# Properties

 Label 400.6.c.f.49.2 Level $400$ Weight $6$ Character 400.49 Analytic conductor $64.154$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ 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 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.6.c.f.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+12.0000i q^{3} -88.0000i q^{7} +99.0000 q^{9} +O(q^{10})$$ $$q+12.0000i q^{3} -88.0000i q^{7} +99.0000 q^{9} -540.000 q^{11} -418.000i q^{13} -594.000i q^{17} +836.000 q^{19} +1056.00 q^{21} +4104.00i q^{23} +4104.00i q^{27} +594.000 q^{29} -4256.00 q^{31} -6480.00i q^{33} +298.000i q^{37} +5016.00 q^{39} +17226.0 q^{41} +12100.0i q^{43} -1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} +19494.0i q^{53} +10032.0i q^{57} -7668.00 q^{59} -34738.0 q^{61} -8712.00i q^{63} +21812.0i q^{67} -49248.0 q^{69} +46872.0 q^{71} +67562.0i q^{73} +47520.0i q^{77} -76912.0 q^{79} -25191.0 q^{81} -67716.0i q^{83} +7128.00i q^{87} -29754.0 q^{89} -36784.0 q^{91} -51072.0i q^{93} +122398. i q^{97} -53460.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 198 q^{9} + O(q^{10})$$ $$2 q + 198 q^{9} - 1080 q^{11} + 1672 q^{19} + 2112 q^{21} + 1188 q^{29} - 8512 q^{31} + 10032 q^{39} + 34452 q^{41} + 18126 q^{49} + 14256 q^{51} - 15336 q^{59} - 69476 q^{61} - 98496 q^{69} + 93744 q^{71} - 153824 q^{79} - 50382 q^{81} - 59508 q^{89} - 73568 q^{91} - 106920 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 12.0000i 0.769800i 0.922958 + 0.384900i $$0.125764\pi$$
−0.922958 + 0.384900i $$0.874236\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 88.0000i − 0.678793i −0.940643 0.339397i $$-0.889777\pi$$
0.940643 0.339397i $$-0.110223\pi$$
$$8$$ 0 0
$$9$$ 99.0000 0.407407
$$10$$ 0 0
$$11$$ −540.000 −1.34559 −0.672794 0.739830i $$-0.734906\pi$$
−0.672794 + 0.739830i $$0.734906\pi$$
$$12$$ 0 0
$$13$$ − 418.000i − 0.685990i −0.939337 0.342995i $$-0.888559\pi$$
0.939337 0.342995i $$-0.111441\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 594.000i − 0.498499i −0.968439 0.249249i $$-0.919816\pi$$
0.968439 0.249249i $$-0.0801839\pi$$
$$18$$ 0 0
$$19$$ 836.000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 1056.00 0.522535
$$22$$ 0 0
$$23$$ 4104.00i 1.61766i 0.588041 + 0.808831i $$0.299899\pi$$
−0.588041 + 0.808831i $$0.700101\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4104.00i 1.08342i
$$28$$ 0 0
$$29$$ 594.000 0.131157 0.0655785 0.997847i $$-0.479111\pi$$
0.0655785 + 0.997847i $$0.479111\pi$$
$$30$$ 0 0
$$31$$ −4256.00 −0.795422 −0.397711 0.917511i $$-0.630195\pi$$
−0.397711 + 0.917511i $$0.630195\pi$$
$$32$$ 0 0
$$33$$ − 6480.00i − 1.03583i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 298.000i 0.0357859i 0.999840 + 0.0178930i $$0.00569581\pi$$
−0.999840 + 0.0178930i $$0.994304\pi$$
$$38$$ 0 0
$$39$$ 5016.00 0.528075
$$40$$ 0 0
$$41$$ 17226.0 1.60039 0.800193 0.599742i $$-0.204730\pi$$
0.800193 + 0.599742i $$0.204730\pi$$
$$42$$ 0 0
$$43$$ 12100.0i 0.997963i 0.866613 + 0.498981i $$0.166292\pi$$
−0.866613 + 0.498981i $$0.833708\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 1296.00i − 0.0855777i −0.999084 0.0427888i $$-0.986376\pi$$
0.999084 0.0427888i $$-0.0136243\pi$$
$$48$$ 0 0
$$49$$ 9063.00 0.539240
$$50$$ 0 0
$$51$$ 7128.00 0.383745
$$52$$ 0 0
$$53$$ 19494.0i 0.953260i 0.879104 + 0.476630i $$0.158142\pi$$
−0.879104 + 0.476630i $$0.841858\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 10032.0i 0.408978i
$$58$$ 0 0
$$59$$ −7668.00 −0.286782 −0.143391 0.989666i $$-0.545801\pi$$
−0.143391 + 0.989666i $$0.545801\pi$$
$$60$$ 0 0
$$61$$ −34738.0 −1.19531 −0.597655 0.801754i $$-0.703901\pi$$
−0.597655 + 0.801754i $$0.703901\pi$$
$$62$$ 0 0
$$63$$ − 8712.00i − 0.276545i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 21812.0i 0.593620i 0.954937 + 0.296810i $$0.0959228\pi$$
−0.954937 + 0.296810i $$0.904077\pi$$
$$68$$ 0 0
$$69$$ −49248.0 −1.24528
$$70$$ 0 0
$$71$$ 46872.0 1.10349 0.551744 0.834014i $$-0.313963\pi$$
