$502$$ 0 0
$$503$$ 5.06588e14 0.701506 0.350753 0.936468i $$-0.385926\pi$$
0.350753 + 0.936468i $$0.385926\pi$$
$$504$$ 0 0
$$505$$ −3.94818e14 −0.534927
$$506$$ 0 0
$$507$$ 3.67512e14i 0.487222i
$$508$$ 0 0
$$509$$ − 8.57534e13i − 0.111251i −0.998452 0.0556254i $$-0.982285\pi$$
0.998452 0.0556254i $$-0.0177153\pi$$
$$510$$ 0 0
$$511$$ 2.45097e13 0.0311188
$$512$$ 0 0
$$513$$ −7.81259e14 −0.970844
$$514$$ 0 0
$$515$$ − 1.09040e15i − 1.32631i
$$516$$ 0 0
$$517$$ 1.43669e15i 1.71066i
$$518$$ 0 0
$$519$$ −2.39498e14 −0.279178
$$520$$ 0 0
$$521$$ −9.27575e14 −1.05862 −0.529312 0.848428i $$-0.677550\pi$$
−0.529312 + 0.848428i $$0.677550\pi$$
$$522$$ 0 0
$$523$$ 2.18187e13i 0.0243820i 0.999926 + 0.0121910i $$0.00388061\pi$$
−0.999926 + 0.0121910i $$0.996119\pi$$
$$524$$ 0 0
$$525$$ − 1.07594e14i − 0.117736i
$$526$$ 0 0
$$527$$ −3.64931e14 −0.391069
$$528$$ 0 0
$$529$$ −6.05238e14 −0.635214
$$530$$ 0 0
$$531$$ 5.89717e14i 0.606211i
$$532$$ 0 0
$$533$$ − 1.78013e14i − 0.179247i
$$534$$ 0 0
$$535$$ 4.35865e14 0.429939
$$536$$ 0 0
$$537$$ −4.23709e14 −0.409458
$$538$$ 0 0
$$539$$ 9.07218e14i 0.858961i
$$540$$ 0 0
$$541$$ 1.69527e15i 1.57273i 0.617765 + 0.786363i $$0.288038\pi$$
−0.617765 + 0.786363i $$0.711962\pi$$
$$542$$ 0 0
$$543$$ 2.51187e14 0.228349
$$544$$ 0 0
$$545$$ 3.54921e14 0.316192
$$546$$ 0 0
$$547$$ 7.52145e14i 0.656706i 0.944555 + 0.328353i $$0.106494\pi$$
−0.944555 + 0.328353i $$0.893506\pi$$
$$548$$ 0 0
$$549$$ − 7.90555e14i − 0.676526i
$$550$$ 0 0
$$551$$ 1.36900e15 1.14834
$$552$$ 0 0
$$553$$ 6.38228e14 0.524793
$$554$$ 0 0
$$555$$ 2.21783e14i 0.178779i
$$556$$ 0 0
$$557$$ − 1.87489e14i − 0.148174i −0.997252 0.0740870i $$-0.976396\pi$$
0.997252 0.0740870i $$-0.0236043\pi$$
$$558$$ 0 0
$$559$$ −9.89417e12 −0.00766681
$$560$$ 0 0
$$561$$ −9.30383e14 −0.706913
$$562$$ 0 0
$$563$$ 2.44971e14i 0.182524i 0.995827 + 0.0912618i $$0.0290900\pi$$
−0.995827 + 0.0912618i $$0.970910\pi$$
$$564$$ 0 0
$$565$$ − 4.11259e14i − 0.300503i
$$566$$ 0 0
$$567$$ −2.78819e13 −0.0199809
$$568$$ 0 0
$$569$$ −1.35243e15 −0.950596 −0.475298 0.879825i $$-0.657660\pi$$
−0.475298 + 0.879825i $$0.657660\pi$$
$$570$$ 0 0
$$571$$ − 1.43223e15i − 0.987447i −0.869619 0.493723i $$-0.835636\pi$$
