# Properties

 Label 256.6.b.c.129.1 Level $256$ Weight $6$ Character 256.129 Analytic conductor $41.058$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.0582578721$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 129.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.129 Dual form 256.6.b.c.129.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-12.0000i q^{3} +54.0000i q^{5} -88.0000 q^{7} +99.0000 q^{9} +O(q^{10})$$ $$q-12.0000i q^{3} +54.0000i q^{5} -88.0000 q^{7} +99.0000 q^{9} -540.000i q^{11} +418.000i q^{13} +648.000 q^{15} +594.000 q^{17} +836.000i q^{19} +1056.00i q^{21} -4104.00 q^{23} +209.000 q^{25} -4104.00i q^{27} +594.000i q^{29} -4256.00 q^{31} -6480.00 q^{33} -4752.00i q^{35} -298.000i q^{37} +5016.00 q^{39} -17226.0 q^{41} +12100.0i q^{43} +5346.00i q^{45} +1296.00 q^{47} -9063.00 q^{49} -7128.00i q^{51} +19494.0i q^{53} +29160.0 q^{55} +10032.0 q^{57} +7668.00i q^{59} +34738.0i q^{61} -8712.00 q^{63} -22572.0 q^{65} +21812.0i q^{67} +49248.0i q^{69} -46872.0 q^{71} -67562.0 q^{73} -2508.00i q^{75} +47520.0i q^{77} +76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} +32076.0i q^{85} +7128.00 q^{87} -29754.0 q^{89} -36784.0i q^{91} +51072.0i q^{93} -45144.0 q^{95} -122398. q^{97} -53460.0i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 176 q^{7} + 198 q^{9}+O(q^{10})$$ 2 * q - 176 * q^7 + 198 * q^9 $$2 q - 176 q^{7} + 198 q^{9} + 1296 q^{15} + 1188 q^{17} - 8208 q^{23} + 418 q^{25} - 8512 q^{31} - 12960 q^{33} + 10032 q^{39} - 34452 q^{41} + 2592 q^{47} - 18126 q^{49} + 58320 q^{55} + 20064 q^{57} - 17424 q^{63} - 45144 q^{65} - 93744 q^{71} - 135124 q^{73} + 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 59508 q^{89} - 90288 q^{95} - 244796 q^{97}+O(q^{100})$$ 2 * q - 176 * q^7 + 198 * q^9 + 1296 * q^15 + 1188 * q^17 - 8208 * q^23 + 418 * q^25 - 8512 * q^31 - 12960 * q^33 + 10032 * q^39 - 34452 * q^41 + 2592 * q^47 - 18126 * q^49 + 58320 * q^55 + 20064 * q^57 - 17424 * q^63 - 45144 * q^65 - 93744 * q^71 - 135124 * q^73 + 153824 * q^79 - 50382 * q^81 + 14256 * q^87 - 59508 * q^89 - 90288 * q^95 - 244796 * q^97

## 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
$$3$$ − 12.0000i − 0.769800i −0.922958 0.384900i $$-0.874236\pi$$
0.922958 0.384900i $$-0.125764\pi$$
$$4$$ 0 0
$$5$$ 54.0000i 0.965981i 0.875625 + 0.482991i $$0.160450\pi$$
−0.875625 + 0.482991i $$0.839550\pi$$
$$6$$ 0 0
$$7$$ −88.0000 −0.678793 −0.339397 0.940643i $$-0.610223\pi$$
−0.339397 + 0.940643i $$0.610223\pi$$
$$8$$ 0 0
$$9$$ 99.0000 0.407407
$$10$$ 0 0
$$11$$ − 540.000i − 1.34559i −0.739830 0.672794i $$-0.765094\pi$$
0.739830 0.672794i $$-0.234906\pi$$
$$12$$ 0 0
$$13$$ 418.000i 0.685990i 0.939337 + 0.342995i $$0.111441\pi$$
−0.939337 + 0.342995i $$0.888559\pi$$
$$14$$ 0 0
$$15$$ 648.000 0.743613
$$16$$ 0 0
$$17$$ 594.000 0.498499 0.249249 0.968439i $$-0.419816\pi$$
0.249249 + 0.968439i $$0.419816\pi$$
$$18$$ 0 0
$$19$$ 836.000i 0.531279i 0.964072 + 0.265639i $$0.0855830\pi$$
−0.964072 + 0.265639i $$0.914417\pi$$
$$20$$ 0 0
$$21$$ 1056.00i 0.522535i
$$22$$ 0 0
$$23$$ −4104.00 −1.61766 −0.808831 0.588041i $$-0.799899\pi$$
−0.808831 + 0.588041i $$0.799899\pi$$
$$24$$ 0 0
$$25$$ 209.000 0.0668800
$$26$$ 0 0
$$27$$ − 4104.00i − 1.08342i
$$28$$ 0 0
$$29$$ 594.000i 0.131157i 0.997847 + 0.0655785i $$0.0208893\pi$$
−0.997847 + 0.0655785i $$0.979111\pi$$
$$30$$ 0 0
$$31$$ −4256.00 −0.795422 −0.397711 0.917511i $$-0.630195\pi$$
−0.397711 + 0.917511i $$0.630195\pi$$
$$32$$ 0 0
$$33$$ −6480.00 −1.03583
$$34$$ 0 0
$$35$$ − 4752.00i − 0.655702i
$$36$$ 0 0
$$37$$ − 298.000i − 0.0357859i −0.999840 0.0178930i $$-0.994304\pi$$
0.999840 0.0178930i $$-0.00569581\pi$$
$$38$$ 0 0
$$39$$ 5016.00 0.528075
$$40$$ 0 0
$$41$$ −17226.0 −1.60039 −0.800193 0.599742i $$-0.795270\pi$$
−0.800193 + 0.599742i $$0.795270\pi$$
$$42$$ 0 0
$$43$$ 12100.0i 0.997963i 0.866613 + 0.498981i $$0.166292\pi$$
−0.866613 + 0.498981i $$0.833708\pi$$
$$44$$ 0 0
$$45$$ 5346.00i 0.393548i
$$46$$ 0 0
$$47$$ 1296.00 0.0855777 0.0427888 0.999084i $$-0.486376\pi$$
0.0427888 + 0.999084i $$0.486376\pi$$
$$48$$ 0 0
$$49$$ −9063.00 −0.539240
$$50$$ 0 0
$$51$$ − 7128.00i − 0.383745i
$$52$$ 0 0
$$53$$ 19494.0i 0.953260i 0.879104 + 0.476630i $$0.158142\pi$$
−0.879104 + 0.476630i $$0.841858\pi$$
$$54$$ 0 0
$$55$$ 29160.0 1.29981
$$56$$ 0 0
$$57$$ 10032.0 0.408978
$$58$$ 0 0
$$59$$ 7668.00i 0.286782i 0.989666 + 0.143391i $$0.0458007\pi$$
−0.989666 + 0.143391i $$0.954199\pi$$
$$60$$ 0 0
$$61$$ 34738.0i 1.19531i 0.801754 + 0.597655i $$0.203901\pi$$
−0.801754 + 0.597655i $$0.796099\pi$$
$$62$$ 0 0
$$63$$ −8712.00 −0.276545
$$64$$ 0 0
$$65$$ −22572.0 −0.662654
$$66$$ 0 0