0.551744 + 0.834014i $$0.313963\pi$$
$$72$$ 0 0
$$73$$ 67562.0i 1.48387i 0.670473 + 0.741934i $$0.266091\pi$$
−0.670473 + 0.741934i $$0.733909\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 47520.0i 0.913376i
$$78$$ 0 0
$$79$$ −76912.0 −1.38652 −0.693260 0.720687i $$-0.743826\pi$$
−0.693260 + 0.720687i $$0.743826\pi$$
$$80$$ 0 0
$$81$$ −25191.0 −0.426612
$$82$$ 0 0
$$83$$ − 67716.0i − 1.07894i −0.842006 0.539468i $$-0.818625\pi$$
0.842006 0.539468i $$-0.181375\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7128.00i 0.100965i
$$88$$ 0 0
$$89$$ −29754.0 −0.398172 −0.199086 0.979982i $$-0.563797\pi$$
−0.199086 + 0.979982i $$0.563797\pi$$
$$90$$ 0 0
$$91$$ −36784.0 −0.465646
$$92$$ 0 0
$$93$$ − 51072.0i − 0.612316i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 122398.i 1.32082i 0.750903 + 0.660412i $$0.229618\pi$$
−0.750903 + 0.660412i $$0.770382\pi$$
$$98$$ 0 0
$$99$$ −53460.0 −0.548202
$$100$$ 0 0
$$101$$ 11286.0 0.110087 0.0550436 0.998484i $$-0.482470\pi$$
0.0550436 + 0.998484i $$0.482470\pi$$
$$102$$ 0 0
$$103$$ 27256.0i 0.253145i 0.991957 + 0.126572i $$0.0403976\pi$$
−0.991957 + 0.126572i $$0.959602\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 122364.i 1.03322i 0.856220 + 0.516612i $$0.172807\pi$$
−0.856220 + 0.516612i $$0.827193\pi$$
$$108$$ 0 0
$$109$$ −99902.0 −0.805393 −0.402697 0.915334i $$-0.631927\pi$$
−0.402697 + 0.915334i $$0.631927\pi$$
$$110$$ 0 0
$$111$$ −3576.00 −0.0275480
$$112$$ 0 0
$$113$$ − 29646.0i − 0.218409i −0.994019 0.109204i $$-0.965170\pi$$
0.994019 0.109204i $$-0.0348303\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 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
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 336512.i 1.85136i 0.378305 + 0.925681i $$0.376507\pi$$
−0.378305 + 0.925681i $$0.623493\pi$$
$$128$$ 0 0
$$129$$ −145200. −0.768232
$$130$$ 0 0
$$131$$ −100980. −0.514111 −0.257056 0.966397i $$-0.582752\pi$$
−0.257056 + 0.966397i $$0.582752\pi$$
$$132$$ 0 0
$$133$$ − 73568.0i − 0.360628i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 317142.i 1.44362i 0.692092 + 0.721809i $$0.256689\pi$$
−0.692092 + 0.721809i $$0.743311\pi$$
$$138$$ 0 0
$$139$$ −148324. −0.651140 −0.325570 0.945518i $$-0.605556\pi$$
−0.325570 + 0.945518i $$0.605556\pi$$
$$140$$ 0 0
$$141$$ 15552.0 0.0658777
$$142$$ 0 0
$$143$$ 225720.i 0.923060i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 108756.i 0.415107i
$$148$$ 0 0
$$149$$ −196614. −0.725519 −0.362759 0.931883i $$-0.618165\pi$$
−0.362759 + 0.931883i $$0.618165\pi$$
$$150$$ 0 0
$$151$$ −74360.0 −0.265398 −0.132699 0.991156i $$-0.542364\pi$$
−0.132699 + 0.991156i $$0.542364\pi$$
$$152$$ 0 0
$$153$$ − 58806.0i − 0.203092i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 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 $$0.0524773\pi$$
−0.986441 + 0.164116i $$0.947523\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 491832.i − 1.36466i −0.731043 0.682332i $$-0.760966\pi$$
0.731043 0.682332i $$-0.239034\pi$$
$$168$$ 0 0
$$169$$ 196569. 0.529417
$$170$$ 0 0
$$171$$ 82764.0 0.216447
$$172$$ 0 0
$$173$$ 707454.i 1.79714i 0.438826 + 0.898572i $$0.355395\pi$$
−0.438826 + 0.898572i $$0.644605\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 92016.0i − 0.220765i
$$178$$ 0 0
$$179$$ 493668. 1.15160 0.575801 0.817590i $$-0.304690\pi$$
0.575801 + 0.817590i $$0.304690\pi$$
$$180$$ 0 0
$$181$$ −559450. −1.26930 −0.634651 0.772799i $$-0.718856\pi$$
−0.634651 + 0.772799i $$0.718856\pi$$
$$182$$ 0 0
$$183$$ − 416856.i − 0.920149i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 320760.i 0.670774i
$$188$$ 0 0
$$189$$ 361152. 0.735420
$$190$$ 0 0
$$191$$ 724032. 1.43607 0.718033 0.696009i $$-0.245043\pi$$
0.718033 + 0.696009i $$0.245043\pi$$
$$192$$ 0 0
$$193$$ 7106.00i 0.0137319i 0.999976 + 0.00686597i $$0.00218552\pi$$
−0.999976 + 0.00686597i $$0.997814\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 530442.i 0.973806i 0.873456 + 0.486903i $$0.161873\pi$$