0.869619 0.493723i $$-0.164364\pi$$
$$572$$ 0 0
$$573$$ − 6.96126e14i − 0.470801i
$$574$$ 0 0
$$575$$ 4.75389e14 0.315410
$$576$$ 0 0
$$577$$ −8.77659e14 −0.571293 −0.285647 0.958335i $$-0.592208\pi$$
−0.285647 + 0.958335i $$0.592208\pi$$
$$578$$ 0 0
$$579$$ 1.37148e15i 0.875907i
$$580$$ 0 0
$$581$$ 4.91187e14i 0.307807i
$$582$$ 0 0
$$583$$ −8.53271e14 −0.524698
$$584$$ 0 0
$$585$$ −3.17118e14 −0.191365
$$586$$ 0 0
$$587$$ 2.43425e15i 1.44164i 0.693124 + 0.720818i $$0.256234\pi$$
−0.693124 + 0.720818i $$0.743766\pi$$
$$588$$ 0 0
$$589$$ 5.63383e14i 0.327469i
$$590$$ 0 0
$$591$$ 7.24775e14 0.413497
$$592$$ 0 0
$$593$$ −3.03318e14 −0.169863 −0.0849313 0.996387i $$-0.527067\pi$$
−0.0849313 + 0.996387i $$0.527067\pi$$
$$594$$ 0 0
$$595$$ 5.58507e14i 0.307033i
$$596$$ 0 0
$$597$$ 1.83555e14i 0.0990619i
$$598$$ 0 0
$$599$$ −1.70198e15 −0.901795 −0.450898 0.892576i $$-0.648896\pi$$
−0.450898 + 0.892576i $$0.648896\pi$$
$$600$$ 0 0
$$601$$ −2.33922e15 −1.21692 −0.608458 0.793586i $$-0.708212\pi$$
−0.608458 + 0.793586i $$0.708212\pi$$
$$602$$ 0 0
$$603$$ − 1.75940e15i − 0.898710i
$$604$$ 0 0
$$605$$ − 2.40689e12i − 0.00120726i
$$606$$ 0 0
$$607$$ 2.49607e15 1.22947 0.614737 0.788732i $$-0.289262\pi$$
0.614737 + 0.788732i $$0.289262\pi$$
$$608$$ 0 0
$$609$$ −5.41810e14 −0.262091
$$610$$ 0 0
$$611$$ − 1.55258e15i − 0.737612i
$$612$$ 0 0
$$613$$ 2.47301e15i 1.15397i 0.816756 + 0.576983i $$0.195770\pi$$
−0.816756 + 0.576983i $$0.804230\pi$$
$$614$$ 0 0
$$615$$ 3.75032e14 0.171892
$$616$$ 0 0
$$617$$ −2.43368e13 −0.0109571 −0.00547854 0.999985i $$-0.501744\pi$$
−0.00547854 + 0.999985i $$0.501744\pi$$
$$618$$ 0 0
$$619$$ − 4.22545e15i − 1.86885i −0.356160 0.934425i $$-0.615914\pi$$
0.356160 0.934425i $$-0.384086\pi$$
$$620$$ 0 0
$$621$$ 1.36616e15i 0.593606i
$$622$$ 0 0
$$623$$ −4.18481e14 −0.178645
$$624$$ 0 0
$$625$$ −4.88896e14 −0.205058
$$626$$ 0 0
$$627$$ 1.43633e15i 0.591947i
$$628$$ 0 0
$$629$$ 1.25835e15i 0.509594i
$$630$$ 0 0
$$631$$ −4.26326e15 −1.69660 −0.848302 0.529513i $$-0.822375\pi$$
−0.848302 + 0.529513i $$0.822375\pi$$
$$632$$ 0 0
$$633$$ 1.71188e15 0.669503
$$634$$ 0 0
$$635$$ 1.26892e15i 0.487731i
$$636$$ 0 0
$$637$$ − 9.80401e14i − 0.370371i
$$638$$ 0 0
$$639$$ 1.11273e15 0.413177