$$67$$ 21812.0i 0.593620i 0.954937 + 0.296810i $$0.0959228\pi$$
−0.954937 + 0.296810i $$0.904077\pi$$
$$68$$ 0 0
$$69$$ 49248.0i 1.24528i
$$70$$ 0 0
$$71$$ −46872.0 −1.10349 −0.551744 0.834014i $$-0.686037\pi$$
−0.551744 + 0.834014i $$0.686037\pi$$
$$72$$ 0 0
$$73$$ −67562.0 −1.48387 −0.741934 0.670473i $$-0.766091\pi$$
−0.741934 + 0.670473i $$0.766091\pi$$
$$74$$ 0 0
$$75$$ − 2508.00i − 0.0514842i
$$76$$ 0 0
$$77$$ 47520.0i 0.913376i
$$78$$ 0 0
$$79$$ 76912.0 1.38652 0.693260 0.720687i $$-0.256174\pi$$
0.693260 + 0.720687i $$0.256174\pi$$
$$80$$ 0 0
$$81$$ −25191.0 −0.426612
$$82$$ 0 0
$$83$$ 67716.0i 1.07894i 0.842006 + 0.539468i $$0.181375\pi$$
−0.842006 + 0.539468i $$0.818625\pi$$
$$84$$ 0 0
$$85$$ 32076.0i 0.481541i
$$86$$ 0 0
$$87$$ 7128.00 0.100965
$$88$$ 0 0
$$89$$ −29754.0 −0.398172 −0.199086 0.979982i $$-0.563797\pi$$
−0.199086 + 0.979982i $$0.563797\pi$$
$$90$$ 0 0
$$91$$ − 36784.0i − 0.465646i
$$92$$ 0 0
$$93$$ 51072.0i 0.612316i
$$94$$ 0 0
$$95$$ −45144.0 −0.513205
$$96$$ 0 0
$$97$$ −122398. −1.32082 −0.660412 0.750903i $$-0.729618\pi$$
−0.660412 + 0.750903i $$0.729618\pi$$
$$98$$ 0 0
$$99$$ − 53460.0i − 0.548202i
$$100$$ 0 0
$$101$$ 11286.0i 0.110087i 0.998484 + 0.0550436i $$0.0175298\pi$$
−0.998484 + 0.0550436i $$0.982470\pi$$
$$102$$ 0 0
$$103$$ −27256.0 −0.253145 −0.126572 0.991957i $$-0.540398\pi$$
−0.126572 + 0.991957i $$0.540398\pi$$
$$104$$ 0 0
$$105$$ −57024.0 −0.504759
$$106$$ 0 0
$$107$$ − 122364.i − 1.03322i −0.856220 0.516612i $$-0.827193\pi$$
0.856220 0.516612i $$-0.172807\pi$$
$$108$$ 0 0
$$109$$ − 99902.0i − 0.805393i −0.915334 0.402697i $$-0.868073\pi$$
0.915334 0.402697i $$-0.131927\pi$$
$$110$$ 0 0
$$111$$ −3576.00 −0.0275480
$$112$$ 0 0
$$113$$ −29646.0 −0.218409 −0.109204 0.994019i $$-0.534830\pi$$
−0.109204 + 0.994019i $$0.534830\pi$$
$$114$$ 0 0
$$115$$ − 221616.i − 1.56263i
$$116$$ 0 0
$$117$$ 41382.0i 0.279477i
$$118$$ 0 0
$$119$$ −52272.0 −0.338378
$$120$$ 0 0
$$121$$ −130549. −0.810607
$$122$$ 0 0
$$123$$ 206712.i 1.23198i
$$124$$ 0 0
$$125$$ 180036.i 1.03059i
$$126$$ 0 0
$$127$$ −336512. −1.85136 −0.925681 0.378305i $$-0.876507\pi$$
−0.925681 + 0.378305i $$0.876507\pi$$
$$128$$ 0 0
$$129$$ 145200. 0.768232
$$130$$ 0 0
$$131$$ 100980.i 0.514111i 0.966397 + 0.257056i $$0.0827524\pi$$
−0.966397 + 0.257056i $$0.917248\pi$$
$$132$$ 0 0
$$133$$ − 73568.0i − 0.360628i
$$134$$ 0 0
$$135$$ 221616. 1.04657
$$136$$ 0 0
$$137$$ 317142. 1.44362 0.721809 0.692092i $$-0.243311\pi$$
0.721809 + 0.692092i $$0.243311\pi$$
$$138$$ 0 0
$$139$$ 148324.i 0.651140i 0.945518 + 0.325570i $$0.105556\pi$$
−0.945518 + 0.325570i $$0.894444\pi$$
$$140$$ 0 0
$$141$$ − 15552.0i − 0.0658777i
$$142$$ 0 0
$$143$$ 225720. 0.923060
$$144$$ 0 0
$$145$$ −32076.0 −0.126695
$$146$$ 0 0
$$147$$ 108756.i 0.415107i
$$148$$ 0 0
$$149$$ 196614.i 0.725519i 0.931883 + 0.362759i $$0.118165\pi$$
−0.931883 + 0.362759i $$0.881835\pi$$
$$150$$ 0 0
$$151$$ 74360.0 0.265398 0.132699 0.991156i $$-0.457636\pi$$
0.132699 + 0.991156i $$0.457636\pi$$
$$152$$ 0 0
$$153$$ 58806.0 0.203092
$$154$$ 0 0
$$155$$ − 229824.i − 0.768362i
$$156$$ 0 0
$$157$$ − 120878.i − 0.391380i −0.980666 0.195690i $$-0.937305\pi$$
0.980666 0.195690i $$-0.0626946\pi$$
$$158$$ 0 0
$$159$$ 233928. 0.733820
$$160$$ 0 0
$$161$$ 361152. 1.09806
$$162$$ 0 0
$$163$$ − 111340.i − 0.328233i −0.986441 0.164116i $$-0.947523\pi$$
0.986441 0.164116i $$-0.0524773\pi$$
$$164$$ 0 0
$$165$$ − 349920.i − 1.00060i
$$166$$ 0 0
$$167$$ −491832. −1.36466 −0.682332 0.731043i $$-0.739034\pi$$
−0.682332 + 0.731043i $$0.739034\pi$$
$$168$$ 0 0
$$169$$ 196569. 0.529417
$$170$$ 0 0
$$171$$ 82764.0i 0.216447i
$$172$$ 0 0
$$173$$ − 707454.i − 1.79714i −0.438826 0.898572i $$-0.644605\pi$$
0.438826 0.898572i $$-0.355395\pi$$
$$174$$ 0 0
$$175$$ −18392.0 −0.0453977
$$176$$ 0 0
$$177$$ 92016.0 0.220765
$$178$$ 0 0
$$179$$ 493668.i 1.15160i 0.817590 + 0.575801i $$0.195310\pi$$
−0.817590 + 0.575801i $$0.804690\pi$$
$$180$$ 0 0
$$181$$ − 559450.i − 1.26930i −0.772799 0.634651i $$-0.781144\pi$$
0.772799 0.634651i $$-0.218856\pi$$
$$182$$ 0 0
$$183$$ 416856. 0.920149
$$184$$ 0 0
$$185$$ 16092.0 0.0345685
$$186$$ 0 0
$$187$$ − 320760.i − 0.670774i
$$188$$ 0 0
$$189$$ 361152.i 0.735420i
$$190$$ 0 0
$$191$$ 724032. 1.43607 0.718033 0.696009i $$-0.245043\pi$$
0.718033 + 0.696009i $$0.245043\pi$$
$$192$$ 0 0
$$193$$ 7106.00 0.0137319 0.00686597 0.999976i $$-0.497814\pi$$
0.00686597 + 0.999976i $$0.497814\pi$$
$$194$$ 0 0
$$195$$ 270864.i 0.510111i
$$196$$ 0 0
$$197$$ − 530442.i − 0.973806i −0.873456 0.486903i $$-0.838127\pi$$
0.873456 0.486903i $$-0.161873\pi$$
$$198$$ 0 0
$$199$$ 56168.0 0.100544 0.0502720 0.998736i $$-0.483991\pi$$