−0.873456 + 0.486903i $$0.838127\pi$$
$$198$$ 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
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 406296.i 0.659047i
$$208$$ 0 0
$$209$$ −451440. −0.714882
$$210$$ 0 0
$$211$$ 339196. 0.524499 0.262249 0.965000i $$-0.415536\pi$$
0.262249 + 0.965000i $$0.415536\pi$$
$$212$$ 0 0
$$213$$ 562464.i 0.849465i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 374528.i 0.539927i
$$218$$ 0 0
$$219$$ −810744. −1.14228
$$220$$ 0 0
$$221$$ −248292. −0.341965
$$222$$ 0 0
$$223$$ − 779360.i − 1.04948i −0.851261 0.524742i $$-0.824162\pi$$
0.851261 0.524742i $$-0.175838\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 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. 0.343692 0.171846 0.985124i $$-0.445027\pi$$
0.171846 + 0.985124i $$0.445027\pi$$
$$230$$ 0 0
$$231$$ −570240. −0.703117
$$232$$ 0 0
$$233$$ − 153846.i − 0.185651i −0.995682 0.0928253i $$-0.970410\pi$$
0.995682 0.0928253i $$-0.0295898\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 922944.i − 1.06734i
$$238$$ 0 0
$$239$$ 1.15474e6 1.30764 0.653820 0.756650i $$-0.273166\pi$$
0.653820 + 0.756650i $$0.273166\pi$$
$$240$$ 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
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 349448.i − 0.364452i
$$248$$ 0 0
$$249$$ 812592. 0.830566
$$250$$ 0 0
$$251$$ −1.34190e6 −1.34442 −0.672211 0.740359i $$-0.734655\pi$$
−0.672211 + 0.740359i $$0.734655\pi$$
$$252$$ 0 0
$$253$$ − 2.21616e6i − 2.17671i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 132354.i − 0.124998i −0.998045 0.0624992i $$-0.980093\pi$$
0.998045 0.0624992i $$-0.0199071\pi$$
$$258$$ 0 0
$$259$$ 26224.0 0.0242912
$$260$$ 0 0
$$261$$ 58806.0 0.0534343
$$262$$ 0 0
$$263$$ − 943272.i − 0.840906i −0.907314 0.420453i $$-0.861871\pi$$
0.907314 0.420453i $$-0.138129\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 357048.i − 0.306513i
$$268$$ 0 0
$$269$$ −967518. −0.815227 −0.407613 0.913155i $$-0.633639\pi$$
−0.407613 + 0.913155i $$0.633639\pi$$
$$270$$ 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.i − 0.358454i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.22273e6i − 1.74055i −0.492566 0.870275i $$-0.663941\pi$$
0.492566 0.870275i $$-0.336059\pi$$
$$278$$ 0 0
$$279$$ −421344. −0.324061
$$280$$ 0 0
$$281$$ −196614. −0.148542 −0.0742709 0.997238i $$-0.523663\pi$$
−0.0742709 + 0.997238i $$0.523663\pi$$
$$282$$ 0 0
$$283$$ 1.55228e6i 1.15213i 0.817403 + 0.576067i $$0.195413\pi$$
−0.817403 + 0.576067i $$0.804587\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 1.51589e6i − 1.08633i
$$288$$ 0 0
$$289$$ 1.06702e6 0.751499
$$290$$ 0 0
$$291$$ −1.46878e6 −1.01677
$$292$$ 0 0
$$293$$ − 1.07217e6i − 0.729616i −0.931083 0.364808i $$-0.881135\pi$$
0.931083 0.364808i $$-0.118865\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2.21616e6i − 1.45784i
$$298$$ 0 0
$$299$$ 1.71547e6 1.10970
$$300$$ 0 0
$$301$$ 1.06480e6 0.677410
$$302$$ 0 0
$$303$$ 135432.i 0.0847451i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 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. −0.194871
$$310$$ 0 0
$$311$$ 730728. 0.428405 0.214203 0.976789i $$-0.431285\pi$$
0.214203 + 0.976789i $$0.431285\pi$$
$$312$$ 0 0
$$313$$ 584858.i 0.337435i 0.985664 + 0.168717i $$0.0539625\pi$$
−0.985664 + 0.168717i $$0.946038\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 1.19882e6i − 0.619992i
$$328$$ 0 0
$$329$$ −114048. −0.0580895
$$330$$ 0 0
$$331$$ −377948. −0.189610 −0.0948052 0.995496i $$-0.530223\pi$$
−0.0948052 + 0.995496i $$0.530223\pi$$
$$332$$ 0 0
$$333$$ 29502.0i 0.0145794i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 639122.i − 0.306555i −0.988183 0.153278i $$-0.951017\pi$$
0.988183 0.153278i $$-0.0489829\pi$$
$$338$$ 0 0
$$339$$ 355752. 0.168131
$$340$$ 0 0
$$341$$ 2.29824e6 1.07031
$$342$$ 0 0
$$343$$ − 2.27656e6i − 1.04483i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 2.90466e6i − 1.29501i −0.762063 0.647503i $$-0.775813\pi$$