$$640$$ 0 0
$$641$$ 1.00830e15 0.368018 0.184009 0.982925i $$-0.441092\pi$$
0.184009 + 0.982925i $$0.441092\pi$$
$$642$$ 0 0
$$643$$ 3.03982e14i 0.109066i 0.998512 + 0.0545328i $$0.0173670\pi$$
−0.998512 + 0.0545328i $$0.982633\pi$$
$$644$$ 0 0
$$645$$ − 2.08447e13i − 0.00735221i
$$646$$ 0 0
$$647$$ 3.43583e15 1.19140 0.595700 0.803207i $$-0.296875\pi$$
0.595700 + 0.803207i $$0.296875\pi$$
$$648$$ 0 0
$$649$$ 2.77421e15 0.945788
$$650$$ 0 0
$$651$$ − 2.22971e14i − 0.0747400i
$$652$$ 0 0
$$653$$ 1.18539e15i 0.390695i 0.980734 + 0.195347i $$0.0625834\pi$$
−0.980734 + 0.195347i $$0.937417\pi$$
$$654$$ 0 0
$$655$$ −3.05028e15 −0.988583
$$656$$ 0 0
$$657$$ −1.66350e14 −0.0530167
$$658$$ 0 0
$$659$$ − 2.26510e15i − 0.709934i −0.934879 0.354967i $$-0.884492\pi$$
0.934879 0.354967i $$-0.115508\pi$$
$$660$$ 0 0
$$661$$ − 5.33012e15i − 1.64297i −0.570232 0.821484i $$-0.693147\pi$$
0.570232 0.821484i $$-0.306853\pi$$
$$662$$ 0 0
$$663$$ 1.00543e15 0.304810
$$664$$ 0 0
$$665$$ 8.62227e14 0.257100
$$666$$ 0 0
$$667$$ − 2.39392e15i − 0.702130i
$$668$$ 0 0
$$669$$ − 1.84839e15i − 0.533272i
$$670$$ 0 0
$$671$$ −3.71902e15 −1.05549
$$672$$ 0 0
$$673$$ 4.74120e15 1.32375 0.661874 0.749615i $$-0.269761\pi$$
0.661874 + 0.749615i $$0.269761\pi$$
$$674$$ 0 0
$$675$$ 1.86856e15i 0.513259i
$$676$$ 0 0
$$677$$ − 1.41307e15i − 0.381880i −0.981602 0.190940i $$-0.938846\pi$$
0.981602 0.190940i $$-0.0611535\pi$$
$$678$$ 0 0
$$679$$ −1.25603e15 −0.333976
$$680$$ 0 0
$$681$$ 3.42680e14 0.0896559
$$682$$ 0 0
$$683$$ 3.03116e15i 0.780359i 0.920739 + 0.390180i $$0.127587\pi$$
−0.920739 + 0.390180i $$0.872413\pi$$
$$684$$ 0 0
$$685$$ 1.43547e15i 0.363660i
$$686$$ 0 0
$$687$$ 2.97975e15 0.742879
$$688$$ 0 0
$$689$$ 9.22102e14 0.226242
$$690$$ 0 0
$$691$$ − 2.74731e15i − 0.663405i −0.943384 0.331703i $$-0.892377\pi$$
0.943384 0.331703i $$-0.107623\pi$$
$$692$$ 0 0
$$693$$ 1.01728e15i 0.241773i
$$694$$ 0 0
$$695$$ 2.88251e15 0.674303
$$696$$ 0 0
$$697$$ 2.12786e15 0.489962
$$698$$ 0 0
$$699$$ 4.42597e15i 1.00319i
$$700$$ 0 0
$$701$$ − 5.72747e15i − 1.27795i −0.769228 0.638974i $$-0.779359\pi$$
0.769228 0.638974i $$-0.220641\pi$$
$$702$$ 0 0
$$703$$ 1.94265e15 0.426718
$$704$$ 0 0
$$705$$ 3.27093e15 0.707345
$$706$$ 0 0