0.0502720 + 0.998736i $$0.483991\pi$$
$$200$$ 0 0
$$201$$ 261744. 0.456969
$$202$$ 0 0
$$203$$ − 52272.0i − 0.0890285i
$$204$$ 0 0
$$205$$ − 930204.i − 1.54594i
$$206$$ 0 0
$$207$$ −406296. −0.659047
$$208$$ 0 0
$$209$$ 451440. 0.714882
$$210$$ 0 0
$$211$$ − 339196.i − 0.524499i −0.965000 0.262249i $$-0.915536\pi$$
0.965000 0.262249i $$-0.0844643\pi$$
$$212$$ 0 0
$$213$$ 562464.i 0.849465i
$$214$$ 0 0
$$215$$ −653400. −0.964013
$$216$$ 0 0
$$217$$ 374528. 0.539927
$$218$$ 0 0
$$219$$ 810744.i 1.14228i
$$220$$ 0 0
$$221$$ 248292.i 0.341965i
$$222$$ 0 0
$$223$$ −779360. −1.04948 −0.524742 0.851261i $$-0.675838\pi$$
−0.524742 + 0.851261i $$0.675838\pi$$
$$224$$ 0 0
$$225$$ 20691.0 0.0272474
$$226$$ 0 0
$$227$$ − 744876.i − 0.959443i −0.877421 0.479722i $$-0.840738\pi$$
0.877421 0.479722i $$-0.159262\pi$$
$$228$$ 0 0
$$229$$ − 272746.i − 0.343692i −0.985124 0.171846i $$-0.945027\pi$$
0.985124 0.171846i $$-0.0549732\pi$$
$$230$$ 0 0
$$231$$ 570240. 0.703117
$$232$$ 0 0
$$233$$ 153846. 0.185651 0.0928253 0.995682i $$-0.470410\pi$$
0.0928253 + 0.995682i $$0.470410\pi$$
$$234$$ 0 0
$$235$$ 69984.0i 0.0826664i
$$236$$ 0 0
$$237$$ − 922944.i − 1.06734i
$$238$$ 0 0
$$239$$ −1.15474e6 −1.30764 −0.653820 0.756650i $$-0.726834\pi$$
−0.653820 + 0.756650i $$0.726834\pi$$
$$240$$ 0 0
$$241$$ 657074. 0.728738 0.364369 0.931255i $$-0.381285\pi$$
0.364369 + 0.931255i $$0.381285\pi$$
$$242$$ 0 0
$$243$$ − 694980.i − 0.755017i
$$244$$ 0 0
$$245$$ − 489402.i − 0.520895i
$$246$$ 0 0
$$247$$ −349448. −0.364452
$$248$$ 0 0
$$249$$ 812592. 0.830566
$$250$$ 0 0
$$251$$ − 1.34190e6i − 1.34442i −0.740359 0.672211i $$-0.765345\pi$$
0.740359 0.672211i $$-0.234655\pi$$
$$252$$ 0 0
$$253$$ 2.21616e6i 2.17671i
$$254$$ 0 0
$$255$$ 384912. 0.370690
$$256$$ 0 0
$$257$$ 132354. 0.124998 0.0624992 0.998045i $$-0.480093\pi$$
0.0624992 + 0.998045i $$0.480093\pi$$
$$258$$ 0 0
$$259$$ 26224.0i 0.0242912i
$$260$$ 0 0
$$261$$ 58806.0i 0.0534343i
$$262$$ 0 0
$$263$$ 943272. 0.840906 0.420453 0.907314i $$-0.361871\pi$$
0.420453 + 0.907314i $$0.361871\pi$$
$$264$$ 0 0
$$265$$ −1.05268e6 −0.920831
$$266$$ 0 0
$$267$$ 357048.i 0.306513i
$$268$$ 0 0
$$269$$ − 967518.i − 0.815227i −0.913155 0.407613i $$-0.866361\pi$$
0.913155 0.407613i $$-0.133639\pi$$
$$270$$ 0 0
$$271$$ 518320. 0.428721 0.214360 0.976755i $$-0.431233\pi$$
0.214360 + 0.976755i $$0.431233\pi$$
$$272$$ 0 0
$$273$$ −441408. −0.358454
$$274$$ 0 0
$$275$$ − 112860.i − 0.0899929i
$$276$$ 0 0
$$277$$ 2.22273e6i 1.74055i 0.492566 + 0.870275i $$0.336059\pi$$
−0.492566 + 0.870275i $$0.663941\pi$$
$$278$$ 0 0
$$279$$ −421344. −0.324061
$$280$$ 0 0
$$281$$ 196614. 0.148542 0.0742709 0.997238i $$-0.476337\pi$$
0.0742709 + 0.997238i $$0.476337\pi$$
$$282$$ 0 0
$$283$$ 1.55228e6i 1.15213i 0.817403 + 0.576067i $$0.195413\pi$$
−0.817403 + 0.576067i $$0.804587\pi$$
$$284$$ 0 0
$$285$$ 541728.i 0.395066i
$$286$$ 0 0
$$287$$ 1.51589e6 1.08633
$$288$$ 0 0
$$289$$ −1.06702e6 −0.751499
$$290$$ 0 0
$$291$$ 1.46878e6i 1.01677i
$$292$$ 0 0
$$293$$ − 1.07217e6i − 0.729616i −0.931083 0.364808i $$-0.881135\pi$$
0.931083 0.364808i $$-0.118865\pi$$
$$294$$ 0 0
$$295$$ −414072. −0.277026
$$296$$ 0 0
$$297$$ −2.21616e6 −1.45784
$$298$$ 0 0
$$299$$ − 1.71547e6i − 1.10970i
$$300$$ 0 0
$$301$$ − 1.06480e6i − 0.677410i
$$302$$ 0 0
$$303$$ 135432. 0.0847451
$$304$$ 0 0
$$305$$ −1.87585e6 −1.15465
$$306$$ 0 0
$$307$$ 1.58589e6i 0.960346i 0.877174 + 0.480173i $$0.159426\pi$$
−0.877174 + 0.480173i $$0.840574\pi$$
$$308$$ 0 0
$$309$$ 327072.i 0.194871i
$$310$$ 0 0
$$311$$ −730728. −0.428405 −0.214203 0.976789i $$-0.568715\pi$$
−0.214203 + 0.976789i $$0.568715\pi$$
$$312$$ 0 0
$$313$$ −584858. −0.337435 −0.168717 0.985664i $$-0.553962\pi$$
−0.168717 + 0.985664i $$0.553962\pi$$
$$314$$ 0 0
$$315$$ − 470448.i − 0.267138i
$$316$$ 0 0
$$317$$ 2.48287e6i 1.38773i 0.720105 + 0.693865i $$0.244094\pi$$
−0.720105 + 0.693865i $$0.755906\pi$$
$$318$$ 0 0
$$319$$ 320760. 0.176483
$$320$$ 0 0
$$321$$ −1.46837e6 −0.795376
$$322$$ 0 0
$$323$$ 496584.i 0.264842i
$$324$$ 0 0
$$325$$ 87362.0i 0.0458790i
$$326$$ 0 0
$$327$$ −1.19882e6 −0.619992
$$328$$ 0 0
$$329$$ −114048. −0.0580895
$$330$$ 0 0
$$331$$ − 377948.i − 0.189610i −0.995496 0.0948052i $$-0.969777\pi$$
0.995496 0.0948052i $$-0.0302228\pi$$
$$332$$ 0 0
$$333$$ − 29502.0i − 0.0145794i
$$334$$ 0 0
$$335$$ −1.17785e6 −0.573426
$$336$$ 0 0
$$337$$ 639122. 0.306555 0.153278 0.988183i $$-0.451017\pi$$
0.153278 + 0.988183i $$0.451017\pi$$
$$338$$ 0 0
$$339$$ 355752.i 0.168131i
$$340$$ 0 0
$$341$$ 2.29824e6i 1.07031i
$$342$$ 0 0
$$343$$ 2.27656e6 1.04483
$$344$$ 0 0
$$345$$ −2.65939e6 −1.20291
$$346$$ 0 0