0.762063 0.647503i $$-0.224187\pi$$
$$348$$ 0 0
$$349$$ 3.99157e6 1.75420 0.877102 0.480304i $$-0.159474\pi$$
0.877102 + 0.480304i $$0.159474\pi$$
$$350$$ 0 0
$$351$$ 1.71547e6 0.743217
$$352$$ 0 0
$$353$$ 1.42922e6i 0.610466i 0.952278 + 0.305233i $$0.0987344\pi$$
−0.952278 + 0.305233i $$0.901266\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 1.08923e6i − 0.422139i −0.977471 0.211069i $$-0.932305\pi$$
0.977471 0.211069i $$-0.0676946\pi$$
$$368$$ 0 0
$$369$$ 1.70537e6 0.652009
$$370$$ 0 0
$$371$$ 1.71547e6 0.647066
$$372$$ 0 0
$$373$$ 3.50577e6i 1.30470i 0.757918 + 0.652350i $$0.226217\pi$$
−0.757918 + 0.652350i $$0.773783\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 248292.i − 0.0899724i
$$378$$ 0 0
$$379$$ 4.04385e6 1.44610 0.723048 0.690798i $$-0.242740\pi$$
0.723048 + 0.690798i $$0.242740\pi$$
$$380$$ 0 0
$$381$$ −4.03814e6 −1.42518
$$382$$ 0 0
$$383$$ − 5.18746e6i − 1.80700i −0.428591 0.903499i $$-0.640990\pi$$
0.428591 0.903499i $$-0.359010\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.19790e6i 0.406577i
$$388$$ 0 0
$$389$$ 950346. 0.318425 0.159213 0.987244i $$-0.449104\pi$$
0.159213 + 0.987244i $$0.449104\pi$$
$$390$$ 0 0
$$391$$ 2.43778e6 0.806403
$$392$$ 0 0
$$393$$ − 1.21176e6i − 0.395763i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 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
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 160920.i − 0.0481531i
$$408$$ 0 0
$$409$$ −2.64051e6 −0.780511 −0.390255 0.920707i $$-0.627613\pi$$
−0.390255 + 0.920707i $$0.627613\pi$$
$$410$$ 0 0
$$411$$ −3.80570e6 −1.11130
$$412$$ 0 0
$$413$$ 674784.i 0.194666i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 1.77989e6i − 0.501248i
$$418$$ 0 0
$$419$$ −4.98020e6 −1.38584 −0.692918 0.721016i $$-0.743675\pi$$
−0.692918 + 0.721016i $$0.743675\pi$$
$$420$$ 0 0
$$421$$ −237994. −0.0654426 −0.0327213 0.999465i $$-0.510417\pi$$
−0.0327213 + 0.999465i $$0.510417\pi$$
$$422$$ 0 0
$$423$$ − 128304.i − 0.0348650i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.05694e6i 0.811368i
$$428$$ 0 0
$$429$$ −2.70864e6 −0.710572
$$430$$ 0 0
$$431$$ 3.88238e6 1.00671 0.503356 0.864079i $$-0.332098\pi$$
0.503356 + 0.864079i $$0.332098\pi$$
$$432$$ 0 0
$$433$$ − 66958.0i − 0.0171626i −0.999963 0.00858129i $$-0.997268\pi$$
0.999963 0.00858129i $$-0.00273154\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 2.35937e6i − 0.558505i
$$448$$ 0 0
$$449$$ −3.77671e6 −0.884092 −0.442046 0.896992i $$-0.645747\pi$$
−0.442046 + 0.896992i $$0.645747\pi$$
$$450$$ 0 0
$$451$$ −9.30204e6 −2.15346
$$452$$ 0 0
$$453$$ − 892320.i − 0.204303i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.18069e6i 0.712412i 0.934407 + 0.356206i $$0.115930\pi$$
−0.934407 + 0.356206i $$0.884070\pi$$
$$458$$ 0 0
$$459$$ 2.43778e6 0.540085
$$460$$ 0 0
$$461$$ 6.68547e6 1.46514 0.732571 0.680691i $$-0.238320\pi$$
0.732571 + 0.680691i $$0.238320\pi$$
$$462$$ 0 0
$$463$$ 4.35122e6i 0.943318i 0.881781 + 0.471659i $$0.156345\pi$$
−0.881781 + 0.471659i $$0.843655\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 7.07994e6i 1.50223i 0.660170 + 0.751117i $$0.270484\pi$$
−0.660170 + 0.751117i $$0.729516\pi$$
$$468$$ 0 0
$$469$$ 1.91946e6 0.402945
$$470$$ 0 0
$$471$$ 1.45054e6 0.301284
$$472$$ 0 0
$$473$$ − 6.53400e6i − 1.34285i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.92991e6i 0.388365i
$$478$$ 0 0
$$479$$ 3.22186e6 0.641604 0.320802 0.947146i $$-0.396048\pi$$
0.320802 + 0.947146i $$0.396048\pi$$
$$480$$ 0 0
$$481$$ 124564. 0.0245488
$$482$$ 0 0
$$483$$ 4.33382e6i 0.845286i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.29710e6i 0.438891i 0.975625 + 0.219446i $$0.0704248\pi$$
−0.975625 + 0.219446i $$0.929575\pi$$
$$488$$ 0 0
$$489$$ −1.33608e6 −0.252674
$$490$$ 0 0
$$491$$ −2.82150e6 −0.528173 −0.264087 0.964499i $$-0.585070\pi$$
−0.264087 + 0.964499i $$0.585070\pi$$
$$492$$ 0 0
$$493$$ − 352836.i − 0.0653816i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 4.12474e6i − 0.749040i