$$707$$ − 1.36870e15i − 0.291409i
$$708$$ 0 0
$$709$$ 6.98326e14i 0.146388i 0.997318 + 0.0731938i $$0.0233192\pi$$
−0.997318 + 0.0731938i $$0.976681\pi$$
$$710$$ 0 0
$$711$$ −4.33171e15 −0.894081
$$712$$ 0 0
$$713$$ 9.85170e14 0.200225
$$714$$ 0 0
$$715$$ 1.49182e15i 0.298561i
$$716$$ 0 0
$$717$$ 1.79917e15i 0.354583i
$$718$$ 0 0
$$719$$ −9.70979e15 −1.88452 −0.942260 0.334882i $$-0.891304\pi$$
−0.942260 + 0.334882i $$0.891304\pi$$
$$720$$ 0 0
$$721$$ 3.78004e15 0.722525
$$722$$ 0 0
$$723$$ − 5.82893e13i − 0.0109731i
$$724$$ 0 0
$$725$$ − 3.27427e15i − 0.607093i
$$726$$ 0 0
$$727$$ 2.46469e15 0.450114 0.225057 0.974346i $$-0.427743\pi$$
0.225057 + 0.974346i $$0.427743\pi$$
$$728$$ 0 0
$$729$$ −3.08202e15 −0.554413
$$730$$ 0 0
$$731$$ − 1.18269e14i − 0.0209568i
$$732$$ 0 0
$$733$$ − 7.91285e15i − 1.38121i −0.723230 0.690607i $$-0.757343\pi$$
0.723230 0.690607i $$-0.242657\pi$$
$$734$$ 0 0
$$735$$ 2.06548e15 0.355173
$$736$$ 0 0
$$737$$ −8.27677e15 −1.40213
$$738$$ 0 0
$$739$$ − 8.40694e15i − 1.40312i −0.712613 0.701558i $$-0.752488\pi$$
0.712613 0.701558i $$-0.247512\pi$$
$$740$$ 0 0
$$741$$ − 1.55220e15i − 0.255239i
$$742$$ 0 0
$$743$$ 1.36287e15 0.220809 0.110404 0.993887i $$-0.464785\pi$$
0.110404 + 0.993887i $$0.464785\pi$$
$$744$$ 0 0
$$745$$ 5.38754e15 0.860065
$$746$$ 0 0
$$747$$ − 3.33373e15i − 0.524405i
$$748$$ 0 0
$$749$$ 1.51100e15i 0.234215i
$$750$$ 0 0
$$751$$ −6.81722e15 −1.04133 −0.520664 0.853762i $$-0.674316\pi$$
−0.520664 + 0.853762i $$0.674316\pi$$
$$752$$ 0 0
$$753$$ 3.27173e15 0.492499
$$754$$ 0 0
$$755$$ − 3.98208e15i − 0.590747i
$$756$$ 0 0
$$757$$ − 6.67049e14i − 0.0975282i −0.998810 0.0487641i $$-0.984472\pi$$
0.998810 0.0487641i $$-0.0155283\pi$$
$$758$$ 0 0
$$759$$ 2.51166e15 0.361936
$$760$$ 0 0
$$761$$ 7.74408e15 1.09990 0.549951 0.835197i $$-0.314646\pi$$
0.549951 + 0.835197i $$0.314646\pi$$
$$762$$ 0 0
$$763$$ 1.23039e15i 0.172250i
$$764$$ 0 0
$$765$$ − 3.79064e15i − 0.523088i
$$766$$ 0 0
$$767$$ −2.99800e15 −0.407809
$$768$$ 0 0
$$769$$ 2.52411e15 0.338465 0.169232 0.985576i $$-0.445871\pi$$
0.169232 + 0.985576i $$0.445871\pi$$
$$770$$ 0 0
$$771$$ 6.03822e15i 0.798197i
$$772$$ 0 0
$$773$$ − 1.11453e16i − 1.45246i −0.687453 0.726229i $$-0.741271\pi$$