$$347$$ 2.90466e6i 1.29501i 0.762063 + 0.647503i $$0.224187\pi$$
−0.762063 + 0.647503i $$0.775813\pi$$
$$348$$ 0 0
$$349$$ 3.99157e6i 1.75420i 0.480304 + 0.877102i $$0.340526\pi$$
−0.480304 + 0.877102i $$0.659474\pi$$
$$350$$ 0 0
$$351$$ 1.71547e6 0.743217
$$352$$ 0 0
$$353$$ 1.42922e6 0.610466 0.305233 0.952278i $$-0.401266\pi$$
0.305233 + 0.952278i $$0.401266\pi$$
$$354$$ 0 0
$$355$$ − 2.53109e6i − 1.06595i
$$356$$ 0 0
$$357$$ 627264.i 0.260483i
$$358$$ 0 0
$$359$$ 1.16186e6 0.475794 0.237897 0.971290i $$-0.423542\pi$$
0.237897 + 0.971290i $$0.423542\pi$$
$$360$$ 0 0
$$361$$ 1.77720e6 0.717743
$$362$$ 0 0
$$363$$ 1.56659e6i 0.624005i
$$364$$ 0 0
$$365$$ − 3.64835e6i − 1.43339i
$$366$$ 0 0
$$367$$ 1.08923e6 0.422139 0.211069 0.977471i $$-0.432305\pi$$
0.211069 + 0.977471i $$0.432305\pi$$
$$368$$ 0 0
$$369$$ −1.70537e6 −0.652009
$$370$$ 0 0
$$371$$ − 1.71547e6i − 0.647066i
$$372$$ 0 0
$$373$$ 3.50577e6i 1.30470i 0.757918 + 0.652350i $$0.226217\pi$$
−0.757918 + 0.652350i $$0.773783\pi$$
$$374$$ 0 0
$$375$$ 2.16043e6 0.793346
$$376$$ 0 0
$$377$$ −248292. −0.0899724
$$378$$ 0 0
$$379$$ − 4.04385e6i − 1.44610i −0.690798 0.723048i $$-0.742740\pi$$
0.690798 0.723048i $$-0.257260\pi$$
$$380$$ 0 0
$$381$$ 4.03814e6i 1.42518i
$$382$$ 0 0
$$383$$ −5.18746e6 −1.80700 −0.903499 0.428591i $$-0.859010\pi$$
−0.903499 + 0.428591i $$0.859010\pi$$
$$384$$ 0 0
$$385$$ −2.56608e6 −0.882304
$$386$$ 0 0
$$387$$ 1.19790e6i 0.406577i
$$388$$ 0 0
$$389$$ − 950346.i − 0.318425i −0.987244 0.159213i $$-0.949104\pi$$
0.987244 0.159213i $$-0.0508956\pi$$
$$390$$ 0 0
$$391$$ −2.43778e6 −0.806403
$$392$$ 0 0
$$393$$ 1.21176e6 0.395763
$$394$$ 0 0
$$395$$ 4.15325e6i 1.33935i
$$396$$ 0 0
$$397$$ 520738.i 0.165822i 0.996557 + 0.0829112i $$0.0264218\pi$$
−0.996557 + 0.0829112i $$0.973578\pi$$
$$398$$ 0 0
$$399$$ −882816. −0.277612
$$400$$ 0 0
$$401$$ 764370. 0.237379 0.118690 0.992931i $$-0.462131\pi$$
0.118690 + 0.992931i $$0.462131\pi$$
$$402$$ 0 0
$$403$$ − 1.77901e6i − 0.545651i
$$404$$ 0 0
$$405$$ − 1.36031e6i − 0.412099i
$$406$$ 0 0
$$407$$ −160920. −0.0481531
$$408$$ 0 0
$$409$$ −2.64051e6 −0.780511 −0.390255 0.920707i $$-0.627613\pi$$
−0.390255 + 0.920707i $$0.627613\pi$$
$$410$$ 0 0
$$411$$ − 3.80570e6i − 1.11130i
$$412$$ 0 0
$$413$$ − 674784.i − 0.194666i
$$414$$ 0 0
$$415$$ −3.65666e6 −1.04223
$$416$$ 0 0
$$417$$ 1.77989e6 0.501248
$$418$$ 0 0
$$419$$ − 4.98020e6i − 1.38584i −0.721016 0.692918i $$-0.756325\pi$$
0.721016 0.692918i $$-0.243675\pi$$
$$420$$ 0 0
$$421$$ − 237994.i − 0.0654426i −0.999465 0.0327213i $$-0.989583\pi$$
0.999465 0.0327213i $$-0.0104174\pi$$
$$422$$ 0 0
$$423$$ 128304. 0.0348650
$$424$$ 0 0
$$425$$ 124146. 0.0333396
$$426$$ 0 0
$$427$$ − 3.05694e6i − 0.811368i
$$428$$ 0 0
$$429$$ − 2.70864e6i − 0.710572i
$$430$$ 0 0
$$431$$ 3.88238e6 1.00671 0.503356 0.864079i $$-0.332098\pi$$
0.503356 + 0.864079i $$0.332098\pi$$
$$432$$ 0 0
$$433$$ −66958.0 −0.0171626 −0.00858129 0.999963i $$-0.502732\pi$$
−0.00858129 + 0.999963i $$0.502732\pi$$
$$434$$ 0 0
$$435$$ 384912.i 0.0975300i
$$436$$ 0 0
$$437$$ − 3.43094e6i − 0.859429i
$$438$$ 0 0
$$439$$ −6.50135e6 −1.61006 −0.805031 0.593233i $$-0.797851\pi$$
−0.805031 + 0.593233i $$0.797851\pi$$
$$440$$ 0 0
$$441$$ −897237. −0.219690
$$442$$ 0 0
$$443$$ 4.60760e6i 1.11549i 0.830012 + 0.557745i $$0.188333\pi$$
−0.830012 + 0.557745i $$0.811667\pi$$
$$444$$ 0 0
$$445$$ − 1.60672e6i − 0.384626i
$$446$$ 0 0
$$447$$ 2.35937e6 0.558505
$$448$$ 0 0
$$449$$ 3.77671e6 0.884092 0.442046 0.896992i $$-0.354253\pi$$
0.442046 + 0.896992i $$0.354253\pi$$
$$450$$ 0 0
$$451$$ 9.30204e6i 2.15346i
$$452$$ 0 0
$$453$$ − 892320.i − 0.204303i
$$454$$ 0 0
$$455$$ 1.98634e6 0.449805
$$456$$ 0 0
$$457$$ 3.18069e6 0.712412 0.356206 0.934407i $$-0.384070\pi$$
0.356206 + 0.934407i $$0.384070\pi$$
$$458$$ 0 0
$$459$$ − 2.43778e6i − 0.540085i
$$460$$ 0 0
$$461$$ − 6.68547e6i − 1.46514i −0.680691 0.732571i $$-0.738320\pi$$
0.680691 0.732571i $$-0.261680\pi$$
$$462$$ 0 0
$$463$$ 4.35122e6 0.943318 0.471659 0.881781i $$-0.343655\pi$$
0.471659 + 0.881781i $$0.343655\pi$$
$$464$$ 0 0
$$465$$ −2.75789e6 −0.591486
$$466$$ 0 0
$$467$$ 7.07994e6i 1.50223i 0.660170 + 0.751117i $$0.270484\pi$$
−0.660170 + 0.751117i $$0.729516\pi$$
$$468$$ 0 0
$$469$$ − 1.91946e6i − 0.402945i
$$470$$ 0 0
$$471$$ −1.45054e6 −0.301284
$$472$$ 0 0
$$473$$ 6.53400e6 1.34285
$$474$$ 0 0
$$475$$ 174724.i 0.0355319i
$$476$$ 0 0
$$477$$ 1.92991e6i 0.388365i
$$478$$ 0 0
$$479$$ −3.22186e6 −0.641604 −0.320802 0.947146i $$-0.603952\pi$$
−0.320802 + 0.947146i $$0.603952\pi$$
$$480$$ 0 0
$$481$$ 124564. 0.0245488
$$482$$ 0 0
$$483$$ − 4.33382e6i − 0.845286i
$$484$$ 0 0