$$498$$ 0 0
$$499$$ −4.13628e6 −0.743634 −0.371817 0.928306i $$-0.621265\pi$$
−0.371817 + 0.928306i $$0.621265\pi$$
$$500$$ 0 0
$$501$$ 5.90198e6 1.05052
$$502$$ 0 0
$$503$$ − 8.33263e6i − 1.46846i −0.678901 0.734230i $$-0.737543\pi$$
0.678901 0.734230i $$-0.262457\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 2.35883e6i 0.407546i
$$508$$ 0 0
$$509$$ −4.34101e6 −0.742670 −0.371335 0.928499i $$-0.621100\pi$$
−0.371335 + 0.928499i $$0.621100\pi$$
$$510$$ 0 0
$$511$$ 5.94546e6 1.00724
$$512$$ 0 0
$$513$$ 3.43094e6i 0.575599i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 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 $$-0.683117\pi$$
−0.544070 + 0.839040i $$0.683117\pi$$
$$522$$ 0 0
$$523$$ 7.72196e6i 1.23445i 0.786787 + 0.617224i $$0.211743\pi$$
−0.786787 + 0.617224i $$0.788257\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.52806e6i 0.396517i
$$528$$ 0 0
$$529$$ −1.04065e7 −1.61683
$$530$$ 0 0
$$531$$ −759132. −0.116837
$$532$$ 0 0
$$533$$ − 7.20047e6i − 1.09785i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 5.92402e6i 0.886504i
$$538$$ 0 0
$$539$$ −4.89402e6 −0.725594
$$540$$ 0 0
$$541$$ −682066. −0.100192 −0.0500960 0.998744i $$-0.515953\pi$$
−0.0500960 + 0.998744i $$0.515953\pi$$
$$542$$ 0 0
$$543$$ − 6.71340e6i − 0.977109i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.15772e6i 0.308337i 0.988045 + 0.154169i $$0.0492699\pi$$
−0.988045 + 0.154169i $$0.950730\pi$$
$$548$$ 0 0
$$549$$ −3.43906e6 −0.486978
$$550$$ 0 0
$$551$$ 496584. 0.0696809
$$552$$ 0 0
$$553$$ 6.76826e6i 0.941161i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 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 $$0.0759113\pi$$
−0.971698 + 0.236228i $$0.924089\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.21681e6i 0.289581i
$$568$$ 0 0
$$569$$ 1.29225e7 1.67327 0.836633 0.547764i $$-0.184521\pi$$
0.836633 + 0.547764i $$0.184521\pi$$
$$570$$ 0 0
$$571$$ 6.08357e6 0.780851 0.390426 0.920634i $$-0.372328\pi$$
0.390426 + 0.920634i $$0.372328\pi$$
$$572$$ 0 0
$$573$$ 8.68838e6i 1.10548i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.58241e7i 1.97869i 0.145579 + 0.989347i $$0.453495\pi$$
−0.145579 + 0.989347i $$0.546505\pi$$
$$578$$ 0 0
$$579$$ −85272.0 −0.0105709
$$580$$ 0 0
$$581$$ −5.95901e6 −0.732375
$$582$$ 0 0
$$583$$ − 1.05268e7i − 1.28269i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.60220e6i 0.551278i 0.961261 + 0.275639i $$0.0888894\pi$$
−0.961261 + 0.275639i $$0.911111\pi$$
$$588$$ 0 0
$$589$$ −3.55802e6 −0.422590
$$590$$ 0 0
$$591$$ −6.36530e6 −0.749636
$$592$$ 0 0
$$593$$ 8.61122e6i 1.00561i 0.864401 + 0.502803i $$0.167698\pi$$
−0.864401 + 0.502803i $$0.832302\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 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 $$-0.305204\pi$$
0.574481 + 0.818518i $$0.305204\pi$$
$$602$$ 0 0
$$603$$ 2.15939e6i 0.241845i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 9.95843e6i − 1.09703i −0.836140 0.548516i $$-0.815193\pi$$
0.836140 0.548516i $$-0.184807\pi$$
$$608$$ 0 0
$$609$$ 627264. 0.0685342
$$610$$ 0 0
$$611$$ −541728. −0.0587054
$$612$$ 0 0
$$613$$ 4.19586e6i 0.450993i 0.974244 + 0.225497i $$0.0724005\pi$$
−0.974244 + 0.225497i $$0.927600\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 9.12551e6i − 0.965038i −0.875885 0.482519i $$-0.839722\pi$$
0.875885 0.482519i $$-0.160278\pi$$
$$618$$ 0 0
$$619$$ 6.45734e6 0.677372 0.338686 0.940900i $$-0.390018\pi$$
0.338686 + 0.940900i $$0.390018\pi$$
$$620$$ 0 0
$$621$$ −1.68428e7 −1.75261
$$622$$ 0 0
$$623$$ 2.61835e6i 0.270276i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 5.41728e6i − 0.550316i
$$628$$ 0 0
$$629$$ 177012. 0.0178392
$$630$$ 0 0
$$631$$ 1.40514e7 1.40490 0.702450 0.711733i $$-0.252090\pi$$
0.702450 + 0.711733i $$0.252090\pi$$
$$632$$ 0 0
$$633$$ 4.07035e6i 0.403759i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 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 $$-0.992583\pi$$