0.687453 0.726229i $$-0.258729\pi$$
$$774$$ 0 0
$$775$$ 1.34746e15 0.173124
$$776$$ 0 0
$$777$$ −7.68847e14 −0.0973922
$$778$$ 0 0
$$779$$ − 3.28500e15i − 0.410279i
$$780$$ 0 0
$$781$$ − 5.23465e15i − 0.644624i
$$782$$ 0 0
$$783$$ 9.40952e15 1.14256
$$784$$ 0 0
$$785$$ 6.35201e15 0.760551
$$786$$ 0 0
$$787$$ 1.32271e16i 1.56172i 0.624705 + 0.780861i $$0.285219\pi$$
−0.624705 + 0.780861i $$0.714781\pi$$
$$788$$ 0 0
$$789$$ − 6.11698e15i − 0.712219i
$$790$$ 0 0
$$791$$ 1.42570e15 0.163703
$$792$$ 0 0
$$793$$ 4.01902e15 0.455112
$$794$$ 0 0
$$795$$ 1.94266e15i 0.216958i
$$796$$ 0 0
$$797$$ − 2.30248e15i − 0.253615i −0.991927 0.126807i $$-0.959527\pi$$
0.991927 0.126807i $$-0.0404730\pi$$
$$798$$ 0 0
$$799$$ 1.85587e16 2.01623
$$800$$ 0 0
$$801$$ 2.84027e15 0.304355
$$802$$ 0 0
$$803$$ 7.82560e14i 0.0827146i
$$804$$ 0 0
$$805$$ − 1.50775e15i − 0.157199i
$$806$$ 0 0
$$807$$ 6.51110e15 0.669653
$$808$$ 0 0
$$809$$ −5.60472e15 −0.568639 −0.284320 0.958730i $$-0.591768\pi$$
−0.284320 + 0.958730i $$0.591768\pi$$
$$810$$ 0 0
$$811$$ 5.08516e15i 0.508968i 0.967077 + 0.254484i $$0.0819056\pi$$
−0.967077 + 0.254484i $$0.918094\pi$$
$$812$$ 0 0
$$813$$ 9.49519e14i 0.0937574i
$$814$$ 0 0
$$815$$ 1.72833e15 0.168368
$$816$$ 0 0
$$817$$ −1.82584e14 −0.0175486
$$818$$ 0 0
$$819$$ − 1.09934e15i − 0.104249i
$$820$$ 0 0
$$821$$ 2.79111e14i 0.0261150i 0.999915 + 0.0130575i $$0.00415644\pi$$
−0.999915 + 0.0130575i $$0.995844\pi$$
$$822$$ 0 0
$$823$$ −1.35265e16 −1.24878 −0.624391 0.781112i $$-0.714653\pi$$
−0.624391 + 0.781112i $$0.714653\pi$$
$$824$$ 0 0
$$825$$ 3.43531e15 0.312946
$$826$$ 0 0
$$827$$ − 2.72544e14i − 0.0244994i −0.999925 0.0122497i $$-0.996101\pi$$
0.999925 0.0122497i $$-0.00389930\pi$$
$$828$$ 0 0
$$829$$ − 1.80459e16i − 1.60077i −0.599486 0.800385i $$-0.704628\pi$$
0.599486 0.800385i $$-0.295372\pi$$
$$830$$ 0 0
$$831$$ 4.13757e15 0.362193
$$832$$ 0 0
$$833$$ 1.17191e16 1.01239
$$834$$ 0 0
$$835$$ 1.33058e16i 1.13440i
$$836$$ 0 0
$$837$$ 3.87230e15i 0.325821i
$$838$$ 0 0
$$839$$ −7.96183e15 −0.661184 −0.330592 0.943774i $$-0.607248\pi$$
−0.330592 + 0.943774i $$0.607248\pi$$
$$840$$ 0 0
$$841$$ −4.28775e15 −0.351440
$$842$$ 0 0
$$843$$ − 5.30100e15i − 0.428851i
$$844$$ 0 0