$$485$$ − 6.60949e6i − 1.27589i
$$486$$ 0 0
$$487$$ 2.29710e6 0.438891 0.219446 0.975625i $$-0.429575\pi$$
0.219446 + 0.975625i $$0.429575\pi$$
$$488$$ 0 0
$$489$$ −1.33608e6 −0.252674
$$490$$ 0 0
$$491$$ − 2.82150e6i − 0.528173i −0.964499 0.264087i $$-0.914930\pi$$
0.964499 0.264087i $$-0.0850705\pi$$
$$492$$ 0 0
$$493$$ 352836.i 0.0653816i
$$494$$ 0 0
$$495$$ 2.88684e6 0.529553
$$496$$ 0 0
$$497$$ 4.12474e6 0.749040
$$498$$ 0 0
$$499$$ − 4.13628e6i − 0.743634i −0.928306 0.371817i $$-0.878735\pi$$
0.928306 0.371817i $$-0.121265\pi$$
$$500$$ 0 0
$$501$$ 5.90198e6i 1.05052i
$$502$$ 0 0
$$503$$ 8.33263e6 1.46846 0.734230 0.678901i $$-0.237543\pi$$
0.734230 + 0.678901i $$0.237543\pi$$
$$504$$ 0 0
$$505$$ −609444. −0.106342
$$506$$ 0 0
$$507$$ − 2.35883e6i − 0.407546i
$$508$$ 0 0
$$509$$ − 4.34101e6i − 0.742670i −0.928499 0.371335i $$-0.878900\pi$$
0.928499 0.371335i $$-0.121100\pi$$
$$510$$ 0 0
$$511$$ 5.94546e6 1.00724
$$512$$ 0 0
$$513$$ 3.43094e6 0.575599
$$514$$ 0 0
$$515$$ − 1.47182e6i − 0.244533i
$$516$$ 0 0
$$517$$ − 699840.i − 0.115152i
$$518$$ 0 0
$$519$$ −8.48945e6 −1.38344
$$520$$ 0 0
$$521$$ 6.74185e6 1.08814 0.544070 0.839040i $$-0.316883\pi$$
0.544070 + 0.839040i $$0.316883\pi$$
$$522$$ 0 0
$$523$$ 7.72196e6i 1.23445i 0.786787 + 0.617224i $$0.211743\pi$$
−0.786787 + 0.617224i $$0.788257\pi$$
$$524$$ 0 0
$$525$$ 220704.i 0.0349472i
$$526$$ 0 0
$$527$$ −2.52806e6 −0.396517
$$528$$ 0 0
$$529$$ 1.04065e7 1.61683
$$530$$ 0 0
$$531$$ 759132.i 0.116837i
$$532$$ 0 0
$$533$$ − 7.20047e6i − 1.09785i
$$534$$ 0 0
$$535$$ 6.60766e6 0.998075
$$536$$ 0 0
$$537$$ 5.92402e6 0.886504
$$538$$ 0 0
$$539$$ 4.89402e6i 0.725594i
$$540$$ 0 0
$$541$$ 682066.i 0.100192i 0.998744 + 0.0500960i $$0.0159527\pi$$
−0.998744 + 0.0500960i $$0.984047\pi$$
$$542$$ 0 0
$$543$$ −6.71340e6 −0.977109
$$544$$ 0 0
$$545$$ 5.39471e6 0.777995
$$546$$ 0 0
$$547$$ 2.15772e6i 0.308337i 0.988045 + 0.154169i $$0.0492699\pi$$
−0.988045 + 0.154169i $$0.950730\pi$$
$$548$$ 0 0
$$549$$ 3.43906e6i 0.486978i
$$550$$ 0 0
$$551$$ −496584. −0.0696809
$$552$$ 0 0
$$553$$ −6.76826e6 −0.941161
$$554$$ 0 0
$$555$$ − 193104.i − 0.0266109i
$$556$$ 0 0
$$557$$ 2.67597e6i 0.365463i 0.983163 + 0.182731i $$0.0584939\pi$$
−0.983163 + 0.182731i $$0.941506\pi$$
$$558$$ 0 0
$$559$$ −5.05780e6 −0.684592
$$560$$ 0 0
$$561$$ −3.84912e6 −0.516362
$$562$$ 0 0
$$563$$ − 3.55331e6i − 0.472457i −0.971698 0.236228i $$-0.924089\pi$$
0.971698 0.236228i $$-0.0759113\pi$$
$$564$$ 0 0
$$565$$ − 1.60088e6i − 0.210979i
$$566$$ 0 0
$$567$$ 2.21681e6 0.289581
$$568$$ 0 0
$$569$$ 1.29225e7 1.67327 0.836633 0.547764i $$-0.184521\pi$$
0.836633 + 0.547764i $$0.184521\pi$$
$$570$$ 0 0
$$571$$ 6.08357e6i 0.780851i 0.920634 + 0.390426i $$0.127672\pi$$
−0.920634 + 0.390426i $$0.872328\pi$$
$$572$$ 0 0
$$573$$ − 8.68838e6i − 1.10548i
$$574$$ 0 0
$$575$$ −857736. −0.108189
$$576$$ 0 0
$$577$$ −1.58241e7 −1.97869 −0.989347 0.145579i $$-0.953495\pi$$
−0.989347 + 0.145579i $$0.953495\pi$$
$$578$$ 0 0
$$579$$ − 85272.0i − 0.0105709i
$$580$$ 0 0
$$581$$ − 5.95901e6i − 0.732375i
$$582$$ 0 0
$$583$$ 1.05268e7 1.28269
$$584$$ 0 0
$$585$$ −2.23463e6 −0.269970
$$586$$ 0 0
$$587$$ − 4.60220e6i − 0.551278i −0.961261 0.275639i $$-0.911111\pi$$
0.961261 0.275639i $$-0.0888894\pi$$
$$588$$ 0 0
$$589$$ − 3.55802e6i − 0.422590i
$$590$$ 0 0
$$591$$ −6.36530e6 −0.749636
$$592$$ 0 0
$$593$$ 8.61122e6 1.00561 0.502803 0.864401i $$-0.332302\pi$$
0.502803 + 0.864401i $$0.332302\pi$$
$$594$$ 0 0
$$595$$ − 2.82269e6i − 0.326867i
$$596$$ 0 0
$$597$$ − 674016.i − 0.0773988i
$$598$$ 0 0
$$599$$ −7.98228e6 −0.908992 −0.454496 0.890749i $$-0.650181\pi$$
−0.454496 + 0.890749i $$0.650181\pi$$
$$600$$ 0 0
$$601$$ −1.01740e7 −1.14896 −0.574481 0.818518i $$-0.694796\pi$$
−0.574481 + 0.818518i $$0.694796\pi$$
$$602$$ 0 0
$$603$$ 2.15939e6i 0.241845i
$$604$$ 0 0
$$605$$ − 7.04965e6i − 0.783031i
$$606$$ 0 0
$$607$$ 9.95843e6 1.09703 0.548516 0.836140i $$-0.315193\pi$$
0.548516 + 0.836140i $$0.315193\pi$$
$$608$$ 0 0
$$609$$ −627264. −0.0685342
$$610$$ 0 0
$$611$$ 541728.i 0.0587054i
$$612$$ 0 0
$$613$$ 4.19586e6i 0.450993i 0.974244 + 0.225497i $$0.0724005\pi$$
−0.974244 + 0.225497i $$0.927600\pi$$
$$614$$ 0 0
$$615$$ −1.11624e7 −1.19007
$$616$$ 0 0
$$617$$ −9.12551e6 −0.965038 −0.482519 0.875885i $$-0.660278\pi$$
−0.482519 + 0.875885i $$0.660278\pi$$
$$618$$ 0 0
$$619$$ − 6.45734e6i − 0.677372i −0.940900 0.338686i $$-0.890018\pi$$
0.940900 0.338686i $$-0.109982\pi$$
$$620$$ 0 0
$$621$$ 1.68428e7i 1.75261i
$$622$$ 0 0
$$623$$ 2.61835e6 0.270276
$$624$$ 0 0
$$625$$ −9.06882e6 −0.928647
$$626$$ 0 0
$$627$$ − 5.41728e6i − 0.550316i
$$628$$ 0 0
$$629$$ − 177012.i − 0.0178392i