0.999729 0.0233004i $$-0.00741743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.48119e6i 0.233023i 0.993189 + 0.116512i $$0.0371713\pi$$
−0.993189 + 0.116512i $$0.962829\pi$$
$$648$$ 0 0
$$649$$ 4.14072e6 0.385891
$$650$$ 0 0
$$651$$ −4.49434e6 −0.415636
$$652$$ 0 0
$$653$$ − 5.29130e6i − 0.485601i −0.970076 0.242800i $$-0.921934\pi$$
0.970076 0.242800i $$-0.0780660\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.68864e6i 0.604539i
$$658$$ 0 0
$$659$$ 4.72468e6 0.423798 0.211899 0.977292i $$-0.432035\pi$$
0.211899 + 0.977292i $$0.432035\pi$$
$$660$$ 0 0
$$661$$ −6.17420e6 −0.549639 −0.274819 0.961496i $$-0.588618\pi$$
−0.274819 + 0.961496i $$0.588618\pi$$
$$662$$ 0 0
$$663$$ − 2.97950e6i − 0.263245i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.43778e6i 0.212168i
$$668$$ 0 0
$$669$$ 9.35232e6 0.807893
$$670$$ 0 0
$$671$$ 1.87585e7 1.60839
$$672$$ 0 0
$$673$$ − 9.40925e6i − 0.800787i −0.916343 0.400394i $$-0.868873\pi$$
0.916343 0.400394i $$-0.131127\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 1.50086e7i − 1.25854i −0.777185 0.629272i $$-0.783353\pi$$
0.777185 0.629272i $$-0.216647\pi$$
$$678$$ 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
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 3.27295e6i 0.264574i
$$688$$ 0 0
$$689$$ 8.14849e6 0.653927
$$690$$ 0 0
$$691$$ −2.26556e7 −1.80501 −0.902506 0.430677i $$-0.858275\pi$$
−0.902506 + 0.430677i $$0.858275\pi$$
$$692$$ 0 0
$$693$$ 4.70448e6i 0.372116i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 1.02322e7i − 0.797791i
$$698$$ 0 0
$$699$$ 1.84615e6 0.142914
$$700$$ 0 0
$$701$$ 1.90169e7 1.46166 0.730828 0.682562i $$-0.239134\pi$$
0.730828 + 0.682562i $$0.239134\pi$$
$$702$$ 0 0
$$703$$ 249128.i 0.0190123i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 993168.i − 0.0747264i
$$708$$ 0 0
$$709$$ −1.51311e7 −1.13046 −0.565231 0.824933i $$-0.691213\pi$$
−0.565231 + 0.824933i $$0.691213\pi$$
$$710$$ 0 0
$$711$$ −7.61429e6 −0.564879
$$712$$ 0 0
$$713$$ − 1.74666e7i − 1.28672i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 1.38568e7i 1.00662i
$$718$$ 0 0
$$719$$ −1.50323e7 −1.08443 −0.542217 0.840238i $$-0.682415\pi$$
−0.542217 + 0.840238i $$0.682415\pi$$
$$720$$ 0 0
$$721$$ 2.39853e6 0.171833
$$722$$ 0 0
$$723$$ 7.88489e6i 0.560983i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 7.41230e6i − 0.520136i −0.965590 0.260068i $$-0.916255\pi$$
0.965590 0.260068i $$-0.0837449\pi$$
$$728$$ 0 0
$$729$$ −1.44612e7 −1.00782
$$730$$ 0 0
$$731$$ 7.18740e6 0.497483
$$732$$ 0 0
$$733$$ − 2.77928e6i − 0.191061i −0.995426 0.0955306i $$-0.969545\pi$$
0.995426 0.0955306i $$-0.0304548\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 1.17785e7i − 0.798768i
$$738$$ 0 0
$$739$$ −1.21046e7 −0.815342 −0.407671 0.913129i $$-0.633659\pi$$
−0.407671 + 0.913129i $$0.633659\pi$$
$$740$$ 0 0
$$741$$ 4.19338e6 0.280555
$$742$$ 0 0
$$743$$ − 4.46926e6i − 0.297005i −0.988912 0.148502i $$-0.952555\pi$$
0.988912 0.148502i $$-0.0474452\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 6.70388e6i − 0.439567i
$$748$$ 0 0
$$749$$ 1.07680e7 0.701345
$$750$$ 0 0
$$751$$ −2.88463e7 −1.86634 −0.933168 0.359442i $$-0.882967\pi$$
−0.933168 + 0.359442i $$0.882967\pi$$
$$752$$ 0 0
$$753$$ − 1.61028e7i − 1.03494i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 9.60868e6i − 0.609430i −0.952444 0.304715i $$-0.901439\pi$$
0.952444 0.304715i $$-0.0985612\pi$$
$$758$$ 0 0
$$759$$ 2.65939e7 1.67563
$$760$$ 0 0
$$761$$ 4.54588e6 0.284549 0.142274 0.989827i $$-0.454558\pi$$
0.142274 + 0.989827i $$0.454558\pi$$
$$762$$ 0 0
$$763$$ 8.79138e6i 0.546696i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.20522e6i 0.196730i
$$768$$ 0 0
$$769$$ 2.15923e7 1.31669 0.658345 0.752716i $$-0.271257\pi$$
0.658345 + 0.752716i $$0.271257\pi$$
$$770$$ 0 0
$$771$$ 1.58825e6 0.0962238
$$772$$ 0 0
$$773$$ − 1.48400e7i − 0.893276i −0.894715 0.446638i $$-0.852621\pi$$
0.894715 0.446638i $$-0.147379\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 314688.i 0.0186994i