$$845$$ 7.04397e15i 0.562478i
$$846$$ 0 0
$$847$$ 8.34387e12 0.000657671 0
$$848$$ 0 0
$$849$$ 4.21172e15 0.327693
$$850$$ 0 0
$$851$$ − 3.39705e15i − 0.260909i
$$852$$ 0 0
$$853$$ − 1.49826e16i − 1.13598i −0.823037 0.567988i $$-0.807722\pi$$
0.823037 0.567988i $$-0.192278\pi$$
$$854$$ 0 0
$$855$$ −5.85201e15 −0.438017
$$856$$ 0 0
$$857$$ 2.22561e16 1.64458 0.822290 0.569068i $$-0.192696\pi$$
0.822290 + 0.569068i $$0.192696\pi$$
$$858$$ 0 0
$$859$$ − 5.44237e15i − 0.397032i −0.980098 0.198516i $$-0.936388\pi$$
0.980098 0.198516i $$-0.0636122\pi$$
$$860$$ 0 0
$$861$$ 1.30011e15i 0.0936403i
$$862$$ 0 0
$$863$$ −1.08110e16 −0.768787 −0.384393 0.923169i $$-0.625589\pi$$
−0.384393 + 0.923169i $$0.625589\pi$$
$$864$$ 0 0
$$865$$ −4.59037e15 −0.322299
$$866$$ 0 0
$$867$$ 3.38185e15i 0.234449i
$$868$$ 0 0
$$869$$ 2.03777e16i 1.39491i
$$870$$ 0 0
$$871$$ 8.94444e15 0.604579
$$872$$ 0 0
$$873$$ 8.52477e15 0.568989
$$874$$ 0 0
$$875$$ − 6.01111e15i − 0.396197i
$$876$$ 0 0
$$877$$ 2.81024e16i 1.82914i 0.404431 + 0.914568i $$0.367470\pi$$
−0.404431 + 0.914568i $$0.632530\pi$$
$$878$$ 0 0
$$879$$ 6.02957e15 0.387568
$$880$$ 0 0
$$881$$ 4.22209e15 0.268016 0.134008 0.990980i $$-0.457215\pi$$
0.134008 + 0.990980i $$0.457215\pi$$
$$882$$ 0 0
$$883$$ 5.16092e14i 0.0323551i 0.999869 + 0.0161776i $$0.00514970\pi$$
−0.999869 + 0.0161776i $$0.994850\pi$$
$$884$$ 0 0
$$885$$ − 6.31609e15i − 0.391075i
$$886$$ 0 0
$$887$$ 5.71906e15 0.349740 0.174870 0.984592i $$-0.444050\pi$$
0.174870 + 0.984592i $$0.444050\pi$$
$$888$$ 0 0
$$889$$ −4.39894e15 −0.265698
$$890$$ 0 0
$$891$$ − 8.90230e14i − 0.0531098i
$$892$$ 0 0
$$893$$ − 2.86510e16i − 1.68832i
$$894$$ 0 0
$$895$$ −8.12109e15 −0.472702
$$896$$ 0 0
$$897$$ −2.71427e15 −0.156061
$$898$$ 0 0
$$899$$ − 6.78541e15i − 0.385388i
$$900$$ 0 0
$$901$$ 1.10223e16i 0.618421i
$$902$$ 0 0
$$903$$ 7.22617e13 0.00400521
$$904$$ 0 0
$$905$$ 4.81442e15 0.263619
$$906$$ 0 0
$$907$$ 8.43778e13i 0.00456445i 0.999997 + 0.00228222i $$0.000726455\pi$$
−0.999997 + 0.00228222i $$0.999274\pi$$
$$908$$ 0 0
$$909$$ 9.28952e15i 0.496468i
$$910$$ 0 0
$$911$$ 1.10091e16 0.581298 0.290649 0.956830i $$-0.406129\pi$$
0.290649 + 0.956830i $$0.406129\pi$$
$$912$$ 0 0
$$913$$ −1.56829e16 −0.818158