$$630$$ 0 0
$$631$$ −1.40514e7 −1.40490 −0.702450 0.711733i $$-0.747910\pi$$
−0.702450 + 0.711733i $$0.747910\pi$$
$$632$$ 0 0
$$633$$ −4.07035e6 −0.403759
$$634$$ 0 0
$$635$$ − 1.81716e7i − 1.78838i
$$636$$ 0 0
$$637$$ − 3.78833e6i − 0.369913i
$$638$$ 0 0
$$639$$ −4.64033e6 −0.449569
$$640$$ 0 0
$$641$$ 8.47168e6 0.814375 0.407188 0.913345i $$-0.366510\pi$$
0.407188 + 0.913345i $$0.366510\pi$$
$$642$$ 0 0
$$643$$ 488564.i 0.0466009i 0.999729 + 0.0233004i $$0.00741743\pi$$
−0.999729 + 0.0233004i $$0.992583\pi$$
$$644$$ 0 0
$$645$$ 7.84080e6i 0.742098i
$$646$$ 0 0
$$647$$ 2.48119e6 0.233023 0.116512 0.993189i $$-0.462829\pi$$
0.116512 + 0.993189i $$0.462829\pi$$
$$648$$ 0 0
$$649$$ 4.14072e6 0.385891
$$650$$ 0 0
$$651$$ − 4.49434e6i − 0.415636i
$$652$$ 0 0
$$653$$ 5.29130e6i 0.485601i 0.970076 + 0.242800i $$0.0780660\pi$$
−0.970076 + 0.242800i $$0.921934\pi$$
$$654$$ 0 0
$$655$$ −5.45292e6 −0.496622
$$656$$ 0 0
$$657$$ −6.68864e6 −0.604539
$$658$$ 0 0
$$659$$ 4.72468e6i 0.423798i 0.977292 + 0.211899i $$0.0679647\pi$$
−0.977292 + 0.211899i $$0.932035\pi$$
$$660$$ 0 0
$$661$$ − 6.17420e6i − 0.549639i −0.961496 0.274819i $$-0.911382\pi$$
0.961496 0.274819i $$-0.0886180\pi$$
$$662$$ 0 0
$$663$$ 2.97950e6 0.263245
$$664$$ 0 0
$$665$$ 3.97267e6 0.348360
$$666$$ 0 0
$$667$$ − 2.43778e6i − 0.212168i
$$668$$ 0 0
$$669$$ 9.35232e6i 0.807893i
$$670$$ 0 0
$$671$$ 1.87585e7 1.60839
$$672$$ 0 0
$$673$$ −9.40925e6 −0.800787 −0.400394 0.916343i $$-0.631127\pi$$
−0.400394 + 0.916343i $$0.631127\pi$$
$$674$$ 0 0
$$675$$ − 857736.i − 0.0724593i
$$676$$ 0 0
$$677$$ 1.50086e7i 1.25854i 0.777185 + 0.629272i $$0.216647\pi$$
−0.777185 + 0.629272i $$0.783353\pi$$
$$678$$ 0 0
$$679$$ 1.07710e7 0.896567
$$680$$ 0 0
$$681$$ −8.93851e6 −0.738580
$$682$$ 0 0
$$683$$ 1.29707e7i 1.06393i 0.846768 + 0.531963i $$0.178545\pi$$
−0.846768 + 0.531963i $$0.821455\pi$$
$$684$$ 0 0
$$685$$ 1.71257e7i 1.39451i
$$686$$ 0 0
$$687$$ −3.27295e6 −0.264574
$$688$$ 0 0
$$689$$ −8.14849e6 −0.653927
$$690$$ 0 0
$$691$$ 2.26556e7i 1.80501i 0.430677 + 0.902506i $$0.358275\pi$$
−0.430677 + 0.902506i $$0.641725\pi$$
$$692$$ 0 0
$$693$$ 4.70448e6i 0.372116i
$$694$$ 0 0
$$695$$ −8.00950e6 −0.628989
$$696$$ 0 0
$$697$$ −1.02322e7 −0.797791
$$698$$ 0 0
$$699$$ − 1.84615e6i − 0.142914i
$$700$$ 0 0
$$701$$ − 1.90169e7i − 1.46166i −0.682562 0.730828i $$-0.739134\pi$$
0.682562 0.730828i $$-0.260866\pi$$
$$702$$ 0 0
$$703$$ 249128. 0.0190123
$$704$$ 0 0
$$705$$ 839808. 0.0636366
$$706$$ 0 0
$$707$$ − 993168.i − 0.0747264i
$$708$$ 0 0
$$709$$ 1.51311e7i 1.13046i 0.824933 + 0.565231i $$0.191213\pi$$
−0.824933 + 0.565231i $$0.808787\pi$$
$$710$$ 0 0
$$711$$ 7.61429e6 0.564879
$$712$$ 0 0
$$713$$ 1.74666e7 1.28672
$$714$$ 0 0
$$715$$ 1.21889e7i 0.891659i
$$716$$ 0 0
$$717$$ 1.38568e7i 1.00662i
$$718$$ 0 0
$$719$$ 1.50323e7 1.08443 0.542217 0.840238i $$-0.317585\pi$$
0.542217 + 0.840238i $$0.317585\pi$$
$$720$$ 0 0
$$721$$ 2.39853e6 0.171833
$$722$$ 0 0
$$723$$ − 7.88489e6i − 0.560983i
$$724$$ 0 0
$$725$$ 124146.i 0.00877178i
$$726$$ 0 0
$$727$$ −7.41230e6 −0.520136 −0.260068 0.965590i $$-0.583745\pi$$
−0.260068 + 0.965590i $$0.583745\pi$$
$$728$$ 0 0
$$729$$ −1.44612e7 −1.00782
$$730$$ 0 0
$$731$$ 7.18740e6i 0.497483i
$$732$$ 0 0
$$733$$ 2.77928e6i 0.191061i 0.995426 + 0.0955306i $$0.0304548\pi$$
−0.995426 + 0.0955306i $$0.969545\pi$$
$$734$$ 0 0
$$735$$ −5.87282e6 −0.400985
$$736$$ 0 0
$$737$$ 1.17785e7 0.798768
$$738$$ 0 0
$$739$$ − 1.21046e7i − 0.815342i −0.913129 0.407671i $$-0.866341\pi$$
0.913129 0.407671i $$-0.133659\pi$$
$$740$$ 0 0
$$741$$ 4.19338e6i 0.280555i
$$742$$ 0 0
$$743$$ 4.46926e6 0.297005 0.148502 0.988912i $$-0.452555\pi$$
0.148502 + 0.988912i $$0.452555\pi$$
$$744$$ 0 0
$$745$$ −1.06172e7 −0.700838
$$746$$ 0 0
$$747$$ 6.70388e6i 0.439567i
$$748$$ 0 0
$$749$$ 1.07680e7i 0.701345i
$$750$$ 0 0
$$751$$ −2.88463e7 −1.86634 −0.933168 0.359442i $$-0.882967\pi$$
−0.933168 + 0.359442i $$0.882967\pi$$
$$752$$ 0 0
$$753$$ −1.61028e7 −1.03494
$$754$$ 0 0
$$755$$ 4.01544e6i 0.256369i
$$756$$ 0 0
$$757$$ 9.60868e6i 0.609430i 0.952444 + 0.304715i $$0.0985612\pi$$
−0.952444 + 0.304715i $$0.901439\pi$$
$$758$$ 0 0
$$759$$ 2.65939e7 1.67563
$$760$$ 0 0
$$761$$ −4.54588e6 −0.284549 −0.142274 0.989827i $$-0.545442\pi$$
−0.142274 + 0.989827i $$0.545442\pi$$
$$762$$ 0 0
$$763$$ 8.79138e6i 0.546696i
$$764$$ 0 0
$$765$$ 3.17552e6i 0.196183i
$$766$$ 0 0
$$767$$ −3.20522e6 −0.196730
$$768$$ 0 0
$$769$$ −2.15923e7 −1.31669 −0.658345 0.752716i $$-0.728743\pi$$
−0.658345 + 0.752716i $$0.728743\pi$$
$$770$$ 0 0
$$771$$ − 1.58825e6i − 0.0962238i
$$772$$ 0 0
$$773$$ − 1.48400e7i − 0.893276i −0.894715 0.446638i $$-0.852621\pi$$