$$778$$ 0 0
$$779$$ 1.44009e7 0.850251
$$780$$ 0 0
$$781$$ −2.53109e7 −1.48484
$$782$$ 0 0
$$783$$ 2.43778e6i 0.142098i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 2.48785e7i − 1.43182i −0.698194 0.715909i $$-0.746013\pi$$
0.698194 0.715909i $$-0.253987\pi$$
$$788$$ 0 0
$$789$$ 1.13193e7 0.647330
$$790$$ 0 0
$$791$$ −2.60885e6 −0.148254
$$792$$ 0 0
$$793$$ 1.45205e7i 0.819970i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 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
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 1.16102e7i − 0.627562i
$$808$$ 0 0
$$809$$ 3.10009e6 0.166534 0.0832669 0.996527i $$-0.473465\pi$$
0.0832669 + 0.996527i $$0.473465\pi$$
$$810$$ 0 0
$$811$$ −1.87180e6 −0.0999328 −0.0499664 0.998751i $$-0.515911\pi$$
−0.0499664 + 0.998751i $$0.515911\pi$$
$$812$$ 0 0
$$813$$ 6.21984e6i 0.330030i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.01156e7i 0.530196i
$$818$$ 0 0
$$819$$ −3.64162e6 −0.189707
$$820$$ 0 0
$$821$$ −2.00184e7 −1.03650 −0.518252 0.855228i $$-0.673417\pi$$
−0.518252 + 0.855228i $$0.673417\pi$$
$$822$$ 0 0
$$823$$ − 1.53118e7i − 0.787999i −0.919111 0.394000i $$-0.871091\pi$$
0.919111 0.394000i $$-0.128909\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.59310e6i 0.487748i 0.969807 + 0.243874i $$0.0784183\pi$$
−0.969807 + 0.243874i $$0.921582\pi$$
$$828$$ 0 0
$$829$$ −2.52209e7 −1.27460 −0.637302 0.770615i $$-0.719949\pi$$
−0.637302 + 0.770615i $$0.719949\pi$$
$$830$$ 0 0
$$831$$ 2.66727e7 1.33988
$$832$$ 0 0
$$833$$ − 5.38342e6i − 0.268810i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1.14883e7i − 0.550234i
$$848$$ 0 0
$$849$$ −1.86273e7 −0.886913
$$850$$ 0 0
$$851$$ −1.22299e6 −0.0578895
$$852$$ 0 0
$$853$$ − 486970.i − 0.0229155i −0.999934 0.0114578i $$-0.996353\pi$$
0.999934 0.0114578i $$-0.00364720\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1.92634e6i 0.0895945i 0.998996 + 0.0447972i $$0.0142642\pi$$
−0.998996 + 0.0447972i $$0.985736\pi$$
$$858$$ 0 0
$$859$$ 2.23538e7 1.03364 0.516820 0.856094i $$-0.327116\pi$$
0.516820 + 0.856094i $$0.327116\pi$$
$$860$$ 0 0
$$861$$ 1.81907e7 0.836258
$$862$$ 0 0
$$863$$ − 1.85838e7i − 0.849390i −0.905337 0.424695i $$-0.860381\pi$$
0.905337 0.424695i $$-0.139619\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 1.28043e7i 0.578504i
$$868$$ 0 0
$$869$$ 4.15325e7 1.86569
$$870$$ 0 0
$$871$$ 9.11742e6 0.407217
$$872$$ 0 0
$$873$$ 1.21174e7i 0.538114i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 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 $$-0.887783\pi$$
0.938499 0.345283i $$-0.112217\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 3.45874e7i − 1.47608i −0.674758 0.738039i $$-0.735752\pi$$
0.674758 0.738039i $$-0.264248\pi$$
$$888$$ 0 0
$$889$$ 2.96131e7 1.25669
$$890$$ 0 0
$$891$$ 1.36031e7 0.574044
$$892$$ 0 0
$$893$$ − 1.08346e6i − 0.0454656i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.05857e7i 0.854248i
$$898$$ 0 0
$$899$$ −2.52806e6 −0.104325
$$900$$ 0 0
$$901$$ 1.15794e7 0.475199
$$902$$ 0 0
$$903$$ 1.27776e7i 0.521471i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.74396e7i 0.703914i 0.936016 + 0.351957i $$0.114484\pi$$
−0.936016 + 0.351957i $$0.885516\pi$$
$$908$$ 0 0
$$909$$ 1.11731e6 0.0448503
$$910$$ 0 0
$$911$$ 2.59589e6 0.103631 0.0518155 0.998657i $$-0.483499\pi$$
0.0518155 + 0.998657i $$0.483499\pi$$
$$912$$ 0 0
$$913$$ 3.65666e7i 1.45180i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 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
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.69834e6i 0.103133i
$$928$$ 0 0
$$929$$ −3.96785e7 −1.50840 −0.754199 0.656646i $$-0.771975\pi$$
−0.754199 + 0.656646i $$0.771975\pi$$
$$930$$ 0 0
$$931$$ 7.57667e6 0.286486
$$932$$ 0 0
$$933$$ 8.76874e6i 0.329787i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 3.93413e7i − 1.46386i −0.681380 0.731930i $$-0.738620\pi$$
0.681380 0.731930i $$-0.261380\pi$$