$$914$$ 0 0
$$915$$ 8.46715e15i 0.436437i
$$916$$ 0 0
$$917$$ − 1.05743e16i − 0.538544i
$$918$$ 0 0
$$919$$ −4.86351e15 −0.244746 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$920$$ 0 0
$$921$$ −3.85840e15 −0.191857
$$922$$ 0 0
$$923$$ 5.65691e15i 0.277952i
$$924$$ 0 0
$$925$$ − 4.64630e15i − 0.225594i
$$926$$ 0 0
$$927$$ −2.56555e16 −1.23095
$$928$$ 0 0
$$929$$ 3.57534e15 0.169524 0.0847620 0.996401i $$-0.472987\pi$$
0.0847620 + 0.996401i $$0.472987\pi$$
$$930$$ 0 0
$$931$$ − 1.80921e16i − 0.847744i
$$932$$ 0 0
$$933$$ 1.25685e16i 0.582017i
$$934$$ 0 0
$$935$$ −1.78323e16 −0.816102
$$936$$ 0 0
$$937$$ −3.86373e16 −1.74759 −0.873795 0.486295i $$-0.838348\pi$$
−0.873795 + 0.486295i $$0.838348\pi$$
$$938$$ 0 0
$$939$$ 2.50692e16i 1.12067i
$$940$$ 0 0
$$941$$ 3.48997e16i 1.54198i 0.636846 + 0.770991i $$0.280239\pi$$
−0.636846 + 0.770991i $$0.719761\pi$$
$$942$$ 0 0
$$943$$ −5.74437e15 −0.250858
$$944$$ 0 0
$$945$$ 5.92634e15 0.255806
$$946$$ 0 0
$$947$$ − 2.85123e16i − 1.21649i −0.793751 0.608243i $$-0.791875\pi$$
0.793751 0.608243i $$-0.208125\pi$$
$$948$$ 0 0
$$949$$ − 8.45688e14i − 0.0356653i
$$950$$ 0 0
$$951$$ 2.10091e16 0.875817
$$952$$ 0 0
$$953$$ −4.00334e16 −1.64973 −0.824863 0.565332i $$-0.808748\pi$$
−0.824863 + 0.565332i $$0.808748\pi$$
$$954$$ 0 0
$$955$$ − 1.33424e16i − 0.543520i
$$956$$ 0 0
$$957$$ − 1.72992e16i − 0.696644i
$$958$$ 0 0
$$959$$ −4.97630e15 −0.198109
$$960$$ 0 0
$$961$$ −2.26161e16 −0.890100
$$962$$ 0 0
$$963$$ − 1.02553e16i − 0.399028i
$$964$$ 0 0
$$965$$ 2.62867e16i 1.01120i
$$966$$ 0 0
$$967$$ 1.84953e16 0.703422 0.351711 0.936109i $$-0.385600\pi$$
0.351711 + 0.936109i $$0.385600\pi$$
$$968$$ 0 0
$$969$$ 1.85540e16 0.697682
$$970$$ 0 0
$$971$$ 2.14877e16i 0.798884i 0.916759 + 0.399442i $$0.130796\pi$$
−0.916759 + 0.399442i $$0.869204\pi$$
$$972$$ 0 0
$$973$$ 9.99271e15i 0.367335i
$$974$$ 0 0
$$975$$ −3.71243e15 −0.134938
$$976$$ 0 0
$$977$$ −8.73880e15 −0.314074 −0.157037 0.987593i $$-0.550194\pi$$
−0.157037 + 0.987593i $$0.550194\pi$$
$$978$$ 0 0
$$979$$ − 1.33615e16i − 0.474844i
$$980$$ 0 0
$$981$$ − 8.35079e15i − 0.293459i
$$982$$ 0 0
$$983$$ 1.18924e16 0.413263 0.206631 0.978419i $$-0.433750\pi$$
0.206631 + 0.978419i $$0.433750\pi$$