0.894715 0.446638i $$-0.147379\pi$$
$$774$$ 0 0
$$775$$ −889504. −0.0531978
$$776$$ 0 0
$$777$$ 314688. 0.0186994
$$778$$ 0 0
$$779$$ − 1.44009e7i − 0.850251i
$$780$$ 0 0
$$781$$ 2.53109e7i 1.48484i
$$782$$ 0 0
$$783$$ 2.43778e6 0.142098
$$784$$ 0 0
$$785$$ 6.52741e6 0.378065
$$786$$ 0 0
$$787$$ − 2.48785e7i − 1.43182i −0.698194 0.715909i $$-0.746013\pi$$
0.698194 0.715909i $$-0.253987\pi$$
$$788$$ 0 0
$$789$$ − 1.13193e7i − 0.647330i
$$790$$ 0 0
$$791$$ 2.60885e6 0.148254
$$792$$ 0 0
$$793$$ −1.45205e7 −0.819970
$$794$$ 0 0
$$795$$ 1.26321e7i 0.708856i
$$796$$ 0 0
$$797$$ − 3.16080e7i − 1.76259i −0.472568 0.881294i $$-0.656673\pi$$
0.472568 0.881294i $$-0.343327\pi$$
$$798$$ 0 0
$$799$$ 769824. 0.0426604
$$800$$ 0 0
$$801$$ −2.94565e6 −0.162218
$$802$$ 0 0
$$803$$ 3.64835e7i 1.99668i
$$804$$ 0 0
$$805$$ 1.95022e7i 1.06070i
$$806$$ 0 0
$$807$$ −1.16102e7 −0.627562
$$808$$ 0 0
$$809$$ 3.10009e6 0.166534 0.0832669 0.996527i $$-0.473465\pi$$
0.0832669 + 0.996527i $$0.473465\pi$$
$$810$$ 0 0
$$811$$ − 1.87180e6i − 0.0999328i −0.998751 0.0499664i $$-0.984089\pi$$
0.998751 0.0499664i $$-0.0159114\pi$$
$$812$$ 0 0
$$813$$ − 6.21984e6i − 0.330030i
$$814$$ 0 0
$$815$$ 6.01236e6 0.317067
$$816$$ 0 0
$$817$$ −1.01156e7 −0.530196
$$818$$ 0 0
$$819$$ − 3.64162e6i − 0.189707i
$$820$$ 0 0
$$821$$ − 2.00184e7i − 1.03650i −0.855228 0.518252i $$-0.826583\pi$$
0.855228 0.518252i $$-0.173417\pi$$
$$822$$ 0 0
$$823$$ 1.53118e7 0.787999 0.394000 0.919111i $$-0.371091\pi$$
0.394000 + 0.919111i $$0.371091\pi$$
$$824$$ 0 0
$$825$$ −1.35432e6 −0.0692766
$$826$$ 0 0
$$827$$ − 9.59310e6i − 0.487748i −0.969807 0.243874i $$-0.921582\pi$$
0.969807 0.243874i $$-0.0784183\pi$$
$$828$$ 0 0
$$829$$ − 2.52209e7i − 1.27460i −0.770615 0.637302i $$-0.780051\pi$$
0.770615 0.637302i $$-0.219949\pi$$
$$830$$ 0 0
$$831$$ 2.66727e7 1.33988
$$832$$ 0 0
$$833$$ −5.38342e6 −0.268810
$$834$$ 0 0
$$835$$ − 2.65589e7i − 1.31824i
$$836$$ 0 0
$$837$$ 1.74666e7i 0.861778i
$$838$$ 0 0
$$839$$ −1.77623e7 −0.871154 −0.435577 0.900151i $$-0.643456\pi$$
−0.435577 + 0.900151i $$0.643456\pi$$
$$840$$ 0 0
$$841$$ 2.01583e7 0.982798
$$842$$ 0 0
$$843$$ − 2.35937e6i − 0.114348i
$$844$$ 0 0
$$845$$ 1.06147e7i 0.511407i
$$846$$ 0 0
$$847$$ 1.14883e7 0.550234
$$848$$ 0 0
$$849$$ 1.86273e7 0.886913
$$850$$ 0 0
$$851$$ 1.22299e6i 0.0578895i
$$852$$ 0 0
$$853$$ − 486970.i − 0.0229155i −0.999934 0.0114578i $$-0.996353\pi$$
0.999934 0.0114578i $$-0.00364720\pi$$
$$854$$ 0 0
$$855$$ −4.46926e6 −0.209084
$$856$$ 0 0
$$857$$ 1.92634e6 0.0895945 0.0447972 0.998996i $$-0.485736\pi$$
0.0447972 + 0.998996i $$0.485736\pi$$
$$858$$ 0 0
$$859$$ − 2.23538e7i − 1.03364i −0.856094 0.516820i $$-0.827116\pi$$
0.856094 0.516820i $$-0.172884\pi$$
$$860$$ 0 0
$$861$$ − 1.81907e7i − 0.836258i
$$862$$ 0 0
$$863$$ −1.85838e7 −0.849390 −0.424695 0.905337i $$-0.639619\pi$$
−0.424695 + 0.905337i $$0.639619\pi$$
$$864$$ 0 0
$$865$$ 3.82025e7 1.73601
$$866$$ 0 0
$$867$$ 1.28043e7i 0.578504i
$$868$$ 0 0
$$869$$ − 4.15325e7i − 1.86569i
$$870$$ 0 0
$$871$$ −9.11742e6 −0.407217
$$872$$ 0 0
$$873$$ −1.21174e7 −0.538114
$$874$$ 0 0
$$875$$ − 1.58432e7i − 0.699555i
$$876$$ 0 0
$$877$$ 2.91048e7i 1.27781i 0.769286 + 0.638905i $$0.220612\pi$$
−0.769286 + 0.638905i $$0.779388\pi$$
$$878$$ 0 0
$$879$$ −1.28660e7 −0.561659
$$880$$ 0 0
$$881$$ −3.14696e6 −0.136600 −0.0683001 0.997665i $$-0.521758\pi$$
−0.0683001 + 0.997665i $$0.521758\pi$$
$$882$$ 0 0
$$883$$ 1.59995e7i 0.690566i 0.938499 + 0.345283i $$0.112217\pi$$
−0.938499 + 0.345283i $$0.887783\pi$$
$$884$$ 0 0
$$885$$ 4.96886e6i 0.213255i
$$886$$ 0 0
$$887$$ −3.45874e7 −1.47608 −0.738039 0.674758i $$-0.764248\pi$$
−0.738039 + 0.674758i $$0.764248\pi$$
$$888$$ 0 0
$$889$$ 2.96131e7 1.25669
$$890$$ 0 0
$$891$$ 1.36031e7i 0.574044i
$$892$$ 0 0
$$893$$ 1.08346e6i 0.0454656i
$$894$$ 0 0
$$895$$ −2.66581e7 −1.11243
$$896$$ 0 0
$$897$$ −2.05857e7 −0.854248
$$898$$ 0 0
$$899$$ − 2.52806e6i − 0.104325i
$$900$$ 0 0
$$901$$ 1.15794e7i 0.475199i
$$902$$ 0 0
$$903$$ −1.27776e7 −0.521471
$$904$$ 0 0
$$905$$ 3.02103e7 1.22612
$$906$$ 0 0
$$907$$ − 1.74396e7i − 0.703914i −0.936016 0.351957i $$-0.885516\pi$$
0.936016 0.351957i $$-0.114484\pi$$
$$908$$ 0 0
$$909$$ 1.11731e6i 0.0448503i
$$910$$ 0 0
$$911$$ 2.59589e6 0.103631 0.0518155 0.998657i $$-0.483499\pi$$
0.0518155 + 0.998657i $$0.483499\pi$$
$$912$$ 0 0
$$913$$ 3.65666e7 1.45180
$$914$$ 0 0
$$915$$ 2.25102e7i 0.888847i
$$916$$ 0 0
$$917$$ − 8.88624e6i − 0.348975i
$$918$$ 0 0
$$919$$ −1.76411e7 −0.689028 −0.344514 0.938781i $$-0.611956\pi$$
−0.344514 + 0.938781i $$0.611956\pi$$
$$920$$ 0 0
$$921$$ 1.90307e7 0.739275