$$938$$ 0 0
$$939$$ −7.01830e6 −0.259757
$$940$$ 0 0
$$941$$ 4.62506e7 1.70272 0.851361 0.524581i $$-0.175778\pi$$
0.851361 + 0.524581i $$0.175778\pi$$
$$942$$ 0 0
$$943$$ 7.06955e7i 2.58888i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 3.79025e7i − 1.37339i −0.726947 0.686693i $$-0.759062\pi$$
0.726947 0.686693i $$-0.240938\pi$$
$$948$$ 0 0
$$949$$ 2.82409e7 1.01792
$$950$$ 0 0
$$951$$ −2.97944e7 −1.06828
$$952$$ 0 0
$$953$$ − 2.66462e7i − 0.950394i −0.879879 0.475197i $$-0.842377\pi$$
0.879879 0.475197i $$-0.157623\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 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
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.09790e7i 1.40927i 0.709568 + 0.704637i $$0.248890\pi$$
−0.709568 + 0.704637i $$0.751110\pi$$
$$968$$ 0 0
$$969$$ 5.95901e6 0.203875
$$970$$ 0 0
$$971$$ 2.72034e7 0.925922 0.462961 0.886379i $$-0.346787\pi$$
0.462961 + 0.886379i $$0.346787\pi$$
$$972$$ 0 0
$$973$$ 1.30525e7i 0.441990i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 2.53555e7i − 0.849839i −0.905231 0.424919i $$-0.860302\pi$$
0.905231 0.424919i $$-0.139698\pi$$
$$978$$ 0 0
$$979$$ 1.60672e7 0.535775
$$980$$ 0 0
$$981$$ −9.89030e6 −0.328123
$$982$$ 0 0
$$983$$ − 1.19139e7i − 0.393252i −0.980479 0.196626i $$-0.937002\pi$$
0.980479 0.196626i $$-0.0629984\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 1.36858e6i − 0.0447173i
$$988$$ 0 0
$$989$$ −4.96584e7 −1.61437
$$990$$ 0 0
$$991$$ −2.91931e7 −0.944268 −0.472134 0.881527i $$-0.656516\pi$$
−0.472134 + 0.881527i $$0.656516\pi$$
$$992$$ 0 0
$$993$$ − 4.53538e6i − 0.145962i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.73001e7i 0.551201i 0.961272 + 0.275601i $$0.0888767\pi$$
−0.961272 + 0.275601i $$0.911123\pi$$
$$998$$ 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 400.6.c.f.49.2 2
4.3 odd 2 100.6.c.b.49.1 2
5.2 odd 4 16.6.a.b.1.1 1
5.3 odd 4 400.6.a.d.1.1 1
5.4 even 2 inner 400.6.c.f.49.1 2
12.11 even 2 900.6.d.a.649.2 2
15.2 even 4 144.6.a.c.1.1 1
20.3 even 4 100.6.a.b.1.1 1
20.7 even 4 4.6.a.a.1.1 1
20.19 odd 2 100.6.c.b.49.2 2
35.27 even 4 784.6.a.d.1.1 1
40.27 even 4 64.6.a.f.1.1 1
40.37 odd 4 64.6.a.b.1.1 1
60.23 odd 4 900.6.a.h.1.1 1
60.47 odd 4 36.6.a.a.1.1 1
60.59 even 2 900.6.d.a.649.1 2
80.27 even 4 256.6.b.g.129.2 2
80.37 odd 4 256.6.b.c.129.1 2
80.67 even 4 256.6.b.g.129.1 2
80.77 odd 4 256.6.b.c.129.2 2
120.77 even 4 576.6.a.bd.1.1 1
120.107 odd 4 576.6.a.bc.1.1 1
140.27 odd 4 196.6.a.e.1.1 1
140.47 odd 12 196.6.e.d.165.1 2
140.67 even 12 196.6.e.g.177.1 2
140.87 odd 12 196.6.e.d.177.1 2
140.107 even 12 196.6.e.g.165.1 2
180.7 even 12 324.6.e.a.109.1 2
180.47 odd 12 324.6.e.d.109.1 2
180.67 even 12 324.6.e.a.217.1 2
180.167 odd 12 324.6.e.d.217.1 2
220.87 odd 4 484.6.a.a.1.1 1
260.47 odd 4 676.6.d.a.337.2 2
260.187 odd 4 676.6.d.a.337.1 2
260.207 even 4 676.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 20.7 even 4
16.6.a.b.1.1 1 5.2 odd 4
36.6.a.a.1.1 1 60.47 odd 4
64.6.a.b.1.1 1 40.37 odd 4
64.6.a.f.1.1 1 40.27 even 4
100.6.a.b.1.1 1 20.3 even 4
100.6.c.b.49.1 2 4.3 odd 2
100.6.c.b.49.2 2 20.19 odd 2
144.6.a.c.1.1 1 15.2 even 4
196.6.a.e.1.1 1 140.27 odd 4
196.6.e.d.165.1 2 140.47 odd 12
196.6.e.d.177.1 2 140.87 odd 12
196.6.e.g.165.1 2 140.107 even 12
196.6.e.g.177.1 2 140.67 even 12
256.6.b.c.129.1 2 80.37 odd 4
256.6.b.c.129.2 2 80.77 odd 4
256.6.b.g.129.1 2 80.67 even 4
256.6.b.g.129.2 2 80.27 even 4
324.6.e.a.109.1 2 180.7 even 12
324.6.e.a.217.1 2 180.67 even 12
324.6.e.d.109.1 2 180.47 odd 12
324.6.e.d.217.1 2 180.167 odd 12
400.6.a.d.1.1 1 5.3 odd 4
400.6.c.f.49.1 2 5.4 even 2 inner
400.6.c.f.49.2 2 1.1 even 1 trivial
484.6.a.a.1.1 1 220.87 odd 4
576.6.a.bc.1.1 1 120.107 odd 4
576.6.a.bd.1.1 1 120.77 even 4
676.6.a.a.1.1 1 260.207 even 4
676.6.d.a.337.1 2 260.187 odd 4
676.6.d.a.337.2 2 260.47 odd 4
784.6.a.d.1.1 1 35.27 even 4
900.6.a.h.1.1 1 60.23 odd 4
900.6.d.a.649.1 2 60.59 even 2
900.6.d.a.649.2 2 12.11 even 2