$$984$$ 0 0
$$985$$ 1.38915e16 0.477365
$$986$$ 0 0
$$987$$ 1.13392e16i 0.385336i
$$988$$ 0 0
$$989$$ 3.19279e14i 0.0107298i
$$990$$ 0 0
$$991$$ −2.34409e16 −0.779056 −0.389528 0.921015i $$-0.627362\pi$$
−0.389528 + 0.921015i $$0.627362\pi$$
$$992$$ 0 0
$$993$$ −1.60232e16 −0.526657
$$994$$ 0 0
$$995$$ 3.51813e15i 0.114363i
$$996$$ 0 0
$$997$$ − 2.14004e16i − 0.688016i −0.938967 0.344008i $$-0.888215\pi$$
0.938967 0.344008i $$-0.111785\pi$$
$$998$$ 0 0
$$999$$ 1.33524e16 0.424571
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 256.12.b.c.129.2 2
4.3 odd 2 256.12.b.e.129.1 2
8.3 odd 2 256.12.b.e.129.2 2
8.5 even 2 inner 256.12.b.c.129.1 2
16.3 odd 4 1.12.a.a.1.1 1
16.5 even 4 64.12.a.f.1.1 1
16.11 odd 4 64.12.a.b.1.1 1
16.13 even 4 16.12.a.a.1.1 1
48.29 odd 4 144.12.a.d.1.1 1
48.35 even 4 9.12.a.b.1.1 1
80.3 even 4 25.12.b.b.24.2 2
80.19 odd 4 25.12.a.b.1.1 1
80.67 even 4 25.12.b.b.24.1 2
112.3 even 12 49.12.c.c.30.1 2
112.19 even 12 49.12.c.c.18.1 2
112.51 odd 12 49.12.c.b.18.1 2
112.67 odd 12 49.12.c.b.30.1 2
112.83 even 4 49.12.a.a.1.1 1
144.67 odd 12 81.12.c.d.55.1 2
144.83 even 12 81.12.c.b.28.1 2
144.115 odd 12 81.12.c.d.28.1 2
144.131 even 12 81.12.c.b.55.1 2
176.131 even 4 121.12.a.b.1.1 1
208.51 odd 4 169.12.a.a.1.1 1
240.83 odd 4 225.12.b.d.199.1 2
240.179 even 4 225.12.a.b.1.1 1
240.227 odd 4 225.12.b.d.199.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 16.3 odd 4
9.12.a.b.1.1 1 48.35 even 4
16.12.a.a.1.1 1 16.13 even 4
25.12.a.b.1.1 1 80.19 odd 4
25.12.b.b.24.1 2 80.67 even 4
25.12.b.b.24.2 2 80.3 even 4
49.12.a.a.1.1 1 112.83 even 4
49.12.c.b.18.1 2 112.51 odd 12
49.12.c.b.30.1 2 112.67 odd 12
49.12.c.c.18.1 2 112.19 even 12
49.12.c.c.30.1 2 112.3 even 12
64.12.a.b.1.1 1 16.11 odd 4
64.12.a.f.1.1 1 16.5 even 4
81.12.c.b.28.1 2 144.83 even 12
81.12.c.b.55.1 2 144.131 even 12
81.12.c.d.28.1 2 144.115 odd 12
81.12.c.d.55.1 2 144.67 odd 12
121.12.a.b.1.1 1 176.131 even 4
144.12.a.d.1.1 1 48.29 odd 4
169.12.a.a.1.1 1 208.51 odd 4
225.12.a.b.1.1 1 240.179 even 4
225.12.b.d.199.1 2 240.83 odd 4
225.12.b.d.199.2 2 240.227 odd 4
256.12.b.c.129.1 2 8.5 even 2 inner
256.12.b.c.129.2 2 1.1 even 1 trivial
256.12.b.e.129.1 2 4.3 odd 2
256.12.b.e.129.2 2 8.3 odd 2