$$922$$ 0 0
$$923$$ − 1.95925e7i − 0.756982i
$$924$$ 0 0
$$925$$ − 62282.0i − 0.00239336i
$$926$$ 0 0
$$927$$ −2.69834e6 −0.103133
$$928$$ 0 0
$$929$$ 3.96785e7 1.50840 0.754199 0.656646i $$-0.228025\pi$$
0.754199 + 0.656646i $$0.228025\pi$$
$$930$$ 0 0
$$931$$ − 7.57667e6i − 0.286486i
$$932$$ 0 0
$$933$$ 8.76874e6i 0.329787i
$$934$$ 0 0
$$935$$ 1.73210e7 0.647955
$$936$$ 0 0
$$937$$ −3.93413e7 −1.46386 −0.731930 0.681380i $$-0.761380\pi$$
−0.731930 + 0.681380i $$0.761380\pi$$
$$938$$ 0 0
$$939$$ 7.01830e6i 0.259757i
$$940$$ 0 0
$$941$$ − 4.62506e7i − 1.70272i −0.524581 0.851361i $$-0.675778\pi$$
0.524581 0.851361i $$-0.324222\pi$$
$$942$$ 0 0
$$943$$ 7.06955e7 2.58888
$$944$$ 0 0
$$945$$ −1.95022e7 −0.710402
$$946$$ 0 0
$$947$$ − 3.79025e7i − 1.37339i −0.726947 0.686693i $$-0.759062\pi$$
0.726947 0.686693i $$-0.240938\pi$$
$$948$$ 0 0
$$949$$ − 2.82409e7i − 1.01792i
$$950$$ 0 0
$$951$$ 2.97944e7 1.06828
$$952$$ 0 0
$$953$$ 2.66462e7 0.950394 0.475197 0.879879i $$-0.342377\pi$$
0.475197 + 0.879879i $$0.342377\pi$$
$$954$$ 0 0
$$955$$ 3.90977e7i 1.38721i
$$956$$ 0 0
$$957$$ − 3.84912e6i − 0.135857i
$$958$$ 0 0
$$959$$ −2.79085e7 −0.979918
$$960$$ 0 0
$$961$$ −1.05156e7 −0.367304
$$962$$ 0 0
$$963$$ − 1.21140e7i − 0.420943i
$$964$$ 0 0
$$965$$ 383724.i 0.0132648i
$$966$$ 0 0
$$967$$ 4.09790e7 1.40927 0.704637 0.709568i $$-0.251110\pi$$
0.704637 + 0.709568i $$0.251110\pi$$
$$968$$ 0 0
$$969$$ 5.95901e6 0.203875
$$970$$ 0 0
$$971$$ 2.72034e7i 0.925922i 0.886379 + 0.462961i $$0.153213\pi$$
−0.886379 + 0.462961i $$0.846787\pi$$
$$972$$ 0 0
$$973$$ − 1.30525e7i − 0.441990i
$$974$$ 0 0
$$975$$ 1.04834e6 0.0353177
$$976$$ 0 0
$$977$$ 2.53555e7 0.849839 0.424919 0.905231i $$-0.360302\pi$$
0.424919 + 0.905231i $$0.360302\pi$$
$$978$$ 0 0
$$979$$ 1.60672e7i 0.535775i
$$980$$ 0 0
$$981$$ − 9.89030e6i − 0.328123i
$$982$$ 0 0
$$983$$ 1.19139e7 0.393252 0.196626 0.980479i $$-0.437002\pi$$
0.196626 + 0.980479i $$0.437002\pi$$
$$984$$ 0 0
$$985$$ 2.86439e7 0.940678
$$986$$ 0 0
$$987$$ 1.36858e6i 0.0447173i
$$988$$ 0 0
$$989$$ − 4.96584e7i − 1.61437i
$$990$$ 0 0
$$991$$ −2.91931e7 −0.944268 −0.472134 0.881527i $$-0.656516\pi$$
−0.472134 + 0.881527i $$0.656516\pi$$
$$992$$ 0 0
$$993$$ −4.53538e6 −0.145962
$$994$$ 0 0
$$995$$ 3.03307e6i 0.0971237i
$$996$$ 0 0
$$997$$ − 1.73001e7i − 0.551201i −0.961272 0.275601i $$-0.911123\pi$$
0.961272 0.275601i $$-0.0888767\pi$$
$$998$$ 0 0
$$999$$ −1.22299e6 −0.0387713
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 256.6.b.c.129.1 2
4.3 odd 2 256.6.b.g.129.2 2
8.3 odd 2 256.6.b.g.129.1 2
8.5 even 2 inner 256.6.b.c.129.2 2
16.3 odd 4 4.6.a.a.1.1 1
16.5 even 4 64.6.a.b.1.1 1
16.11 odd 4 64.6.a.f.1.1 1
16.13 even 4 16.6.a.b.1.1 1
48.5 odd 4 576.6.a.bd.1.1 1
48.11 even 4 576.6.a.bc.1.1 1
48.29 odd 4 144.6.a.c.1.1 1
48.35 even 4 36.6.a.a.1.1 1
80.3 even 4 100.6.c.b.49.1 2
80.13 odd 4 400.6.c.f.49.2 2
80.19 odd 4 100.6.a.b.1.1 1
80.29 even 4 400.6.a.d.1.1 1
80.67 even 4 100.6.c.b.49.2 2
80.77 odd 4 400.6.c.f.49.1 2
112.3 even 12 196.6.e.d.177.1 2
112.13 odd 4 784.6.a.d.1.1 1
112.19 even 12 196.6.e.d.165.1 2
112.51 odd 12 196.6.e.g.165.1 2
112.67 odd 12 196.6.e.g.177.1 2
112.83 even 4 196.6.a.e.1.1 1
144.67 odd 12 324.6.e.a.217.1 2
144.83 even 12 324.6.e.d.109.1 2
144.115 odd 12 324.6.e.a.109.1 2
144.131 even 12 324.6.e.d.217.1 2
176.131 even 4 484.6.a.a.1.1 1
208.51 odd 4 676.6.a.a.1.1 1
208.83 even 4 676.6.d.a.337.1 2
208.99 even 4 676.6.d.a.337.2 2
240.83 odd 4 900.6.d.a.649.2 2
240.179 even 4 900.6.a.h.1.1 1
240.227 odd 4 900.6.d.a.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 16.3 odd 4
16.6.a.b.1.1 1 16.13 even 4
36.6.a.a.1.1 1 48.35 even 4
64.6.a.b.1.1 1 16.5 even 4
64.6.a.f.1.1 1 16.11 odd 4
100.6.a.b.1.1 1 80.19 odd 4
100.6.c.b.49.1 2 80.3 even 4
100.6.c.b.49.2 2 80.67 even 4
144.6.a.c.1.1 1 48.29 odd 4
196.6.a.e.1.1 1 112.83 even 4
196.6.e.d.165.1 2 112.19 even 12
196.6.e.d.177.1 2 112.3 even 12
196.6.e.g.165.1 2 112.51 odd 12
196.6.e.g.177.1 2 112.67 odd 12
256.6.b.c.129.1 2 1.1 even 1 trivial
256.6.b.c.129.2 2 8.5 even 2 inner
256.6.b.g.129.1 2 8.3 odd 2
256.6.b.g.129.2 2 4.3 odd 2
324.6.e.a.109.1 2 144.115 odd 12
324.6.e.a.217.1 2 144.67 odd 12
324.6.e.d.109.1 2 144.83 even 12
324.6.e.d.217.1 2 144.131 even 12
400.6.a.d.1.1 1 80.29 even 4
400.6.c.f.49.1 2 80.77 odd 4
400.6.c.f.49.2 2 80.13 odd 4
484.6.a.a.1.1 1 176.131 even 4
576.6.a.bc.1.1 1 48.11 even 4
576.6.a.bd.1.1 1 48.5 odd 4
676.6.a.a.1.1 1 208.51 odd 4
676.6.d.a.337.1 2 208.83 even 4
676.6.d.a.337.2 2 208.99 even 4
784.6.a.d.1.1 1 112.13 odd 4
900.6.a.h.1.1 1 240.179 even 4
900.6.d.a.649.1 2 240.227 odd 4
900.6.d.a.649.2 2 240.83 odd 4